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EC2002

Mock Exam No. of Pages: 5 No. of Questions: 7

Subject ECONOMICS

Title of Paper EC2002: Intermediate Microeconomics II

Time Allowed Two Hours (2h)

Instructions to candidates

Answer all questions. The use of a non-programmable calculator is allowed.

SECTION A-

This section carries 50% of the exam mark.

Question 1

The price elasticity of demand for oatmeal is constant and equal to −1. When the price ofoatmeal is 10 per unit, the total amount demanded is 6, 000 units.

Part 1 (5 points)Write an equation for the demand function.

Solution:

q = 60, 000/p.

Part 2 (5 points)If the supply is perfectly inelastic at 5,000 units, what is the equilibrium price?.

Solution:

12.

Question 2

Banana Computer Company sells Banana computers in both the domestic and foreignmarkets. Because of differences in the power supplies, a Banana purchased in one marketcannot be used in the other market. The demand and marginal revenue curves associatedwith the two markets are as follows:

Pd = 20, 000− 20Q

Pf = 25, 000− 50Q

MRd = 20, 000− 40Q

MRf = 25, 000− 100Q

Banana’s production process exhibits constant returns to scale and it takes 1, 000, 000 toproduce 100 computers.

Part 1 (5 points)If Banana is maximizing its profits, how many computers and at what prices will it sellin both markets?

Solution:

it will sell 250 computers in the domestic market at 15, 000 dollars each and 150 com-puters in the foreign market at 17, 500 dollars each.

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Part 2 (5 points)What are Banana’s total profits?

Solution:

2, 375, 000

Question 3

Dean Foster Z. Interface and Professor J. Fetid Nightsoil exchange platitudes and bro-mides. When Dean Interface consumes TI platitudes and BI bromides, his utility is givenby UI(BI , TI) = BI + 2

√TI . When Professor Nightsoil consumes TN platitudes and BN

bromide, his utility is given by UN(BN , TN) = BN + 4√TN . Dean Interface’s initial endow-

ment is 12 platitudes and 8 bromides. Professor Nightsoil’s initial endowment is 4 platitudesand 8 bromides.

Part 1 (5 points)If Dean Interface consumes TI platitudes and BI bromides, and Professor Nightsoilconsumes TN platitudes and BN bromides, what will there marginal rates of substitu-tion be?

Solution:

−T−1/2I and −2T−1/2

N .

Part 2 (5 points)On the contract curve, Dean Interface’s marginal rate of substitution equals ProfessorNightsoil’s. Write an equation that states this condition.

Solution:√TI =

√TN/2

Question 4

Consider a pure exchange economy with two consumers and two goods. At some givenPareto efficient allocation it is known that both consumers are consuming both goods andthat consumer A has a marginal rate of substitution between the two goods of 3.

Part 1 (5 points)What is consumer B’s marginal rate of substitution between these two goods?

Solution:

3.

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Part 2 (5 points)In a competitive equilibrium, what should be the ration between the prices of bothgoods if initial endowments where (3, 4) and (1, 2) for consumers 1 and 2 respectively?

Solution:

3.

Question 5

A chemical works makes 1, 000 profit but pollutes a fish farm. The fish farm makes 400 profitbut would make 600 without the pollution. The production process could be made clean byinstalling a certain technology at a cost of 300. The chemical works could move elsewhereat a cost of 600. Assume that negotiation is possible and that the Coase Theorem applies.

Part 1 (5 points)Assume that the chemical works has the right to pollute, will the fish farm install theclean technology?

Solution:

No, cost of clean technology is greater than the benefit from installing it. Optimaloutcome is the status quo.

Part 2 (5 points)Assume that the chemical works does not have the right to pollute, will the chemicalworks install the clean technology to avoid paying compensation to the fish farm?

Solution:

As optimal outcome is the status quo, if the chemical works does not have the right topollute then it is better off by paying 200 compensation and continue to pollute.

SECTION B-

This section carries 50% of the exam mark.

Question 6

Consider a Cournot setting where there are n > 1 identical firms producing with constantmarginal cost MC = c. The inverse demand is given by

P = a− bQ

where Q is the aggregate demand and a and b are two positive numbers.

