mathematical economics - university of calicut

MATHEMATICAL ECONOMICS (MECICO1)

STUDY MATERIAL

I SEMESTER

B.Sc. MATHEMATICS (2019 Admission)

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION

CALICUT UNIVERSITY P.O. MALAPPURAM - 673 635, KERALA

19552

School of Distance Education University of Calicut

Study Material

I Semester

Complementary Course

B.Sc. Mathematics

MECICO1: MATHEMATICAL ECONOMICS

Prepared by:

Dr. K.X. JOSEPH Director Academic Staff College University of Calicut. Scrutinised by:

Sri. C.P. MOHAMED (Retd.) Poolakkandy House Nanmanda (P.O.) Kozhikode

DISCLAIMER

“The author(s) shall be solely responsible

for the content and views

expressed in this book”.

Printed @ Calicut University Press

CONTENTS

MODULE I DEMAND AND SUPPLY

ANALYSIS

1-36

Unit 1 Demand Analysis 1

Unit 2 Demand Curve 10

Unit 3 Determinants of Demand 15

Unit 4 Elasticities of Demand 22

Unit 5 Supply Function and Curves 32

Revision Exercises 34

MODULE II COST AND REVENUE

FUNCTIONS

37-54

Unit 1 Cost Function and Curves 37

Unit 2 Long run Cost function 43

Unit 3 Cost Elasticity 46

Unit 4 Revenue Function and Curves

Revision Exercise

48

MODULE III THEORY OF CONSUMER

BEHAVIOUR

55-72

Unit 1 Utility Analysis 55

Unit 2 Indifference Curve Analysis 61

Unit 3 Methods of Maximisation of

Utility

66

Revision Exercises 70

MODULE IV ECONOMIC APPLICATIONS

OF DERIVATIVES

73-203

Unit 1 Marginal Average and Total

Concepts

73

Unit 2 Maxima and Minima 98

Unit 3 Economic Applications of

Maxima and Minima

110

Unit 4 Functions of Several Variables 132

Unit 5 Differentials and Total

Differentials

147

Unit 6 Optimisation with Equality

Constraint

155

Unit 7 Comparative Static Analysis 172

Unit 8 Optimisation of Multivariable

Functions

185

Revision Excercises 199

MECICO1: Mathematical Economics

Shool of Distance Education, University of Calicut 1

MODULE I

DEMAND AND SUPPLY ANALYSIS

UNIT 1

DEMAND ANALYSIS

Introduction

Demand refers to the quantities of goods that

consumers are willing and able to purchase at various prices

during a given period of time. For your demand to be

meaningful in the marketplace you must be able to make a

purchase; that is, you must have enough money to make the

purchase. There are, no doubt, many items for which you have

a willingness to purchase, but you may not have an effective

demand for them because you don’t have the money to

actually make the purchase. For example, you might like to

have a 3600 square foot flat in Kochi, an equally large beach

house in Goa, and a private jet to travel between these places

on weekends and between semesters. But it is likely that you

have a budget constraint that prevents you from having these

items.

For demand to be effective, a consumer must also be

willing to make the purchase. There are many products that

you could afford (that is, you have the ability to buy them), but

for which you may not be willing to spend your income. Each

of us has a unique perspective on our own personal satisfaction

and the things that may enhance that satisfaction. The

important point is that if you do not expect the consumption of

something to bring you added satisfaction, you will not be

willing to purchase that good or service. Therefore, you do not



have a demand for such things despite the fact that you might

be able to afford them.

When we discuss demand, we are always referring to

purchases made during a given period of time. For example,

you might have a weekly demand for soft drinks. If you are

willing and able to buy four soft drinks at a price of Rs 5.00

each, your demand is four soft drinks a week. But your

demand for shoes may be better described on a yearly basis so

that, at an average price of Rs. 800.00 a pair, you might buy

three pairs of shoes per year. The important point here is that

when we refers to a person’s demand for a product, we usually

mean the demand over some appropriate time period, not

necessarily over the rest of the person’s life.

Think about the last time you spent money. It could

have been spent on a car, a computer, a new tennis racquet, or

a ticket to a movie, among literally thousands of other things.

No matter what you purchased, you decided to buy something

because it would please you. You are not forced to make

purchases. You do so because you expect them to increase

your personal satisfaction.

If these things give us satisfaction, we say that they

have value to us. Used in this way, value implies value in use.

Air has a value in use, because we benefit from breathing air.

But air is free. If air has value to us, why is it free? We

certainly would be willing to pay for air rather than do without

it. But air is available in such abundance that we treat it as a

free good. We also get satisfaction from using petrol. Petrol

has value is use. But unlike air, we must pay for the petrol we

use. That is, petrol has value in exchange as well as value in

use. We are willing to exchange something-usually money-for

the use of some petrol.

Why is air free, but petrol is costly? One important

reason is that petrol is scarce, whereas air is abundant. This



should start making you think about the role that scarcity plays

in the economy. But be careful as you do so. Just because

something is scarce does not necessarily mean it will have

value in exchange. Another reason that something may not

have value in exchange is because it has no value in use. That

is, people just do not get any satisfaction from possessing or

using it.

We all have a limited amount of money that we can

exchange for goods and services. The limit varies from

individual to individual. For example, a school teacher

typically has far less money to spend than a successful

investment banker. An unskilled labourer has less money to

exchange for goods and services than a skilled labourer.

However, we all (even the richest among us) have a limited

amount of money for buying things that can bring us

satisfaction. As a result, we all make decisions about how we

will spend, save, and/ or borrow money. This implies that how

we choose to allocate our money is an important factor in

determining the demand for various goods and services in the

economy.

The Demand Function

The demand function sets out the variables, which are

believed to have an influence on the demand for a particular

product. The demand for different products may be determined

by a range of factors, which are not always the same for each

of them. The presentation in this section is of a generic

demand function which includes some of the most common

variables that affect demand. For any individual product,

however, some of these may not apply. Thus, any attempt by

the firm to predict demand for a product on the basis of the

demand function will require some initial knowledge, or at

least informed guesswork, about the likely influences on it.

Generally,



The demand function can be written as:

Qd = I (Po, Pc, Ps, Yd, T, A, CR, R, E, N, 0). We can illustrate

the variables one by one as explained below.

1. Po, Pc and Ps — Price related variables

The first three variables in the function relate to price.

They are the own price of the product (Po), the price of

complements (Pc) and the price of substitutes (Ps)

respectively. In the case of the own price of a good, the

expected relationship would be, the higher the price the lower

the demand, and the lower the price the higher the demand.

This is the law of demand which is explained in greater detail

in the next section. In the case of complements, if the price of

complementary goods increases, we would expect demand to

fall both for it and for the good that it is complementary to.

This is the case as fewer people would now wish to buy either

good given that the complementary good is now more

expensive and this has the effect of reducing demand for the

other good as well. In contrast, if the price of a substitute good

rises, then demand for the good that it is a substitute for would

be expected to rise as people switched to buying the latter

rather than its more expensive substitute. Complements and

substitutes are also explained in detail later on.

2. Yd-Disposable Income

The fourth variable in the demand function, Yd stands

for disposable income, that is the amount of money available

to people to spend. The greater the level of disposable income.

The more people can afford to buy and hence the higher the

level of demand for most products will be. This assumes of

course that they are ‘normal’ goods, purchases of which

increase with rising levels of income, as opposed to ‘inferior

'goods that are purchased less frequently as income rises. The



use of disposable income rather than just income is justified on

the grounds that people do not have total control over their

gross incomes. There will, for example be deductions to be

made in the form of taxes. Thus the level of disposable income

can change over time, for example changes in tax rates.

3. T-Taste

The effect of changes in disposable income on the

demand for individual products will of course be determined

by the ways in which it is spent. This is where the fifth

variable, tastes (T), needs to be taken into account. Over a

period of time, tastes may change significantly, but this may

incorporate a wide range of factors. For example, in case of

food, greater availability of alternatives may have a significant

effect in changing the national diet. Thus, in India for instance,

the demand for bajra has fallen over the past 10 years as people

have switched to eating rice and wheat instead. Social

pressures may also act to alter tastes and hence demand. For

example, tobacco companies have been forced to seek new

markets as smoking has become less socially acceptable, thus

reducing demand in these areas. Changes in technology may

also have an impact. For example, as the demand for colour

televisions increased, the demand for black and white

televisions fell as tastes changed and the latter were deemed to

be inferior goods. Thus there are a number of ways in which

tastes may change overtime.

4. A-Level of Advertising

The next set of variables, the A variable, relates to

levels of advertising, representing the level of own product

advertising, the advertising of substitutes and the advertising of

complements respectively. The relationships here are as

follows. In general, the higher the level of own advertising for

a good, the higher the demand for that good would be

expected, other things being constant. Likewise, the higher the



level of advertising of a complimentary good, the higher the

demand for it and the good(s) which it is complementary to

will be, given their symbiotic relationship. Conversely,

however, the higher the level of advertising of a substitute

good, the lower the demand for the good for which it is an

alternative and people buy more heavily promoted good. The

overall effect of advertising will depend on the extent to which

each of these forms of advertising is used at any given point of

time as they may, at least in part, cancel each other out. This is

something the firm will also need to know in order to

determine its optimal advertising strategy.

5. CR and R-Credit and Rate of interest

The variables CR and R are also related. The former

represents the availability of credit while the latter represents

the rate of interest, that is the price of credit. These variables

will be most important for purchases of consumer durable

goods, for example cars. Someone’s ability to buy a car will

depend on his or her ability to raise money to pay for it. This

means that the easier credit is to obtain, the more likely they

are to be able to make the purchase. At the same time credit

must be affordable, that is the rate of interest must be such that

they have the money to pay. These two variables have

traditionally been regarded as exogenous to the firm that is,

they cannot be ‘controlled’ by it. In recent years, however,

major car manufacturers have increasingly sought to bring

them under the control through the provision of finance

packages.

6. E-Expectations

The letter E in the demand function stands for

expectations. This may include expectations about price and

income changes. For example, if consumers expect the price of

a good to rise in future then they may well bring forward their

purchases of it in order to avoid paying the higher price. This



creates an increase in demand in the short term, but over the

medium term, demand may fall in response to the higher price

charged. The firm will need to adjust its production

accordingly. An example of this might be when increased

taxes are expected to be levied on particular goods, for

example an increase in excise duties on alcohol or cigarettes,

as is usually the case after the Central Budget. Consumers of

these products may buy more of them prior to the

implementation of the duty increases in order to avoid paying

the higher prices arising from the higher level of duties.

Alternatively, expectations about incomes may be important.

For example, people who expect their incomes to rise may buy

more goods, whereas those who expect their incomes to fall

will buy less. At the level of the individual consumer this may

not be significant but when aggregated across a country’s

population it can be. Thus during a boom in the economy the

additional expected purchasing power of consumers will lead

to increases in demand for a significant number of products.

Conversely, the expectation that incomes will fall, perhaps as a

result of redundancy during a recession, will reduce demand as

consumers become more cautions.

7. N-Number of Potential Customers

The variable N stands for the number of potential

customers. Each product is likely to have a target market, the

size of which will vary. The number of potential customers

may be a function of age or location. For example, the number

and type of toys sold in a particular country will be related to

its demographic spread, in this case the number of children

within it and their ages.

8. O-Other Miscellaneous Factors

Finally, we come to O which represents any other

miscellaneous factors which may influence the demand for a

particular product. For example, it could be used to represent



seasonal changes in demand for a particular product if demand

is subject to such fluctuations rather than spread evenly

throughout the year. Examples of such products might include

things such as umbrellas, ice creams and holidays. In sum, this

is a ‘catch all’ variable which can be used to represent

anything else which the decision maker believes to have an

effect on the demand for a particular product.

Thus each product will have its own particular demand

function depending on which of the above variables influence

the demand for it. The ways in which the level of demand can

be estimated on the basis of this demand function will be

discussed later.

THE LAW OF DEMAND

For most goods, consumers are willing to purchase

more units at a lower price than at a higher price. The inverse

relationship between price and the quantity consumers will buy

is so widely observed that it is called the law of demand. The

law of demand is the rule that people will buy more at lower

prices than at higher prices if all other factors are constant.

This idea of the law of demand seems to be a pretty logical and

accurate description of the behaviour we would all expect to

observe and for now, this will suffice.

The law of demand states that consumers are willing or

able to purchase more units of a good or service at lower prices

than at higher prices, other things being equal. Have you ever

thought about why the law of demand is true for nearly all

goods and services? Two influences, known as the income

effect and the substitution effect, are particularly important in

explaining the negative slope of demand functions. The income

effect is the influence of a change in a product’s price on real

income, or purchasing power. If the price of something that we

buy goes down, our income will go farther and we can

purchase more goods and services (including the goods for



which price has fallen) with a given level of money income.

The substitution effect is the influence of a reduction in a

product’s price on quantity demanded such that consumers are

likely to substitute that good for others that have thus become

relatively more expensive.



UNIT 2

DEMAND CURVE

The concept of demand is often depicted in a graphic

model as a demand curve. A demand curve is a graphic

illustration of the relationship between price and the quantity

purchased at each price. When plotting a graph for demand, the

price is measured along the vertical axis and the quantities that

would be purchased at various prices are measured along the

horizontal axis. The demand curve shows the relationship

between the own price of a good and the quantity demanded of

it. Any change in own price causes a movement along the

curve as shown in the Figure. In this case, a rise in price from

P1 to P2 results in a fall in quantity demanded from Q1 to Q2 ie.,

a move from B*to A*in the figure.



Demand schedule

The same information can also be given in a table or

demand schedule, such as given in the following table or by an

equation for the demand function such as the following:

P=100-0.25Q

where P is price and Q is quantity. The advantage of

the equation is that it is compact to work with, and rely on

economists such functions. The following is an example of a

demand schedule.

Demand Schedule

Price(Rs) Quantity(Units)

90 40

70 120

50 200

30 280

10 360

The Market Demand Curve

The market demand curve is the total of the quantities

demanded by all individual consumers in an economy (or

market area) at each price. Economic theory supports the

proposition that individual consumers will purchase more of a

good at lower prices than at higher prices. If this is true of

individual consumers, then it is also true of all consumers

combined. This relationship is demonstrated by the example in

the following Figure, which shows two individual demand

curves and the market demand that is estimated by adding the

two curves together.



The Market Demand Curve

A market demand curve is the sum of the quantities

that all consumers in a particular market would be willing and

able to purchase at various prices. If we plotted the quantity

that all consumers in this market would buy at each price, we

might have a market demand curve such as the one shown the

above figure. The market demand curve in the figure shows

that at a price of Rs.15, the market demand would be 4 for the

first consumer and 2 for the second consumer, giving a total of

6 units as market demand. Analogously, at Rs.10.00 the total

market demand is 13 units.

Another way of showing the derivation of the market

demand curve is through equations representing individual

consumer demand functions. Consider the following three

equations representing three consumers' demand functions:

Consumer 1:P = 12 - Q1

Consumer 2:P = 10 - 2Q2

Consumer 3:P = 10 - Q3

You should substitute some value of Q (such as Q=4)

in each of these equations to verify that they are consistent

with the data in the Table given below. Now, add these three

demand functions together to get an equation for the market

demand curve. Be careful while doing this. There is



sometimes a temptation to just add equations with out thinking

about what is to be aggregated. From the table, it is easy to see

that the quantities sold to each consumer at each price have

been added. For example, at a price of Rs.6, consumer number

1 would buy six units (Q1=6), consumer number 2 would buy

two units (Q2=2), and consumer number 3 would buy four

units (Q3=4). Thus, the total market demand at a price of Rs.6

is 12 units (6+2+4=12). The important point to remember is

that the quantities are to be added; not the prices. To add the

three given demand equations, we must first solve each for Q

because we want to add the quantities (that is, we want to add

the functions horizontally, so we must solve them for the

variable represented on the horizontal axis). Solving the

individual demand functions for Q as a function of P (for

consumes 1, 2 and 3), we have

Q1 = 12–P

Q2 = 5 –0.5 P

Q3 = 10–P

Adding these equations results in the following:

Q1+ Q2+Q3+=27–2.5P

And letting QM=Q1+Q2+Q3 where QM is market

demand.

QM=27–2.5P

QM is the total quantity demanded

This is the algebraic expression for the market demand

curve. We could solve this expression for P to get the inverse

demand function:

P=10.8 – 0.4 QM

Now, check to see that this form of expression the

market demand is consistent with the data shown in the Table



given below.

Derivation of a Market Demand Schedule

Price Q1 Q2 Q3 QM

10 2 0 0 2

8 4 1 2 7

6 6 2 4 12

4 8 3 6 17

2 10 4 8 22

The market demand curve shows that the quantity

purchased goes up from 12 to 22 as the price falls from

Rs.6.00 to Rs.2.00. This is called a Change in quantity

demanded. As the price falls, a greater quantity is demanded.

As the price goes up, a smaller quantity is demanded. A

change in quantity demanded is caused by a change in the

price of the product for any given demand curve. This is true

of individual consumers' demand as well as for the market

demand. But what determines how much will be bought at

each price? Why are more paperback books bought today than

in previous years, even though the price has gone up?

Questions such as these are answered by looking at the

determinants of demand.



UNIT 3

DETERMINANTS OF DEMAND

Introduction

Many forces influence our decisions regarding the

bundle of goods and service we choose to purchase. It is

important for managers to understand these forces as fully as

possible in order to make and implement decisions that

enhance their firms' long-term health. It is probably

impossible to known about all such forces, let alone be able to

identify and measure them sufficiently to incorporate them into

a manager's decision framework. However, a small subset of

these forces is particularly important and nearly universally

applicable. As stated above, the overall level of demand is

determined by consumers' incomes, their attitudes or feelings

about products, the prices of related goods, their expectations,

and by the number of consumers in the market. These are

often referred to as the determinants of demand. Determinants

of demand are the factors that determine how much will be

purchased at each price. As these determinants change over

time, the overall level of demand may change. More or less of

a product may be purchased at any price because of changes in

these factors.

Such changes are shown by a shift of the entire demand

curve. If the demand curve shifts to the right, we say that there

has been an increase in demand. This is shown as a move

from the original demand D1, D1 to the higher demand D2 D2 in

the figure given below.

The original demand curve can be thought of as being

the market demand curve for soft drinks. At a price of

Rs.15.00, given the initial level of demand, consumes would

purchase 6,000 soft drinks. If demand increases to the higher



demand, consumers would purchase 13,000 soft drinks rather

than the 6,000 along the original demand curve.

A decrease in demand can be illustrated by a shift of

the whole demand curve to the left. In the second figure given

above, this is represented by a move from the original demand

D1, D1 to the lower demand D2, D2. At the price of Rs.13

initially 8,000 soft drinks are purchased, while following the

decrease in demand only 7,000 soft drinks are bought.

It is important to see that these changes in demand are

different from the changes in quantity demanded. We

discussed how changes in price cause a change in quantity

demanded. As price changes, people buy more or less along a

given demand curve. Movement from A* to B* in the demand

curve given earlier shows the change in quantity demanded as

price changes. It is not a shift in the whole demand curve.

Such as that shown in the two figures given above. When the

whole demand curves changes, there is a change in demand.

Some of the variables that cause a change in demand are

changing incomes, changing tastes of consumers, changes in

other prices, changes in consumer expectations and changes in

the number of consumers in the market etc. These variables



that cause a change in demand are also known as shifter

variables.

`The following are the important determinants of

demand.

1. The Product's Price as a Determinant of Demand

It has already been noted that consumers are expected

to be willing and able to purchase more of a product at lower

prices than at higher prices. In evaluating a demand or sales

function for a firm or an entire industry, one of the first things

an economist will consider is the price of a product. If

inventories have built up, a firm may consider lowering the

price to stimulate quantity demanded. Rabates have become a

popular way of doing this. Rebate programmes of one type or

another have appeared for cars, home appliances, toys and

even food products. Such rebates constitute a way of lowering

the effective purchase price and thereby increasing the quantity

that consumers demand without the negative repercussions of

realising the price back to its normal level, the firm simply

allows the rebate programme to quietly come to an end. As

has been stated above, this is called a change in quantity

demanded. As the effective price falls, a greater quantity is

demanded.

2. Income as a Determinant of Demand

On the other hand, shifter variables, as the name

implies cause the demand curve to shift ie., there is a change in

demand. Nearly all goods and services are what economists

refer to as normal goods. These are goods for which

consumption goes up as the incomes of consumers rise, and the

converse is also true. In fact, it is rare to find a demand

function that does not include some measure of income as an

important independent variable. Goods for which

consumption increases as the incomes of consumers rise are



called normal goods. Goods for which consumption decreases

as the incomes of consumes rise are called inferior goods.

This relationship between product demand and income

is one of the reasons that so much national attention is given to

the level of Gross Domestic Product (GDP) and changes in the

rate of growth of GDP. The GDP is the broadest measure of

income generated in the economy. In demand analysis, other

more narrowly defined measures, such as personal income or

disposable personal income, are often used; but these measures

are highly correlated with GDP. Thus, looking at the changing

trends in GDP is helpful for understanding what may happen

to the demand for a product.

3. Tastes and Preferences as Determinants of Demand

We all like certain things and dislike others. A pair of

identical twins brought up in the same environment may have

different preferences in what they buy. Exactly how these

preferences are formed and what influences them is not easy to

know. Psychologists, Sociologists and social Psychologists

have a lot to offer in helping economists and other business

analysts understand how preferences are formed and altered.

Even if we do not have a thorough understanding of

preference structures, one thing is clear. Preferences and

changes in preferences affect demand for goods and services.

All have observed how such changes in tastes and preferences

have influenced various markets. For example, consider the

automobile market. In the United States, people appeared to

have a preference for big, powerful cars throughout the 1950s

and 1960s. During the 1970's the preference structure started

to change in favour of smaller, less-powerful, but more fuel

efficient cars. In part, the change in preference structure for

cars may also have been related to lifestyle factors, such as

being sportier and more concerned with resource conservation.

Convenience factors, such as ease of driving and parking, may



also have been important. Demographic changes, especially a

trend toward smaller families, may have had some effect as

well. In terms of the theory, the change in preference toward

fuel-efficient cars will shift the demand curve for smaller cars

to the right. On the other hand, social attitudes towards

smoking has changed and thus one would expect that the

demand curve for cigarettes has shifted to the left. Likewise,

the growing awareness in respect of noise and environmental

pollution has resulted in a decline in the demand for crackers

during Diwali celebrations.

4. Other Prices as Determinants of Demand

How much consumers buy of a product may be

affected by the prices charged for other goods or services as

well. The figures given earlier show the effect on the demand

curve following a change in the price of a related good or

service. Both graphs are self explanatory. Earlier, it was

noted that the rise in the price of gas online during the 1970s

had some effect on the demand for large versus small cars in

the United States. Gasoline and cars are complementary

goods; they are used together and complement one another.

