ECON 5113 Advanced Microeconomics

Transcription

1 Test 1 February 1, 008 carefully and provide answers to what you are asked only. Do not spend time on what you are not asked to do. Remember to put your name on the front page. 1. Let be a preference relation on R n +. (a) Define the relations and induced by. (b) Show that for any x R n +, (x) and (x) form a partition of R n +.. Suppose that U : R n + R + is a continuous, increasing, and quasi-concave utility function which represent a preference relation. Show that is (c) Find the expenditure (d) Find the Hicksian demand 6. Suppose that U(x) is a linearly homogeneous (a) Show that the indirect utility function can be expressed as V (p, y) = v(p)y. (b) Use the above result to show that the expenditure function can be expressed as E(p, u) = e(p)u. (a) transitive, (b) continuous, (c) convex. 3. State and prove Roy s identity. 4. Let U be an utility function on a consumption set R n +. (a) Define the marginal rate of substitution of good i for good j. (b) For the two-good case (n = ), if x 1 = 0 and x > 0, what is the relation between MRS 1 and the price ratio of good 1 and good? (c) Illustrate your answer in Part 4b with a diagram. 5. A consumer s utility function is given by u(x 1, x ) = x 1. (a) Find the Marshallian demand (b) Find the indirect utility Local non-satiation

2 Test February 9, 008 to put your name on the front page. 1. Prove the Slutsky equation: d i (p, y) = h i(p, u) d j (p, y) d i(p, y), p j p j y where d i and h i are the ordinary and Hicksian demand functions respectively. Hint: h i (p, u) = d i (p, E(p, u)).. (a) Define price elasticity of demand for good i with respect to good j, ɛ ij, and income elasticity of good i, η i. (b) Prove that n ɛ ij + η i = 0, i = 1,..., n. j=1 Supply the missing numbers. Does the resulting matrix possess all the properties of a Stutsky matrix? 5. Let the observed consumption bundles of a consumer in period 0 and period 1 be x 0 and x 1, given the price vectors p 0 and p 1 respectively. (a) State the weak axiom of revealed preferences (WARP). (b) Naomi claims that she prefers Pepsi to Coke. Last week she bought five bottles of Pepsi when both drinks were $1 per bottle. This week Coke is on sale for 90 cents per bottle while the price of Pepsi remains the same. Naomi buys six bottles of Coke instead. Is Naomi s behaviour consistent with WARP? Explain. 3. Let U(x) be a continuous, increasing, and quasiconcave utility (a) With a consumer s income normalized to y = 1, show that p = U(x) U(x) T x. (b) Let U(x) = Ax α 1 x 1 α for 0 < α < 1, find the consumer s inverse demand functions, p 1 (x 1, x ) and p (x 1, x ). 4. The Slutsky matrix below belongs to a consumer with a regular utility function of three goods, with market price p = (1,, 6) T : 10??? 4? 3??.

3 Test 3 March 8, 008 to put your name on the front page. 1. Let f(x 1, x ) be a constant returns-to-scale production (a) Define average products and marginal products. (b) Suppose that the average product of input factor 1 is rising. Show that the marginal product of input factor is negative.. Let π(p, w) be the profit function of a competitive firm that produces a single product with production function f : R n + R + that is continuous, strictly increasing, and strictly quasiconcave on R n +, and f(0) = 0. (a) Find the supply (b) Prove that w π(p, w) = d(p, w), where d is the input demand (c) Prove that π is increasing in p. 3. A new highway project changes the rental rate of apartment in a suburb from p 0 to p 1. A consumer with a well-defined preference structure has income y 0. (a) Define the equivalent variation (EV) of the consumer due to the price change using the indirect utility (b) Express EV explicitly using the expenditure (c) Express EV as a function of the Hicksian demand (d) The cost of living index between period 0 and period 1 can be defined as I(p 0, p 1, u 1 ) = E(p1, u 1 ) E(p 0, u 1 ) where u 1 is the utility level in period 1. Find an expression for I(p 0, p 1, u 1 ) in terms of EV and y A firm j in a competitive industry has constant returns-to-scale technology. Let w be the vector of factor prices and p be the output price. (a) What form does the conditional cost function have? (b) Write down the objective function of the firm using the conditional cost function above. (c) Can you find the supply function? Why or why not? (d) Suppose all the firms in the industry have the same technology and face the same factor prices. Explain why in the long-run the number of firms in the industry is indeterminate. 5. An industry consists of J identical firms each with cost function c(q) = q + 1. Each firm faces an identical inverse market demand p = 10 15q (J 1)q where q is the output produced by each of the other firms. (a) Find the Cournot equilibrium output of a typical firm. (b) Find the profit of each firm. (c) What would happen if entry to the industry is possible?

4 Final Examination April 14, 008 CB4056 9:00 AM 1:00 PM Instructor: Kam Yu books provided. Use the right-side pages for formal answers to put your name on the front page of every answer book. 1. Suppose that U : R n + R + is a continuous, increasing, and quasi-concave utility function which represents a preference relation. Show that is continuous.. A consumer has an utility function given by { x1 U(x 1, x ) = min, x }, a 1 a where a 1, a > 0. (a) Find the demand functions for both goods. (b) Find the indirect utility (c) Find the expenditure 3. Suppose that a consumer s utility function is given by U(x 1, x ) = f(x 1 ) + x where f is a differentiable, strictly increasing and strictly concave (a) Show that the marginal rate of substitution is independent of x. (b) Prove that the income elasticity of demand for good 1, η 1 = Suppose that a consumer has wealth W. There is a probability p that she will lose the amount L in a year. An insurance company offers the consumer a contract that will fully compensate her loss for a cost of I per year. (a) List the four outcomes in the set A of this risky situation. (b) If the consumer is risk-averse and I = pl, will the consumer accept the insurance offer? Explain. 5. Suppose a firm produces a single product with production function f : R n + R + that is continuous, strictly increasing, and strictly quasiconcave on R n +, and f(0) = 0. (a) Define the conditional input demand function h(w, y). (b) Show that h is homogeneous of degree zero in w. 6. The market demand for a product is Q = α βp. The product is provided by two firms with identical marginal cost C (q) = c > 0 and no fixed cost. (a) What is the profit function of each firm if they engage in Bertrand competition. (b) Find the market price of the product. (c) Find the Bertrand equilibrium if firm 1 has higher marginal cost than firm, that is, c 1 > c > Consider a two-good, two-consumer exchange economy, with utility functions and endowments U 1 (x 1, x ) = x 1/ 1 x 1/, e 1 = (1, 0), U (x 1, x ) = x 1/ 1 x 1/, e = (0, 1). (a) Find the set of Pareto-efficient allocations. (b) Find the core of the economy. (c) Find a Walrasian equilibrium for this economy and the associated WEA.

5 8. Suppose that a competitive firm has production technology represented by a production set Y R n. (a) Define a strongly convex set. (b) Suppose Y is strongly convex and market prices p R n ++. Show that the optimal production plan is unique. 9. Given a production economy where e i R n +. E = {(U i, e i, θ ij, Y j ) i I, j J } (a) Define the aggregate excess demand function for good k, z k (p). (b) Prove Walrus s Law in a production economy, that is, for all p R n ++, z(p) T p = Given a production economy as in Question 9. (a) State the Second Welfare Theorem for a production economy. (b) Let Y R n be the production set of a competitive firm with 0 Y. Let ŷ Y and define a production set Ȳ = Y {ŷ}. Show that 0 Ȳ.