Quantitative Methods for Economics

Katherine Eyal Tutorial 5

TUTORIAL 5

23rd-27th August 2010

ECO3021S

PART 1

1. A monopolist sets a price p = q−1/β where p is the price, q is the quantity of thegood being produced, and β is the elasticity of demand for the good. We assume thiselasticity of demand is constant. The monopolist faces a cost function TC = c.q.

(a) What does the firm’s cost function imply about the firm’s marginal cost?

(b) Find the equilibrium price and quantity that the monopolist should charge inorder to maximize profits.

2. Find the extreme values of the function z = 8x3 + 2xy − 3x2 + y2 + 1. Classify theextrema as local (relative) or global (absolute) and as minima or maxima.

3. A South African car manufacturer has the following total revenue and total coststructures:

TR = 8Q−Q2

TC =1

3Q3 − 10Q2 + 80Q+ 500

How many cars should the firm produce if its goal is to maximize profits?

4. The production of Gautrain rolling stock (Q) requires two inputs, capital (K) andlabour (L):

Q = f(K,L) = K1/4L1/4

A unit of capital costs R8 and a unit of labour costs R4.

(a) What is the cost minimising bundle of capital and labour to produce any fixedQ = Q?

(b) Confirm that your answer to (a) represents a constrained minimum.

(c) Give an economic interpretation of the Lagrangian multiplier.

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PART 2

5. Every individual faces a decision about how to allocate their time between leisureactivities (R) and work activities (L). On any day, there are only 24 hours in whichthe individual can either be working or enjoying leisure activities i.e. L + R = 24.If the individual works, he earns an income (I) where I = wL. In other words, theincome earned is equal to the wage paid (w), multiplied by the number of hoursworked (L). Suppose the individual’s utility function is given by:

U(I,R) = 4IR2

The hourly wage rate is R4. How many hours should the individual devote to leisure,and how many should be devoted to work?

6. A perfectly competitive firm sells its output for R12 per unit. This firm has a totalrevenue function and total cost function given by:

TR = P.Q

TC = Q3 − 4.5Q2 + 18Q− 7

(a) What level of output should this firm produce in order to maximize profits?

(b) According to microeconomic theory, perfectly competitive firms will maximizeprofit by producing where P = MC, and where the slope of the MR curve isless than the slope of the MC curve. Show that theory holds in this example.

7. A monopolist can produce quantities x and y of two products X and Y respectively,at cost 4x2 + xy + 2y2. The inverse demand functions are

pX = 150 − 5x+ y

pY = 30 + 2x− 2y

where pX and pY are the prices charged for X and Y .

(a) Find the values of x, y, pX and pY which maximise profit, and the maximalprofit.

(b) Confirm that your answer to (a) represents a maximum.

8. The utility a consumer derives from consuming two goods, A and B, is determinedby the following utility function:

U = 40A0.25B0.5

If A costs R4 per unit, and B costs R10 per unit, and the consumer has R600 tospend, what combination of A and B will maximize the consumer’s utility?

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TUTORIAL 4 SOLUTIONS

ECO3021S

PART 1

1. A monopolist sets a price p = q−1/β where p is the price, q is the quantity of thegood being produced, and β is the elasticity of demand for the good. We assume thiselasticity of demand is constant. The monopolist faces a cost function TC = c.q.

(a) What does the firm’s cost function imply about the firm’s marginal cost?

MC(q) =d

dq[TC(q)] = c.

This implies that marginal cost is constant for a monopolist.

(b) Find the equilibrium price and quantity that the monopolist should charge inorder to maximize profits.

The monopolist will set its price where MC = MR. Here

TC(q) = cq

TR(q) = pq = q(q−(1/β)) = q1−(1/β) = q(β−1)/β.

So now

MR = MC

⇒(β − 1

β

)q−(1/β) = c

q−(1/β) =

(β

β − 1

)× c

q =

[(β

β − 1

)× c]−β

And

p = q−(1/β)

=

([(β

β − 1

)× c]−β)−(1/β)

=

(β

β − 1

)× c

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2. Find the extreme values of the function z = 8x3 + 2xy − 3x2 + y2 + 1. Classify theextrema as local (relative) or global (absolute) and as minima or maxima.

