ECON 5113 Microeconomic Theory

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1 Test 1 January 30, 2015 Time Allowed: 1 hour 20 minutes phones or calculators 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. You can keep the question sheet after the test. 1. Suppose that a consumer s preference relation % on a consumption set X R n + is complete and transitive. (a) Define the following induced relations on X: (a) is strictly preferred to,, (b) is indi erent to,. (b) Explain if is complete, reflexive, transitive, circular, symmetric, asymmetric, or antisymmetric. (c) Show that is an equivalence relation. 2. Suppose that a consumer s preference relation satisfies A1, A2, A3, A4, and A5 (but not A5, see appendix A). (a) Explain if the solution to the utility maximization problem still exists. (b) If yes, is the choice for the optimal bundle unique? (c) Illustrate the answers above with a diagram with a two-good case. (a) Set up Aminata s grocery shopping as a Kuhn- Tucker optimization problem. (b) Find the optimal bundle. (c) State the condition of the marginal rate of substitution at Aminata s optimal bundle in relation with the market prices. 4. A consumer s utility function is given by U(x 1,x 2 )=Ax 1 x 1 2, A > 0, 0 apple apple 1. (a) Set up the expenditure minimization problem and find the expenditure function E(p,u). (b) A cost of living index is defined as I(p 0, p 1,u 0 )= E(p1,u 0 ) E(p 0,u 0 ), where p 0 and p 1 are prices in period 0 and 1 respectively, and u 0 is the utility level in period 0. Show that the index for our consumer is independent of the utility level. 5. Consider the Slutsky i (p,y) i(p,u) d j State and prove the law of demand. You can assume the properties of the expenditure function listed in appendix B. 3. Aminata buys two goods, xigua and yam every week. Her utility function is given by U(x, y) =2x + log(y + 4). The market prices for xigua and yam are $2 and $1 per kg respectively. Aminata s weekly budget for food is $5.

2 Appendix A: 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 Local Non-Satiation: For any >0, there exists an x 2 B (a) \ X such that x a. A4 Strict Monotonicity: If a b, thena % b. If a b, thena b. Local Non-Satiation? A5 Convexity: If a % b, thenta +(1 t)b % b for all t 2 [0, 1]. A5 Strict Convexity: If a 6= b and a % b, thenta + (1 t)b b for all t 2 (0, 1). Appendix B: Properties of Expenditure Function Suppose that the consumer s preference relation satisfies Axioms A1 to A4. Then the expenditure function E(p,u) 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. 2

3 Test 2 February 27, 2015 Time Allowed: 1 hour 20 minutes phones 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. You can keep the question sheet after the test. 1. Suppose that in period t = 0, 1, a consumer buys bundle x t when the observed price vector is p t. (a) State the weak axiom of revealed preference. (b) Determine if the following observation satisfies WARP: p 0 =(2, 6), x 0 = (19, 9), p 1 =(3, 5), x 1 = (17, 4). 2. Suppose that a consumer s expenditure function is E(p 1,p 2,u)= up 1p 2 p 1 + p 2. (b) If the firms are competitive, what restrictions should the analyst impose on the parameters? (c) What extension do you suggest to make the functional form more flexible? 4. A competitive firm produces a single output y with a vector x of n inputs with technology represented by the production function y = f(x). (a) Define the profit function of the firm, (p, w). (b) State and prove Hotelling s lemma. (c) What additional properties can you infer from the Hotelling s lemma? 5. Consider a competitive firm with a production function y = f(x). (a) Define global decreasing returns to scale in production. (b) Explain why we assume that technology exhibits decreasing returns to scale when we study the profit function of a firm. (a) Find the indirect utility function. (b) Use the indirect utility function to find the direct utility function. 3. An analyst has collected cost data on firms to study the cost structure of an industry. She uses a log-linear regression model as follows: log c = 0 + nx i=1 i log w i + log y, where c is the observed total cost, y is the output quantity, and w i is the market price of input i. The parameters estimated in the regression are 0, 1,..., n and. (a) What is the functional form of the cost function the analyst is using? We re still the same great company we ve always been, only we ve ceased to exist.

