ECON 5113 Microeconomic Theory

Transcription

1 Test 1 February 3, Let % be a consumer s preference relation on a consumption set X R n +. Suppose that x 2 X. (a) Define the sets (x) and (x). (b) Show that (x) \ (x) =?. 2. Suppose that a consumer s preference relation % on a consumption set X R n + is complete, transitive and strictly monotonic. Let e =(1,...,1) and define U : X! R + such that U(x)e x. (a) Show that U is a well-defined function. (b) Show that the function U represents the consumer s preference relation % on X. 3. Suppose that a consumer s preference relation on a consumption set satisfies the axioms of a rational consumer listed in the appendix. 5. Suppose that a consumer s utility function is di erentiable, increasing, and quasi-concave. (a) Define, h ij, the elasticity of Hicksian demand for good i with respect to the price of good j. (b) Show that nx h ij =0, j=1 i =1,...n. Appendix: Axioms of Consumer Choice For all a, b, c in the consumption set X, the relation % satisfies the following axioms: A1 Completeness: Either a % b or b % a. A2 Transitivity: If a % b and b % c, thena % c. A3 Continuity: The upper contour set % (a) and the lower contour set -(a) are closed. A4 Strict Monotonicity: If a b, thena % b. If a b, thena b. A5 Strict Convexity: If a 6= b and a % b, thenta + (1 t)b b for all t 2 (0, 1). (a) Define the consumer s expenditure function E(p,u). (b) Show that E is concave in p. 4. A consumer s utility function is given by U(x 1,x 2 )=x 1 x 1 2. (a) Set up the utility maximization problem and find the ordinary demand function. (b) Find the indirect utility function. (c) Derive the expenditure function from the indirect utility function.

2 Test 2 March 10, Suppose that when the market prices of three goods in period 1 are p 1 =(2, 3, 3). Khawla buys quantities x 1 =(3, 1, 7). In period 2, prices and quantities are p 2 =(3, 2, 3) and x 2 =(7, 3, 1). Does Khawla s behaviour satisfy the weak axiom of revealed preference? Explain. 2. Suppose that a function E : R n ++ R +! R + satisfies the seven properties of an expenditure function (see the appendix). (a) Define the optimization problem that can recover the utility function directly for any given consumption bundle x 2 R n +. (b) Show that the resulting utility function U(x) is unbounded above. 3. Suppose that a consumer s indirect utility function is given by y V (p,y)=. p 1/2 1 p 1/2 2 Derive the consumer s utility function U(x). 4. Let y = f(x) be the production function of a competitive firm that produces one output with n inputs with a constant returns to scale technology. (a) Define the average product AP i and marginal product MP i of input i. (b) Show that nx f(x) = (MP i )x i. i=1 (c) Suppose that n = 2. Show that if the average product of input 2 is rising, then the marginal product of input 1 is negative. 5. Suppose that a competitive firm produces one output with n inputs with a production function y = f(x). Let p be the market price of the output and w be the vector of input prices. (a) Define the profit function of the firm. (b) Show that the profit function is increasing in p and decreasing in w. Appendix: Properties of an Expenditure Function Suppose that a consumer s utility function is continuous, increasing, strictly quasi-concave. Then the expenditure function has the following properties: 1. E(p,u m ) = 0, where u m is the minimum value in its domain, that is, u m = U(0). 2. E is continuous on its domain. 3. For all p 0, E is strictly increasing and unbounded above in u. 4. Shephard s lemma: If E is di erentiable in p, then r p E(p,u)=h(p,u)=x. 5. E is an increasing function of p. 6. E is linearly homogeneous in p. 7. E is concave in p.

3 Test 3 March 31, Consider a market with two firms. Each firm has identical cost function C(q j )=1+q 2 j, j =1, 2. The inverse market demand function is p = 10 q 1 q 2, (i) What are the Cournot equilibrium outputs of the two firms? (ii) What is the total profit of the duopoly? (iii) If the government allows the duopoly to merge, what will be the profit of the monopoly? 2. A market with a Stackelberg structure has one leader and five followers. The market demand function is given by p = 100 q. The cost function of each of the followers is C(q j ) = q 2 j, j =1,...,5. (i) Define the equivalent variation (EV) of the price change for a consumer by her indirect utility function. (ii) Express EV in terms of the Hicksian demand function. (iii) Suggest a method to measure EV in practice. 4. Consider a two-person, two-good exchange economy with utility functions and endowments as follows: U 1 (x 1,x 2 ) = x 1/2 1 x 1/2 2, e 1 =(2, 0), U 2 (x 1,x 2 ) = x 1 x 2, e 2 =(0, 2). (i) Given market prices p =(p 1,p 2 ), derive the excess demand function for each consumer. (You can write down the ordinary demand functions directly if you recognize the utility functional forms.) (ii) Find the aggregate excess demand function, z(p). (iii) Is p =(1, 1) a Walrasian equilibrium? Explain. 5. Consider a set of consumers I = {1, 2,...,I} that forms an exchange economy E = (% i, e i ):i 2I, in which each consumer has a rational preference relation % i on n goods. (i) Carefully state the second welfare theorem. (ii) Explain the economic meaning of the theorem. (i) Find the total supply function of the followers. (ii) Find the demand function facing the leader. (iii) If the cost function of the leader is given by C(q) = q, find the market equilibrium price. 3. A government project changes the market price of a good from p 0 to p 1.