Dz =

[∂z/∂x∂z/∂y

]=

[24x2 + 2y − 6x

2x+ 2y

]FOCS

24x2 + 2y − 6x = 0 (1)

2x+ 2y = 0 (2)

Solve simultaneously:

From (2)y = −x (3)

Substitute (3) into (1)

24x2 + 2 (−x)− 6x = 0

⇒ 24x2 − 8x = 0

⇒ x (24x− 8) = 0

⇒ x = 0,1

3

Therefore, candidates for extrema are x = 0, y = 0, z = 0 and x =1

3, y = −1

3, z =

23

27.

SOCS:

H = D2z =

[∂2z/∂x2 ∂2z/∂x∂y∂2z/∂y∂x ∂2z/∂y2

]=

[48x− 6 2

2 2

]The definiteness of the Hessian matrix depends on the value of x, so it cannot beeverywhere positive or negative definite.

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We must determine the definiteness of the Hessian matrix at the candidate extrema:

At x = 0, y = 0, z = 0 :

H =

[−6 22 2

]|H1| = |−6| = −6 < 0

|H2| =

∣∣∣∣−6 22 2

∣∣∣∣ = −16 < 0

At this point, H is neither positive nor negative definite. So this point is neither aminimum nor a maximum.

At x =1

3, y = −1

3, z =

23

27:

H =

[10 22 2

]|H1| = |10| = 10 > 0

|H2| =

∣∣∣∣10 22 2

∣∣∣∣ = 16 > 0

At this point, H is positive definite. So this point is a local minimum.

3. A South African car manufacturer has the following total revenue and total coststructures:

TR = 8Q−Q2

TC =1

3Q3 − 10Q2 + 80Q+ 500

How many cars should the firm produce if its goal is to maximize profits?

To maximize profits, set up the objective function and find the stationary points bysetting the first derivative equal to zero and solving:

π(Q) = TR− TC

= 8Q−Q2 − (1

3Q3 − 10Q2 + 80Q+ 500)

= −1

3Q3 + 9Q2 − 72Q− 500

π′(Q) = 0

⇒ −Q2 + 18Q− 720 = 0

⇒ (Q− 12)(Q− 6) = 0

3

Thus Q = 12 or 6.

Now we must check whether these stationary points are maxima or minima by eval-uating the second derivative at the stationary points:

π′′(Q) = −2Q+ 18

So, we can see that there is a relative maximum where production is equal to 12units. Therefore, the firm maximizes its profits where it produces 12 cars.

4. The production of Gautrain rolling stock (Q) requires two inputs, capital (K) andlabour (L):

Q = f(K,L) = K1/4L1/4

A unit of capital costs R8 and a unit of labour costs R4.

(a) What is the cost minimising bundle of capital and labour to produce any fixedQ = Q?

Problem:minK,L

8K + 4L subject to Q = K1/4L1/4

Lagrangian:

= 8K + 4L+ λ(Q−K1/4L1/4

)FOCs:

∂

∂K= 8− λ1

4K−3/4L1/4 = 01 (4)

∂

∂L= 4− λ1

4K1/4L−3/4 = 02 (5)

∂

∂λ= Q−K1/4L1/4 = 03 (6)

(4)÷ (5)

2 =L

K⇒ L = 2K4 (7)

Sub (7) into (6)

Q−K1/4 (2K)1/4 = 0

K∗ =Q

2

√2

4

L∗ =√

2Q2

(b) Confirm that your answer to (a) represents a constrained minimum.

H =

0 −14K−3/4L1/4 −1

4K1/4L−3/4

−14K−3/4L1/4 3

16λK−7/4L1/4 − 1

16λK−3/4L−3/4

−14K

1/4L−3/4 − 116λK

−3/4L−3/4 316λK

1/4L−7/4

∣∣H∣∣ = − 1

32λK−5/4L−5/4

< 0

⇒ minimum

(c) Give an economic interpretation of the Lagrangian multiplier.

It is the marginal cost.