When the price of gasoline rose, there were at least two effects

on the automobile market. First, the higher price of gas

increased the cost of driving, and thus reduced the total

number of miles individuals tended to drive. Second, smaller,

more fuel-efficient cars became more attractive relative to big

cars.

This relationship can be stated in more general terms.

Suppose that we observe two goods, A and B, and B is

complementary to A. If the price of B goes up, we can expect

the quantity demanded for A to be reduced. Why? Because as

the price of goods B increases, its quantity demanded

decreases according to the law of demand. But now, some

individuals who would have purchased B at the lower price are



not longer making those purchases. These same individuals

now no longer have any use for A, because A was a good

useful only in conjunction with B. Thus, the quantity

demanded of A goes down as well. The reverse is also true: if

the price of B falls, the demand for A will rise. It should be

clear why business analysts are concerned not only about the

effect that their product's price has on sales but also with the

effect of the prices of complementary products.

Demand Curves for Substitutes and Complements

5. Other Determinants of Demand

It would be a monumental task to identify everything

that might have some influence on the demand for any product.

So far, the four most important influences have been

identified: a product's price, income, tastes and preferences,

and the price of complementary or substitute products. A

number of others were identified in earlier section, which also

affect demand. By now you will be able to establish the

direction of the effect i.e., whether it will increase or decrease

demand. For example, population growth obviously causes the

potential demand for nearly all products look at individual



components of the population as determinants of demand. The

changing age distribution, for example, may have differential

effects on different markets. The growing proportion of

people over 65 in the population has important ramifications

for demand for such things such as healthcare products.

Changes in other characteristics and in the geographical

distribution of the population may also be important. You may

think of a variety of other effects on consumer demand as well.



UNIT 4

ELASTICITIES OF DEMAND

What is Elasticity?

We studied that when price falls, quantity demanded

would increase. While we know this qualitative effect exists

for most goods and services, managers and business analysts

are often more interested in knowing the magnitude of the

response to a price change ie., by how much? There are many

situations in which one might want to measure how sensitive

the quantity demanded is to changes in a product's price.

Economists and other business analysts are frequently

concerned with the responsiveness of one variable to changes

in some other variable. It is useful to know, for example, what

effect a given percentage change in price would have on sales.

The most widely adopted measure of responsiveness is

elasticity. Elasticity is a general concept that economists,

business people, and government officials rely on for such

measurement. For example, the finance minister might be

interested in knowing whether decreasing tax rates would

increase tax revenue. Likewise, it is often useful to measure

the sensitivity of changes in demand in one of the determinants

of demand, such as income or advertising.

Elasticity is defined as the ratio of the percentage

change in quantity demanded to the percentage change in some

factor (such as price or income) that stimulates the change in

quantity. The reason for using percentage change is that it

obviates the need to know the units in which quantity and price

are measured. For example quantity could be in kilograms,

grams, litres or gallons and price could be in dollars, rupees,

euro etc. A measure of elasticity based on units would lead to

confusion and misleading comparisons across different

products. The use of percentage change makes the measure of



elasticity independent of units of measurement and hence easy

to understand. Elasticity is the percentage change in some

dependent variable given a one-percent change in an

independent variable, centeris paribus. If we let Y represent

the dependent variable, X the independent variable and E the

elasticity, their elasticity is represented as

E = % change in Y / % change in x

There are two forms of elasticity: arc elasticity and

point elasticity. The former reflects the average

responsiveness of the dependent variable to change in the

independent variable over some interval. The numeric value of

arc elasticity can be found as follows:

2 1 2 1

2 1 2 1

2 1 2 1

2 1 2 1

/ 0.5 ( )changeinY/average Arc elasticity

change in X/average X / 0.5 ( )

*

Y Y Y YY

X X X X

Y Y X X

X X Y Y

Where the subscripts refer to the two data points

observed, or the extremes of the interval for which the

elasticity is calculated.

Point elasticities indicate the responsiveness of the

dependent variable to the independent variable at one

particular point on the demand curve. Point elasticity's are

calculated as follows: It is denoted by e or .

So, 1

1

Y

X*

δx

δYe

This form works well when the function is bivariate:

Y = f(X). However, when there are more independent

variables, partial derivatives must be used. For example,

suppose that Y= f (W, X, Z) and we want to find the elasticity's



for each of the independent variables. We would have

Y

Z*

Z

Ye

Y

X*

X

Ye

Y

W*

W

Ye

z

x

w

Although economists use a great variety of elasticities,

the following three deserved particular attention because of

their wide application in the business world: price elasticity,

income elasticity, and cross-price elasticity. We discuss these

in detail in the subsequent sections.

1. The Price Elasticity of Demand

Price elasticity of demand measures the responsiveness

of the quantity sold to changes in the product's price, ceteris

paribus. It is the percentage change in sale divided by a

percentage change in price. The notation Ep will be used for

the arc price elasticity of demand, and ep will be used for the

point price elasticity of demand. If the absolute value of Ep

(or ep) is greater than one, a given percentage decrease

(increase) in price will result in an even greater percentage

increase (decrease) in sales. In such a case, the demand for the

product is considered elastic; that is, sales are relatively

responsive to price changes. Therefore, the percentage change

in quantity demanded will be greater than the percentage

change in the price. When the absolute value of the price

elasticity of demand is less than one, the percentage change in

sales is less than a given percentage change in price. Demand

is then said to be inelastic with respect to price. Unitary price

elasticity results when a given percentage change in price

results in an equal percentage change in sales. The absolute

value of the coefficient of price elasticity is equal to one in



such cases. These relationships are summarized as follows:

If |ep| or Ep| > 1, demand is elastic

If |ep| or Ep| < 1, demand is inelastic

If |ep| or Ep| = 1, demand is unitarily elastic

2. ARC Price Elasticity

Consider the hypothetical prices of some product and

the corresponding quantity demanded, as given in the

following table. We could calculate the arc price elasticity

between the two lowest prices ie., between Rs.30 and Rs.10 as

follows:

25.03010

3010

280360

280360

pE

Thus, demand is inelastic in this range. This value of

Ep= 0.25 means that a one percent change in price results in a

0.25% change in the quantity demanded (in the opposite

direction of the price change) over this region of the demand

function.

Demand Schedule to Demonstrate Price Elasticities

Price Rs. (P) Quantity

(Units) (Q)

Arc

Elasticity

Point

Elasticity

90

70

50

30

10

40

120

200

280

360

–4.00

–1.50

–0.67

–0.25

–9.00

–2.33

–1.00

–0.43

0.11

If we calculate the arc price elasticity between the

prices of 50 and 70, we have



51. - 7050

7050

120200

120200

pE

We would say that demand is price elastic in this range

because the percentage change in sales is greater than the

percentage change in price. You can calculate arc elasticity

over any price range. As an exercise estimate the arc elasticity

between the extremes of the demand function shown the table

ie., between Rs.90 and Rs.10. Satisfy yourself that the

absolute value of arc elasticity between these two points is 1.

3. Point Price Elasticity

The algebraic equation for the demand schedule given

in the above table is

P=100–0.25Q or Q = 400–4P

We can use this demand function to illustrate the

determination of point price elasticities. Let's select the point

at which P=10 and Q = 360.

11.0

)360/10)(4(

*

p

p

p

e

e

Q

P

dP

dQe

Because |ep|<1, we would say that demand is inelastic

at a price of Rs.10. Now, consider a price of Rs.70:

33.2

)120/70)(4(

*

p

p

p

e

e

Q

P

dP

dQe

Here |ep|>1, and demand is price elastic.



This example shows that the price elasticity of demand

may (and usually does) vary along any demand function,

depending on the portion of the function for which the

elasticity is calculated. It follows that we usually cannot make

such statements as "the demand for product X is elastic"

because it is likely to be elastic for one range of price and

inelastic for another. Usually at high prices demand is elastic,

while at lower prices demand tends to be inelastic. Intuitively,

this is so because lowering price from very high levels is likely

to stimulate demand much more compared to lowering prices

when price is already low. As an illustration, consider the

prices of cellular phones (handsets) when these were first

introduced in the Indian market at prices ranging between

Rs.25,000 to Rs. 30,000 per handset. Demand was limited to

the higher end of the market. As these prices fell, demand was

stimulated and resulted in increasing penetration in the middle

and lower end segments, indicating an elastic response.

4. Cross-Price Elasticity

The sales volume of one product may be influenced by

the price of either substitute or complementary products.

Cross-price elasticity of demand provides a means to quantify

that type of influence. It is defined as the ratio of the

percentage change in sales of one product to the percentage

change in price of another product. The relevant arc (Ec) and

point (ec) cross-price elasticities are determined as follows.

2 1 2 1

2 1 2 1

*b b a ac

a a b b

a bc

b a

Q Q P PE

P P Q Q

Q Pe

P Q

Where the alphabetic subscripts differentiate between

two products involved. A negative coefficient of cross-price

elasticity implies that a decrease in the price of product. A



results in an increase in sales of product B, or vice versa, we

can conclude that the products are complementary to one

another (such as cassette tape players and cassette tapes).

Thus, when the coefficient of cross-price elasticity for two

products in negative, the products are classified as

complements.

A similar line of reasoning leads to the conclusion that

if the cross-price elasticity is positive, the products are

substitutes. For example, an increase in the price of sugar

would cause less sugar to be purchased, but would increase the

sale of sugar substitutes. When we calculate the cross-price

elasticity for this case, both the numerator and the denominator

(%change in Q of sugar substitutes and % change in P of

sugar, respectively) would have the same sign, and the

coefficient would be positive.

If we goods are unrelated, a change in the price of one

will not affect the sales of the other. The numerator of the

cross-price elasticity ratio would be 0, and thus the coefficient

of cross-price elasticity would be 0. In this case, the two

commodities would be defined as independent. For example,

consider the expected effect that a 10% increase in the price of

eggs would have on the quantity of electronic calculator sales.

These relationships can be summarized as follows:

If ec or Ec > 0, goods are substitutes

If ec or Ec < 0, goods are complementary

If ec or Ec 0, goods are independent

Cross price elasticities may not always be symmetrical.

For example, consider two dailies, Time of India and the

Hindustan Times competing in the Delhi market. Most

analysts will agree that the two products are substitutes ie., the

cross price elasticity is positive. However, there is no reason

to believe that the change in demand for the Times of India



following a one percent change in the price of Hindustan times

will be equal to the change in demand for Hindustan Times

following a one percent change in the price of the Times of

India.

Determinants of Price Elasticity of Demand

The following are the important determinants of price

elasticity.

(i) Availability of substitutes: If close substitutes are

available then the elasticity of demand will be high.

Other wise it will be less elastic.

(ii) Position of a commodity in a consumers budget: The greater the proportion of income spent on a

commodity, the greater will be its elasticity of demand

and vice versa. Eg: Salt, clothing.

(iii) Nature of the need that a commodity satisfies:

Luxury goods are price elastic while necessities are

price inelastic.

(iv) Number of uses to which a commodity can be put:

The more the possible uses of a commodity the greater

will be its price elasticity.

For example, Milk can be put to several uses. If its

price decreases its demand will increase drastically and vice

versa.

(v) The period: In long run the demand will be more

elastic compared to short run elasticity of demand.

(vi) Consumer habits: Habitual consumption makes the

demand for a good inelastic.

(vii) Tied demand: The demand for those goods which are

tied to others is normally inelastic compared to

autonomous goods.



(viii) Price Range: Goods which are in very high range or

in very low price range have inelastic demand but those

in middle range have elastic demand.

Properties of Price Elasticity of Demand

Theorem 1: The elasticity of demand at different points on

the same demand curve is different.

Point p

PQe

p Q

Suppose we want to measure elasticity of demand at a

particular point A in the figure. For this purpose, draw a

straight line MN tangent to A. Line MN has the same slope

throughout.

Hence at A an B the slope is the same.

X

P

dP

dx

P

dP

X

dx

P

dP

X

dxe

OQ

OP

PP

QQ

Q

P

p

Qe

y

yy

y

y

y

cd .//

..1

1

21

31



Where X = quantity demanded of X, Py = price of

commodity Y.

The cross elasticity of demand for X may be positive or

negative depending upon the nature of relationship between X

and Y commodities.

If two goods X and Y are substitutes, then ecd > 0, the

higher the value of ecd the more close will be the substitutes.

If ecd < 0 then X and Y are complements

If ecd = 0 then X and Y are unrelated or independent



UNIT 5

SUPPLY FUNCTION AND CURVES

The supply of a product refers to different quantities

that the producer is willing to offer at given levels of prices.

Supply also depends on a number of variables applying

the ceteris paribus condition. Here we write supply function as

S= f (p) where S= Supply of the product, P= Price of

the product

Supply is positively related to price

Elasticity of Supply

The elasticity of supply can be defined as a percentage

change in quantity supplied divided by a percentage change in

the price

q

p

dp

dq

Q

P

e s

p

..P

q

pricein change %

suppliedquantity in %change



dq = absolute change in supply, dp = absolute change in price.

p = Price; q = quantity supplied

s

p will always be positive. The elasticity of supply is

different at different points on the supply curve.



Market Equilibrium

When the demand of a commodity is equal to the

supply of that commodity we say that the equilibrium is

attained. So to obtain the equilibrium price and quantity

demanded we will equate the demand function to the supply

function.

REVISION EXERCISES

I. Very Short Answer Questions

1. What is a demand function?

2. What is the law of demand?

3. What is a demand curve?

4. What is a demand schedule?

5. Define elasticity of demand

6. Define arc price elasticity of demand

7. Define a supply function

8. Define elasticity of supply

9. Define market equilibrium

10. Explain shift in demand



II Short answer Questions

11. Explain a demand function. State the variables

involved it.

12. Explain market demand curve.

13. Distinguish between arc price elasticity and point price

elasticity.

14. What is cross price elasticity?

15. Define demand schedule and demand function with the

help of a diagram.

16. What do you mean by expansion of demand? Illustrate

it with the help of an example.

17. What do you mean by contraction of demand? Explain

the concept with the help of a diagram.

18. How do you measure the responsiveness of demand to

the changes in price?

19. Explain the concept of supply with the help of a

diagram.

20. What are the various degrees of elasticities of supply?

III Long Answer Questions

21. Explain demand function in detail

22. What are the determinants of demand? Explain.

23. Describe the various elasticities of demand.

24. What are determinants of price elasticity of demand?

25. What are the properties of price elasticity of demand?

26. Explain the five degrees of elasticities of demand.

Explain each term briefly.

27. The elasticity of demand of different points on the

same demand curve is different, prove?



28. Distinguish between point elasticity and arc elasticity

of demand. Indicate their analytical significance.

29. Give the nature and property of a demand function for

a normal good.

30. Explain the four factors which are required to specify

the demand function and demand curve.

31. If x = 25 – 3p– p2 be a demand function, find the price

elasticity of demand at p=3



MODULE II

COST AND REVENUE FUNCTIONS

UNIT I

COST FUNCTION AND CURVES

It explains the relationship between the output of a

commodity and the expenditures incurred in its production.

Prof: Marshall has made a distinction between the cost of

production and the expenses of production.

Total cost = Explicit Cost + Implicit Cost,

The relation between cost and output is called cost

function.

The cost function of a firm depends upon production

function and the prices of factors used for production.



To express the problem of cost of production the

following assumptions are made.

1. Some of the factors of production are employed in

fixed amounts irrespective of the output of the firm.

2. The expenditure on these factors are fixed and known.

3. Remaining factors variable, and the condition of their

supply are known.

4. Technical condition under which production is carried

out are known and fixed.

5. The output of the firm is obtained with the lowest

possible total cost.

Mathematically, the cost function can be expressed as

C = f(q)

Where C–Total Cost, q – Output

Short run Cost Functions

During short run a firm is unable to change its inputs of

production.

In the short run the firm's decisions are confined only

to the variable inputs.



Hence short-run cost function may be stated, as an

explicit function of the level of output plus the cost of the fixed

inputs as given below

C = f(q)+a

Where a = fixed cost which is independent of the level

of output.

If q = 0, it means that the firm is not employing any

variable inputs in the short run.

Thus the above equation becomes C = 0 + a.

It clearly states that even at zero output level the firm

has to incur fixed costs.

Total Average and Marginal Cost Functions

Special cost functions can be derived from eq: (2) We

have C = f(q) + a = TVC + TFC

TVC = f(q)

where TVC=Total variable cost function, TFC = total

fixed cost function.

q

af(q)

q

CAC

where AC = Average cost function



q

f(q)

q

TVCAVC where AVC = Average variable

cost function.

q

a

q

TFCAFC where AFC = Average fixed cost

function.

(q)f"dq

dCMC where MC = Marginal cost function.

The various types of cost functions can be graphically

represented as follows.



The above specific cost functions may take many

different shapes.

Total cost (TC-function) is a qubic function of output.

Some of the important TC functions may be defined as

follows.

ayq

βqαqC

ayq

βqαqC

aeqC

2

βqa

Where C=Total cost function, q = output, a = fixed cost

, , are the parameters (constants) AC, AVC and MC are all

second degree curves which first decline and then increase as

output is expanded. MC reaches its minimum before AC and

AVC functions. AVC function reaches its minimum before

AV functions.

The flow of MC, AVC and AC functions can be

expressed to 3 stages.

I) MC function reaches minimum

II) AVC function - reaches minimum

III) AC function reaches minimum

MC curve cuts AVC and AC curves at the minimum

point which states that

AC = AVC = MC

The AFC function in the figure 2 is a rectangular

hyperbola. The AFC function will never be zero in the short

run. The vertical distance between AC function and AVC

function equals the AFC function. It decreases as output is

expanded.



Traditionally, economic theory determines the output at

the point MC = MR. It implies that

i) FC generally has no effect upon firms optimizing

decisions during the period of short-run. FC has to be

paid regardless of the level of firms output and it

merely adds a constant terms to its profit () equations.

ii) The FC term 'a' vanishes upon differentiation and MC

is independent of its level.

iii) The maximum loss that a firm would be ready to bear

in the short run, must not be greater than this constant

'a'.

If loss > FC then, it is in the interest of the firm to

discontinue production and accept a loss equal to its fixed cost.



UNIT 2

LONG RUN COST FUNCTION

In the long run the firm can change the size of its plant.

ie., all factors of production are variable. This means

that in the long run a firm will go on increasing its size of plant

if it adds to its total profit or it can produce at the minimum

cost. The following figure shows the long run Average Cost

curve of the firm.

The firm has '6' short run average cost curves as seen in

the figure. SAC1 to SAC6. By joining the minimum points of

these short run Average Cost curves we get the long run

average cost curve.

The long run total cost (LTC) function can be derived

from the short run total cost function.

LTC= minimum cost of producing each output level

when the plant size can be freely varied.



Let us assume that there are three different plant sizes

a1, a2 and a3. There are three short-run total cost curves

corresponding to each plant size (shown as a1, a2 and a3. in the

following figure).

Joining the minimum points of the short run cost curves

we get the LTC.

The firm can produce the output level OX1 in any of

the plants. Its total cost will be T1X1 for plant size a1, T2,X1

for a2 and T3X1 for plant size a3.

Here a1 gives minimum cost for output OX1. Hence T1

lies as the LTC.

For output OX2,V1 gives minimum cost, it lies on LTC.

For output OX3, R1 gives minimum cost, it lies on

LTC.

Thus long run total cost curve is defined as the locus or

path indicating the minimum cost points to produce various

output levels as shown in figure.



If the firms fixed inputs = 'a' then total cost function of

the firm is

C = f(q) + (a) = C (q,a)

where, f (q) = TVC and (a) = T + C

In the long run TFC is not a constant but a variable

term. Hence any change in 'a' will affect 'C'.

Different values of 'a' will yield a family of short run

cost curves. The LAC is the envelope of the short run cost

curves. Hence it is also called as envelope curve.

Equation for family of short run cost functions in

implicit form

C-f(q) –(a) = C(C, q, a) = 0

Setting the partial derivative w.r.t. a gives zero.

C' (c,q,a) = 0

Then equation of long run cost function is obtained by

eliminating 'a' from the above equations and solving for 'C' as a

function of q we get.

C=f(q)

LTC is a function of output level, given the condition

that each output level is produced in a plant of optimum size.



UNIT 3

COST ELASTICITY

Definition

It is the measure of responsiveness of cost to change in

output.

outputin change ateProportion

cost in total change ateProportionElasticityCost

c

q

dq

dc

c

q

q

q

c

c

qq

cc..

q

c.

/

/Ec

Where Ec = Cost elasticity, C=Total cost, q= output

or d = change. Defining cost elasticity in terms of marginal

and Average Cost.

c

dC q MCE .

dq C AC

Cost Averageq

CAC

andCost Marginaldq

dCMC where

Elasticity of Average Cost

outputin change ateProportion

cost averagein change ateProportion ACE

2

/q

.

C C C Cd

q q q qq

q Cq dq Cq

q



dq

q

Cd

C

q

Cq

C

dq

d

.

q -

22

On differentiation we get

1EE

EdC

dq.

C

q Here 1, -

dC

dq.

C

q

C - dq

dCq

q

1 .

C

q

q

C -

dq

dC

q

1

C

q

CAC

C

2

2

2

2

Hence elasticity of average cost can be calculated by

subtracting one from the elasticity of total cost. If we know

the value of the coefficient Ec at different levels of output (q)

we may easily predict the stage in which a firm is operating.

Important points; Regarding Cost Elasticity

i) if Ec >1 relative change in cost is greater than relative

change in output. ie., increasing cost operates where

MC > AC. Both AC and MC curves are positively

sloped.

ii) if Ec < 1 relative change in cost is less than relative

change in output. ie., diminishing conditions operates

where MC < AC.

iii) if Ec=1, proportionate change in cost leads to

proportionate change in output. ie., constant return

operates. Here AC = MC when average cost is

minimum.



UNIT 4

REVENUE FUNCTION AND CURVES

Revenue is the sale proceeds of a firm. This depends

mainly upon the demand for the product.

When Q is the demand and P is the price, the product

TR = PQ is called the total revenue obtainable from this

demand and price. It represents the total money revenue of the

producers and the total money expenditure of the consumers.

Since the demand function can be expressed in the two

alternative forms Q = f (p) or p = (Q), total revenue can be

considered either as a function of price or as a function of

demand. The latter is more convenient in most cases and the

function TR = PQ is called the total revenue function of the

given demand curve, Q = f(P).

Example

If p = a–bq is the demand curve

then TR = qp = q (a–bq)

= aq–bq2 which can be reduced to the form

22

2b

a-q b-

4b

aR

The graphical presentation of this revenue function is

shown below.



We can see that the total revenue curve is thus a

parabola with axis vertical and pointing upwards as shown in

the figure. The highest point of the curve occurs where

q = a/2b. Total revenue increases as output increases at first,

reaches a maximum value a2/4b at the output q = a/ab and then

decreases as output increases further. The height of the total

revenue curve measure the total revenue obtainable from the

output indicated.

