# Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.

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1 Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price taker in the factor markets. A) Is this production function homogeneous? If so, of what degree? It is homogeneous of degree 1. B) Is this production function concave? Prove your answer. Yes. You could prove this by showing that the Hessian is negative semidefinite. Alternatively, you can show that it is quasi-concave (you need to actually show this) and homogeneous of degree 1 and using the result that such a function is concave. Irving Fernandez offered an even neater proof. Note that ( x 1 + x 2 ) 2 = x x 1 x 2 + x 2. Show that this is the sum of concave functions and hence a concave function. C) Define the elasticity of substitution between two factors and calculate the elasticity of substitution between x 1 and x 2 for the firm in this problem. Elasticity of substitution for this function is 2. D) Where the prices of the two inputs are p 1 and p 2, find the amount of each factor that would be used to produce one unit of output in the cheapest possible way. x 1 (p, 1) = x 2 (p, 1) = ( p2 ( p1 ) 2 ) 2

2 E) Where the prices of the two in puts are p 1 and p 2, find the amount of each factor that would be used to produce y units of output in the cheapest possible way. ( ) 2 p2 x 1 (p, y) = y ( ) 2 p1 x 2 (p, y) = y F) Find the cost function for producing this good. p 1 x 1 (p, y) + p 2 x 2 (p, y) = Question 2. A consumer has utility function ( p1 p 2 U(x 1, x 2 ) = ( ) x x defined over the set {(x 1, x 2 ) x 1 > 0, x 2 > 0}. A)Is this utility function strictly increasing in both goods at all points in its domain? (Prove your answer.) Yes, take derivatives and show that they are positive. B) Find this consumer s Marshallian demand functions for goods 1 and 2. x 1 (p, m) = m p 1 + p 1 p 2 x 2 (p, m) = m p 2 + p 1 p 2 C) Find this consumer s indirect utility function. V (p, m) = m + p 1 p 2 = D) Verify that Roy s identity applies. This is a matter of straightforward calculation. Question 3. A consumer has utility function ) y m ( p 1 + p 2 ) 2 V (x 1, x 2, x 3 ) = x a 3U(x 1, x 2 ) 1 a

3 where 0 < a < 1 and U(x 1, x 2 ) = ( ) x x A) Solve for the Marshallian demand functions for goods 1, 2, and 3, using a two step procedure in which you first find the highest utility that a consumer can achieve if he spends a total amount of money m on goods 1 and 2. (Hint: use the answer from Question 2 to help you.) Now if the consumer has total income M, find the best way to divide his expenditure, spending M m on good 3 and m on goods 1 and 2. If the consumer spends a total of m on goods 1 and 2 and M m on good 3, her utility will be ( ) a ( M m x a 3V (p 1, p 2, m) 1 a m = p 3 + p 1 p 2 ) 1 a The consumer will choose m to maximize this function. This is maximized when M m = am and m = (1 a)m. But then x 3 (p, M) = M m p 3 = am p 3 and (using the answer to problem 2) we have x 1 (p, m) = x 2 (p, m) = (1 a)m p 1 + p 1 p 2 (1 a)m p 2 + p 1 p 2 BWrite down the indirect utility function for this consumer. V (p, m) = Ma a (1 a) 1 a p a 3 ( + p 1 p 2 ) a 1 Question 4. A ) State the weak axiom of revealed preference.

4 Let x 0 be the bundle chosen at prices p 0 and x 1 the bundle chosen at prices p 1. If p 0 x 1 p 0 x 0, then p 1 x 0 > p 1 x 1. B ) Prove that a utility-maximizing consumer who has strictly monotonic and strictly convex preferences will necessarily satisfy the weak axiom of revealed preference. If preferences are strictly convex and if x 0 is chosen at prices p 0, it must be that u(x 0 ) > u(x) for all x such that px p 0. (You should show why this is true.) Suppose that p 0 x 1 p 0 x 0. Then it must be that u(x 0 ) > u(x 1 ). If x 1 is chosen at price vector p 1, it must be that u(x 1 ) > u(x) for all x such that p 1 x p 1 x 1. Since u(x 0 ) > u(x 1 ), it therefore cannot be that p 1 x 0 p 1 x 1. It follows that if p 0 x 1 p 0 x 0, it must be that p 1 x 0 > p 1 x 1. C) Show that a utility-maximizing consumer with weakly convex preferences might violate the weak axiom of revealed preference. Consider the following example. This is weakly convex but not strictly convex. Let p 0 = (1, 1) and also let p 1 = (1, 1). Then the commodity bundle (x 0 1, x 0 2) = (2, 0) could be be chosen by a utility maximizing consumer with utility function u(x 1, x 2 ) = x 1 + x 2 and income 2. The bundle x 1 = (1, 1) could also be chosen by the same utility maximizing consumer with income 2. Now p 0 x 1 = p 0 x 0 = p 1 x 1 = p 1 x 0 = 2. Therefore we have p 0 x 1 p 0 x 0 and also p 1 x 0 p 1 x 1 in violation of the weak axiom of revealed preference. Question 5. An economy has two consumers and two goods. Consumer A has the utility function U(x A 1, x A 2 ) = α ln x A 1 + (1 α) ln x A 2 and an initial endowment of ω1 A has the utility function units of good 1 and no good 2. Consumer B U(x B 1, x B 2 ) = β ln x B 1 + (1 β) ln x B 2 and an initial endowment of ω B 2 units of good 2 and no good 1. A) Let good 2 be the numeraire with a price of 1, and denote the price of good 1 by p. Solve for the demand for good 1 by each consumer as a function of p. Consumer A s income is pω A 1. Solving A s maximization problem, we find that A spends the fraction α of her income on good 1. Thus we have D A 1 (p) = αω A 1.

