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MICROECONOMIC THEORY IV

ECO 6122

FINAL EXAM

Question 1 (14 marks)

An individual consumes two goods : 1 and 2. Her utility function is u(x1 , x2) = min{ax1  , bx2 }. Prices are p1 and p2, and the individual's income is m.

a) Find the Marshallian demand functions.

b) Find the indirect utility function.

c) Find the expenditure function.

d) Find the Hicksian demand functions.

e) Find the money metric utility function.

f) Find the money metric indirect utility function.

g) Verify Roy's identity for good 1.

h) Verify Shephard's lemma for good 2.

i) Assume that p1 increases. Verify the Slutsky equation (you must verify two equations, one for the effect on x1, and one for the effect on x2).

j) Why, in this problem, are the compensated (Hicksian) demand functions not affected by prices?

k) Are these goods net complements? Prove your answer.

l) Are these goods gross complements? Prove your answer.

Question 2 (6 marks)

Let x, y and z be consumption bundles, and let the function f represent the optimal choice of the consumer among affordable bundles. For example if we write f(x,y,z) = x, this means that at a certain date the three bundles x, y and z were all affordable (were all within the budget set of the consumer), and the consumer chose bundle x, and did not chose any of the other bundles at that date.

Similarly, if we write f(x,y) = y,  this means that at a certain date the two bundles x and y were affordable (were all within the budget set of the consumer), and the consumer chose bundle y, and did not chose bundle x.

a) You are given the following information about the choices made by individual A: f(x,y) = x        ;          f(y,z) = y        ;          f(x,z) = z

Does this individual  satisfy the Weak Axiom of Revealed Preference? Does  she  satisfy the Generalized axiom of Revealed Preference? Show the analysis underlying your answer.

b) You are given the following information about the choices made by individual B: f(x,y,z) = x     ;          f(x,y) = x        ;          f(x,z) = z         ;          f(y,z) = z

Does this individual  satisfy the Weak Axiom of Revealed Preference? Does  she  satisfy the Generalized axiom of Revealed Preference? Show the analysis underlying your answer.

c) You are given the following information about the choices made by individual C: f(x,y,z) = x     ;          f(x,y) = x        ;          f(y,z) = y        ;          f(x,z) = x

Does this individual  satisfy the Weak Axiom of Revealed Preference? Does  she  satisfy the Generalized axiom of Revealed Preference? Show the analysis underlying your answer.

Question 3 (6 marks)

An individual has the utility function u(w) = w, where w is her wealth. She owns an asset X that will be worth $69 with a probability of 50%, and will be worth $0 with a probability of 50%.

a) Suppose this person has no other assets. What is the lowest price P at which she would agree to sell the asset X?

b) Suppose now that in addition to that asset, she has $100 safely saved in a bank account. What is the lowest price P at which she would agree to sell the asset X?

c) Without doing any further calculations, explain, based on your answers to parts (a) and (b), whether, as her wealth increased, this person became more risk averse, or less risk averse.

Question 4 (6 marks)

An individual has an asset that is worth $50 000 . The asset may be stolen with a probability of 1%. She can buy an insurance that will pay her back the amount $k if the asset is stolen. The insurance premium is 5% ($0.05 for each dollar of coverage). Her utility function is u(w) = ln(w), where w is her wealth. The individual chooses k to maximize her expected utility.

a) Set up her maximization problem and find how much coverage she will buy.

b) Calculate the expected profit of the insurance company.

Question 5 (7 marks)

Jack has the utility function M(w) = w 0. 5 , where w is his wealth.

Linda has the utility function X(w) = w 0.25 , where w is her wealth.

a) Show that   > −  .

b) Show that X(w) = K(M(w)), with K increasing and strictly concave (hint : all you have to do is find the function K and determine the signs of its first and second derivatives).

c) Each of these two persons has a wealth of $100, and faces the risk of losing $30 with a probability of 90%. Show that Linda is willing to pay more than Jack to avoid this risk (for this question, use 4 decimal places for all calculations).

Question 6 (7 marks)

A Cournot industry is composed of 8 identical firms. Industry demand is given by p = 3000 – Y

where Y is industry output.

All firms have a constant marginal cost of $14.

a) Find Y in the Cournot equilibrium (for part a you can use the general formula given in the class notes).

b) Show that this equilibrium can be obtained as the solution to the maximization of this particular welfare function :

 

Question 7 (12 marks)

Two firms use the Trigger strategy to try to enforce collusion. The firms meet over a potentially infinite horizon. If both firms cooperate (collude), each firm makes a profit of π* per period. If both firms play Nash (punishment), each firm makes a profit of πC per period. If one firm cheats (deviates), it makes a profit of πd  in the current period, but gets punished by both firms playing Nash after that as long as the game continues. The discount factor is δ . There is uncertainty : the game may end at any time. Given that firms are meeting (playing) today, there is a probability p that they will meet again at the next period, and a probability 1-p that they will never meet again.

a) Calculate the present value of cooperation profits (assuming the other firm cooperates).

b) Calculate the present value of deviation profits (assuming the other firm cooperates in the current period, and then firms play Nash from the next period).

c) Find the condition under which firms will choose to cooperate as long as the game continues.

d) Is cooperation more likely or less likely in this model than in the model without uncertainty? Why?

Question 8 (10 marks)

Consider the market for used cars with asymmetric information, with a large number of buyers and sellers. Each seller has exactly one car, and each buyer would like to buy at most one car. Let p represent the market price, and let k represent the quality of a car. Quality is private information to the seller. Buyers only know that quality follows a uniform distribution on the following interval : k ~ U[4000 , 17000]. The utility of a buyer if she buys a car is

Ub = 1.3k p

and zero otherwise.

The utility of a seller if he sells his car is

Us = p k

and zero otherwise.

a) Find which cars will be sold in equilibrium.

b) Find the equilibrium price.

c) Find the expected quality of cars sold in equilibrium.

Question 9 (12 marks)

A firm (the principal) is trying to develop a new product. For that, it will hire a scientist. There are two types of scientists (agents) : authentic scientists with authentic diploma, and fake scientists with fake diploma.

If the firm hires an authentic scientist, the probability of success of the product is 20%. If the firm hires a fake scientist, the probability of success of the product is 2%.

It is impossible for the firm to distinguish between authentic and fake scientists.

Let Us represent the benefit to the firm from success, and let Uns represent the benefit to the firm

in case of no success. When the firm pays a wage of w, this reduces its utility (benefit) by w. The firm is risk neutral.

The utility of any scientist from the wage received is

u(w) = w

where w is the wage received. The wage can take two values :

ws  : wage received if the product succeeds.

wns  : wage received if the product does not succeed.

The reservation utility of the authentic scientist is 10.

The reservation utility of the fake scientist is 1.

The firm wants to design a contract which attracts only authentic scientists, meaning that it must be unattractive to fake scientists.

a) Is this a problem of adverse selection or moral hazard?

b) Write the objective function of the principal.

c) Write the individual rationality (participation) constraint of an authentic scientist.

d) Write the individual rationality (participation) constraint of a fake scientist (hint : it must be such that the expected utility of the fake scientist is lower or equal than his reservation utility, since the firm wants this kind of scientist to refuse the contract).

e) The two constraints you wrote in parts (c) and (d) are binding and characterize the solution. Solve them to find the equilibrium values of ws and wns (you do NOT need to set up the Lagrangian and take first order conditions).