Average and Marginal Revenue

Price can be obtained from the total revenue curve. If p

is any point on the total revenue curve, the price.

OP ofGradient OM

MP

Q

TRP

Price can be regarded, infact, as 'average revenue' i.e.,

as the revenue per unit of the output concerned. Average

revenue is measured by the gradient or slope of the line joining

the origin to the appropriate point on the total revenue curve.

Since price decreases as demand increases, the line OP



becomes less and less steep as we move to the right along the

total revenue curve.

Another important concept in the theory of firm is the

Marginal Revenue. The marginal revenue is the change in the

total revenue resulting from selling an additional unit of the

commodity. Graphically, the marginal revenue is the slope of

the revenue curve at any one level of output. Mathematically

dQ

TRd is MR

Relation between AR, MR and Elasticity of Demand

The relationship between elasticity and total revenue

can be shown using calculus. Total revenue is price times

quantity. Taking the derivative of total revenue with respect to

quantity yields marginal revenue.

TR = P*Q

dQ

dpQP

dQ

d(PQ)

dQ

d(TR)MR

The equation states that the additional revenue resulting

from the sale of one more unit of a good or service is equal to

the selling price of the last unit (P), adjusted for the reduced

revenue from all other units sold at a lower price (QdP/dQ).

This equation can be written as,

pdQ

QdP1PMR

But note that (Q/P) dP/dQ=1/P. Thus



p

p

p

1MR P 1

η

1MR AR 1 where P AR Average Revenue

η

MR 1 1 MR-AR1 , - 1

AR AR

AR η

AR MR

p p

MR

AR

This equation indicates that marginal revenue is a

function of the elasticity of demand. For example, if demand

is unitary elastic, p = –1 then

01-

11

PMR

Because marginal revenue is zero, a price change

would have no effect on total revenue. In contrast, if demand

is elastic, say p = –2 marginal revenue will be greater than

zero. This implies that a price reduction, by stimulating a

considerable increase in demand would increase total revenue.

This equation also implies that if demand is inelastic, say p =

–0.5, marginal revenue is negative, indicating that a price

reduction would decrease total revenue.

REVISION EXERCISES

I Very Short Answer Questions

1. Define a cost function

2. Define average cost function

3. Define marginal cost function

4. Define total cost function



5. Define cost elasticity

6. Define elasticity of average cost

7. Define a revenue function

8. Define marginal revenue

9. Define average revenue

10. Establish the relationship between AR, MR and

elasticity of demand

II Short Answer Questions

11. What are the assumptions on the problem of cost of

production?

12. Distinguish between short run and long run cost

functions.

13. Define cost function and cost curve

14. Distinguish between cost of production and expenses of

production

15. Explain the following terms

1. c >1 2. c <1 3. c =1

16. What do you mean by total revenue function?

17. Show that MRAR

ARd

18. If the demand law is c, - x

ap show that the total

revenue decreases as output increases, MR being a

negative constant.

19. What is the nature of short run cost functions?

20. What are the different forms of cost functions? Give

examples.



21. Define elasticity of total cost. Show that the elasticity

of total cost K = MC/AC.

22. The demand curve of a monopolist is q = 400–20p and

the average cost function is .50

5q

AC Find the

equilibrium output and price.

23. Find MR for the demand function Q = 36–2p, evaluate

at Q = 4.

24. Find the MC function for the average cost function

.46

45.1q

qAC


25. Define cost function, cost curve and state their

properties.

26. The cost function is given by = a + bq + cq2, where q

is the quantity of output produced. Obtain the relation

between AC and MC.

27. Explain various cost functions of short run with the

help of a diagram.

28. If the demand function of a monopolist is p = 15 – 2x,

where x is the number of units demanded. Determine

the total revenue and sketch its graph.

29. Suppose the price p and quantity q of a commodity are

related by the equation q = 30 – 4p – p2. Find

i) Elasticity of demand at p = 2 and

ii) Marginal revenue MR

30. Let = a + bq + cq2, when q is the quantity of output

produced and a, b, c are constants. Find the expression



for AC, MC and prove that )(1)(

ACMCqdq

ACd

31. Show that average cost and marginal cost are equal

when average cost is minimum.

32. How will you derive long run cost curve from a

combination of short run cost curves.



MODULE III

THEORY OF CONSUMER BEHAVIOUR

UNIT 1

UTILITY ANALYSIS

Principle of consumption is based on fundamental

economic problem arising out of the existence of unlimited

ends and scarce means which have alternative uses. Human-

wants are unlimited but the goods and services necessary for

satisfying human wants are scarce. Every individual

consumer, group and community makes choice at different

levels and aims at maximizing his satisfaction. In a free

market economy, consumer is, however, free to choose, what

goods he will buy and to what quantum? This freedom of

choice enables the consumers maximizing their total

satisfactions. If various combinations are available to the

consumer, he will choose that combination which maximizes

his total satisfaction. This process of optimization constitutes

the subject matter of consumer's behaviour.

Theory of consumer's demand, which studies the

behaviour of a consumer confronted by an economic situation,

has undergone various vicissitudes during the last few decades.

It took about a century for the utility approach to develop in

1870 when a revolutionary change took place in economic

thinking. The utility approach developed simultaneously in

England, France, Austria, in the hands of William Stanley

Jevonns, Leon Walras and Carl Menger respectively.

Subsequent developments and refinements in the theory were,

however, made by Marshall, Clark an Fisher. The theory of

demand as developed can be divided in (a) the cardinal utility

analysis; (b) the indifference curve or ordinal utility analysis;

(c) the revealed preference analysis and (d) the cardinal utility

analysis involving risky choices. We confined ourselves to the



study of two methods (a) and (b).

Cardinal Utility Analysis or Utility Approach

'Utility' ia an attribute possessed by a commodity to

satisfy a human want, to yield satisfaction to consumer. It is

defined as the want satisfying power of a commodity.

Marshall contended that utility can be measured and developed

a cardinal utility analysis and observed, "the price which a

person is willing to pay for a commodity is the utility for that."

Thus, consumer was capable of assigning to every commodity

a number representing the amount of utility associated with it.

We assume that the utility of Qx is 5 units and that of Qy is 20

units. The consumer would like Qy four times as compared to

Qx.

Marshall defined Marginal utility, as, derived by the

marginal unit of the commodity. Total utility is the aggregate

of marginal utilities. Marshall assumed that Marginal utility

keeps on diminishing. The total utility shall increase to a

certain point, consequent upon every increase in the

consumption of a commodity. The point at which total utility

becomes maximum, is known as saturation point for that

commodity.

According the Marshall

MU=Marginal Utility

= Utility derived from the consumption of additional

unit of a commodity TU.

TU = Total utility = sum of MU's

Quantity U MU TU

Q1 10 10 10

Q2 40 30 40

Q3 50 20 50



The utility of a commodity diminishes with more

consumption of the same. Marginal utility first increase,

reaches the maximum and then diminishes. In the figure 'Z' is

called the Saturation point for a commodity because after

reaching 'Z' any additional consumption of a good will not give

more satisfaction to the consumer.

Maximization of Utility

While maximizing utility we assume the consumer to

behave rationally. He has to maximize his utility function

taking certain constraints into consideration. It was Marshall,

who tried to explain this problem theoretically through the Law

of Equi-Marginal Utility, but modern economists have given a

mathematical exposition to it. Let us suppose that Marshallian

assumptions regarding the measurement of utility exist in the

economy.

The problem of the consumer is to

Maximize U= f (q1, q2, ...... qn) (1)

Subject to Y = P1q1 + P2q2 + ..... + Pnqn (2)

where U = Consumers total utility function

Y= Consumers income



q1,q2,..,qn=quantities of various commodities

consumed.

P1, P2....Pn = prices of n commodities each.

For ease of exposition we take two commodity case

then

U= f (q1 q2) (3)

Y = p1 q1 + P2q2 (4)

from (4) 2

212

P

qpYq

(5)

Put (5) in (3) we get

2

111

P

qPYq fU (6)

Two conditions for maximization are

01

dq

dU (first order condition) (7)

02

1

2

dq

Ud (second order condition) (8)

Taking the Ist

order derivative of (6) we get

0P

Pff

dq

dU

2

121

1

(9)

11 2 1 2 1 2

2

pf f where f and f MU of q and q

p

2

1

2

1

P

P

f

f (10)

ie., 2

1

2

1

P

P

MU

MU (11)



MU1 = Marginal utility of first commodity

MU2 = Marginal utility of second commodity

For maximization of total utility the ratio of the

marginal utilities of the two commodities must be equal to the

ratio of their prices.

Rewriting eg; (11) we get

2

2

1

1

P

MU

P

MU (12)

Generalizing (12)

n

n

3

3

2

2

1

1

P

MU....

P

MU

P

MU

P

MU (13)

For second order condition

1

2

1

2

1

2

2

122

2

121

1

21212

1

2

dq

P

P d

fdq

dq

p

P f

P

Pf

dq

dqff

dq

Ud

2

1 111 12 22

2 2

P Pf 2 f f 0

P p

where 2112

2

1

1

2 ff and P

P

dq

dq

Multiplying by ,2

2P a positive number, on both sides of

the above we have 0PfPP2fPf 2

1222111

2

211 (14)

Hence the maximum satisfaction shall be obtained

when the two equations (13) and (14) hold true.

Criticisms of the Utility Approach

The criticisms of this approach are based on the

unrealistic assumptions.



Assumptions: (i) Utility can be measured cardinally

Criticism (i) Utility cannot be measured by price

because two persons who pays the same

price for a commodity have different

utilities

Assumptions: (ii) Utilities are independent

Criticism (ii) In real life commodities are either

substitutes or compliments. Hence their

utilities are interdependent

Assumptions: (iii) Diminishing MU law holds in the

economy

Criticism (iii) Sometimes even this is not true eg: For

greedy people MU of money never

diminishes

Assumptions: (iv) MU of money is constant

Criticism (iv) MU of money is different for different

people



UNIT 2

INDIFFERENCE CURVE ANALYSIS

Indifference Curves and Indifference Map

A popular alternative to the theory of consumer's

demand is indifference curve analysis; developed by

Edgeworth.

According to this theory there can be a number of

combinations of two or more commodities yielding the same

level of satisfaction. Hence all the combinations are equally

desirable or preferable to the consumer. The curve joining all

commodity combinations giving the consumer the same level of

satisfaction is called an indifference curve.

An indifference map is a collection of indifference

curves at different levels of satisfaction. Combinations of

goods situated on an indifference curve yield the same utility.

Combinations of goods lying on a higher indifference curve

give higher level of satisfaction and are preferred.

Combinations of goods on a lower indifference curve yield a

lower utility.

The total utility function in the case of two

commodities x and y is u = f(x,y). The equation of an

indifference curve is, u = f(x,y) = k where k is a constant. An

indifference map can be obtained by assigning to k every

possible value. An indifference curve is shown in fig (i) and a

partial indifference map is depicted in fig (ii).



Properties of the Indifference Curves

1. Indifference curve slopes downward to the right.

2. The father away from the origin an indifference curve

lies, the higher the level of utility it denotes

3. Indifference curves do not intersect.

4. Indifference curves are convex to the origin

Marginal Rate of Substitution (MRS)

The concept of marginal rate of substitution (MRS) is

an important tool of indifference curve analysis. The rate at

which the consumer is prepared to exchange one commodity

for another is known a marginal rate of substitution.

The marginal rate of substitution of x for y is defined as

the number of units of commodity y that must be given up in

exchange for an extra unit of commodity x so that the

consumer maintains the same level of satisfaction. That means

the rate at which the consumer trades off y for x is called

Marginal Rate of Substitution (MRS). As the consumer slides

down the indifference curve, he is willing to give up less and

less y for a given gain in x and therefore MRS is negative and

it is the slope of indifference curve at any one point.

Therefore MRS of two commodities x and y is given

by



MRSxy = Slope of the indifference curve at any given

point.

x

y -

Indifference Curve Approach for Maximizing Utility

It was for the first time in 1881 that a British

economist, Edgeworh introduced the concept of indifference

curve. In 1906, Vilfredo Pareto, an Italian Economist

modified the 'Edgeworth Technique'. However, the main

credit of developing this concept goes to British Economists

J.R. Hicks and R.G.D. Allen. Now, this technique is employed

by the consumer to solve his problem. It is based on ordinal

number. Indifference curve is however, the locus of all those

combinations of two commodities q1 and q2 which yield equal

satisfaction to the consumer.

In our analysis, we are assuming two commodities q1,

q2 and the utility function and the constraints are given below:

We know U=f (q1 q2) - Objective function

Y= P1 q1 + P2q2 - constraint

P1 q1 = expenditure incurred on Ist commodity

P2 q2 = expenditure incurred on 2nd

commodity

Where P1, P2 and y are exogenous variable and q1 and

q2 are endogenous variables

We know that for maximum satisfaction 2

1

2

1

P

P

M

MU

In the following figure

MN = Price line (Budget line)

OM = Maxim good q1 that can be purchased if entire

income is spent on it.



IC = Indifference curve

E = Equilibrium point

Slope of the indifference curve 2

1qq

MU

MUMRS

12

Slope of budge line = 2

1

P

P

For equilibrium the marginal rate of substitution (MRS)

must be equal to the slope of the price line.

In the diagram at 112

2

PE,MRS

P

At this point indifference curve (IC) is tangent to price

line MN.

Proof:

U= f(q1q2)

dU = f1dq1 + f2 dq2 = 0



f1dq1 = –f2dq2

MRSdq

dq

f

f

1

2

2

1

= negative of the slope of the

indifference curve.

–f1 q1 is the amount of satisfaction given up by a

consumer to have more utility or satisfaction from q2 which is

equal to +f2 q2.

–f1 q1 = + f2q2 in utility terms



UNIT 3

METHODS OF MAXIMIZATION OF UTILITY

Method 1

We have

2

111

2211

21

p Y

fqU

p

qpYq

qpq

q

Conditions of Maximization

(ii) 0U

(i)2

1

2

1 q

U

q

<0

11 2

1 2

pU f 0

pf

q

2

1

2

1

2

121

p

p

f

for

p

pf f

Here 2

1

2

1

2

1

f

f

p

p MRS thereforeMRS,

f

f

Now denoting the second partial derivates of the utility

function by f11and f22 and the second cross partial derivatives

by f11and f22 the second order condition for maximization

requires that

1

222 1 1 2

11 12 21 22 22

1 1 2 2 1 1

p

pp pf 0

dq dqU

f d f fq q p p dq dq



2

1 1 111 12 21 22

2 2 2

p p p f f f f

p p p

<0

Where

1

22 1

1 2 1

0

p

pq pand

q p q

because prices are assumed to be

constant, or

2

1 111 12 22

2 2

f 2 0p p

f fP p

Multiplying by 2

2p on both sides we have

02_ 2

1222112

2

211 pfppfpf

This is the required condition for maximization of utility

Method II

Lagrange's method

we form a new function as

V = f (q1q 2) + (Y–p1q1–p2q2). It can be written as

V = f (q1,q 2, )

where = Lagrange’s multiplier to budget constraint.

To maximize V1 we calculate the partial derivatives of V with

respect to the three variable q1, q2 and and set them equal to

zero.

(3) 0

(2) 0

(1) 0

2211

22

1

11

1

qpqpYV

pfq

V

pfq

V



From (1) and (2)

1 2

1 2

1 2 1 1

1 2 1 2

and

f f f por

p p f p

f f

p p

For second order condition we find out the second

order derivatives of (1), (2) and (3) and the relevant bordered

Hessian determinant be positive.

11 1 12 2 1 1

21 1 22 2 1 2

1 1 1 1 2 2 2 2

f dq f dq p dλ λ dp 0

f dq f dq p dλ λ dp 0

dy-p dq -q dp dq q dp 0p

(5)

Rewriting (5) i.e., constants on right hand side and

variables on the left hand side we have

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2 1 1 2 2

f dq f dq p dλ λ dp

f dq f dq p dλ λ dp

-p dq -p dq - dy dp +q dp 6q

(6)

Converting to matrix notation

11 12 1 1 1

21 22 2 2 2

1 2 1 1 2 2

f _p dq dp

f _p dq

_ _p 0 _

f

f dp

p d dy q dp q dq

(7)

from (7) we observe the relevant bordered Hessian determinant

as

11 12 1

21 22 2

1 2

f f _p

f _p 0

_p _p 0

f

(8)



Expanding (8) we get

0___f __p _f _p 12221212121

2

111 pfpppf

or 0 _ 2

12221122121

2

211 pfppfppfpf

or 0 2_ 2

1222112

2

211 pfppfpf

or 0 2_ 2

1222112

2

211 pfppfpf (9)

Equation (9) is the same as we proved in the first method

Similarity of Utility approach and IC approach

1. Both approaches aims to prove that consumer tries to

maximize the satisfaction he gets from his expenditure.

2. According to Prof. Leftwich the indifference curve

analysis can be transformed into the utility analysis.

We have, 2

112

p

pMRS

In utility analysis maximum utility is at

2

2

1

1

p

MU

p

MU

MRS12 = units of q2 the consumer is willing to give up

of 1 unit of q1 and remains at the same level of total

satisfaction.

That is

1122 MU x qMU x dq d (3)

1

2

2

1

dq

dq

MU

MU (4)

Now 1

2

dq

dq= MRS12 (5)



Put (5) in (4) we get 12

2

1 MRSMU

MU (6)

We know 2

112

p

pMRS

Substituting in (6) we get

2

1

2

1

p

p

MU

MU (7)

or 2

2

2

1

P

MU

P

MU

Thus both approach lead to the same conclusion.

REVISIONEXERCISES

I. VeryShortAnswerQuestions

1. Define utility function.

2. State the law of equi-marginal utility

3. What is an indifference curve?

4. What is an indifference map?

5. Define marginal rate of substitution.

6. Explain the concept of utility function.

7. What are budget constraints?

8. What is ordinal utility function?

9. What is cardinal utility function?

10. Write a note on Lagrange’s multiplier



II Shoty Answer Questions

11. Distinguish between cardinal and ordinal utility

analysis

12. What do you mean by utility? Explain the utility

approach developed by Alfred Marshal.

13. Explain indifference curve approach of utility analysis.

14. The utility function is given by u = 3x2y

2 + y

2, prove

that the rate of change of marginal utility of x w.r.t.y

and vice versa are equal.

15. What are criticisms against utility approach?

16. What are the properties of indifference curve?

17. What is the significance of budget constraint in

consumption theory?

18. Explain the concept of 'rate of commodity substitution'

19. Explain the equilibrium of the consumer under

indifference curve analysis.

20. What are the maxima and minima conditions of

consumer's equilibrium?


21. Give mathematical exposition of consumer behaviour

22. Give mathematical approach of consumer behaviour

from indifference curve analysis.

23. How will a consumer maximize his utility according to

Marshallion utility approach?

24. Explain Lagrangian Method of utility maximization.

25. What are the similarities between utility approach and

indifference curve approach?



26. Define the concept of utility function. Examine its

properties and role in the theory of consumer

behaviour.

27. Explain the theory of utility maximization of a

consumer.

28. A utility function u = f (q1,q2) with total income

M = p1q1 + p2p2. Obtain the conditions for maximum

utility for given p1 and p2.

29. Given an utility function u = xy where x and y are

quantities. If the budgetary constraints is 2x+y = 6.

What is the maximum value of u.

30. The total utility function is given by u = q1 q2 when q1

and q2 are quantities. If the prices are p1 = Rs.4/- , p2 =

Rs.20/- and the consumers income y = Rs.100/- find the

equilibrium level of each commodity q1 and q2.



MODULE IV

ECONOMIC APPLICATIONS OF

DERIVATIVES

UNIT I

MARGINAL, AVERAGE AND TOTAL CONCEPTS

In economics, variation of one quantity y with respect

to another quantity x usually described in terms of two

concepts.

i) Average concept and

ii) Marginal concept

The average concept expresses the variation of y over a

whole range of values of x. It is usually measured from zero to

a certain selected value, say from 5 to 10. Whereas marginal

concept concerns with the instantaneous rate of change in the

dependent variable for every small variation of x from a given

value of x.

Therefore a marginal concept is precise only when

variation in x are made smaller and smaller i.e., considering

limiting value only. Hence dx

dyis interpreted as the marginal

value of y.

Few applications of the derivative are discussed below:

1 Average and Marginal Cost

Suppose the total cost C of producing and marketing x

units of an item is represented by the function C=C(x). Then

the average cost which represents the cost per units is given by



Average Cost x

xCor

x

CAC

Now, if the output is increased from x to x + x , and

corresponding total cost becomes C+ C then the average

increase in cost per unit output is given by the ratio ΔC

Δx and

the marginal cost is defined as:

Marginal Cost = dx

dc

Δx

ΔC

0x

Lt

That is, marginal cost is the first derivative of the total

cost C with respect to output .

Example I

The total cost, C(x) associated with producing and

marketing x units of an item is given by

C(x)= 0.005 x3 - 0.02 x

2 – 30 x + 3000

find i) total cost when output is 4 units

ii) average cost of output of 10 units

iii) marginal cost when output is 3 units

Solution

i) Given that

C(x)=0.005x3 – 0.02x

2 – 30x +3000

For x = 4 units, the total cost C(x) becomes

C(x)= 0.005(4)3 – 0.02(4)

2 – 30 (4)+ 3000

= 0.32 – 0.32 –120 + 3000 = Rs.2880

ii) Average Cost x

xCAC



= 3 20.005 0.02x 30x 3000

x

= 0.005x2 –0.02x–30+

x

3000

Average Cost at x = 10 units becomes

AC = 0.005 (10)2 – 0.02 (10) –30+

10

3000

= 0.5–0.2–30+300 = Rs.270.3

iii) Marginal cost at x is given by dx

dc

Therefore differentiating both sodes of C(x) with

respect to x, we have dx

dcC(x)= 0.005 x 3x

2–0.02 x 2x–30

Marginal cost at x=3 becomes

dx

dc = 0.015(3)

2 –0.04(3)–30

= 0.135–0.120–30 = Rs.30.015

2. Total Revenue (TR), Marginal Revenue (MR) and

Average Revenue (AR)

Let p be price per units and q is the number of units of

an item sold. Then the total revenue (R) is given by

R = p.q

The demand function is P=f(q). therefore R becomes

R= q.f(q)

Now average revenue (AR) or revenue per unit which

represents the price per unit is given by

X=3



(price) pq

p.q

q

RR A

This shows that the average revenue and price are

identical. Since total revenue is given by R = p.q, therefore

marginal revenue (MR) is defined as:

1

dR dp q dp. p 1

dq dq p dqMR p q

Example 2

The demand for a certain product is represented by the

equation p = 20 + 5q – q2 where q is the number of units

demanded and p is the price per unit. Find marginal revenue

function. What is the marginal revenue at q = 2.