PART 2

5. Every individual faces a decision about how to allocate their time between leisureactivities (R) and work activities (L). On any day, there are only 24 hours in whichthe individual can either be working or enjoying leisure activities i.e. L + R = 24.If the individual works, he earns an income (I) where I = wL. In other words, theincome earned is equal to the wage paid (w), multiplied by the number of hoursworked (L). Suppose the individual’s utility function is given by:

U(I,R) = 4IR2

The consumer has to maximize utility (which depends on either working and earningmoney or engaging in leisurely pursuits) subject to a time constraint:

⇒ max U(I,R) = 4IR2

subject to L+R = 24

Our objective function is in terms of I and R, so we have to re-write our constrainti.t.o. I and R as well.

We know I = wL and that w = 4 (as given). Therefore, L =I

w=I

4and so the

constraint becomesI

4+R = 24

5

. We can now set up our Lagrangian function:

maxI,R

= 4IR2 + λ(24− 1

4I −R)

F.O.C.s:

∂

∂I= 0⇒ 4R2 − 1

4λ = 01 (8)

∂

∂R= 0⇒ 8IR− λ = 02 (9)

∂

∂λ= 0⇒ 24− 1

4I −R = 03 (10)

Solve for the independent variables. To do this we can divide (8) by (9)

⇒ 4R2

8IR=

14λ

λ

⇒ R

I=

2

4

⇒ R =1

2I

Substitute for R in equation (10) to get

24− 1

4I − 1

2I = 0

⇒ 3

4I = 24

⇒ I = 32

∴ R =1

2I =

1

2(32) = 16

And L =1

wI =

1

4(32) = 8

The consumer therefore maximizes utility when he/she spends 16 hours on leisureactivities and 8 hours on work (which add up to 24, as per the constraint). The valueof the utility function for these values is equal to:

U∗(I , R) = 4(32)(16)2 = 32768

6

Is this definitely a maximum? To answer this question we have to evaluate theS.O.C.s by constructing and interpreting the Bordered Hessian:

H =

0 gx gygx xx xy

gy yx yy

=

0 −14 −1

−14 0 8R−1 8R 8I

We are now interested in the signs of the second and subsequent principal minors:(here there will be only a second principal minor).∣∣H2

∣∣ =1

4

[(−1

4)(8I)− (−1)(8R)

]− 1

[(−1

4)(8R)− 0

]=

1

4[−2I + 8R] + 2R

= −1

2I + 4R

Which is a maximum if −12I + 4R > 0.

6. A perfectly competitive firm sells its output for R12 per unit. This firm has a totalrevenue function and total cost function given by:

TR = P.Q

TC = Q3 − 4.5Q2 + 18Q− 7

(a) What level of output should this firm produce in order to maximize profits?

maxQ

π = P.Q−Q3 + 4.5Q2 − 18Q+ 7

You are told that P = 12, substitute this in and re-write the expression as:

maxQ

π = 12Q−Q3 + 4.5Q2 − 18Q+ 7

F.O.C.:

∂π

∂Q= 12− 3Q2 + 9Q− 18 = 0

⇒ −3Q2 + 9Q− 6 = 0

⇒ (3Q− 6)(Q− 1) = 0

7

So Q = 1 or 2.

Now, to be a maximum we need the second derivative to be negative:

∂2π

∂Q2= −6Q+ 9 < 0

if Q >9

6

Thus, Q = 2 gives us our maximum. (Alternatively, you could just plug in thetwo values into your S.O.C. and see which one is negative).

(b) According to microeconomic theory, perfectly competitive firms will maximizeprofit by producing where P = MC, and where the slope of the MR curve isless than the slope of the MC curve. Show that theory holds in this example.

Derive an expression for MR and MC

MR =∂TR

∂Q= 12

MC =∂TC

∂Q= 3Q2 − 9Q+ 18

The firm maximizes its profits where

MR = MC

⇒ 12 = 3Q2 − 9Q+ 18

and with a bit of re-arranging you will see that you have the exact same expres-sion as you derived earlier for the first derivative. So, again, you’ll solve for thevalues of Q = 1 and Q = 2.

Now you are asked to show that the firm only max. profits if the slope of MRcurve is less than the slope of the MC curve. Essentially, you’re being asked (ina roundabout way) to find the S.O.C.. To find the slope of the MR curve, takethe second derivative of the TR:

TR′′(Q) =d

dQ(MR) = 0

TC ′′(Q) =d

dQ(MC) = 6Q− 9

If the slope of the MR curve is less than the slope of the MC curve, then thismeans that

6Q− 9 > 0

⇒ Q >9

6

which is again exactly what you showed earlier with your S.O.C.!!!

