ECON 100AProf. Kamala Review ProblemFall 2014 Consider Melissa’s preference over goods X and Y which can be modeled by the following utility function: ( , ) (1 )U X Y X Y= + . She has 11 as her income. Part 1:

a) Solve her utility maximization problem by deriving her demand functions X and Y as a function of prices ( , )X Yp p and income, M.

b) Suppose that the prices are given as ( , ) (1,1).X Yp p = Solve for her demands for X and Y.

c) If the price of good X increases to 2, what will be her optimal consumption then?

d) Decompose the changes in quantity demanded of X when Xp changes from 1 to 2 into the (Hicksian) substitution and income effects.

e) What is the Hicksian compensation required to keep Melissa as well off before the price change as she was after the price change?

f) What is the Slutsky compensation required to keep Melissa as well off before the price change as she was after the price change? Why is this larger than your answer in d)?

Part 2:

g) Now, solve for Melissa’s demands for X and Y when the price of good X is: 50 cents, $1, $2 and $3. In all cases, fix the price of good Y at $1 and keep income at $11. Draw her price expansion path and Marshalian (inverse) demand curve.

h) Now fix prices at (1,1) but change her income from $11 to $15, then to $21, $27 and finally to $31. Calculate her demands at these prices. Draw the income expansion path and use this information to derive her Engel curve.

Part 3:

i) Set up and solve the expenditure minimization problem. The arguments you derive that minimize expenditure, X* and Y* are the compensated demand functions.

j) On the same set of axes, draw her Marshalian demand curve and Hicksian (compensated) demand curve for good X. Set initial prices at (1,1) and income at 11. Vary prices as in (g). Comment on which curve has the higher relative elasticity and why. If the curves cross, comment on where the two curves cross, and explain why this is.

I

v::) (v\ (A )\, v ext '-/) -= '!- c ( .,_ Y ) ::- x + Y ---r

c *- - If'~- X +- P'-~ Y ; M

'i: X t- X '1 +- \ [M- f~ f- f'-f 't ]

J- ~ I l + '1 ~ ;\ p(( :0 c A-:J ?'f.

u. 'f.. - A fy ~ u c~J ~y

• C) ~ . M- fy X - fy Y --- 0 [cJ -'JA ' t-

+

I +- '-( '(>'f----- -'1- "'

""' ~ ~ ~) - -

l -t' '{ f'l-

1"'\~ {C- J +

+ M -r'f- [~ Lt+Y)J- p'y 'I : \)

M- Py -~ r ~ ~ - f, y ;;:::- 0

M - f'/ - 2 P1 y =-D

M- f't ::. L. f1 Y ;;j rt:t. ~ 'Y ~) -

+

r

r

t

6)

M - p ?- X -fy( M - f tJ ::::- 0 L~ 2ft

M - 2 f'J. y._ + f 1

..: o

M t-fy = ?_('~ f ~ ['{._If-~ M'?.-~:'1 ..)

"':4--::: tl + (l t-"'

;)._

Yw-: 1/-( - S ::t

A+ L p .,_ I p '() ;; ( 2 I I ) c:::~.V\J /Jl ::::: I I

----X ~ - t I + Lt) _ l'l t *_- ~. I ,f--= ... ..-- ~

).(1..} '-{

y -4- - 5 (_ OV\ \1 ~"· L+ UV\ vC- r, ) -

r}.) S .fl.-e_ ~ f' J sol t-(-h\M. p~ ') f- (' q +~ t-4 d

s]

d) Decompose the changes in quantity demanded of X when p x changes from 1 to 2 into the substitution and income effects.

We would now like to decompose this into the substitution and income effects. Recall that the

substitution effect is the change in demand, holding utility constant (the so-called "Hicksian"

substitution effect). The income effect is the change in demand due to the change in purchasing

power.

First, we should recall the general MRS. It is easy to see that this is simply 1 + y = Px . X Py

Now, we would like to figure out the original level of utility before the price change. Plug in the

first set of (old) prices into the utility function:

U=X(l+Y)

u = 6(1 + 5) = 36 Now, using the MRS condition and the utility function, we solve for the values of X andY

required to maintain consumption :

36 ~ X(l + 7.5) Now I I so that

X"'"4.2

U = X(l+Y)

u = [ p,, o + n] [ o + n] Pr

u = l. (1 + Y) 2

2

36 = l.o + Y)2

2

\\

72 = (1 + Y) 2 => JTi = 1 + y

Y=JTi-1 lr ~ 7.sl

Hence, the substitution effect is 6- 4.2 = [1]] and the income effect is 4.2- 3 = [11].

3

I ~

<?) To --h'"J ~L f1-,· J~/OL., ~~tJftr'?f&.-h UVJ, [,v.e_

V\ S .e_ ~ e new o p-hy., c.. ( b'""' ""'d l c... VVl --KL t:3 LPf IV\.c.L.C.~c..e.. Cu.rve. q-J- ~ f\..tw P'"' u.s ~

"*'-L V\e w sJ~ 0 -t 1"h.e.. !V._w bvt ~ t--1- l (~' - 2_ .