Solution

The total revenue is given by

Revenue, R = (demand) (price)

= q(20+5q–q2) = 20q+5q

2–q

3

The marginal revenue (MR) at q = 2 is given by

23q10q20

dq

dRMR

= 20+10(2)–3(2)2

= 20+20–12 = 28

The marginal revenue (MR) at q = 2 is given by

23q10q20

dq

dRMR

= 20+10(2)–3(2)2= 20 + 20 – 12 = 28

Hence, the marginal revenue when two units are

demanded is Rs.28.



3. Elasticity

The elasticity of a function y = f(x) at a point x is

defined as the ratio of the rate of proportional change in y per

units to the proportinal change in x. That is,

dx

dy.

y

x

dx/x

dy/yEyx

The elasticity of a function is independent of the units

in which the variables are measured because its definition is in

terms of proportional changes. Notations usually used to

denote elasticity are: ey, or ny or yx.

The above definition can also be expressed as:

Function Average

Function Marginal

y/x

dy/dx

dx/x

dy/yn y

The crucial value ny=1. However the sign of ny

depends upon the sign of dx

dy. It may be positive, negative or

zero. Apart from the sign, we are also concerned about the

absolute value |ny| of ny.

a. Price Elasticity of Supply

Let q be the supply and p be the price and the function

is expressed q = f (p). Then the formula for elasticity of

supply is same as that of ny. That is ns= dp

dq.

q

p

The sign of ns will be positive because slope of supply

curve is positive.

b. Price Elasticity of Demand

The price elasticity of demand at price 'P' is derfined

as:



dqdpdp

dq

q

p

p

q

q

pn Ltd

/

1

q

p- ._

0x

The sign of nd is negative, because, in general the slope of

demand dp

dqis negative.

c. Marginal Revenue and Elasticity of Demand

You know that the total revenue (R) is given by R = p.q

Where p is the price and q is the quantity sold.

Also the average revenue (AR) and marginal revenue

(MR) are defined as

Average revenue (AR) pq

qp

q

R

.

Marginal revenue (MR) ).( qpdq

d

dq

dR

dq

dp

p

qp

dq

dpqp 1 .1.

)1.....(1

11

1 .dd

ARp

Since | d | = dp

dq

q

p.

From this definition of MR, it follows that

i) If | d | = 1, then AR = 0 and hence MR = 0, i.e., total

revenue remains constant with a fall in price.



ii) If d | >1, then AR=0 and hence MR > 0, i.e., total

revenue increases with an increase in demand or with a

fall in price.

iii) If | d | < 1, then AR < 0 and hence MR < 0, i.e., total

revenue decreases with an increase in demand or with a

fall in price.

iv) d

d

1 hint: MR AR 1

AR

AR MR

Example 3

Suppose the price p and quantity q of a commodity are

related by the equation q = 30–4p–p2.

Find: i) Elasticity of demand d at p = 2, and

ii) Marginal revenue (MR).

Solution

i) Elasticity of demand, d is defined as:

2

d 430../

/pp

dp

d

q

p

dp

dq

q

p

pdp

qdq

2

2

2 430

2424.

430 pp

ppp

pp

p

Thus at p = 2, 9

8

18

16

)2(2430

)2(2242

2

d

x

x

ii) Marginal revenue (MR) is defined as:

dq

dp

dp

dR

dq

dRMR .



dpdqdp

qpd

dpdqdq

dR

/

1.

).(

/

1.

p

pppdp

d

24

.1430. 2

22 1 30 8 3

30 8 30 4 2 4 2

p pp p

p p

4. Relationship between Production, Costs and

Revenue Functions

The chain rule of differentiation can be used to

establish certain economic relationship between production,

cost and revenue functions.

Let us consider a production function where output

depends on employment of labour with fixed capital and other

inputs such that

Q = g(L) ......(1)

Where Q is output and L is labour employment.

Similarly, the total revenue function of the firm depends on the

quantity sold such that R = f (Q) .....(2)

Where R is total revenue

In order to know marginal revenue product of labour,

or in other words in order to know what is the increment of

total revenue as a result of additional employment of labour,

we are to find out the derivative of R with respect to L, i.e.,

dL

dR.

Now following chain rule of differentiation

(L) g' x (Q)f'dL

dQx

dQ

dR

dL

dR



Since dQ

dRis marginal revenue (MR) and

dL

dQis marginal

physical product of labour (MPP) and since dL

dRrepresent

marginal revenue product of labour (MRP).

MRP = MR x MPP .....(3)

Note: The same relationship given by (3) can be established if

we consider a demand function.

P = f (Q) ....(4)

instead of total revenue function R = f (Q).

5. Relationship between Marginal Product (MP) and

Marginal Cost (MC)

In a single explanatory variable production function

(short run), only one of the factors of production is assumed to

be variable and all other inputs are taken to be fixed. Most

common form of such production functions assume fixed

capital equipment and labour as the only variable input such

that Q = f(L). If we represents the wage rate, the total variable

cost will be V = w.L.

The marginal cost of labour is the additional variable

cost (V) due to employment of an additional labour (L) which

is given by dL

dV. But marginal cost of production is the

additional total cost due to increase in an additional output (Q).

Since the total fixed cost remains constant, the marginal cost

will be additional variable cost due to increase in an additional

output.

Thus dQ

dVMC

Using chain rule of differentiation



dQ

dL

dL

dV

dQ

dV

dQ

dVMC

or dV 1 w

MC dLdL dQ f '(L)

product Marginal

factor variableofcost MarginalMC

The slope of the MC curve will be given by the

derivative of MC with respect to output (Q). So the slope of

MC is

(L)f'

w

dQ

d

dQ

Vd2

2

Assuming that wage rate (w) is constant, We substitute

Z= f ' (L) to apply chain rule to find out 2

2

dQ

Vdwhich is given

by 3

(L)' f

(L)wf"- on differentiation. Since both w and f '(L) are

positive, the sign of 2

2

dQ

Vdwill depend on the sign of f "(L).

2

2

dQ

Vd>0 if and only if f " (L)<0. Thus if the marginal product

of labour is diminishing { f "(L)<0}, the slope of MC

2

2

dQ

Vdwill be positive. On the other hand if the marginal

product of labour is increasing {f "(L)>0}, the slope of

2

2

dQ

VdMC will be negagive.

a. Marginal Utility



In the theory of economic behaviour, normally utility

function relates to total utility (u) obtained from the

consumption of a given quantity (Q). Thus given the utility

function u = u(Q) the additional derivative from an additional

infinitesimal consumption of Q is given by derivative

'du

u QdQ

which is called marginal utility. Further, the

change in marginal utility due to infinitesimal change in Q is

given by the second order derivative.

2

n 2

d du d uu"(Q)

dQ dQ dQ

If marginal utility (MU) declines as increases, then

u"(Q) < 0 indicating the operation of the law of diminishing

marginal utility.

b. Marginal Product

In short-term analysis the production process is

assumed to have fixed inputs and only one variable, say labour

input (L). Therefore, in short-run while formulating a

production function, the output produced (Q) is a function of

labour employed (L) only such that Q = f(L). The change in

output due to infinitesimal change in the employment of labour

is given by the derivative (L)f'dL

dQ which is called marginal

product of labour. Again, the change in the marginal product

(MP) of labour due to infinitesimal change in L is given by the

derivative of MP with respect to L. That is

(Q)f"dL

Qd(MP)

dL

d2

2

Since MP declines as L increases so f"(Q) < 0 implying

the operation of the law of diminishing marginal product.



c. Marginal Propensity to Consume (MPC)

In macro-economics, we frequently use the concept

marginal propensity to consume (MPC) in the formulation of

consumption function. In such a consumption function, total

consumption expenditure (C) is assumed to be dependent on

aggregate income (Y) such that C = C(Y). The MPC is

defined as the change in consumption expenditure due to an

infinitesimal change in the level of income. So MPC is given

by the derivative of C with respect to Y such that

)(1 YCdy

dCMPC

If saving is denoted by S = Y – C = Y – sC(Y) = S(Y)

So the marginal propensity to save (MPS) is

).('dY

dSYS In the same way, if the import (M) is considered

to be the function of national income (Y) such taht M = M(Y),

then the derivative )('dY

dMYM is the marginal propensity to

import.

Example 4

If T = 20+x+ 20 x,x

100 and T is the total cost of

producing x chairs, find

i) Total cost for producing 100 chairs and 101 chairs

ii) The cost of 101 at chair

iii) The marginal cost at 100 chairs and 101 chairs

Solution

20,100

20 xx

xT

USER

Sticky Note

=



i) Total cost for producing 100 chairs is

112100

10010020100)T(at x

Total cost for producing 101 chairs is

99.112101

10010120101)T(at x

ii) The cost of 101 st chair = The cost for 101 chairs-the

total cost for 100 chairs = 0.99

iii) Marginal cost is

x

100x20

dx

d

dx

dT=

2 2

100 1000 1 1

x x

Marginal cost at x = 1000 is 0.9900 and Marginal

cost at x=101 is 0.9902.

Example 5

Find the average cost and the marginal cost functions

from the total cost function.

C = 60+10 x + 15x2

Solution

Given C = 60 +10 x 15x2

Average cost = x

15x10x60

x

C 2

= 15x10x

60

Marginal cost, 30x1015(2x)10dx

dC

USER

Sticky Note

+



Example 6

Find the slope of average cost curve in terms of

average cost and marginal cost.

Solution

Average cost = C/x

Slope of AC curve = /d

C Xdx

2

. 11 1

dCx C

dC Cdx MC ACx x dx x x

Note:

The above example provides the following useful

results

i) Slope< 0 if MC < AC. (This is a situation in which the

manufacturer will try to increase the production).

ii) Slope > 0 if MC > AC. (This is a situation in which the

manufacturer will try to decrease the production).

iii) Slope = 0 if MC = AC. (This is a situation in which the

manufacturer will try to maintain).

Production Function

The production function for two input variable, say

labour L and capital K, is given by

q = f(L,K) where q is the quantity of output, L is the

labour input, and K is the capital input.

Marginal product of factors of L and K are

K

qMP and

L

qMP kL

Marginal rate of technical substitution is



,

/

/L K

q LMRTS

q K

Factor intensity = the capital labour ratio K/L.

Average product of factors are k

q

L

qAPL kAP and

Example 7

Find out the marginal revenue function (MR) given the

average revenue function AR=10–0.5 q.

Solution

MR is the derivative of total revenue (TR) function and

TR is the product of AR and quantity (q). Now

TR = (AR). q = (10–0.5q) q = 10 q = 0.5q2

qqxTR 105.0210)(dq

dMR

Example 8

The total cost C of a firm is given by

32

3

1801001000 qqqC

Where q is the quantity produced

i) Find the marginal cost of production

ii) At what value of q does marginal cost equal average

variable cost?

Solution

(i) Marginal cost is given by the differentiation of total cost

function.



2

3

138021000

dq

dCMC qxqx

(ii) The average variable cost = TVC/q

3

22

q3

180q100

q

q 3

180q100q

The value of q at which MC is equal to average

variable cost (AVC) is given by the solution of the equating

MC with AVC such that

22 q

3

180q100q160q-100

or 080q160qq3

1q 22

or 22 2

q 80q 0 or q 80 i.e.,q 1203 3

Example 9

The average revenue function is given by AR=100 –3q.

Find out the elasticity of demand when q=5.

Solution

The elasticity of demand is given by the relation

AR

ηAR MR

To find out M R, we are to obtain the T R function

6q100(TR)dq

dMR

3q-100q3q)q-(100(AR)qTR 2



When q = 5 AR =100 - 3 x 5=85

and MR = 100- 6 x 5 = 70

66.515

85

70-85

85

MR- AR

AR

Since >1, the commodity is likely to be a luxury

good

Example 10

If the total cost function is given by

TC = 100 - 2q + 0.5q2

show that the slope of average cost (AC) curve is negative

when output is less than 10.

Solution

q

qq

q

TCAC

25.02100

Slope of AC is given by the differentiation of AC with

respect to quantity (q).

2

22 )5.02100()5.02100(.

)(q

dq

dqqqqq

dq

dq

ACdq

d

2

25.02100)25.02(

q

qqqxq

22

2

2

22 1005.0

1005.05.021002

qq

q

q

qqqq

2q

100-0.5AC of slope the



When q=10, the slope of AC is – 0.5. When q < 10, the

slope of AC will be less than -0.5.

Example 11

The total cost function of a firm is given by

TC = 625-5q+q2

show that optimum size of output of the firm is 25 units.

Solution

The optimum level of output of a firm is given by the

level of output at which average cost is minimum. The

average cost is minimum when the slope of AC curve is zero.

Now q

q5q625

q

TCAC

2

2 2

2

dq (625 5 ) (625 5 )

dq the slope of AC

dqq q q q

dq

q

2

2

q

q5q6252q)5q(

2

2

2

22 625

q

q5q6252q5q-

q

q

or slope of AC=1- 625/q2

When q = 25, the slope of AC = 0 and so 25 is the

optimum level of output. Similarly when q >25, the AC is

positively sloped as .1625

2

q

Example 12

Given the price equation, p = 100-2Q where Q is

quantity demanded, find



i) The Marginal Revenue

ii) Point Elasticity of demand when Q=10

iii) Nature of the commodity

Solution

i. Since marginal revenue (MR) is obtained by

differentiating the total revenue function with respect to output

(Q), we find out total revenue first, which is defined as

TR = AR x Q

or TR = (100 – 2Q) Q =100 Q – 2Q2

4Q - 100 MR

ii. Point Elasticity of demand is obtained from the

following relation

when Q 10d

AR

AR MR

MR = 100 – 4 x 10 = 60

P = AR=100–2 x 10=80

80

480 60

d

iii. Since d >1, the commodity is elastic in demand and

is supposed to be a luxury good.

Example 13

Given the production function Q = 5½ and the price

equation P = 200 – 2Q obtain the marginal revenue product of

labour (L) when L = 25.

Solution

Marginal revenue product of labour is given by the

differentiation of total revenue function with respect to labour



employment.

Now total revenue (R) is given by the product of price

and quantity so that

R = P x Q or R=(200-2Q) Q = 200 Q – 2Q2

Now using chain rule,

)(5LdL

d x)2Q(200Q

dQ

d

dL

dQx

dQ

dR

dL

dR ½2

-½5200 4

2Q L

5050100505

50050

(25)

500

2

20x5

2L

1000

2L

5)20L(200

½½½

½

Example 14

Given a consumption function

Y3

50001000C(Y)C

i. Find marginal propensity to consume when Y= 97

ii Find marginal propensity to save when Y = 97

iii Determine whether MPC and MPS move in the same

direction when Y changes.

Solution

MPC is given by the differentiation of the function

Y respect towith Y3

50001000C

Now C=1000 – 5000 (3+Y)-1



22 )3(

5000

)3(

5000)1(0

dY

dCMPC

YY

5.01000

5000

)100(

5000

)973(

500022

ii. Saving function is defined as S=Y-C

or S = Y – 1000 + 5000 (3+Y) –1

10000

50001

)3(

5000)1(01

dY

dSMPS

2

Y

=1-0.5=0.5

iii. In order to verify whether MPC and MPS move in the

same direction or not, we are to find out the rate of

growth of MPC and MPS. That means we are to find

out the derivatives of MPC and MPS.

Now 0Y)(3

50002)(

dY

Cd(MPC)

dY

d32

2

and 0)3(

25000

Y)(3

50002)(

dY

Sd(MPS)

dY

d332

2

y

x

Since 0.dY

Sd0and

dY

Cd2

2

2

2

MPC and MPS move in the

opposite direction as Y changes.

Exercises

1. Calculate the elasticity of demand of the following

functions.

a) D=10P-5

b) D=D=10+0.25p

1

2. Calculate the elasticity of demand of the following

functions when price p=10



a) D.100 – 2P + 0.004 p2 b) D=720 – 6.5p

3. Calculate the elasticity of supply of the following

supply functions when price p=10

a) S= -100 + 2P b) S = -77 + 4P + P2

4. Given the consumption function

Y)(5

60002000C(Y)C

i) Find out marginal propensity to consume (MPC)

and marginal propensity to save (MPS) when Y =

95.

ii) Also show that MPC and MPS move in the

opposite direction when income (Y) changes.

5. Given the production Q = 2.5L2 where Q and L are

quantity and labour and the given total revenue

function R=120 Q – 0.2 Q2 find marginal revenue

product of labour (L) when L=10.

6. A monopolist's demand curve is given by P=200 – 2.5q

Where P is price and q is quantity demanded

a) Find the marginal revenue function

b) Establish the relationship between the slopes of

average and marginal revenue curves.

c. Find out the price at which marginal revenue is

zero.

7. The total cost function of a firm is given by TC=200q –

005q3 where q is the quantity produced. Find out the

output at which marginal cost (MC) is equal to average

cost (AC).

8. A firms production is given by



23 10LL3

2f(L)Q

Where L is the labour employed.

Show that diminishing marginal product of labour

operates when employment of labour is 6 or more.

9. For the following total functions, state whether

marginal cost is increasing or decreasing function of

quantity produced (Q).

a) C=1000+120Q – 10Q2 +2Q

3

b. C=5050 +90Q -2Q2

10. The demand function of a firm is given by p = 50 –

0.005 q and the cost function c = 30q + 10,000 where q

is the number of outputs. Find the optimum output

level.

11. A firm has the demand function P =12 – 3 x and the

cost function T=x2 +2x. Find average cost, average

revenue, marginal cost, marginal revenue and profit for

the firm at the point of equilibrium.

12. Suppose the two demand curve of the monopolist are

P1=100-2x1 and P2 = 80 –x2 and the total cost is C=20

(x1+x2) where P1 and P2 are price and x1 and x2 are the

quantities demanded in market I and market II

respectively. Determine the equilibrium level of output

and prices in both the markets.

13. When the price of a good is x, its demand u and supply

v are given by 2x vand

2x

8xu

. Find the rate of

change of demand and the rate of change of supply at

equilibrium price. What is the elasticity of demand at

the equilibrium price?



14. The total cost, T of producing q units is T=100 + 7q

+ q find the marginal cost at (i) output of 100 units

(ii) output of 400 units.

15. The total cost T of manufacturing q units of a product

is 2q3q225q

3T . Find the cost of 11

th unit.

16. If the cost of selling (q units) is T and

27q4q500q

5T find the selling cost for (i)

100 units (ii) 101st units.

17. If the total cost of making q litres of an acid in Rs. is T

= -30 +80q½ . Find the number of units at which the

marginal cost is Rs.0.25.

18. Given the sales revenue function R and total cost

function T where R=1000x2 + 1000 x and T = –10000

x + 30,000 where x is the price of an item.

i) Find dR dT

and dx dx

ii) dR dt

Find x for which and dx dx

iii) Find the profit function

iv) What can you say about the value of x obtain ....

v) What is the total profit at that value of x.

vi) What can you say about this profit.

19. If the total cost is T = 50 + 10q + 25q2 find the average

and marginal cost when q =13.

20. Given the demand function q=165- 3p- 2p2 find the

elasticity of demand at the price p =5.



21. The supply y of a commodity at the price x is given by

y=4x2 – x+3. Find the rate at which supply is changing

when the price is 2.

22. The total money deposited in bank, y, upto the end of x

years is given by y=2x2 +x – 79. Find the rate at which

the deposit increases at the end of 5 years.

23. At the price q, the demand function y is

2

127

x

xy . Find the marginal revenue as x = 3.

24. What is the marginal revenue function for the demand

p a bx .

25. If q=100 + 10K –K2 is a production function where K

represents the capital. Find the marginal productivity

when the capital is 2.

26. If the demand function is x =25 – 4p + p2 where x is the

demand for commodity at price p, find the elasticity of

demand.

27. If the demand law is p=a – bx find the total revenue

function and marginal revenue function,

28. If the demand law is c,x

ap show that the total

revenue decreases as the output increases, MR being a

non-negative constant.

29. The demand function of a monopolist is p=15 –2x and

cost function is C(x)=x2 + 2x. Find (i) Marginal cost

(ii) Marginal Revenue (iii) Average Cost (iv) AC

when the output is 4 units.

30. For the demand law p =10–x/2 find (i) d (ii) TR

(iii) MR.

31. If the marginal revenue is 25 and elasticity of demand

w.r.f price is 2. Find the average revenue.



UNIT 2

MAXIMA AND MINIMA

Increasing and Decreasing Functions

We recall the definition of an increasing functions.

Function is an increasing function if the value of the function

increases with an increasing the value of the argument and

decreases with a decrease in the value of the argument in.......

f(x) is an increasing function if the following is true.

ε x,for x ),f(x)f(xxx 212121 domain of the

function. Likewise, the value of a decreasing function

increases with a fall in the value of its argument and vice-

versa.

i.e., ε x,for x ),(x f)f(xxx 212121 domain of the

function.

This section explains show the notation of the

derivative of a function can be applied to check whether given

function is an increasing function or a decreasing function in a

given interval. We have learnt that the first derivative can be

interpreted as the rate of change of the function with respect to

it argument. If the sign of the first derivative is positive, it

means that the value of the function increases as the value of

the argument increases and decreases as the value of the

argument decreases. But this is precisely the definition of an

increasing function.

Definition

A function y = f(x) differentiable in the interval (a,b) is

said to be an increasing function if and only if its derivative on

(a, b) is non negative.



0dx

dy i.e., in the interval (a, b).

A function y = f (x) differentiable in the interval (a,b) is

said to be a decreasing function if and only if its derivative on

(a,b) is non positive.

0dx

dy i.e., in the interval (a, b).

The derivative of a curve at a point also measures the

slope of the tangent to the curve at that point. If derivative is

positive, then it means that the tangent has a positive slope and

the function (curve) in question increases as the value of the

argument increases through the point in the neighbourhood of

this point. Similar interpretation is given to the decreasing

function and negative slope of the tangent. This is illustrated

in the following figures.

Example 1

Write down the derivative of 3x3 + 3x

2 + x – 1 and

show that this function is monotonic increasing.



Solution

2223 1)(3x16x9x1)x3x(3x

dx

dy

1)x3x(3xdx

d 23 is always positive since (3x+1)2

is always positive. Square of a real number is always positive.

Therefore, 3x3 +3x

3+ x – 1 is monotonic increasing

Example 2

Are the following functions monotonic?

a) y = –x6 +5 b) y=4x

5 + x

3 +3x

Solution

a) 5dy dy

= 6x . 0dx dx

when x < 0 and dy

0dx

when x > 0,

therefore, the function is not a monotonic function.

b) 4 2dy

=20 x 3 3dx

x is positive of all values of x

therefore, the function is not a monotonically

increasing function.