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7. A monopolist can produce quantities x and y of two products X and Y respectively,at cost 4x2 + xy + 2y2. The inverse demand functions are

pX = 150− 5x+ y

pY = 30 + 2x− 2y

where pX and pY are the prices charged for X and Y .

(a) Find the values of x, y, pX and pY which maximise profit, and the maximalprofit.

π = xpX + ypY −(4x2 + xy + 2y2

)= x (150− 5x+ y) + y (30 + 2x− 2y)− 4x2 − xy − 2y2

= 150x− 5x2 + xy + 30y + 2xy − 2y2 − 4x2 − xy − 2y2

= −9x2 + 2xy + 150x− 4y2 + 30y

FOC:

Dπ (x, y) =

[∂π/∂x∂π/∂y

]= 0

⇒[−18x+ 2y + 150

2x− 8y + 30

]= 0

Solving simultaneously:

x∗ = 9

y∗ = 6

Therefore:

p∗X = 150− 5 (9) + 6 = 111

p∗Y = 30 + 2 (9)− 2 (6) = 36

And:

πmax = xpX + ypY −(4x2 + xy + 2y2

)= 9 (111) + 6 (36)−

(4 (9)2 + 9 (6) + 2 (6)2

)= 765

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(b) Confirm that your answer to (a) represents a maximum.

SOC:

H = D2π (x, y) =

[∂2π/∂x2 ∂π/∂x∂y∂π/∂y∂x ∂2π/∂y2

]=

[−18 2

2 −8

]

|H1| = |−18| < 0

|H2| =

∣∣∣∣−18 22 −8

∣∣∣∣ = 140 > 0

Thus, H is negative definite and hence we have a (global) maximum.

8. The utility a consumer derives from consuming two goods, A and B, is determinedby the following utility function:

U = 40A0.25B0.5

If A costs R4 per unit, and B costs R10 per unit, and the consumer has R600 tospend, what combination of A and B will maximize the consumer’s utility?

U = 40A0.25B0.5

pA = 4, pB = 10, m = 600

Set up the Lagrangian function:

maxA,B

= 40A0.25B0.5 + λ(600− 4A− 10B)

F.O.C.s:

∂

∂A= 0 =⇒ 10A−0.75B0.5 − 4λ = 01 (11)

∂

∂B= 0 =⇒ 20A0.25B−0.5 − 10λ = 02 (12)

∂

∂λ= 0 =⇒ 600− 4A− 10B = 03 (13)

Now solve for the variables:

10

Divide (11) by (12)

⇒ 10A−0.75B0.5

20A0.25B−0.5=

4λ

10λ

⇒ B

2A=

2

5

⇒ B =4A

5

Now substitute B into equation (13) and solve for A

600− 4A− 10B = 0

⇒ 600− 4A− 10(4A

5) = 0

⇒ 600− 12A = 0

⇒ 12A = 600

⇒ A = 50

∴ B =4(50)

5= 40

The consumer therefore maximizes utility when he/she consumes 50 units of A and40 units of B. However, we must now check if this is indeed a maximum. To do thiswe check the S.O.C.s by constructing the Bordered Hessian and evaluating it:

H =

0 gx gygx xx xy

gy yx yy

=

0 −4 −10−4

(−30

4 A−1.75B0.5

) (5A−0.75B−0.5

)−10

(5A−0.75B−0.5

) (−10A0.25B−1.5

)

∣∣H2

∣∣ = 4

[(−30

4A−1.75B0.5)(−10A0.25B−1.5)

]− 10

[(5A−0.75B−0.5)

]2= 4

[300

4A−1.5B−1

]− 10

[25A−1.5B−1

]= 300A−1.5B−1 − 250A−1.5B−1

= 50A−1.5B−1 > 0

∴∣∣H2

∣∣ > 0

Because∣∣H2

∣∣ > 0 ∀A,B ∈ N+ ⇒ the Hessian is negative definite and the functionhas a maximum.

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