5 Consumer B s income is ω2 B good 1. This implies that and B spends the fraction β of his income on D B 1 (p) = ωb 2 p. B) Write an equation for excess demand for good 1 as a function of price p of good 1 when good 2 is the numeraire. At what price or prices p is this excess demand for good 1 equal to zero. E 1 (p) = D A 1 (p) + D A 2 (p) ω A 1 = αω A 1 + βωb 2 p Therefore E 1 ( p) = 0 when ω A 1. βω B 2 p = (1 α)ω which is the case when p = βω B 2 (1 α)ω A 1. C) If the price of good 1 is the price you found in Part B, at what price will excess demand for good 2 be zero? The same price, by Walras Law. D) Is there a competitive equilibrium in which the price of good 2 is 3? If so, what must the price of good 1 be for there to be a competitive equilibrium with the price of good 2 equal to 3? Yes, multiplying all prices by a positive constant preserves competitive equilibrium, so p 2 = 3 and βω B 2 p 1 = 3 p = 3 (1 α)ω1 A is also a competitive equilibrium. Since as we saw in our solution, there is only one competitive equilibrium with p 2 = 1, there also is only one competitive equilibrium in which p 2 = 3.

6 E) Find the quantities of good 1 and good 2 consumed by person A in competitive equilibrium. Find the quantities of good 1 and good 2 consumed by person B in competitive equilibrium. x A 1 = αω1 A x B 1 = (1 αω1 A x A 2 = βω2 B x B 2 = (1 β)ω2 B F) Suppose that person B s initial endowment of of good 2 is increased by one unit. What is the effect on the equilibrium consumption of good 1 by person B? What is the effect on the equilibrium consumption of good 2 by person B? Differentiate your answers to Part E to find: x B 1 ω B 2 = 0 x B 2 ω B 2 = 1 β Question 6. A pure exchange economy has 1000 people and one consumption good. There are two possible states of the world. If State A occurs, everyone will have an initial endowment of 10 units of the consumption good. If State B occurs, each person will have an initial endowment of 20 units of the consumption good. Each person is an expected utility maximizer, with von Neumann-Morgenstern (Bernoullian) expected utility π i ln c i a + (1 π i ) ln c i B where π i is person i s subjective probability that Event A happens and 1 π i is i s subjective probability that B happens, and where c i A and c i B are i s consumption contingent on these events. Before it is known which state will occur, persons can buy or sell either of two kinds of securities. A type A security will pay one unit of the consumption good if State A occurs and nothing if state B occurs. A type B security will pay one unit of the consumption good if state B occurs and nothing if state A occurs. The price of type A securities is p A per unit and the price of type B securities is p B per unit. Let c i A and c i B be i s consumption in events A and

7 B, respectively. The difference between the value of i s consumption in event A and i s endowment in event A is equal to the number of type A securities that i purchases if this difference is positive and sells if this difference is negative. Similarly for type B securities. For each person, the total value of securities purchased must be equal to the total value of securities sold. Thus p A (c i A 10) + p B (c i B 20) = 0 for all i. A) Let the type B security be the numeraire. Find the demand of person i for type A securities as a function of p A and of i s subjective probability π i. c A i = πi p A (10p A + 20) B) If everybody has the same subjective probability for π i = π for event A, what is the competitive equilibrium price for type A securities? In this case, excess demand for type A securities is 1000( π p A (10p A + 20) 10). Therefore excess demand is zero when Solving this equation we have π p A (10p A + 20) 10) = 0. p A = 2 1 π. C) Solve for the competitive price for type A securities where different people have different subjective probabilities for event A. In equilibrium, we must have total excess demand for consumption contingent on event A equal to zero. Thus we need 1000 i=1 π i p A (10p A + 20) = 10, 000. (1)

8 Define pi = π i Thus pi is just the mean of the individual probabilities of event A. Divide both sides of equation 1 by 1000 to get i=1 pi p A (10p A + 20) = 10. Solving this one finds the equilibrium price is p A = 2 1 π. D) Suppose that π i π j, what is the ratio of person i s competitive equilibrium consumption in event A to that of person j in event A? c i A c j A = πi π j.

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