Convex and Concave Functions

Given a monotonic function, we know that the function

increases or decreases. What we do not know is whether the

function increases or decreases at an increasing rate or at a

decreasing rate. The sign of the second derivative of this

function gives us this knowledge. Second derivative of a

function is the rate of change of the tangent gradient. If this is

positive the function increases or decreases (depending on the

sign of the first derivative) at an increasing rate and the

function is said to be a convex function.



Convex Function

If second derivative of a function f(x) is positive i.e., f"

(x) > 0 b a,εx then, the function is said to be a convex

function in the given interval.

Concave Function

If b a,εx , the second derivative of a function f (x) is

negative i.e., f"(x) < 0 then, the function is said to be a concave

function in the given interval.

We notice that the slope of the tangent increases as x

increases [fig (i) ] in case of a convex function and decreases

as x increases in case of a concave function [fig I(ii)].

Example 3

Show that the curve of y=2x –3+1/x convex from

below for all positive values of x. Is same true for y=ax +

b+c/x?

Solution

We have 32

2

2 x

2

dx

yd,

x

1 - 2

dx

dy,

x

13-2x y



Since 2

2 3

d y 2 0dx x

for x > 0, therefore, by a

definition the curve 1

y 2x 3 x

is convex from below for

all positive value of x.

Now consider y = ax+b+c/x.

Since 02

dx

dy and

dx

dy

32

x

c

x

ca if c is positive and

in that case x

c bax y is convex from below for all

positive values of x.

Example 4

Show that the curve 3 xy is convex from below for

all negative values of x and concave from below for all

positive values of x.

Solution

We have 3 xy =x1/3

3 52

22/3

x

1

9

2

dx

ydandx

3

1

dx

dy

When 0dx

yd0,x

2

2

and the curve is concave from below.

When 2

2

d yx 0, 0

dx and the curve is concave from below.

Example 5

Show that the demand curve cb)(x

ap

is

downward sloping and convex from below. Do the same



properties hold of the marginal revenue curves?

Solution

We have cb)(x

ap

dx

dp since

b)(x

a-

dx

dp2

is negative (if x >0) therefore,

the demand curve is downward sloping.

b)(x

2a

dx

pd32

2

is positive for x > 0 and therefore, the

demand curve is convex from below.

Total Revenue cxb)(x

axp.xR

Marginal Revenue cb)(x

ax-b)a(x

dx

dRM

2

= cb)(x

ab2

3

2

( )

dM ab

dx x b

is negative 0x (assuming a, b > 0) and

therefore, marginal revenue curve is also downward sloping.

42

2

b)(x

6ab

dx

Md

is positive, therefore marginal revenue curve

is also convex from below.

Example 6

If the supply of a good is related to its price by

cbpax , where a, b, c are positive constants, show

that the supply curve is upward sloping and concave to axis OP



at all points.

Solution

We have b-pac-x c,bpax

2

a

c-xb-p ,bp

a

cx

ba

c-xp

2

02

dx

pd and 0c)-(x

a

2

dx

dp

22

2

2

a

Since 0dx

pd and c for x 0

dx

dp

2

2

, therefore,

supply curve is upward sloping and convex from below (i.e.,

concave to the OP axis).

Stationary Values - Maxima and Minima

When we talked about the sign of first derivative, we

showed that if the derivative of a function is positive

(negative), then the function is an increasing (decreasing)

function. We did not consider the possibility where the

derivate was equal to zero. We discuss this case and relate it

to the maximum and minimum value of a function.

If the first derivative of a function at some point is

zero, we say that the function is neither increasing nor

decreasing and is momentarily stationary at that point. The

value of the function at that point is known as the stationary

value. The stationary point may be a point of maximum value

or minimum value or a point of inflexion. We discuss the first

two, postponing the discussion on the third until a later section.



We begin by defining the maximum and minimum values of a

function.

Here we are interested in the case when the derivative

is zero at a given point say, If f '(a)=0 then f(x) is neither

increasing nor decreasing. The value of the function is

momentarily stationary and the curve has a tangent parallel to

the x-axis. The value of the function at such a point is called

Stationary value.

Maximum and Minimum Values

Consider the curve y = f(x) given above. Consider the

points A, B and C. We can see that the curve ceases to

increase through these points and beings to decrease. Where

as through the points D, E and F the curve ceases to decrease

and beings to increase. Such points where the curve ceases to

increase and beings to decrease and vice versa are called

turning points or stationary points on the curve and the value

of the function at these points are called turning values or

optimum values.

Definition

If, for a differentiable function y = f(x), x = x1, is a

point of maximum or minimum value, then the derivative of

the function at x=x1, is equal to zero i.e., 0dx

dyor 0(x)f' .



This is a necessary condition for maxima or minima of

functions and not a sufficient condition. If f ' (x) at some value

of x is equal to zero, we can not say that the function has

reached a maximum or a minimum. What we know from this

is merely that the function is stationary at these points and they

can be either points of maximum, or minimum or points of

inflexion. Therefore, we need something more than the

necessary conditions.

Proposition

A sufficient condition for a function to have a

maximum value (a minimum value) at the point x=x1 is that

the derivative of the function changes sign from positive

(negative) to negative (positive) as we move from the left of x1

to the right of x1 through x1.

Now, we are in a position to give a complete criterion

for maximum (minimum) value of a given function.

Definition

A point x1 is said to be a point of maximum value of a

function y=f(x) if the value of the function, in the

neighbourhood of the point x1 is smaller than

Δx)f(x)f(x i.e., f(x) 111 sufficiently close, in absolute value,

x1.

Definition

A point x2 is said to be a point of maximum value of a

function y=f(x) if the value of the function, in the

neighbourhood of the point x2 is greater than

)f(xΔx)f(xi.e., f(x i.e., f(x) 2222 for Δx sufficiently close,

in absolute value, x2.

f(x1) and f(x2) are known as maximum and minimum

values, respectively and extremum values, collectively.



Creterion

Given a function, y = f(x),

a) A necessary condition for extremum at x=x1 is f ' (x1)=

0, ie., 0dx

dy

b) If the sign of the second derivative at x =x1 is positive,

x1 is a point of minimum and if it is negative, x1 is a point of

maximum (sufficient condition).

Before giving examples on extreme values, we give a

working rule to evaluate the minimum or maximum values of a

function.

Working Rule to find Minimum and / or Maximum Values

Step 1: Given a function y = (x), find the first derivative f ' (x)

and solve the equation f '(x)=0, 0dx

dy to get x = x1, x2...., as

solutions [(f '(x1)=f ' ((x2) = ....0)]

Step 2: Get the second derivative at these values of x.

i) 0d

or ,0)(x f" If2

2

1 dx

y we have a maximum at x

=x1.

ii) 0d

or ,0)(x f" If2

2

2 dx

y we have a minimum at x

=x2.

Example 6

Find the maximum and minimum value of the function

2x4x1Y

Solution

Given y=1+4x–x2

dy 4 2x

dx



For maximum or minimum,

2 x4,2x 0,2x-4 i.e., 0 dx

dy

Now, ve- 2,- dx

yd

2

2

Y has a maximum at x = +2

The maximum value of the function is

y = 1 + 4 x 2 –(2)2 = 1 + 8–4 = 5

Example 7

Find the maximum and minimum values of

y = 2x3–3x

2 – 12 x +4

Solution

Let y = 2x3 – 3x

2 –12x + 4

For maximum or minimum, we have 0. dx

dy

Here 126x6x dx

dy 2

0126x6x demands 0. dx

dy 2

6(x2–x–2) = 0, i.e., x

2–x–2=0;

i.e., 2 1 0 2 0x x x or 1 0x

i.e., x = 2 or x= –1

Now 2

2

d y 12x -6dx



When 2

2

d y x 2, 12 2 6 18, ve.

dx Therefore y

has minimum when x=2.

Minimum value of y = 2 23–3.2

2–12.2 +4

= 16–12–24+4= –16

When 2

2

d y x 1, 12 (-1)-6 -18, -ve

dx

Therefore y has maximum at x= –ve

Therefore y has maximum at x = –1

Maximum value of y =2 (–1)3 –3 (–1)

2 –12(–1)+4 = 11



UNIT 3

ECONOMIC APPLICATIONS OF MAXIMA

AND MINIMA

1. Cost Minimization

One of the basic problems of a producer is to find out

the level of output at which the average cost of production is

minimum or the average variable cost of production is

minimum. We can apply the conditions of minimization as

given in the last section.

TC = aq2 + bq + C ....(1)

Where q is the quantity and C is the total fixed cost and

all parameters are positive.

The average cost is given by

q

Cbaq

q

TC AC ....(2)

To find out the output at which the average cost (AC)

will be minimum, we have to satisfy the following first order

and second order conditions such that

0dq

(AC)d and 0

dq

d(AC)

2

2

Now a

Ca 2

2q0r 0

q

C - 0

dq

d(AC)

a

C -or either ,

a

Cq

a

C



Since output cannot be negative, we choose a

Cq

Now 2

2 1

2 3

d (AC) 20 - (-2) Cq

dq

C

q

When 2

2 3

( ) 2, 0,

C d AC Cq

a dq q since a > 0 and C > 0

the average costs will be minimum at C/aq

If the average cost is given by the function

AC=aq2 + bq + C, (a > 0; b < 0; c > 0)

Then the determination of output at which the average

cost (AC) will be minimum requires that

0dq

(AC)d and 0

dq

d(AC)2

2

Now 2a

b-q 0, b aq 2

dq

d(AC)

and 0 a as 0a 2dq

(AC)d2

2

Thus the average cost will be minimum when the

output is –b/2a.

It may be noted that marginal cost curve cuts the

average cost curve at the minimum point of AC curve as

shown in figure. We take the total cost function (1). The

marginal cost is given by

b aq 2dq

d(TC)MC

Thus at minimum cost,



AC = MC

b2abq

Cbaq

or C/aq C/aqor aqC/q 2

Since output cannot be negative, therefore the average

cost will be minimum when C/aq . This is the same value

of output we derived using first and second order conditions of

minimization.

Example I

Find out the output at which the average cost is

minimum from the total cost function TC=2q2 + 5q + 18.

Solution

The average cost is given by

q

1852q

q

TCAC



Minimization of AC requires that the first order

differentiation of (1) with respect to q should be equal to zero.

So we have

0q

1802

dq

d(AC)2

or 39q 92

182 q

We take q = +3

The second order condition of maximization requires

that

2

20

d AC

dq

Now 2

2 3 3

d (AC) 18 36( 2) 0 if q 3

dq q q

The output at which average cost is minimum is 3.

Example 2

Given the short-run total cost function

C=2q3 – 15q

2 + 30q + 16

a. Find out the level of output at which average variable cost

(AVC) is minimum and also show that MC = AVC at the

total output.

b. Show that when output q=4, the average cost is minimum

and MC=AC.

Solution

a. The total variable cost is given by

TVC = 2q3 –15q

2 + 30q



3015q2qq

TVC AvC 2

The minimization of AVC requires

0; dq

(AVC)d and0(AVC)

dq

d2

2

Now 0(AVC)dq

d gives 4q – 15 = 0.

04dq

(AVC)d 3.75,

4

15q

2

2

q = 3.75 is the level of output at which AVC is minimum.

Now

3 2 2dC dMC 2q 15q 30q 16 6q 30q 30

dq dq

or MC = 6q2 – 30q + 30

When q=3.75, MC=1.87 and AVC = 1.87

MC = AVC at minimum point of AVC curve.

b. The average cost (AC) function is given by

q

163015q2q

q

CAC 2

The minimum value of AC requires that

0dq

(AC)d and 0

dq

d(AC)2

2

Now 2

d(AC) 160 gives 4q -15 0 - 0

dq q



and 2

2 3

d (AC) 324 0

dq q

when q=4, 0dq

d(AC)

the average cost is minimum when q = 4

Now at q = 4

MC = 6q2 – 30q + 30 = 6 (4)

2 – 30 (4) + 30 = 6

AC = 2q2 – 15 q + 30 + 16/q

= 2(4)2 – 15 (4) + 30 + 4 = 6

MC = AC when q = 4

2. Profit Maximization

In producer's equilibrium or in the theory of firm, the

basic problem is to choose the combination of price and

quantity in order to maximize profits (the question of choosing

price does not arise for a firm under perfect competition as the

price is given). The optimum level of output which maximizes

profit of a firm is arrived at when

a. Marginal revenue equals marginal cost and

b. Marginal cost curve cuts marginal revenue curve from

below.

C(q) - (q) R or CR

....(1)

So finally profit () is also function of quantity (q)

In order to obtain the level of output at which the profit

will be maximum, we follow the procedure of maximizing a

function in which the first derivative is zero and the second

derivative is negative.



Thus

MC MRor (q) ' C (q) ' Ror 0 (q) ' C - (q) ' dq

d gives 0 R

dq

d

....(2)

The second order condition states

2

2"( ) " (q) 0 or R" (q) C" (q)

dR q C

dq

or

slope of MR < slope of MC.

The conditions set by (2) and (3) imply that for profit

maximization, MR=MC and MC should cut MR from below.

The first order and second order - conditions of profit

maximization under imperfect competition as well as under

perfect competition can be more clearly seen from the figures

(1) and (2) respectively.

Figures 1 (a), 1 (b) and 1 (c) show that at equilibrium

output b q gap between total revenue and total cost is

maximum and so the profit function attains the highest point of

the profit curve and MC=MR with MC cutting MR from

below. At output q1, total cost over total revenue is maximum

and so the profit attains the minimum point with MR=MC but

MC cuts MR from above.

The same is the condition under perfect competition as

shown in figures 2 (a), 2 (b) and 2 (c).



Example 3

In a perfectly competitive market, the total revenue and

total cost of a firm are given by R=20q and C= q2 + 4q +20.

Find profit maximizing output and maximum profit.

Solution

By definition profit () is the difference between total

revenue (R) and total cost (C)

= R – C or = 20 q – q2 – 4q –20

Since = f (q) , the maximum of profit requires

0. d

and 02

2

dqdq

d

Now 8 qor 16 2qor 0 4- 2q - 20 dq

d

Again 022

2

dq

d

Since the second order derivative is negative, q=8 will

maximize profit of the firm. The maximum profit is obtained

by substituting q=8 in the profit function.



Maximum profit = 20 x 8 – (8)2 – 4 x 8 – 20 = 44

Example 4

A monopolists has the following total revenue (R) and

total cost (C) functions R = 30 q – q2,

C= q3 – 15 q

2 + 10 q +

100

Find

a) Profit maximizing output

b) Maximum Profit

c) Equilibrium price

d) Point elasticity of demand at equilibrium level of output.

Solution

a) The profit function is given by

= R – C = 30q – q2 – q

3 + 15q

2 – 10 q – 100

or = – q3 + 14q

2 + 20 q – 100

For profit maximization, 0 d

and 02

2

dqdq

d

dq

d= 0 gives –3q

2 + 28q +20 =0 or 3q

2 – 28q – 20 =

0

2x3

20)4x3x((28)28q

2

= 28 1024 28 32 2

10 or - 6 6 3

Now 2

26 28

dq

dq



When 2

210, 32 0

dq

dq

the profit maximizing output is q = 10.

b. Substituting q=10 in profit function we get maximum

profit = – (10)3 + 14 (10)

2 + 20 x 10 - 100 = 500.

c. The price equation or average revenue function is

obtain as .q

R

q30q`

q2-q 30 Price

For equilibrium output q=10, Price = 30 –10 = 20

d. Point Elasticity of demand is define as MRAR

AR

Now MR= 30 – 2q = 30 –2x10=10

and AR = Price = 20 21020

20

Example 5

A firm has the total cost (C) function C=7q2 + 5q + 120

and demand function P=180 – 0.5q and also a subsidy of Rs.5/-

per unit of output is paid by the government. Find

a. Profit maximizing output and price

b. Maximum Profit

c. Impact of subsidy on equilibrium output and prices.

Solution

a. When q units of output is produced, the total cost of

subsidy will be 5q. Likewise total revenue is given by

R=p.q = (180 – 0.5q) q=180q –0.5q2



So profit with subsidy is defined as

= TR – TC + subsidy

= 180q –0.5q2 – 7q

2 –5q–120 + 5q

or = 180q –0.5q2 – 7q

2 –120

Profit maximization requires that

0d

and 02

2

dqdq

d

Now

12 q

180 15qor 014q - q - 180or

0 14q - q (0.5) 2180

dq

d

Now 01514102

2

dq

d

with q = 12, the profit will be maximum. The

profit maximizing price is obtained by substituting q=12 in the

price equation.

P = 180 –0.5q = 180 –(0.5) x 12 = 180 –6=174

b. Maximum profit is obtained by putting q = 12 in (1)

= 180 x 12 –0.5 (12)2 – 7 (12)

2 – 120 = 960

c. In order to study the impact of subsidy on equilibrium

price and quantity, we are to find out equilibrium values

without subsidy. So profit without subsidy (*) is given by

* = 180q –0.5q2 – 7q

2 – 5q–120

Now 05-14q- (0.5)q 2-180 gives *

dp

d

or 175–q= –14q=0 or 15q = 175 q = 11.67



015*

2

2

dp

d

So profit maximizing output without subsidy, q=11.67

Substituting q = 11.67 in the price equation, we have

P=180–0.5q = 180 –(0.5) (11.67)

= 180 – 5.83 = 174.17

So the equilibrium profit without subsidy.

* = 180 (11.67) –0.5 (11.67)2 – 7(11.67)

2 – 120=900

Thus equilibrium price, output and profit with and

without subsidy indicate that output increases, price falls and

also profit increases as a result of provision of subsidy to the

firm.

3. Revenue Maximization

As we stated in the previous section that the profit

maximization is the objective of the producing firm, the

revenue maximization instead of profit may also be the

objective for an imperfectly competitive firm. However, a

firm cannot pursue revenue maximization, irrespective of what

happens to profit. Even with the objective of revenue

maximization, the firm must earn a certain minimum amount

of profit which is sufficient enough to satisfy its shareholders.

So profit maximizing output without subsidy, q=11.67

Substituting q=11.67 in the price equation, we have

P=180–0.5q=180 – (0.5) (11.67)

= 180 – 5.83 = 174.17

So the equilibrium profit without subsidy

* = 180 (11.67) –0.5 (11.67)2 – 7(11.67)

2 –120=900



Thus equilibrium price, output and profit with and

without subsidy indicate that output increases, price falls and

also profit increases as a result of provision of subsidy to the

firm.

3. Revenue Maximization

As we stated in the previous section that the profit

maximization is the objective of the producing firm, the

revenue maximization instead of profit may also be the

objective for an imperfectly competitive firm. However, a

firm cannot pursue revenue maximization, irrespective of what

happens to profit. Even with the objective of revenue

maximization, the firm must earn a certain minimum amount

of profit which is sufficient enough to satisfy its shareholders.

Example 6

A firm has a total revenue function R=20q –2q2 where

q is quantity and a total cost (C) function C=q2–4q+20.

Find the revenue maximizing output level and the

corresponding value of profit, price and total revenue.

Solution

The maximization of total revenue function R=20q –

2q2 requires that 0

dq

Rd and 0

dq

dR2

2

Now 5. qor 0 4 dq

d 0,4q-20 gives 0

dq

dR2

2

the revenue maximizing output is 5.

With the total cost, C=q2–4q +20, the profit function is

= (20q –2q2) – (q

2 –4q + 20) or

= 20q –2q2 – q

2 + 4q – 20



the level of profit at revenue maximizing output q =

5 is given by

= (20x5 –2(5)2 – (5

2) +4x5– 20–25

The price if the product, when q=5, is

10 2x5- 20 2q- 20q

RP

The maximum revenue = 20q–2q2=20x5–2(5)

2=50

Example 7

The total revenue (R) and total cost (C) functions of a

firm are given as

R=26q – 3q2 where q=quantity, C=2q

2 -4q + 10

Find (a) the profit maximizing output and

corresponding profit, price and total revenue at the level of

output.

(b) The revenue maximizing output and corresponding

profit, price and total revenue at the level of output.

Solution

a. The profit function is given by

=R–C=26q – 3q2

– 2q2 + 4q –10

Profit maximization requires that

......(1)

0 dq

d and 0

dq

d2

2

1044q-6q-26 gives 0dq

d

30 10 10q



10 20q

2q

or 10q=20 2,q 2

2

d10 0

dq

q=3 is profit maximizing output

So maximum profit is given by

2 2π 26x3 3(3) 2(3) 4x3 10 35

Price at profit maximizing output

173x3263q26q

Rp

Total revenue R=Pq = 17 x 3=51

b. The revenue maximization requires that

0 dq

d and 0

dq

d2

2

RR

Now 33.4 3

1 4q 06q-26 gives 0

dq

d

R

At the revenue maximizing level of output q=4.33

=26(4.33)–3(4.33)2–2 (4.33)

2 + 4 (4.33)–10=26.15

133(4.33)263q26q

Rp

R=pq =13 (4.33) = 56.25

c. Since profit at the revenue maximizing output level is

26.15, the profit constraint 30 will prevent the firm to

attain revenue maximizing output. If profit is set at = 30,

then the solution of the profit function.



26q–3q2–2q

2 + 4q–10=30 or

5q2 – 30q +40=0 or q=4

gives the highest revenue fetching output.

Points of Inflexion

A point of inflexion for a single valued function y=f(x)

is said o exist if the function changes its curvature at the point.

The function may become concave from convex or convex

from concave at the point of inflexion. The value of the

function corresponding to this point is known as inflexion

value. We discuss some features of points of inflexion and see

how the idea of extrema can be extended in the case of

inflexion points.

The following figures illustrate points of inflexion for

the two types of inflexion cases.

Notice that, in both the types of inflexion points the

tangent to the curve at the point of inflexion crosses the curve.

In figure (i), the inflexion points is x1 where the curve changes

its curvature from convex to concave. To the left of x1 the

tangent at any point on the curve lies below the curve, to the

right of x1, the tangent to any point on the curve lies above the

curve and only at the point of inflexion does the tangent cross

the curve. The same is true for the other types of the point of

inflexion.



The following figure, also show the characteristics of

point of inflexion.



Criterion (Inflexion Points)

Given a single valued function y=f(x),

a) A necessary condition for the point of inflexion at x=x1

is f " (x1) = 0

b) The sufficient condition for the point of inflexion at

x=x1 is f "' (x1) 0

If f "(x1) > 0, we have an inflexion point for concave to

convex function f "(x1) we have an inflexion point for convex

to concave function.

Example 8

Show that the curve 1

22

x

xy has three points of

inflexion separated by a maximum and a minimum point.

We have 1

22

x

xy

2 2

2 22 2

x 1 2 2x.2x 2 1 xdy

dx x 1 1 x



42

2222

2

2

x1

2xx1 2x-1 24x x1

dx

yd

42

222

x1

x-1 41 2-2x x1

x

32

3

32

22

x1

3 x4x

x1

2x2x-1- 4x

3or 0 either x 0dx

yd2

2

a) Case (i) x=0

we have 2

2

dx

yd>0 for x < 0

and 2

2

dx

yd < 0 for x > 0 in the neighbourhood of x=0

therefore, x = 0 is a point of inflexion.

b) Case (ii) 3x

3for x 0 dx

yd2

2

2

2

d y

dx3for x 0 in the neighbourhood of

x + 3 therefore 3 x is a point of inflexion.