4 Final Examination April 21, 2017 Time: 1:00 pm 4:00 pm devices are allowed. Please write your answers on the answer book provided. Use the right-side pages for formal answers and the left-side pages for your rough work. Answers should be provided in complete and readable essay form, not just in mathematical formulae and notations. Remember to put your name on the front page of every answer book. You can keep the question sheets after the exam. 1. Let % be the preference relation of a rational consumer on a consumption set X R n +. (i) Define the following induced relations on X: (a) is strictly preferred to,, (b) is indi erent to,. (ii) Show that is a transitive relation. 2. Suppose that a consumer s utility function U(x) on a consumption set X R n + is continuous, increasing, quasi-concave, and linearly homogeneous. (i) Show that the ordinary demand function is separable in market price p and income y, that is, x = d(p)y. (ii) Suppose that consumers in an economy have linearly homogeneous but not identical utility functions. Show that market demands depend on income distribution. 3. Consider the Slutsky i j j d j where d i and h i are the ordinary and Hicksian demand functions respectively for good i. (i) Explain i (p,u)/@p i apple 0. (ii) Define a normal good. (iii) State the law of demand. 4. Consider the von Neumann-Morgenstern utility function U(w) = + log w. (i) What restrictions if any must be placed on the parameters and to display risk aversion? Explain. (ii) Find the absolute risk aversion. (iii) Find the relative risk aversion. 5. Nikki s only asset is her $500,000 house in Fort William. Each year there is a probability that a fire will destroy the house completely. (i) Find the expected value of Nikki s asset. (ii) Suppose Nikki s utility function on wealth is U(w) = p w.findherexpectedutility. (iii) What is the certainty equivalence of Nikki s expected utility. (iv) An insurance company o ers to fully insure Nikki s house with an annual premium of $500. Will she accept the o er? 6. Consider a software company which has considerable market power over one of its products. The company sells its products online so the marginal cost is practically zero. (i) Show that the profit maximizing price-quantity combination is at the point on the demand curve that the price elasticity of demand is equal to 1. (ii) Suppose that the inverse demand function is given by p = a bq. Show that unitary elasticity is at the mid-point of the curve. 7. The diagram below shows the sum of the marginal costs of all the competitive firms and the demand curve in a market. (i) Define market consumer surplus in integral form.

5 (ii) Define market producer surplus in integral form. (iii) Show that a competitive market achieves maximum e ciency. Price mc(q) p CS PS q p(q) q 8. Consider a set of consumers I = {1, 2,...,I} and a set of firms J = {1, 2,...,J} that forms a production economy with n goods: E = (U i, e i, ij,y j ):i 2I, j 2J. (1) Each consumer has a di erentiable, increasing, and quasi-concave utility function U i on the n goods. Each firm s compact and strongly convex production set Y j R n contains the zero net output vector. (i) Carefully state the second welfare theorem of the above production economy. (ii) Explain the economic meaning of the theorem. 9. Consider the production economy in equation (1). Assume that each of the n goods is produced by one competitive firm only. (i) State the utility maximization problem of a typical consumer i. (ii) State the profit maximization problem of a typical firm j. (iii) Pick any two goods k and l, with market equilibrium prices p k and p l respectively. Show that the MRS kl of all consumers and the MRTS kl of all firms are the same. (iv) What is the economic meaning of the above result? 10. The production economy described in question 8 above provide an analytical framework for the aggregate economy. Many economists employ this Walrasian approach over time to study the business cycles of the macroeconomy. Many so-called dynamic stochastic general equilibrium models reduce I = J = 1 for simplicity. Write a short critique of the DSGE approach in macroeconomics. 2