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Part 1 (5 points)Show that the best response of each firm is given by

qi = (a− c− b(q1 + . . .+ qi−1 + qi+1 + . . .+ qn))/2b

where qi is the quantity produced by firm i.

Solution:

Each firm solves

maxqi

(a− b

n∑i=1

qi

)qi − cqi

which results in the equation given.

Part 2 (5 points)Consider the symmetric equilibrium: i.e. q1 = q2 = . . . = qn = q. What is the quantityproduced by each firm q as a function of the number of firms n?

Solution:

Substitute qi for q for all i ∈ {1, . . . , n} in the equation from part 1 and solve to obtain:

q =a− c

b(n+ 1).

Part 3 (5 points)Assume now that n = 1, i.e. there is only one firm. What is the optimal quantity the firmwill produce? How does this quantity compare with the quantity a monopolist wouldproduce?

Solution:

Substitute n = 1 in part 2 to obtain q = a−c2b

, which equals the monopolist profitmaximizing quantity as expected.

Part 4 (5 points)What happens to the price if n tends to infinity? Is this result surprising?

Solution:

If n → ∞ then P → c. With infinity many firms price equals marginal cost, asexpected given that infinitely many firms imply perfect competition.

Part 5 (5 points)Assume that n = 2. Does an asymmetric equilibrium (i.e. and equilibrium with q1 6= q2)exist?

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Solution:

Substitute n = 2 in part 1 to find a linear system of two equations, two unknowns anda unique solution with q1 = q2. Hence, the answer is no.

Question 7

Jim and Tammy are partners in Business and in Life. As is all too common in this imperfectworld, each has a little habit that annoys the other. Jim’s habit, we will call activity X, andTammy’s habit, activity Y . Let x be the amount of activity X that Jim pursues and y bethe amount of activity Y that Tammy pursues. Due to a series of unfortunate reverses,Jim and Tammy have a total of only 1, 000, 000 a year to spend. Jim’s utility function isUJ = cJ + 500 lnx− 10y, where cJ is the money he spends per year on goods other thanhis habit, x is the number of units of activity X that he consumes per year, and y is thenumber of units of activity Y that Tammy consumes per year. Tammy’s utility function isUT = cT +500 ln y−10x, where cT is the amount of money she spends on goods other thanactivity Y , y is the number of units of activity Y that she consumes, and x is the number ofunits of activity X that Jim consumes. Activity X costs 20 per unit. Activity Y costs 100 perunit.

Part 1 (5 points)Suppose that Jim has a right to half their joint income and Tammy has a right to theother half. Suppose further that they make no bargains with each other about howmuch activity X and Y they will consume. How much of activity X will Jim choose toconsume?

Solution:

25 units.

Part 2 (5 points)How much of activity Y will Tammy consume?

Solution:

5 units.

Part 3 (5 points)Because Jim and Tammy have quasilinear utility functions, their utility possibility fron-tier includes a straight line segment. Furthermore, this segment can be found by max-imizing the sum of their utilities. Notice that

UJ(cJ , x, y) + UT (cT , x, y) = cJ + 500 lnx− 20y + cT + 500 ln y − 10x

= cJ + cT + 500 lnx− 10x+ 500 ln y − 10y.

But we know from the family budget constraint that cJ + cT = 1, 000, 000− 20x− 100y.Therefore we can write

UJ(cJ , x, y) + UT (cT , x, y) = 1, 000, 000− 20x− 100y + 500 lnx− 10x+ 500 ln y − 10y

= 1, 000, 000 + 500 lnx+ 500 ln y − 30x− 110y.

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Let us now choose x and y so as to maximize UJ(cJ , x, y) + UT (cT , x, y). Settingthe partial derivatives with respect to x and y equal to zero, we find the maximum isachieve at what values of x and y?

Solution:

x = 16.67 and y = 4.54

Part 4 (5 points)If we plug these numbers into the equation UJ(cJ , x, y) + UT (cT , x, y) = 1, 000, 000 +500 lnx+500 ln y− 30x− 110y, Show that the equation of the utility possibility frontieris given byUJ + UC = 1, 001, 163.86.

Solution:

UJ + UC = 1, 001, 163.86.

Part 5 (5 points)Along this frontier, what is the total expenditure on the annoying habits X and Y byJim and Tammy given by?

Solution:

787.34.

END OF PAPER

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