4 Test 3 March 20, 2015 Time Allowed: 1 hour 20 minutes phones 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. You can keep the question sheet after the test. 1. Consider an economy with I consumers. (a) Define a homothetic preference relation. (b) How does homotheticity characterize a utility function? (c) If all consumers in the economy have identical homothetic preferences. Discuss if the aggregate demand function depends on income distribution. 2. Suppose that a monopoly faces a market demand function p = a bq. (a) Derive the marginal revenue function of the firm. (b) The cost function of the firm is given by C(q) = cq. Derive the profit maximizing output q m. (c) What will be the price p m the firm charges? 3. Suppose that an industry is supplied by a duopoly, each has a cost function 4. A competitive industry consists of 100 identical firms. The short-run cost function of each firm is given by C(q) = 200q + 15q 2. (a) What is the supply function for each firm? (b) What is the market supply function? (c) Suppose that the market demand function is p = q. Calculate the market equilibrium price and quantity. (d) Calculate the market consumer surplus and the producer surplus. 5. The utility functions and endowments of Aicheng and Bailin on two goods are given by U a (x 1,x 2 )=x 1 x 2, e a =(2, 2), U b (x 1,x 2 )=x 1/2 1 x 1/2 2, e b =(1, 1). (a) Characterize the contract curve of this exchange economy. (b) Find C(e), the core of the economy. (c) Draw an Edgeworth box of this exchange economy and show the contract curve and the core as accurate as possible. C(q j )=cq j, c > 0, j =1, 2. Market demand is an a ne function p = a b(q 1 + q 2 ), a,b > 0, a > c. (a) Set up the profit maximization problem of the firms and find the necessary condition. (b) Find the output for each firm. (c) Find the market equilibrium price. I ll stop when it isn t fun anymore.

5 Final Exam April 21, 2015 Time: 9:00 am 12:00 pm phones 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. You can keep the question sheets after the exam. 1. A consumer s preference relation is represented by a di erentiable and strictly increasing utility function U(x) on R n +. Suppose that the consumer s income is y>0 and the market prices of x are p 0. (a) Define the indirect utility function V. (b) State Roy s identity. (c) Prove Roy s identity. 2. A consumer s preference relation is represented by a di erentiable, strictly quasi-concave, and strictly increasing utility function U(x) on R n +. (a) Define the consumer s expenditure function E. (b) Show that E is concave in the market prices p. 3. Suppose that E is the expenditure function in question 2. (a) Define the compensated demand function h. (b) State Shephard s lemma (you do not need to prove it). (c) Prove Hick s Third Law: nx i (p,u) p j =0, 1=1,...,n. 4. Suppose that the ordinary demand function of a consumer is d(p 1,p 2,y)=( y/p 1, (1 )y/p 2 ) T,where 0 < <1. (a) Does d exhibit the budget balancedness property? (b) Find the Slutsky matrix S (see Appendix). (c) Normalize p 2 to 1. Write down the di erential equation you would need to solve to find the expenditure function. 5. Let the set of all possible outcome be A = {w 1,w 2,...,w n }, where w i 0 are wealth level of a consumer. The consumer has a von Neumann-Morgenstein utility function U on G s, the set of simple gambles on A. Let g =(p 1 w 1,p 2 w 2,...,p n w n ). Define the following terms: (a) Expected value of g. (b) Expected utility of g. (c) Certainty equivalent value of g. (d) Risk premium of g. 6. Faisal s von Neumann-Morgenstein utility function is given by U(w) =a + bw + cw 2. (a) What restrictions if any must be placed on parameters a, b, and c if Faisal is risk averse. (b) Over what domain of wealth can this utility function be defined? (c) Find Faisal s Arrow-Pratt measure of absolute risk aversion, R a (w). 7. A propose government project, if carried out, will change the market price of a product from p 0 to p 1. (a) Define a consumer s equivalent variation (EV) with income y 0. (b) Show that EV = Z p 0 p 1 h(p, u 1 )dp, where h is the compensated demand function.

6 (c) Let the Paasche-Konüs cost of living index be P K (p 0,p 1,u 1 )= E(p1,u 1 ) E(p 0,u 1 ), where E is the expenditure function. Show that P K (p 0,p 1,u 1 )= y 0 y 0 +EV. 8. Consider an exchange economy with a set of consumers I = {1, 2,...,I} and n goods: E = (% i, e i ):i 2I. Let p 2 R n ++ be a price vector. It s my mobile wallet. (a) Define aggregate excess demand z(p) function of the economy. (b) Show that z is homogenous of degree zero. 9. Consider the exchange economy in question 8. State and prove Walras law. 10. Suppose that a production economy has a set of firms J = {1, 2,...,J}, each with a net output vector y j. Each firm s technology is represented by a production set Y j R n with the following properties: (a) 0 2 Y j. (b) Y j is a compact set. (c) Y j is strongly convex: For all y 1 6= y 2 2 Y j and 2 (0, 1), there exists a y 2 Y j such that y > y 1 +(1 )y 2. Suppose that for all j 2J, ŷ j is the profit maximizing output vector. Let Ȳ j = Y j {ŷ j }. Show that (a) 0 2 Ȳ j, (b) Ȳ j is compact, (c) Ȳ j is strongly convex. Appendix Slutsky i (p,y) i(p,u) d j 2