2

2 2

dy 2(1-x )0 0 1

dx (1 )x

x

USER

Sticky Note

USER

Sticky Note

x2



maximum, ofpoint a is 1 xe, therefor0, 1)(1

3)-(1 )1(4

maximum, ofpoint a is 1 xe, therefor0, 1)(1

3)-(1 )1(4

3

1

2

2

3

1

2

2

x

x

dx

yd

dx

yd

Thus, we have seen that 1

22

x

xy has three points of

inflexion 3,0,3- separated by a point of minimum at

x= –1 and a point of maximum at x=+1

Example 9

Prove that the curve y=x3 has a single stationary point

which is a point of inflexion.

Solution

We have 23 3x

dx

dyxy

0x 0 6xdx

yd2

2

and

2

2

d y6x 0 x 0

dx Therefore, y = x

3 has a single

stationary point which is a point of inflexion.

Exercises

1. Find the minimum value of the cost function y=5+2x2–x

3

2. Find the maximum and minimum value of the function

3x2–36 x +10

3. Find the maxima and minima 4x3–21x

2 +18 x + 20

4.

Determine the maxima and minima 2x3–3x

2 –36 x + 20



5. Determine the maximam value of y = xe-x

6. Find the maximum and minimum values of the function

Y = (x–1) (x + 2)2

7. Find the maximum profit that a company can make if the

profit function is given by p(x) = 41 –24x –18x2

8. Find the maximum value of y=6x4 –10x

3 +6x

2+5

9. Find the maximum and minimum values of the function

Y = x3 –9x

2 +15x +3

10. Show that the function x

xy

2

2 has a minimum and a

maximum and that the former is greater than the later.

11. A firm has revenue function R=600q – 0.003q2 and the

cost function. C=150q + 60000.

12. A firm sells all the product it makes at Rs.9/- per unit.

The cost of making x units is C=0.1 x3+3x+8. Find the

maximum profit for the firm and also the number of units

to be sold for securing maximum profit.

13. A firm sells a product at Rs.3/- per unit. The total cost of

the firm for producing x units is given by C=20+0.6x +

0.01 x2. How many units should be made to achieve

maximum profit. Verify that the condition for a

minimum is satisfied.

14. The amount (in lakhs of rupees) A invested in a firm and

the profit p are expected to have a relation p=400+200A–

50A2. For what value of A, p attains the maximum.

15. Suppose the equation connecting the profit p in rupees

and the number of units n produced in a single lot of a

factory is given by p=247 + 1243n –0.025n2. Determine

the optimum lot size of the factory?



UNIT 4

FUNCTIONS OF SEVERAL VARIABLES AND

PARTIAL DIFFERENTIATION

Introduction

We have already studied the rules of differentiation

relate to the functions of a single independent variable. But in

most economic problems a particular economic variable

depends on a number of other independent variables. In such

cases, some of the independent variables may have positive

effect on the dependent variable while others may have

negative effect. Let us refer to a demand function where

quantity demanded (Qd) depends on the price of the product

(P), price of related goods (PR), income of consumer (1) and

say, the size of the family (S) such that

Qd = f (P, PR, I, S)

The impact of rise in the price of the product and the

price of the complementary goods on demand is negative. But

the increase in income of the consumer and increase in the size

of the family of the consumer will raise the demand for the

product. Further, the increase in the price (s) of the substitute

(s) is likely to raise the demand of the product. In such a

situation, when all the variables like price, prices of

complementary and substituted goods, income, family size etc.

increase simultaneously, it is impossible to trace out or to

quantify the effect of an individual independent variable on the

dependent variable. Say, for instance, if the price increases

from Rs.10 to Rs.13 per unit of output and income of

consumer increases from Rs.3000 to Rs.3500 per month

simultaneously and the net demand increases form 20 kg to 21



kg. We cannot trace out the individual effect of rise in price by

Rs.3 and increase in incom by Rs.500 on the change in the

demand since both price and income increase at the same time.

In such a situation, we are interested to find out or quantify the

effect on individual independent variable on the change in the

value of dependent variable. This can be done only when we

assume that when a particular independent variable changes,

the other independent variables don't change at the same time.

In terms of the above example, when we want to quantify the

effect of change in price on demand we must assume that the

income of the consumer, prices of related goods and size of

family of consumer don't change. Similarly, in order to trace

the effect of change in income on demand, price of the

product, prices of related goods and size family of the

consumer are assumed to remains same. When we find out

such effects by using the technique of differentiation, it is

known as "partial derivatives". It is called "Partial derivatives"

in the sense that the effect of individual independent variable

on the dependent variable is "partial" as the other independent

variable are assumed to be unchanged.

But when all the independent variables change

simultaneously, we may be interested to find out the total net

effect on the dependent variable. This can be done with the

help of the concept of total differentiation.

Partial Differentiation

In order to define the concept of partial derivative, let

us consider a function having 'n' independent variables.

y = f (x1, x2, x3, ..... , xn) (1)

where the variables x1, x2,...., xn are all independent of each

other so that each can vary itself without resulting in any

variation in the other variables. A change in the value of x1

with all other independent variables x1, x2,...., xn remaining



unchanged, will bring about corresponding changes in the

value of y. If x1, denotes the change in the value of x1, the

new value of x1 becomes (x1 +x1). If the corresponding

change in the value of y is denoted by x1 the value of y will

change to (y +y).

n3211 x...,x,x,Δxx f Δy y or

yx...,x,x,Δxx f y n3211 or

x

n21n3211

1

x.., ,x,x fx...,x,x,Δxx f

x

y

1x

y

represents the rate of change in y with respect to a

change in the value of x1 assuming other independent variables

x2, x2, ....xn to be constant.

Taking the limiting value of x1 equal to zero, the limit

of the quotient 1x

y

is called the "partial derivative" of y with

respect to x1 and is denoted by

1

Δx1 Δx

Δylim

x

y

01

(2)

similarly, the partial derivative of y with respect to the second

variable is denoted by

2

Δx2 Δx

Δylim

x

y

0 2

Assuming that other variable x2, x2, ....xn remain

unchanged. In the same way, the partial derivatives of y with

respect to the other independent variables can be defined. The

process of finding the partial derivative is called partial

differentiation.



Partial derivates are assigned distinctive symbols.

Instead of using the letter d as in the case of ordinary

derivative dx

dy, the symbol (delta) is employed to indicate

partial derivative such as nx

y

,...,

x

y,

x

y

21

etc. The partial

derivatives ae also denoted f1, f2, .... fn etc. to express partial

derivative respect to x1.

Thus partial derivative 1x

y

measures an instantaneous

rate of change of y with respect to an infinitesimal change in

x1. So 1x

y

will then provide information about both the

direction and magnitude of change in y resulting from an

infinitesimal change in the value of x1.

Example 1

Given

2 2

1 2 1 1 2 2

1 2

y yy f(x ,x ) 2x x x 3x find out and

x x

Solution

In the process of finding 1x

y

, we treat x2 as a constant.

So in the process of differentiation the additive constant 2

23x

will vanish and the multiplicative constant (in terms of x1 x2)

will be retained. Similarly in finding 1

2

,x

yx

will be

considered as constant and usual rules of differentiation will

follow. Thus



21211

1

x4x0x4xfx

y

21212

2

x6xx6x0fx

y

It appears from the above results that the partial

derivatives are also functions of x1 and x2. Thus we can write

f1 = f2 (x1, x2)

f2 = f2 (x1, x2)

Example 2 Find f1 and f2 given

3

2

2

21

3

121 3x x2xx)x,(x f y

Solution 02x3xfx

y 2

2

2

11

1

constant) x(since 2x3x 2

2

2

2

1

Similarly 2

2 1 2 2

2

yf 0 2x 2x 9x

x

= 4x1 x2 +6x2 (since x1= constant)

Example 3 Find f1 and f2 given

2 2

1 2 1 1 2y f x ,x x 5 2x x

Solution: Since the function y is the product of two functions,

the partial derivatives are derived by using product rule of

differentiation.

Now

5)(x x

)x(2x)x(2xx

5)(xfx

y 2

1

1

2

21

2

21

1

2

11

1

2 2

1 1 2 1(x 5) (2-0) (2x x ) (2x 0) (since x1 is constant)



= 1

2

21

2

1 )2xx(2x5)2(x

2

21

2

1

2

1 xx24x102x

2 2

1 1 2 6x 2x x 10

5)(x x

)x(2x)x(2xx

5)(xfx

y 2

1

2

2

21

2

21

2

2

12

2

2 2

1 2 1 2 1(x 5) (-2x ) (2x x ) x 0 (since x is constant)

2

1 2 22 10x x x

Example 4 Given 212

2

1

2

21

x

y and

x

y find

)3x(x

)x(2xy

Solution

Since the function is a quotient of two functions, the

quotient rule of differentiation shall be applied to derive the

partial derivatives 21 x

y and

x

y

Now

2

2

2

1

2

2

1

1

2

21

2

21

1

2

2

1

1 )3x(x

)3x(xx

)x(2x)x(2xx

)3x(x

x

y

2 2

1 2 1 2 1

222

1 2

x 3x .2 2x x 2x (since x is constant)

x 3x

=2

2

2

1

2

21

2

12

2

1

)3x(x

x2x4x6x2x



2 2

1 2 1 2

2 2

1 2

2x 6x 2x x

(x 3x )

Similarly

2

2

2

1

2

2

1

2

2

21

2

21

2

2

2

1

2 )3x(x

)3x(xx

)x(2x)x(2xx

)3x(x

x

y

constant) is x(since )3x(x

3 )x-x2()2x (- )3x(x12

2

2

1

2

2122

2

1

2

2

2

1

2

21

2

22

2

1

)3x(x

3x6x6x2x-

x

2 2

1 2 2 1

2 2

1 2

-2x x 3x 6x

(x 3x )

Example 5

Give

21

3

2

2

21

2

121x

yand

x

y find ),xx5x(x)x,f(xy

Solution

We use here modified version of chain rule of

differentiation. So taking ,xx5xxu 3

2

2

21

2

1

We have y = (u)5.

11 x

u x

x

y

du

dy

= constant) is xsince ( 05x2xx 5u 2

2

21

4



2

21

43

2

2

21

2

1 52x 5 5 xxxxx

Similarly,

22 x

ux

du

dy

x

y

2

221

4 3x x2x5x0 x 5u

2

221

43

2

2

21

2

1 3xx10x xxxx 5

Example 6

Given y=log 21

4

221

2

1x

y and

x

y find,xx2x50

Solution

Here also Chain rule of differentiation is applied in a modified

way

4

221

2

1 xx2xx50u

)log(uy

constant is xsince 02x2x0x u

1

x

ux

du

dy

x

y221

11

4

221

2

1

21

250

22

xxxx

xx

Similarly

2 2

y dy ux

x du x

3

1 2 1

1 0 0 2x 4x since x is constant

u



4

221

2

1

3

21

x250

42

xxx

xx

Example 7

A consumer consumes two commodities x1 and x2 and

the utility function is given by 221

2

1 5xx3xxu

Fin out marginal utilities of x1 and x2.

Solution

The marginal utility is noting but the increase in total

utility as a result of consumption of additional unit and is given

by the derivatives. Since the utility function involves two

variables x1 and x2, the marginal utility of x1 and x2 will given

by the partial derivative of u with respect to x1 and x2

respectively.

Marginal utility of x1 is given by

constant is xsince 03x2xx

u221

1

=2x1 + 3x2

Similarly, marginal utility of x2 is given by

constant is xsince 53x0x

u11

2

= 5+3x1

Example 8

Given a demand function of Engel's curvetype

D=AP

N

Where D is demand, P is price, N is income and A,,



are parameters. Find the partial derivatives N

D and

P

D and

also interpret the values of , and .

Solution

In the function D=APN, when we differentiate D will

respect to P, N taken to be constant.

β1α NAPα,

P

D

or P

D

P

NAP α

P

D βα

Similarly,

1-βα NAP .

N

D

constant) is P (since N

NAP

βα

or N

D

N

D

From the above partial derivative

P

P

D

D

P

D

P

D

//

pricein change ateProportion

demandin change ateProportion

= Price elasticity of demand

Similarly,

N

N

D

D

N

D

P

D

//



incomein change ateProportion

demandin change ateProportion

= Income elasticity of demand

and represent price elasticity and income

elasticity of demand respectively.

Exercise

21 x

and x

y Find

y of the following functions:

i) 2

2

2

11 10x

x

x3xy

ii) 2

2

3

121 2x4xy xx

iii) 21

212

1

3x5

yxx

exx

iv) 2

2

1

3

1

2

221 5x 2 xxxxxy

v) 2

2

2

12

2

1 2x 23 xxxy

vi) 4

2

2

21

2

1 x10x3xy x

vii) 1072

12

2

2

21

3

1

xxxy

viii) 50x3xx2xlogy 2

2

1

3

21

ix) 21

2

2

3

1

2

1

5100

12

xx

xxxy

x)

2

2

2

11

3

2

2

21

2

1

15

3xx8x2x10

xxxy



Second order Partial Derivative

We can observe, from the above, that in a function

y=f(x), the derivative f ' (x) or dx

dy is also a function of x.

Similarly in case of a function y= f (x1, x2), the partial

derivatives 1 2

1 2

fy y

f andx x

are also functions of x1 and

x2. This implies that 21

and x

y

x

y

can be differentiated

partially again with respect to x1 and x2 to yield second order

partial derivatives.

So in case of a function

y=f (x1, x2) (1)

the first order partial derivative 1

1

fx

y

can be further

differentiated partially with respect to x1 to give

112

1

2

11

fx

y

x

y

x

(2)

f11 in equation (1) measures the rate of change of f1

with respect to x1 assuming x2 to be constant. Similarly the

second order partial derivative of the function (1) with respect

to x2 is defined as

222

2

2

22

fx

y

x

y

x

(3)

So, f22 in (3) measures the rate of change f2 with

respect to x2 while x1 is assumed to be constant.

We have already stated that f1 is a function of x1 and x2

just like f2 is a function of x1 and x2. When f1 is differentiated



partially further with respect x2 or when f2 is differentiated

partially with respect to x1 ; both the second order partial

derivation are known as "cross-second order partial

derivatives" and denoted by

12

21

2

12

fx

y

x

y

x

x (4)

and 21

12

2

21

fx

y

x

y

x

x (5)

Example 9

Given

.f and f,d,f,f,f Find .3xxx2x)x,f(xy 222112112 1

2

221

2

121

Solution: Here f1 = 4x1 + x2, f2 = x1 + 6x2

11

1 1

y

x xf

4044x

21

1

xx

22

2 2

y

x xf

6606x

21

2

xx

12

12x

y

xf

1104x

21

2

xx



21

21x

y

xf

21

1

6x

xx

=1+0=1.

Example 10

Find f11, f12, f22, and f21 with reference to the example 3

above

Solution

1026x )(x

2

21

2

1

1

1

1

11

xx

xff

2

21 212 xx

2 2

12 1 1 1 2

2 2

6x 2 10x

f f x xx

21 2 .2 xx

= – 4x1 x2

22

2

1

2

2

2

22 102x- )(x

xxx

ff

2

112 10x

22

2

1

1

2

1

21 10 2x- )(x

xxx

ff

= –4x1 x2

The concept of second order partial derivative can be

extended to a function having more than two independent

variables say, in

y=f (x1, x2,x3)

USER

Highlight

-2



We can have nine second order partial derivatives f11,

f12, f13, f21, f22, f23, f31, f32 and f33.

Example 11. Given sthe function

2 2 2

1 2 3 1 1 2 2 3y f x ,x ,x x 10 2x x 3x x

find all nine second order partial derivatives

Solution In the function 2

3

2

22

2

11 x3xx2x10x

2

233323231

3223

2

322121

13112211

213

2

23

31

2

32

2

12

32211

x6f , x12x f ,0f

x12xf , 6x f ,4xf

0f , 4x f ,4xf

constant) as xand x(assuming xx6 f

constant) as xand x(assuming x6x2xf

constant) as xand x(assuming x4xf

Exercises

1. Find the four second order partial derivatives of the

following functions.

2

2

2

121

2

2

1

2

21

2

3

21

2

1

xxx2)(x log 3 y c)

5x3xxx2y b)

xx3x2xy a)

2. Find the nine second order partial derivatives of the


2

3

2

2

2

13

2

1

31

2

322

3

1

xxx3xx15y b)

xxxx3xxy a)



UNIT 5

DIFFERENTIALS AND TOTAL DIFFERENTIALS

We have already stated that in a function y= f (x1, x2),

partial derivative measures the rate of change in y with respect

to an infinitesimal change in the value of x1 assuming x2 to be

constant. Similarly f2 holding x1 to be constant. When both

x1 and x2 change simultaneously the resultant total change in y

is known as total differential. Let us now explain and define

the concept of differentials and then the concept of total

differentials.

Differentials

Let us consider a function y =f (x) where an arbitrary

change in the value of x (denoted by x) will bring about a

corresponding change in the value of y. (denoted by x), the

rate of change in y being the difference quotient Δx

Δy such that

ΔxΔx

ΔyΔy

(1)

But when we assume that the change in the value of x

is infinitesimal and corresponding change in the value of y is

infinitesimal, then the difference quotient will become the

derivative dx

dy and the infinitesimal change in x and y are

denoted by dx and dy respectively. Thus equation (1) can be

rewritten as

.dxdx

dydy



or dy = f ' (x).dx (2)

Where dx and dy represent what are known as

differentials of x and y respectively.

As shown in equation (2) since the differential dy is

simply equal to f ' (x) times dx, the process of finding dy is

straight forward. For example y =10x3, then the differential of

y is

dy = f ' (x) .dx =30x2 .dx

For a given value of x and dx, we can easily evaluate

the dy expression above.

Total differentials

The concept of total differentials is applied to functions

of more than one explanatory variables. Thus in a function y=f

(x1, x2) , the total change in y from simultaneous infinitesimal

change in both x and y is given by

2

2

1

1

.dx x

ydx

x

ydy

(3)

or dy = f1.dx1+f2dx2 which is called total

differential of the y function.

Here we consider a simple example in economics. A

saving function can be formulated as depending on national

income and the rate of interest such that

S=S(Y,i) (4)

Where S is savings, Y is national income and i is the

rate of interest. The partial derivative 1x

y

measures the rate of

change in S with respect to an infinitesimal change in Y

assuming that the rate of interest in constant. So the change in

S due to change in Y may be represented by the expression



dy.x

y

1

. Similarly effect of change in i assuming Y to be

constant can be expressed as ..i

Sdi

The total change in S or

total differentials will be equal to the sum of the differentials

of S with respect to Y and i such that.

..diS.dYs dSor

.di i

SdY

Y

SdS

iy

(5)

The process of finding such total differentials is known

as total differentiation.

We can consider another economic example where the

function depends on a large number of explanatory variables.

Say, in a utility function where the consumer consumes in

commodities, the function relation is given by

u=u (x1, x2, ... , xm). (6)

The total differential of the utility function is given by

m

m

2

2

1

1

.dxx

u......dx

x

u.dx

x

udu

or du = u1.dx1 + u2 .dx2 + ... um .dxm (7)

Each term on the right hand side of equation (7)

indicates the amount of change in u as a result of infinitesimal

change in one of the explanatory variables. The economic

interpretation is that the first term u, dx1 implies the marginal

utility of the first commodity times the increment in the

consumption of first commodity and similarly the other terms



also interpreted. The sum of all these terms in equation (6)

represent the total change in utility due to change in the

consumption of m commodities x1, x2, .... , xm.

The rules of differentials are broadly the same as the

rules of derivatives as studied earlier.

Example 1

a) Find differential dy of the function y = 10+2x + 3x2

Solution .dx 6x)(2 dy

).3210( 2

or

dxxxdx

ddy

b) Find differential of the function x10

2xy

2

Solution

.dx

x10

2x40xor

.dxx10

2x -4x x10dyor

.dxx10

2x

dx

ddy

2

2

2

2

2

c) Find total differential dy of the

function 2

2

2

121 2xxx2xy

Solution 1 2

1 2

y ydy .dx

x xdx

2 1 1 1 2 2or dy 2x 2x .dx 2x 4x dx

d) Find total differential of the function

2

21

2

1 xx10 xlog 2y



Solution

2211

2

2

1

2211

2

212

1

2

2

1

1

.dxx2xdxxx

4dyor

dxx2xdxx2xx x

2dyor

.dxx

ydx

x

ydy

Exercises

1. Find the differential dy for each of the following

functions

22

32

23

2

2

23x

10xy e)

2xx 2x y d)

4x y c)

3xx x 110 y b)

810x5xy a)

2. Find the total differential dy for each of the following

functions

2

21

2

21

2121

3

2

2

1

2

2

3

12

2

1

xx log 2xx2y c)

3x10xxxx2x y b)

x2xxxy a)

Total derivatives

While defining total differential in the previous section,

we considered a function with two independent variables such

that y=f (x1, x2). But if we consider a situation when x1 and x2

are not independent then we have the function.



y = f (x1, x2) where x2 = g (x1).

(1)

Since x1 is related to x2 (via function g), a change in x,

will not only affect y directly (via the function f), but also

affected y indirectly by changing y (via the function g). Thus

the rate of change in y with respect to x1 is given by

1

2

211

1

1

1

221

1

1

2

2

1

11

1

dx

x

y

x

y

dx

dyor

1dx

since dx

dyor

dx

dx

dx

dx

dx

dy

dx

dxdx

dxff

ff

(2)

1dx

dy in equation (2) is called total derivative of y with

respect to x1. The total derivative is obtained from the total

differential dy= f1dx1 + f2dx2 by dividing both sides by dx1.

1

22

1

11

1 dx

dxf

dx

dxf

dx

dy

The first term on the right hand side of equation (2)

indicates the direct effect of change in x1 on y. But the second

term 1

2

2 dx

dx.

x

y

measure the change in y resulting from change

in x2 with the change in x2 resulting from initial change in x1.

It means that the second terms of (2) measures the indirect

effect of x1 on y.