Po,V\+-- sf~ h~~

f - 't 1 " M ex- /( I) y ---7·~ ::- -2. ( Y..- ~. 2-)

y- 7. ~.: - 2 '{.. + s. y -t-1.r -r- 1 ' .s-

Y ..:; - 21< + I 5:"'. ~

R e. VI r, --h' A ~ fi._,' s. t\.S bv.) j e-1- ~s-/vz.. l/) f1

?_"/_ +y - I~,~ +-z.ll ,· ~ s uS ..f-'4~+ ~~ 11 ~'--' .-)

r f: f. LA' re-c:.{ ' I~. 1, 111 t.a Me.- I .S

Ovr oleA 1 /leo r'l e v/eA.S l I .

S o I s-. q _, r { ~ 4 r '{ 0; LJ h ,· c. k ; s ~ e._

j-{; c..fc.~; ~"' C-o ~ f .e_...., ~~ ~ ~ .

. ...P) nL SCA> "'-y co,.,.pt>'l.tt.fl '~ iS 3. t/&, br ~ft V) j 1h e_ Drt ~;"' G l bi.Ail J I fL t b' s- ) ~" J .e.s -h'r-1. a-f,,._J'

~t. b \AJ.) .t-.f- I'~ 1/,rvtAJh ...f1...L /1-t..t.J fr/ c..e.J,

"{ - s ::- - 2 ( 1- - ~ ) y-r...: -2 x. +- 112_,

y ::: -2_;<.+-17

..=:>-) 2 'f -r Y ::::- I 7 :::-/ MR,._, f?/1. l i f1 tr:>V'I-e rP?, I.A 1 rc.J :: $ tl

$17- $fl::

A l 5o V1 o -1-e ~"' f l.-1-t.. c_ ovt l J h e-. v.e.. <J ot~ ~ ro \.) J' V\ j ~L w~ ..... , e._

b M := X .. 6 fr~· c...e..s ~b .. ().-r) :::- C·/~1£.

--17. ~e J IAJ ~ &t "'J he,r k~L-L 1.+ is ~'5h~

S { uf~ ~ C--o W1 f e-'1 Sc. +,cYl ~ i VI ) : f.(4e_"""'_ c_ .f?_ c v rvR.

1 : ~&>tV\ teJ. t

--J. s\vkS~ :r~

g)

Price of X

Price of Y Income

Demand for X

Demand for Y

0.5 1 11 12 5 1 1 11 6 5 2 1 11 3 5 3 1 11 2 5 6 1 11 1 5

Price Expansion Path (draw indifference curves/budget constraints yourself)

Marshalian Demand Curve:

0

2

4

6

8

10

12

0 5 10 15

Cons

umpt

ion

of Y

Consumption of X

Price Expansion Path

Series1

0.5, 12

1, 6

2, 3 3, 2

6, 1 0

2

4

6

8

10

12

14

0 2 4 6 8

Pric

e

Quantity

Demand for X

Demand for X

h) Income expansion path (draw the indifference curves/budget lines yourself!)

Price of X

Price of Y Income

Demand for X

Demand for Y

1 1 11 6 5 1 1 15 8 7 1 1 21 11 10 1 1 27 14 13 1 1 31 16 15

Engel curve

6, 5

8, 7

11, 10

14, 13

16, 15

0

2

4

6

8

10

12

14

16

0 5 10 15 20

Dem

and

for Y

Demand for X

Income Expansion Path

6, 11 8, 15

11, 21

14, 27 16, 31

0

5

10

15

20

25

30

35

0 5 10 15 20

Inco

me

Consumption of X

Engel Curve

~,

' ' c~,...,~s ... ~ o/e ..... ,.,j cv.-vu ") ive... ~ J -) } '()') fV\ 1 t'\ fx y._ ~ fy 'i-I .$ . + . X +- 'j 'i .: I)

~ ~ I P~ /\ - ;\ \( ::: O C A :J ~ - · -C).'/-

;) I, I Py - 1\ 'f - 0 Crs J ___..

';Jj

· ~ ,( · V - 'f- Xi ~ 0 {(.,] c2 (\ /

+ CAl I [YJ] ·.

(1+'1) - --- -)

"'

lJ - ,f((l+Y J ~ D

,-ti -* C I r YJ C I t- 'I) 2 0

-\) - L--~ c l-tV) '2- =- 0 f'f- t

5

[5'A ME.. Mfl-S

cDV\/~Ln~J

I

- T u ,; f_j I c 1-r y) 7. ' J 1\ :;7 rr:t ;; (;"")-;_ c \ +-'{) ' :; )

+

t --t u .::: 3G l I t

t t I

t

+

Table for (j)

Final Hicksian demand function for X*:

* 1

y

y x

x y

x

PU

P PX

P PP

− = +

, with U=36 at the given prices and income.

Price of X

Price of Y Income

Demand for X

Demand for Y

Hicks Demand X

Hicks Demand Y

0.5 1 11 12 5 8.485281 3.242641 1 1 11 6 5 6 5 2 1 11 3 5 4.242641 7.485281 3 1 11 2 5 3.464102 9.392305 6 1 11 1 5 2.44949 13.69694

Cross at the optimum (X=6, p=1).

Hicksian is more inelastic less responsive to price changes after we “compensate” her for the loss in purchasing power.

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14

Marshalian vs. Hicksian Demand Curves

Marshalian Demand

Hicksian Demand