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UNIVERSITY OF TORONTO SCARBOROUGH DEPARTMENT OF MANAGEMENT

MGEB02: Price Theory: A Mathematical Approach

Instructor: A. Mazaheri Sample Test-1.1 (Solutions)

Instructions: This is a closed book test. You have 2 Hours. Good Luck! Last Name: First Name: ID FOR MARKERS ONLY:

Q1 Q2 Q3 Q4 Q5 Total Marks Earned

Maximum Marks

Possible 40 15 15 18

12 100

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Answer all following 5 questions: Question-1 [40 Points] Answer the following Short Questions: a) [4 points] You are analyzing the market for Crude oil in the last decade or so, you know that the price has risen from $20 or so to $100+ during this period. Show what must have happened to the demand and supply to lead to such an equilibrium. Demand shifts right, P & Q increase

b) [6 Points] In the following, the initial equilibrium is given. Suppose price of X declines. Assuming X is Giffen, draw the new equilibrium. On the same graph show the income and the substitution effects.

SE IE

X

Y

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c) [6 Points] Draw representative indifference curves for the followings:

i) Sandra has a strange habit and she insists on it; she likes to eat an apple and two bananas together. Perfect complement: ii) Adam does not care about orange juice or apple juice as long as he has juice.

Perfect Substitute:

Banana

Apple

Orange Juice

Apple Juice

1

1

2

1

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d) [6 Points] Assume you have a fixed budget of $10. Further assume that you spend your entire budget. Both good X and good Y cost $1 each. You are spending all your money on X. At this bundle, your marginal utility of X is 10 while your marginal utility of Y is 5. Are you optimizing your utility? Why or why not? Using a graph explain your answer. Solution:

(1) spend your entire income because you are on the budget line (2) The MRS = 10/5=2 > 1, or (MUx/ Px) > (MUy/ Py).

That is, the marginal utility of X per dollar spent is higher than that of Y. However, you already gave away all Y and cannot get more X. Therefore your optimal consumption bundle is a corner solution where you consume no Y.

X

Y

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e) [8 Points] Assume a utility function that is given by U(X,Y) = X0.5Y0.5. Further assume a budget of $50. When the prices where PX=1 and PY=1, you consumed X= 25 and Y = 25, while when the prices changed to PX=2 and PY=1, you consumed X= 12.5 and Y = 25. With the help of the following graph decompose the total effect of the price change into the substitution and income effects. Solution: We know TE = 25-12.5. We need to find the SE. Having SE we can solve for IE as TE=SE+IE. SE is the change in the quantity of x demanded, if (1) the individual remains at the same indifference curve and (2) if MRS is equal to the new price ration.

18.568.175.12 32.72568.17

68.17)2(25)2(),1(

2 1 2

:2

252525:1

5.05.0

5.05.05.05.0

−=−= −=−=

==>==>

==>==

==×=

IE SE

XXX

XY X Y

MRS

YXU

Graphically:

12.5

50

25

25

50 17.68

IE

SE

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f) [5 Points] You have 5 spent on two products, a composite product (Y) with a price of py = 1 and coffee (X) with a price of px = 1. If you purchase three cups of coffee, you will be offered one coffee for free and a 50% discount on each additional cup of coffee purchased. Show graphically how this affects your budget line.

8

5

3 4

y

x

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g) [5 Points] Ahmed only consumes hamburger (Y) and coffee (X). He wants two cups of coffee. If he gets less than two cup of coffee he will not care about anything else and if he is given more than two cups of coffee he will discard it. Graph his representative indifference map.

Coffee

Hamburger

2

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Question-2 [15 Points] As a manufacturer you are interested in obtaining quick estimates of the supply and demand curves for your product. You have done some research and you know that for your product the elasticity of supply is 2, the elasticity of demand is -1.5. You also know that the current price and quantity are $50 and 1,000, respectively. Assume that both demand and supply are linear. a) [6 Points] What is the supply and demand curves at the current price and quantity. b) [4 Points] What impact would a 10% decline in demand have on the equilibrium price and quantity? d) [5 Points] Ignore part (b). Suppose the government subsidizes your product by 5 dollars per unit. What would be the new equilibrium price? Use a graph to show your answer.