We can take an example to find out total derivative

considering a function



5x x where2xxy 3

122

2

1

Thus applying the definition in terms of

(2)

1

2

211 dx

dx.

x

y

x

y

dx

dy

=2

11 2x3x2x

or 2

11

1

6x2xdx

dy

If we have another function

4-x x wherexx3y 2

12

2

2

2

1

then

4x4x6xdx

dyor

)(2x 4)2(x6x

)(2x )(2x6x

dx

dx.

x

y

x

y

dx

dy

2

111

1

1

2

11

121

1

2

211

Now we consider a function y=f (x1, x2) where

x1 = g (t)

(3)

X2 = h (t)

To find out the total derivative of (3) withe respect to t,

first we find the total differential dy = f1dx1+f2dx2

Now dividing both sides by dt, we get

dt

dxf

dt

dxf

dt

dy 22

11



or dt

dx

x

y

dt

dx .

x

y

dt

dy 2

2

1

1

For example, let us take a function

10-2t x x wherex3x2y 2

2

12

2

1 t

6 t2t 8dt

dyor

62t 2t 4

23 2t 4x

dt

dx

x

y

dt

dx .

x

y

dt

dy

2

2

1

2

2

1

1

Exersises

1. 2

221

2

121dx

dy derivative out total find 5 x3 x wherexx4yGiven xx

2. 2

2

21

2

2211dx

dy derivative out total find 5 x wherex2x45yGiven xxx

3. 2

21

2

21dx

dy derivative out total find 2x5 x where3- x 2xyGiven

4. Find the total derivative

2t.5x,3t x where6xx5x2xy wheredt

dy2

2

1

2

221

2

1

5. Find the total derivative 2dx

dy given

.5x9 x where3x2x 5xy 211

2

2

2

1



UNIT 6

OPTIMIZATION WITH EQUALITY CONSTRAINT

Concept of Constrained Extrema

We have already examined the technique of

unconstrained (or free) extrema, where the optional value of a

function with more than one explanatory variable was derived

without having any constraint on the optional choice of the

values of the explanatory variables. There was no question of

choice inn determining the optimal value. But in practice,

there are many economic problems which involve finding an

optimal solution with one or more constraints. For example, in

the theory of consumer behaviour, a consumer has to choose

the combination of goods in order to maaximize his total utility

keeping in view his total income or budget of the consumer as

a constraint. So the budget limitation will restrict the choice of

the consumer in deciding the purchase of the basket of goods.

Similarly, in case of equilibrium of a firm in the factor market,

the cost of production is minimized choosing a combination of

the factor inputs keeping in view the production function as a

constraint. These types of problems are called constrained

optimization. In this chapter, we intend to discuss the

technique of finding the extreme value of a function with

equality constraint(s).

Let us consider a function having two explanatory

variables x1 and x2 such that y = f (x1 , x2) (1)

We want to find out the combination of x1, x2 that will either

maximize or minimize the function (1) subject to the

satisfaction of an equality constraint.

g (x1, x2) = C (2)

Where C is a constant



Whatever the values of x1 and x2 chosen optimally in

(1), they must satisfy the equation (2). Equation (1) is called

the "objective function" and the equation (2) is known as

equality constraint. The effect of an equality constraint (equ.2)

on the optimal solution of a function with two explanatory

variables (1) can be explained graphically in a three-

dimensional graph as shown in figure (1).

The unconstrained (or free) extremum of the function

is at the peak of the entire dome while the constrained

extremum is at the peak of the inverse U shaped curve lying

directly above the constraint line AB. Normally the

constrained maximum is expected to have a lower value than

the unconstrained (on free) maximum, although by

coincidence, both may happen to have the same, value. But it

should be noted that constrained maximum can never exceed

unconstrained maximum.

Lagrange Multiplier Method

As we have discussed in detail in earlier chapters when

we have a problem of unconstrained (free) extrema,

determination of optimal values requires the satisfaction of



straight forward first order and second order conditions. Just

for reference, when we have a function.

y=f (x1, x2, x3, ...., xn)

maximization of y requires that

f1 = f2 = f2 =f3 = .... =fn = 0

and |H1|<0,|H2|>0,|H3|<0, ..., (–1)n |Hn|>0

Similarly, for minimization of y, we need to have

f1 = f2 =f3 = .... =fn = 0

and |H1|>0,|H2|>0,|H3|>0, ..., |Hn|>0 where |Hk| is

called Hessain determinant of order k.

But when we have an objective function to be

maximized or minimized subject to the satisfaction o an

equality constraint, Lagrange multiplier method seeks to

convert the constrained extremum problem into a form to

which the first order and second order conditions of

unconstrained extremum can be still applied.

First Order Conditions

With the objective function (1) and the equality

constraint (2) in a optimizing problem with two explanatory

variables x1 and x2 , when we want to obtain the extreme value

of the objective function, we first construct the Language

function which is the modified version of the objective

function incorporating the constraint as follows,

2121 x,x gcλx,x fL (7)

The symbol (the Greek letter lambda) representing

some undetermined value is called a Lagrange multiplier. If

we have the objective of maximizing (1) subject to (2), then

the maximization of Lagrange function (7) will provide the

optimal values of x1 and x2 which will maximize.



y=f (x1, x2) = c

Satisfying the equality constraint

g(x1, x2) = c

Since the language function given by (7) is now a

function of three variables x1, x2 and , the maximization of L

in equation (7) requires to satisfy the first order and second

conditions of maximization. In this case, the first order

condition requires.

0λ

L and 0

x

L 0,

x

L

21

Now

0λgfLx

L111

1

0λgfLx

L222

2

(8)

0x,x gcLλ

L21λ

(since c is constant, its derivative is zero). From the first two

equations of (8)

2

2

1

1

g

f

g

fλ (9)

Since the last equation of the system of equations (8) is

simply a restatement of the equality constraint (2), Lagrange

function L will automatically satisfy the constraint of the

original objective function (1).

The same first order conditions given by (8) are true for

the minimization problem where we want to minimize the

objective function (1) subject to satisfaction of the equality

constraint (2).



The first order conditions of optimization (8) obtained

by using Lagrange multiplier method, can also be derived by

using total differential approach. The total differential of the

objective function y=f(x1, x2) is given by

dy = f1dx1 + f2dx2 = 0 (10)

or 1

2

2

1

f

f

dx

dx (11)

The statement (10) is still valid even if we add the

constraint g(x1, x2) = c. However, we cannot consider dx1 and

dx2 as ”arbitrary" changes is view of the constraint. Now the

total differential of the constraint is

g1dx1 + g2dx2=dc = c (12)

or 1

2

2

1

g

g

dx

dx (13)

From (11) and (13),

`1

2

1

2

2

1

f

f

dx

dx

g

g

or 1

2

1

2

g

g

f

f (14)

Thus, first order condition of optimization with an

equality constraint given by (14) using total differential

approach is the same as the first order constrained optimization

condition given by (9) using Lagrange multiplier method.

Second Order Condition

The introduction of an additional variable in the form

of in Lagrange function (7) makes it possible to apply the

same first order condition of unconstrained extremum problem

in constrained extremum problem also. However, we should

not apply the second order sufficient condition of



unconstrained (free) extremum in form of Hessain determinant

|H1|, |H2|, |H3|, in the Lagrange function (7). This is because of

the fact that although the optimal solution L depends on the

optimal choice of x1, x2 and , but unlike 21 x and x the choice

meaningless. Even if is replaced by any other value, no

effect will be produced on L since [c–g(x1, x2)] is identically

zero for any value of Lagrange multiplier. Thus, while it is

extremum, we should not blindly follow the second order

condition of unconstrained extremum in the present case of

Lagrange function. So we will restate the second order

conditions of constrained extremum in terms of total

differential and derive a new set of second order conditions for

constrained extremum.

With a constraint g(x1, x2)=c, the choice of x1 and x2 in

the objective function y=f (x1, x2) is not independent or

arbitrary and so dx1 and dx2 are also not arbitrary. Since from

(12), 1

2

12 dx

g

gdx , for specific value of dx1 and dx2 depends of

g1 and g2 which in turn depend x1 and x2. So dx2 will depend

on x1 and x2. The second order condition of extremum

depends on the value of second order total differential d2y. To

find the appropriate value d2y in view of the equality constraint

g(x1, x2) = cwe have.

2

2

1

2

2 dxx

(dy)dx

x

(dy)d(dy)yd

2

22212212112

2

111

22221211212111

22211

2

12211

1

dxfdxdxfdxdxfdxf

dx dxfdxfdx dxfdxf

dx

dxfdxf

xdxdxfdxf

x

or 2

2222112

2

111

2 dxfdxdxf2dxf yd (15)



Now from equality constraint g(x1, x2) = c, dg=0 and so d2g =

d(dg)=0

Following 1(15), in a similar way

0dxgdxdx2gdxggd 2

2222112

2

111

2 (16)

2

2

2222112

2

1112

2

g

dxgdxdx2gdxgxd (17)

2

2

2222112

2

1112

2

2222112

2

111

2

g

dxgdxdx2gdxg f-dxfdxdx2fdxfyd

or

g

dxgdxdx2gdxg fdxfgdxdxf2gdxfgyd

2

2

2222112

2

1112

2

222221122

2

11122

2

222

2

2222112

2

212

2

111

2

211 dx g

g

ff dxdx g

g

ff 2dx g

g

ff

2

2 2222211212

2

11111 dx λgf dxdx λgf 2dx λgf (18)

(14) from f

2

2 g

Now partially differentiation (8) with respect to x1, x2 and we

have

L11 = f11 – g11

L12=f12 – g12=L21

L22 = f22 – g22 (18)

Substituting (18) in (17), we get



2

2222112

2

111

2 dxLdxdx2LdxL yd

or 2

22221212112

2

111

2 dxLdx dxLdxdxLdxL yd (19)

When we have to determine the constrained extremum

of y = f (x1, x2) subject to g (x1, x2) c, the second order

sufficient conditions will depend on the algebraic sign of the

second order total differential d2y. While determining the sign

of d2y, the value of dx1 and dx2 (not both zero) should satisfy

the linear constraint.

g1dx1 + g2dx2 = 0 from (12)

i) For maximum value of y, d2y should be negative

definite subject dg = 0

ii) For maximum value of y, d2y should be positive

definite subject dg = 0

In case of unconstrained extremum the sufficient

conditions are expressed in form of Hessian determinant |H|,

but in case of constrained extremum, we will express the

sufficient conditions in terms of bordered Hessian determinant

|H| .

Now from (13)

1

2

12 dx

g

gdx

and substituting it in (19) , we get (20)

2

12

2

2

122

2

1

2

112

2

111

2 dx g

gLdx

g

g2LdxLyd

or 2

2

2

12

1222112

2

211

2 dx g L 2LgLyd

ggg (21)

From (21), it is distinctly clear that d2y is positive

(negative) definite if value within bracket is positive



(negative). Now the value within bracket of (21) can be

expressed as a 3x3 symmetric determinant such that

2

122

2

2112112

2

1222112

2

211 L - g L 2L Lgg 2LgL gggg

L L

L L

g g 0

-

2221 2

12111

21

g

g (22)

Thus for maximization d2y0 subject to dg = 0 requires

that

0 Lgg 2LgL 2

1222112

2

211 g

or 0

L L

L L

g g 0

-

2221 2

12111

21

g

g (23)

The determinants in (22) and (23) are known as

bordered Hessian determinant and symbolized by |H| and in

this case it is specifically denoted by |H| 2 as there are two

explanatory variables in the optimization problem.

Three Variable Case

When we have an objective function with a constraint

having three explanatory variables x1, x2 and x3 in the form,

say,

Maximize y = f (x1, x2, x3) (1)

Subject

g (x1, x2, x3) = c (2)

then we have the Lagrange function as

L = f (x1, x2, x3) + [c–g (x1, x2, x3) ]



Using the total differentials, the first order condition

requires

0λ

L L and 0

x

L L 0,

x

LL 0,

x

L

3

3

2

2

1

1

L

The second order condition requires

0

L L L

L L L

L L L

g g g 0

|H| and 0

L L

L L

g g 0

|H|

3332313

232221 2

1312111

321

3

2221 2

12111

21

2

g

g

g

g

g

For minimization of the function (1) subject to the

constraint (2), the first order conditions are the same as in case

of maximization and the second order condition requires

.0|3H| and 0|H| 2 In this manner we can generalize the

result with n explanatory variables.

Example I

Maximize y = x1x2+2x1 (1)

subject to x1+2x2=20 (2)

Solution

Now the Lagrange function is given by

L=x1x2 + 2x1+(20–x1–2x2) (3)

The first order condition of maximization requires

0λ

L L and 0

x

LL 0,

x

L

2

2

1

1

L

Now



L1 = x2 +2– =0

L2 = x1 +2=0 (4)

L = 20–x1 – 2x2=0

From first two equations of (4) , we have

42or x 2

2 211

2 xx

x (5)

Adding now equations (2) and (5), we get

12x 242x 11

Now substituting x1=12 in 5), we have 2x2 = 8

4x1

To test whether 12x1 and 4x 2 will maximize the

objective function (1 or not, we have to verify the value of

|H| 2 as second order condition.

Now from

L11 = 0 L12 = 1

L21 = 0 L22 = 0

04

0 1 2

1 0 1

2 1 0

L L

L L

g g 0

|H|

2221 2

12111

21

2

g

g

Since 4x and 12x ,0H| 212 will maximize the

objective function (1) subject to the constraint (2) the

maximum value will be

y = x1x2 +2x1 = 48+24=72



Example 2

Maximize a function y = 5x1x2 subject to x1 + 2x2 = 8

Solution The Lagragean function is given by

L=5x1x2 +(8–x1–2x2) (1)

The first order condition requires

0λ

L L and 0

x

LL 0,

x

L

2

2

1

1

L

Now

L1 = 5x2 – =0

L2 = 5x1 –2=0 (2)

L = 8 –x1 – 2x2=0

From first two equations of (2), we have

010x - 5xor 2

55 21

12

xx (3)

From the third equation of (2), we get

x1+2x2=8 (4)

Now multiplying both sides of (4) by 5 and adding to (3), we

get

4

401010

5

0105

1

2

1

1

21

x

xx

x

xx

Substituting x1 = 4 in (4) we get

2x2 =8 –4=4

x2=2

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The second order condition for maximization requires that

0H| 2 .

From (2) L11 = 0., L12= L21=5 and L22 =0

Again from (4) g1=1 and g2 =2

020

0 5 2

5 0 1

2 1 0

L L

L L

g g 0

|H|

2221 2

12111

21

2

g

g

4x 1 and 2x 2 will maximize the function y = 5x1x2

and the maximum value will be 40)2( )4( 5 y .

Example 3.

Minimize 124x2x subject to2xxxxy 21221

2

1

Solution

With the objective function to minimize 221

2

1 2xxxxy

Subject to 12 – 2x1–4x2 = 0, the Lagrange function is defined

as

21221

2

1 4x2x-12 2xxxxL

(1)

For minimization, first order condition requires

0λ

L L and 0

x

LL 0,

x

L

2

2

1

1

L

Now

L1 = 2x1 – x2 – 2=0

L2 = x1 + 2 – 4 = 0 (2)

L = 12 –2x1 – 4x2=0

From the first two equations of (2), we get

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-x1



1 2 12 x 2

2 4

x x

or 8x1–4x2= –2x1+4

or 10x1 – 4x2 = 4 (3)

From the third equation of (2), we have

2x1+ 4x2 =12 (4)

Adding (3) and (4), we get

3

4 x 1612x 11

Now substituting get we(4),in 3

41 x

2 1

2

4 284 12 2 12 2

3 3

7 x

3

x x

To test whether 3

7 and

3

421 xx will maximize or

minimize the objective function or not, we find out the second

order condition |H| 2

Now from the set of equation (2),

L11=2; L12 = –1 =L21 ; L22 = 0 and g1 = 2; g2=4.

048

0 1- 4

1- 2 2

4 2 0

L L

L L

g g 0

|H|

2221 2

12111

21

2

g

g

3

7x and

3

4,0|H| 222 x



Will minimize the objective function

221

2

1 2xxxxy

the minimum value will be

3

10

3

72

3

7

3

4

3

42

y

Example 4

Using Lagrange multiplier method, find the extreme

value of the function

2

221

2

1 x2

3xxxy

Subject to x1+ 2x2 = 14

Solution The Lagrangean function in case of the above

problem is defined as

21

2

221

2

1 2xx14 λx2

3xxxL (1)

Now from first order condition extremum

L1 = 2x1 – =0

L2 = x1+3x2 –2=0 (2)

L = 14 –x1 – 2x2=0

From (2)

2

32 21

21

xxxx

or 4x1 = x1+3x2

or 3x1 – 3x2 = 0

or x1 – x2 = 0 (3)

From the third equation of (2) x1+2x2=14

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Multiplying (3) by 2 and adding to (4), we get

7x1 =14, x1=2

Substracting (3) and 3rd

equation in (2)

2 2

1

3 14 14

14

x x

x

Now substituting x1 = 2 in eqution (4) we have x2 = 6.

To test whether x1 = 2 and x2=6 will maximize or

minimize the objective function we are to find out the value of

the bordered Hessian determinant |H| 2 for which we need to

find out.

L11 = 2; L12=1 = L21 ; L22=3 and g1; g2=2

Now -76- 1-

3 1 2

1 2 1

2 1 0

L L

L L

g g 0

|H|

2221 2

12111

21

2

g

g

Since ,0|H| 2

x1 = 2 and x2 = 6 will minimize the function

2

221

2

1 x2

3 xxxy

subject to x12x2 =14

The minimum value of y is given by

70)6(2

3 (6) )2()2( 22

Exercises

Find the extreme values of the following functions

subject to the given constraint.



1. 122x3x subject to xx2x2xy 212211

2. 22x xsubject to x2x4x3xy 21221

2

1

3. 24x62x subject to x2x5xx20y 21

2

221

2

1

4. 20x24x subject to x2x5xx2

1y 21

2

221

2

1

5. 9x53x subject to x3x2x2xy 21

2

221

2

1

6. 96x22x subject to 20 x2

1x2xx

2

1y 21

2

221

2

1



UNIT 7

COMPARATIVE STATIC ANALYSIS

Marginal Productivity

The marginal physical product of capital (MPPk) is

defined as the change in output brought about by a small

change in capital when all the other factors of production are

held constant. Given a production function such as

Q=36KL –2K2 –3L

2

the MPPk is measured by taking the partial derivative ./ KQ

Thus,

4K36LK

QMPPK

Similarly, for labor, MPPL= 6L.36KLQ/

Example Find the marginal physical productivity of the

different inputs or factors of production for the production

functions

i) Q=6x2+3xy+2y

2

ii) Q=x2+2xy+3y

2+1.5yz+0.2z

2

Solution

i)

222

K 0.2z1.5yz3y2xyxQ3y 12xx

QMPP

4y3xY

QMPP4y 3x

Y

QMPP yY

ii) MPPx = 2x+2y

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MPPy = 2x+6y+1.5z

MPPz=1.5y +0.4z

Multiplies

1. The concept of multiplier which occupies a very

significant place in the filed of economic analysis and which

become a powerful tool of economic analysis in the hands of

J.M Keynes.

The concept of multiplier was first of all introduced as

Employment Multiplier by Cambridge Economist R.F.Khan.

The theory of investment multiplier was developed by

J.M.Keynes in his book entitled 'General Theory'.

Theory of multiplier explains that increase in the total

income brought about through increased investment spending

will be a certain multiple of the original investment outlay

depending on the propensity to consume.

The investment multiplier which is defined as pro-rata

change in total income (Y) to a given initial change in

investment (for consumption) spending I with which the total

change in income is associated, as related to marginal

propensity to consume in a way such that higher the marginal

propensity to consume, larger is the multiplier and vice versa.

Since multiplier is related to marginal propensity to consume,

hence it is essential to make a reference to it.

Types of Multiplier

On the basis of factors influencing the level of income

multiplier can be classified in following categories. Some of

the specific factor multipliers are,

i. Employment multiplier

ii. Consumption multiplier

iii. Export multiplier



iv. Foregin investment multiplier

v. Tax multiplier

vi. Government expenditure multiplier

vii. Transfer payments multiplier

viii. Balanced budget multiplier

Employment Multiplier

The ratio between overall increase in employment and

subsequent primary employment is called employment

multiplier.

p

EmpN

NK

Where,

KEmp = Employment Multiplier

N = Change in total employment

Np = Change in the level of initial employment

Export Multiplier

Changes in the volume of exports are well manifested

through income of the country. Let us assume that amount of

export is increased by Ex, which in turn causes an increase in

the level of income. Let the increase in income be denoted by

Y. Export Multiplier (KExport) can be expressed

mathematically as:

Y = Ex/(1-b)

or Y = KExportEx

or KExport = Y/Ex

Thus, export multiplier shows the relationship between

change in income corresponding to the volume of exports.



Tax Multiplier

Variations in the amount of tax are very well reflected

on income. Let us suppose, an additional dose of tax causes

tax to rise by Tax, which causes a change in income

equivalent to Y. Tax multiplier which is denoted by (KTax)

can be represented mathematically as:

Y = -bTax/1-b

-b/1-b=KTax

Y = -KTax.Tax

KTax = b

b

Tax

Y

1

Thus, Tax multiplier is the ratio between change in

aggregate income and amount of tax. It is always negative.

Government Expenditure Multiplier

Alike other factors an increase in government

expenditure causes fluctuations in the level of income. If

initial government expenditure G changes by a small amount

G and Income by Y; then Government expenditure

multiplier (KGov,Expen.) can be represented mathematically as:

Y = Gb

1

1

or Y = KGov.Expn. G

ie., KGov.Expn= Y / G

Thus, Government Expenditure Multiplier is the ration

between change in aggregate income and government

expenditure.

Comparative Static Analysis in a National Income Model



In a macro-economic model of income determination,

we know that the growth of national income is affected by the

change in certain exogenous factors like the government

consumption expenditure of autonomous investment and some

other instruments like marginal propensity to consume, rate of

income tax, etc. The concept or technique of partial

differentiation enables us to determine the direction and

magnitude of change in the national income as a result of

changes in the above mentioned instruments like government

expenditure, autonomous investment, m.p.c., m.p.s., rate of

income tax, etc.

Let us consider a simple national income model with

three endogenous variables Y (national income), C

(consumption expenditure) and T (taxes) with three equations

such that

Y = C+I+G (1)

C = + (Y-T) (>0; 0<<I) (2)

T=r+ Y (r>0;0< <I) (3)

where I and G are exogenous variables symbolizing

autonomous investment and government expenditure and , ,

, are the parameters having appropriate economic

interpretations.

The first of the above three equations is an equilibrium

condition for national income while the second and third

equations show how consumption and taxes are determined in

the model respectively.

In the comparative static analysis, we need to find out

how the equilibrium national income Y or equilibrium

consumption expenditure C or equilibrium tax revenue T

changes as a result of changes in the autonomous investment

and government expenditure as well as the parameters , ,



,and . Therefore, we are to find out the equilibrium values

of Y, C and T. This is done by substituting equations (2) and

(3) in (1). Thus

Y = +(y-T) + I+G substituting (2)

or Y = +Y-T+I+G

or Y = +Y- Y +1+G substituting (3)

or Y = +Y- GY 1

or Y-Y+ Y=- +I+G

or (1-+ )Y=- +I+G

I

GIY

Now substituting (4) in (3) we get

I

GIT

or

I

GIT

or

I

GIIT

Similarly C is derived by substituting (4) and (5) in

equation (2)

I

GII

I

GIC



I

GIIGII 2

I

GIGI 22

I

IGI 1

or

I

GIC

1

In order to investigate the effect of an infinitesimal

change in one of the exogenous variables or in one of the

parameters on the equilibrium value of the endogenous

variables T C ororY we are to find out the partial

differentiation of T C oror with respect to I, G, , , and .

It may be noted that the parameters , ,, and simply

autonomous consumption, marginal propensity to consume,

non-income tax revenue and income tax rate respectively. Let

u now see what happens to Y

if I, G, , , and change.

0

I

I

I

Y (7)

0

I

I

G

Y (8)

0

I

IY



2.

I

IGIIY

2

I

GIGI

or

02

I

GIIY (9)

Similarly

20

I

GIIY

or )4 .(0 byI

YY

(10)

In the same way

20

I

GIyIY

or 0

I

Y

The restrictions of the above partial derivatives simply

that investment multiplier is positive (equation 7),

government-expenditure multiplier is also positive (equation

8). The effect of increase in the marginal propensity of

consume on national income is also positive (equation 9) while

the income tax rate multiplier as well as non-income tax

multiplier are negative (equation 10 and 11).

Similarly the effect of change in I, G, , , and on

equilibrium consumption and equilibrium tax can also be

investigated by taking partial derivatives of C and T with



respect to the exogenous variables and the parameters. This

has been left to the readers as an exercise.

Example

The partial derivative can also be used to derive the

various multipliers of an income determination model. Given

Y = C+I+G+ (X-Z)

where C-C0+bY G=G0 Z=Z0

I=I0+aY X=X0

As in the previous example the equilibrium level of

income can be obtained as

00000 ZXGICabI

IY

(1)

Taking the partial derivative of (1) with respect to any of the

variables or parameters gives the multiplier for that variable or

parameter. Thus, the government multiplier is given by

abI

I

G

Y

0

The import multiplier is given by

abI

I

Z

Y

0

Determination of Partial Elasticises

In earlier chapters we discussed the technique of

determining the elasticity of demand of a function where

demand is a function of price alone. But with the knowledge

of partial derivative, we can now define the partial elasticities

of a demand function having more than one independent

variable.

Let us consider a demand function of commodity Q1. It



is assumed to be a function of the price of commodity I(P1)

price of its substitute good (P2) and the income of the

consumer (Y). Hence the demand function may be expressed

as

Q1+ = Q1(P1, P2, Y) (1)

We can now define three partial elasticities for the

above demand function -(a) the own price elasticity of

demand, (b) the cross price elasticity of demand and (c) the

income elasticity of demand.

The own price elasticity of demand is defined as the

ration of proportionate change in quality demanded of good I

to proportionate change in the price of good I assuming that

the price of the second good and income remain constant. So

the own price elasticity of demand is expressed as

1

1

1

1

1

1

1

111 ./

Q

P

P

Q

P

P

Q

QE

where 1P

Q

is the partial differentiation (1) with respect to rp

Similarly, the cross elasticity of demand is defined as

the ratio of the proportionate change in quantity demanded of

good 1 to proportionate change in price of good 2 assuming the

price of good 1 and income remain constant. So the cross

elasticity is expressed as

1

2

2

1

2

2

1

112 /

Q

P

P

Q

P

PQE

If 12E is positive, goods 1 and 2 are substitutes of each other

and if 12E < 0, they are complementary of each other.

It may be noted that if 12E is approximately equal to

zero, it implies that there is hardly any relationship between



goods 1 and 2.

Finally income elasticity of demand is defined as

1

1

1

11 /

Q

Y

Y

Q

Y

Y

Q

QE Y

which measures the ratio of proportionate charge in the

demand for good 1 to proportionate change in the income of

the consumer. Here Y

Q

1 represents the partial derivative of

the function (1) with respect to y. We take an example of a

demand function in linear form to demonstrate the derivation

of the price elasticity ( 11E ) cross elasticity ( 12E ) and income

elasticity ( )( 1YE . Let the demand function be

YPPQ 2.03250 211

For a given income Y = 500, and price 21 p and 2p =

5, we have

161

22)2(

1

1

1

1

1

111

Q

P

Q

P

P

QE

since

1

2

2

112

u

1

Q

P

P

QESimilarly

161

4Eor

1615002.053x250Q

161

53

Q

P3

1

2



161

15Eor 12

Now

.substitues are 2 and 1 goods that theindicatesit ,0E Since

161

100Eor

161

5002.0

12

1Y

1

11

Q

Y

Y

QE Y

Exercises

1. Find the cross partial derivatives for

.1,4at evaluated3 32 yxyxz

2. Given the demand YPPQ ba 1.05.154850 with

200,000,100 aPY and the price of another good

100bP calculate (i) income elasticity of demand, (ii)

cross elasticity of demand.

3. For the above problem, calculate the percentage change

in the demand for good a resulting from a 10% increase

in the price of good b.

4. Find the marginal physical productivity of the different

inputs or factors of production for the following

production functions.

(i) 22 25.0 LKLKQ

(ii) 232 2.05.132 zyzyxyxQ

5. Given Q=700-2P+0.02Y where P=25 and Y=5000



Find (a) Price elasticity of demand (b) Income

elasticity of demand

6. Given a three sector income determination model in

which

TYGCY d00 Y ,1

tYbYCC d 00 TT ,

1t0 1,b0 ,0,,, 0000 TGIC

Calculate

(i) government multiplier

(ii) autonomous tax multiplier

(iii) tax rate multiplier

and comment on the result

7. Given TYGICY d00 Y

tYbYCC d 00 TT ,

75.0 ,90,100 00 bIC

20.0,240T ,330 00 tG

(a) What is the equilibrium level of income?

(b) What is the effect on Y of a 50 increase in

government spending.

(c) What is the effect on Y of a 50 in autonomous

taxation to

8. Given YPPPQ 0075.025.075.0100 3211

At 170Q 10,000,Y and 40P ,20,10 1321 PP

Find the different cross elasticities of demand



UNIT 8

OPTIMIZATION OF

MULTIV ARIABLE FUNCTIONS

For a multivariable function such as z=f(x, y) to be at a

relative minimum or maximum, the following three conditions

must hold good.

1. The first order partial derivatives must be equal to zero

simultaneously. This indicates that at the given point

(a, b), called critical point, the function is neither

increasing nor decreasing.

2. The second order partial derivatives, when evaluated, at

the critical point (a, b) must both be positive for a

minimum and negative for a maximum.

3. The product of the second order direct partials

evaluated at the critical point must exceed the

product of cross parties evaluated at the critical point.

The above statements can be symbolically expressed as

follows.

For relative maximum (i) 0,0 yx ff

(ii) 0,0 yyxx ff

(iii) 2xyyyxx fff

For relative minimum (i) 0,0 yx ff

(ii) 0,0 yyxx ff

(iii) 2xyyyxx fff



Note

(i) If 2xyyyxx fff , when fxx and fyy have the same

signs the function is at an inflexion point.

(ii) If 2xyyyxx fff , when fxx and fyy have different

signs the function is at a saddle point.

(iii) If 2xyyyxx fff , the test is inconclusive

Example 1

For the cubic function f(x, y)= 3x3 + 1.5y

2 - 18xy + 17

1) Find the critical points

2) Determine if at these points the function is at a

relative maximum, relative minimum, inflexion point or saddle

point.

Solution

1) Set the fist order partial derivatives equal to zero

Here fx=9x2

- 18 y = 0

fy = 3y - 18x = 0

Solving for critical points, we have

18y = 9x2, 3y = 18x

or 6xy ,2

1 2 xy

Equating the above, we have x6

2

1 2

i.e., x2 - 12x=0, ie., x(x-12)=0

i.e., x=0, x=12

Substituting x=0 and x=12 in y=6x, we have



y=60=0 and y=612=72

Therefore, the critical points are (0, 0) and (12, 72).

(2) Now find the second order partial derivatives

fxx = 18x, fyy = 3

Evaluate them at critical points and note the signs.

fxx (0, 0)=180=0, fyy (0, 0) = 3>0

fxx (12, 72) = 1872=216>0 fyy (12, 72) = 3>0

Also fxy = -18=fyx

Evaluate it at the critical points and test the third

condition

fxx(a, b). fyy (a, b) > [fxy(a, b)]2

At (0, 0) 03<(-18)2

At (12, 72) 2163>(-18)2 ie., 648 > 324

With fxx fyy > (fxy)2 and fxx, fxy > 0 at (12, 72), f (12, 72)

is a relative minimum.

With fxx fyy <(fxy)2 and fxx and fyy are of the same sign at

(0, 0), f(0,0) is an inflexion point.

Example 2

A monopolist sells two products x and y for which the

demand function are x=25-0.5Px, y=30-Py and the combined

cost function is C=x2 + 2xy+y

2+20.

Find (a) Profit maximising level of output of each product

(b) Profit maximising price for each product and

(c) Macimum profit

Solution

(a) We know that the profit function is given by



=TRx + TRy - TC

Here = Pxx + Pyy - C

From x=25-0.5Px Px=50-2x

y=30-Py Py = 30-y

Then =(50-2x)x+(30-y)y-(x2+xy+20)

=50x-3x2+30y-2y

2-2xy-20

x = 50-6x-2y = 0

y = 30-4y-2x=0

Solving the above, we get of x=7 and y=4

Testing the second order conditions,

xx=-6, yy=-4 and xy= -2

With both direct partials negative and xx yy > (xy)2,

we conclude that is maximised.

(b) Substituting 7x

and 4y in Px and Py, we have Px

= 50-27=36, py 30-4 = 26.

(c) Substituting x =7 and y =4 in , we get

=507-372 + 30 4-24

2- 274-20=215.

Economic Applications of Constrained Optimization of

Multivariable functions

We have already studied that the problem of choice

normally has to be solved within the limitation of certain

constraint(s). We now take up certain economic problems

when choices are to be made subjecte to satisfaction of certain

constraint(s).



Utility Maximization and Consumer's Behaviour

When a consumer has to maximize his total utility from

the consumption of a basket of goods, the Marshallian concept

of consumer behaviour states that the consumer has to allocate

his total budget for the consumption of the various

commodities in such a way that the ration of marginal utility to

price of each of the commodities is equal. The Hicksian

concept of consumer behaviour states that in case of a pair of

consumer goods, the utility maximization requires that the

combination of goods should be chosen in such a way that the

slope of the budget line is equal to the slope of the utility

function (or indifference curve), In other words, the budget

line should be tangent to the indifference curve as well as the

indifference curve must be convex to the origin.

Let us assume that a consumer consumes two

commodities x and y and so the utility function of the

consumer is given by

u=u(x, y) (1)

where we assume that

0,

0),

yxuudy

duand

yxuudx

du

yy

xx

(2)

in playing that both marginal utility of x and y. uy respectively,

are functions of x and y.

It is also assumed that the consumer has a fixed budget,

, and the while income spent on the consumption of x and y

such that

=xpx + uPy

where px and py are the price of x and y.



Now it becomes a problem of maximizing total utility function

(1) subject to the budget constraōint (3). This requires to

formulate the Lagrange function as

L=u(x, y) + (-xpx- ypy)

where is Lagrange multiplier

The first order condition of maximization of (1) subject

to (3) requires

0L

and 0,0

LL

y

LL

x

Lyx

Now

Lx=ux-px = 0

Ly=uy -py=0 (5)

Ly=-xpx-ypy=0

From first two equations of (5), we get

y

y

x

x

p

u

p

u (6)

The first order condition of utility maximization

requires that the ratio of marginal utilities of x and y (ux and

uy) to their prices should be equal which is exactly the

Marshallian condition of consumer's equilibrium.

Equation (6) can be re-arranged as

y

x

y

x

p

p

u

u (7)

When we consider the utility function (1) in terms of

indifference curve, there is no change in total utility even

though the combination of x and y change. Thus the change in

x and y gives.



du-uxdx+uydy (8)

Since there is no change in total utility, du=0. Thus

uxdx+uydy=0

y

x

u

u

dx

dy (9)

which represents the slope of indifference curve.

The budget equation (3) can be re-arranged in terms of

y as a function of x as

xp

p

py

y

x

y

(10)

Indicating that

y

x

P

pis the slope of the budget line. Thus,

equation (7) represents that

Slope of indifference curve = slope of budget line.

The second order condition for maximization of (1)

subject to (3) requires that

2H >0

Now

yyyx

xyxx

yx

2

L L

L L

g g 0

y

x

g

gH and it should be greater than zero

From (5),

Lxx = uxx; Lxy = uxy;

Lyx = uyx and Lyy = uyy

From the constraint (3)



gx = px and gy = py, since is constant.

yyyx

xyxx

yx

2

L L

L L

g g 0

y

x

g

gH

or 2H =2pxpyuxy - 022 yyxxxy upup (11)

[since uxy=uyx]'

Since the indifference curve should be convex to the

origin at the point of equilibrium as shown in the figure, the

necessary condition in terms of derivative is that 02

2

dx

yd

Now

)9( 2

2

fromu

u

dx

d

dx

dy

dx

d

dx

yd

y

x



or rulequotient using22

2

dx

duu

dx

duuy

u

I

dx

yd y

xx

y

(12)

Since ux and uy are both functions of x and y.

ux=ux (x, y) and uy = uy (x, y)

dux = uxx dx +uyxdy

or dxby sidesboth dividing)(

dx

dyu

dx

dxu

dx

udyxxx

x

= uxx+uyx dx

dy

of

mequilibriuat

p

p-

dx

dy

y

x

y

xyxxx

x

p

puu

dx

du (13)

Similarly,

y

xyyxy

y

p

puu

dx

du

Now substituting (13) and (14) in (12), and taking ux =

uy y

x

p

pfrom (6), we get

y

xyyxy

y

xy

y

xyxxxy

y p

puu

p

pu

p

puuu

u

I

dx

yd22

2

2

2

2

12

y

xyyy

y

xxyy

y

xyxyxxy

y p

puu

p

puu

p

puuuu

u

2

22

y

yyxxyyxyxyxxxy

y p

upuppuppup

u

I



2

222

yy

yyxxxyxyyx

pu

upupupp

or )11( 2

2

2

2

frompu

H

dx

yd

yy

Since 0.p and 0,0 y2 yuH (15)

02

2

dx

yd

Thus at the point of equilibrium, the shape of

indifference curve is convex to the origin.

Example 3. Given the utility function u=2+x+2y+xy and the

budget constraint 4x + 6y = 94, find out equilibrium purchase

of x and y in order to maximize total utility.

Solution. The maximization u= 2+x+2y+xy subject to

4x+6y=94 requires to formulate the Lagrange function as

L = (2+x+2y+xy)+(94-4x-6y)

The first order condition of maximization in this case

requires

0L

and 0;0

LL

y

LL

x

Lyx

Now

Lx=I+y-4=0

Ly = 2+x=6=0 (1)

L= (94-4x-6y)=0

From the first two equations of (1), we have

USER

Sticky Note

-6y



6

2

4

1 xy

or 6+6y=8+4x

or –4x+6y=94 (2)

Adding (2) and (3), we have

12y = 96 y = 8

Now substituting y=8 in (3), we get

4x=94–6y=94–48=46

x=11.5


that 0|| 2 H

Now from (1) Lxx =0; Lxy = 1=Lyx and Lyy = 0

From the equality constraint, g1=4 and g2 = 6

8 y and 11.5x ,0||Since 2 Hwill maximize the utility

function subject to his limited budget of Rs. 94.

Example 4. The utility function of a consumer is given by

u = 5 log x1 + 2 log x2

Find out the combination of x1 and x2 which will

maximize the utility function subject to the satisfaction of the

budget constraint.

4x1 + 2x2=28, the Lagrangean function is defined as

)24x-(28 xlog 2log5 2121 xxL

The first order condition of maximization requires

0

LL and L ;0

2

x1

1

1

x

L

x

LLx



Now

2

1

1

2

1 2

5 4 0

2 2 0

28 4 2 0

x

x

Lx

Lx

L x x

` (1)

From first two equations of (1) we have

21 2

2

4

5

xx

(2)

or 4x1 –5x2 = 0 (3)

From the third equation of (1)

4x1+2x2= 28

Subtracting (2) from (3) we get

7x2 = 28

42 x

Now substituting 42 x

in (3), we get

5x 204 11 x


that 0|| 2 H

Now from (1) and the budget constraint

2;4g and x

2- ;0;

x

5- 212

2

2

1

gLLLL yyyxxyxx

USER

Sticky Note

1 small



x2- 0 2

0 5/x 4

2 4 0

L L g

L L

g g 0

|H|

2

2

2

2

yyxy2

xyxx1

22

2 g

For x1 = 5 and x2 = 4, we get

05

14

5

42

81 0 2

0 5

1- 4

2 4 0

|H| 2

4x and 5x ,0||Since 212 H

will maximize the utility of the consumer. The maximum

utility will be

4.72log(4)5log(5)u

Exercises

1. For the quadratic function z=3x2 –xy+2y

2 –4x–7y+12

a) Find critical points at which the function may be

optimized.

b) Determine whether at these points the function is

maximized, is minimised, is at an inflexion point or is

at a saddle point.

2. For the cubic function z(x,y) = 3x3 – 5y

2 –225 x + 70y

+23.

i) Find the critical points



ii) Determine if at these points the functions is at a

relativemaximum, relative minimum, inflexian point or

saddle point.

3. Find the profit maximising level of (a) output (b) price

and (c) profit for a monopolist with the demand

functions.

x=50–0.5Px, y=76–Py and the total cost function is

C=3x2 + 2xy+2y

2+55

4. Use Lagrange multiplers to optimize the function

z=4x2–2xy+6y

2 subject to x+y=72.

5. Find the extreme value of the function

y=2x1+2x1x2+x2 subject to 3x1+2x2=12

REVISION EXERCISES

I. Very short Answer Questions

1. Define an increasing function

2. Define a decreasing function

3. What do you mean by optimisation of a function?

4. What do you mean by points of inflexion?

5. Define concave and convex functions

6. What are the conditions for a maximum or a minimum?

7. What do you mean by partial differentiation?

8. Define total derivative

9. Define multipliers.

10. Define employment multiplier.

11 Define tax multiplier.

12. Define government expenditure multiplier.



13. Define consumption multiplier.

14. Define income elasticity of demand.

15. Define cross elasticity of demand.

16. What is lagranges multiplier

17. State the condition of utility maximisation of a

consumer.

II Short answer Questions

18. Derive the conditions for maximum or minimum

19. What are the conditions for getting a relative maximum

or relative minimum.

20. What are the important multipliers?

21. Explain cross elasticity of demand.

22. Explain income elasticity of demand.

23. Find the maximum profit and the level of output for

maximum profit if the revenue and cost function of the

firm is given by R=200x10.01x2 and T=50x +20000.

24. A monopolist's cost function is given by T=x2+4x and

his average revenue is given by A.R=28–5x. Show that

x=2 is a point for maximum profits which equals 24.

25. A radio manufacturer produce x sets per week at a total

cost of Rs. (x2/25) + 3x+100. He is monopolist and the

demand of his market is x=75–3p, when the price is Rs.

P per set. Show that the maximum net revenue is

obtained when about 30 sets are produced per week.

What is the monopoly price?

26. A firm produces x units of output at a total cost of Rs,

300x –10x2+(x

2/3). Find (i) output at which marginal

cost is minimum (ii) output at which average cost is



minimum; and (iii) output at which average cost is

equal to marginal cost.

27. The total cost function of a firm is given by C=q3-

6q2+2q+50. Find the leve of output of which the

average variable cost is minimum. Also show that

AVC=MC at that level of output.

28. A firm has the following total cost function

C=q3–7q

2+2q+16 where q is the output produced.

a. Derive average variable cost (AVC) function and show

that when AVC is minimum, AVC =MC.

29. In a perfectly competitive market, the total revenue (R)

and total costs (C) of a firm are given by R=4q, C=q2–

6q+10.

Find (a) Profit maximizing output, (b) Maximum

profit.

30. In a perfectly competitive market, the market price of a

commodity is Rs.3 per unit of output(q). The total cost

function of a firm is given by

1010q4qq

3

1 23 C

Find (a) Profit maximizing output (b) Maximum profit

III. Long Answer Questions

31. A monopolist average revenue (AR) and total cost (TC)

functions are given by AR=16–2q, TC=20+4q–q2

Find a) Profit maximizing output

b) Equilibrium price

c) Maximum profit



32. A monopolists has the following average revenue (AR)

function and total cost (TC) functions.

32 q3

16q-2q40TC 3q,29AR

Find a) Profit maximizing output and price

b) Equilibrium profit

c) Point elasticity of demand at equilibrium

output level.

33. A firm has the following average revenue (AR) and

total cost functions AR=160 –q, TC=200 +4q+7q2. A

subsidy of 4 per unit of output is paid.

Find a) Profit maximizing output

b) Equilibrium profit

c) Effect of subsidy on equilibrium price

34. A monopolist has the following demand and total cost

functions P=100–q, TC= 50–2q+4q2 where P is price

and q is quantity of output. The government imposes

excise duty at a rate of Rs.2 per unit of output.

Find a) Profit maximizing output after payment of

tax

b) Show the effect of tax on equilibrium

output, price and profit.

35. A firm has the following production function q=f(L) =

–3L3+18L

2 where L is labour employment.

36. Find thefirst and second order partial derivatives of the


i) z=x3–9xy–3y

2



ii) z=x4+x

3y

3–3xy

3–3y

2

iii) z=(7x+3y)3

iv) z=3x2+12xy+5y

2

v) z = 18x2y –11xy

3

37. For the following functions

a) Find the critical points at which the function may be

optimised.

`b) Determine whether at these points the function is

maximised, is at an inflexion point, or is at a saddle

point.

i) z=3x2-xy+2y

2–4x–7y+12

ii) z=60x+34y–4xy–6x2–3y

2+5

iii) z=5x2–3y

2–30x+7y+4xy

38. Find the profit maximising level of

a) output b) Price and c) Profit for the monopolistic

producer with the demand functions

22112

1 - 36Q ,

3

2 -

3

149 PPQ

and the joint cost

function 120QQ2QQC 2

221

2

1

39. Do the above problem, for the functions

Q1=5200–10P1, Q2= 8200–20P2 and

3252.01.01.0 2

221

2

1 QQQQC

40. Minimise costs for a firm with the cost function

C+5x2+2xy+3y

2+800 subject to the production quota

X+y=39.



41. Maximize utility U=Q1Q2, subject to P1=10, P2=2 and

B=240. What is the marginal utility of money?

42. Find the extreme value of the function

.9622x subject to 20

2

12

2

121

2

221

2

1 xxxxxxy

mathematical economics - university of calicut

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