Solution: a) Demand curve Q = a0 + b0P Ed = b0 × P/Q = -1.5 = b0 × 50/1000 b0 = - 30 => 1000 = a0 - 30(50) => a0 = 2500 Qd = 2500 - 30P Next, we estimate the supply curve Q = a1 + b1P E1 = b1 × P/Q = 2 = b1 × 50/1000 b1 = 40 => 1000 = a1 + 40(50) => a1 = -1000 Qs = -1000 + 40P b) Multiply demand equation by 0.9 Qdʹ = 0.9 (2500 - 30P) Qdʹ = Qs and solve 2250 - 27P = -1000 + 40P P = 48.51, => Qdʹ = 2250 - 27(48.5) Qdʹ = 940.3 c) PD = PS -5, QD =QS = Q Qd = 2500 - 30P => PD = 2500/30-(1/30)Q Qs = -1000 + 40P => PS = 25 +(1/40)Q PD = PS -5 => 2500/30-(1/30)Q = 25 +(1/40)Q –5 => Q =1085.7=> PD = 47.14

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Question-3 [15 Points] Suppose your preferences for Gasoline (X) and a composite good (Y) is set in accordance to U(X,Y) = 2X0.25 + Y0.25. You have an annual income of $30,000 and that the price of the composite good is $1. a) [5 Points] If the government introduces a rationing system such that you can only consume 15,000 liters a year at $1 a liter. What would be your optimal consumption bundle? b) [6 Points] If the government removes the rationing system and the free market price of gasoline jumps to $2. What would be your new optimal consumption bundle? Are you better off with or without the rationing? c) [4 Points] Illustrate your solution in a clearly labeled graph.

a)

11.8523 89.21476

000,303969.0 3969.05.0

2 1 1

25.0 5.0

75.075.0

75.0

75.0

75.0

75.0

==> ==>

=+=> ==>==>

=== −

−

Y X

XX XYXY

X Y

Y X

MRS

But cannot consume more than 15000 L, therefore corner solution:

000,15 000,15

000,30

==> =

=+

Y X

YX

b) Without the rationing and with the new price:

000,10 000,10

000,302

2 2

75.0

75.0

==> ==>

=+=> ==>

==

Y X

XX XY X Y

MRS

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You are better off with the rationing because:

withoutRation

without

Ration

UU U U

> =+×=

=+×=

30000,10000,102 20.33000,155000,12

25.025.0

25.025.0

c)

30

30

10 15

15

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Question-4 [18 Points] Mary has the following utility function: U(x, y) = 4y + 2x0.5. Let px and

py be the corresponding prices and I her income.

a) [6 Points] Setup the Lagrangian function and find the first order conditions (FOCs). Use these FOCs to find the expression for the marginal rate of substitution (MRS). Use the MRS to graph the indifference map. What is “special” about these indifference curves?

4

0)(

04

0

)(24

5.0

5.0

5.0

−

−

=

=−+−= ∂ ∂

=−= ∂ ∂

=−= ∂ ∂

−+−+=

x MRS

IYpXp L

p y L

px x L

IYpXpxyL

yx

y

x

yx

λ

λ

λ

λ

The indifference curve can cross the horizontal line since MRS does not depend on y.

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b) [6 Points] Find the demand functions for x and y. Graph the Engle curve for x.

0)(P if

16 4

if 4

44

* x

2

* 2

2 *

2/1

==>>=

−=  

  

 −

= −

==>≤ 

  

 =

 

  

 ==>=

−

yIx P I

x

p p

p I

p p

p pI

p xpI

yIxP p

p x

p p

x p px

x

x

y

yy

x

y x

y

x X

x

y

x

y

y

x

(Py/4Px)2

Slope = Px

I

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c) [6 Points] Suppose Mary lives in a city where px = 1, py = 4 and her job offers her I = 0.5. Find the optimal consumption levels for x and y? Graph your solution.

1 4 4

4

22

=  

  ==>



  

 = x

p p

x x

y

cannot afford this so hr consumption will be 0.5 X and no y.

0.5

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Question-5 [12 Points] There are 50 consumers in the economy. Half of them live in city A and demand Orange according to the individual inverse demand curve P = 2− Q. The other half live in city B and demand Orange according to the individual inverse demand curve P = 6−3Q. Suppose that the market-clearing price for Orange is $1. a) [6 Points] Write down the market demand for Orange in this economy and then graph it? b) [6 Points] Suppose the price increases from $1 to $2, how does the consumer surplus change? Solution:

[ ]

PQ

PP Q

PPQ

M

B

A

33.33100 3

25 50

3 22525

255022525

−=

−= 

  −=

−=−=

But demand seize to exist in city A when the P >= 2 while in city B the demand is zero when P>=6 therefore the demand will be kinked at P=2 or:

)2(33.33100

)2( 3

25 50

≤−=

>−=

PPQ

P P

Q

M

M

33.33

2

6

100

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b)

[ ]

505.375.12 5.37

67.66) 3 4

45.0(25

3 4

3 2:2

17.104) 3 5

55.0(25

3 5

3 2:1

25 0

0,2 5.12)5.011(25

12:1

−=−−=∆ −=∆

=×××=

= 

  −==

=×××=

= 

  −==

−=∆ =

== =×××=

=−==

TS CS

CS

P QP

CS

P QP

CS CS

QP CS

PQP

A

B

B

B

B

A

A

A

A

A