Exercises - SOLUTIONS UEC Advanced Microeconomics, Fall 2018 Instructor: Dusan Drabik, de Leeuwenborch 2105

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1 Eercises - SOLUTIONS UEC-5806 Advanced Microeconomics, Fall 08 Instructor: Dusan Drabik, de Leeuwenborch 05. A consumer has a preference relation on R which can be represented by the utility function u() = Is this function quasi-concave? Briefly eplain. Is there a concave utility function representing the consumer's preferences? If so, display one; if not, why not? Without loss of generality, consider any two points du d 0 for, R, such that R, we have u u, from which u min u, u t form t t for 0, t min,. Because. Now, t t. Because and du d 0, it must be the case that u u u u, hence u() is a quasi-concave function. k, for k 4 Alternatively, the set 0 : u k 0 : 4 4 k is 0, for k 4 a conve set for all k R. Yes, there is such a concave utility function, for eample: v. A consumer has Leicographic preferences on whenever, or and rational, i.e., complete and transitive., or R if the relationship v. satisfies. Show that leicographic preferences on R are

2 3. A consumer with conve, monotonic preferences consumes non-negative amounts of and. a.) If, u represents those preferences, what restrictions must there be on the value of parameter? Eplain. b.) Given those restrictions, calculate the Marshallian demand functions. a.) Monotonicity requires u 0 0and u 0. Because the utility function is homogeneous of degree ½, it is strictly concave, hence also quasiconcave and quasi-concave functions have conve superior sets (i.e., preferences are conve). So no further restrictions on are required. y b.), p y. p 4. In a two-good case, show that if one good is inferior, the other must be normal. p p y p p y y y p y p y y y y s s where i y pi i i, si ; si ; i, y y i i Without loss of generality, assume is the inferior good. Then, we must have 0, which means that 0 because >0. Thus, must be a normal good. 5. How would you determine whether the function pi X p p I p p, y, could be a demand function for commodity of a utility maimizing consumer with preferences defined over the various combinations of and y? Is it a demand function? y

3 One has to check all the properties of the Marshallian demand function, as well as the negative semi-definiteness of the Slutsky matri. The function above is not a Marshallian s p, p y,i entry of the Slutsky matri demand function because the I s p, p,i 0. This entry is non-positive for a well-behaved Marshallian y p py demand function. 6. A firm produces output y from two inputs (, ) using the production function y = f(, ). The output price is given by p(y), the price of input one is w per unit and the price of input two is w per unit. That is, if the firm sells y units of output, the price it receives per unit is p(y). Assume that f: R Ris strictly concave and increasing and that p: R Ris decreasing and conve. Both f and p are twice differentiable. Note that this firm is a price taker in the input market; its choices do not affect the input prices (w,w). a.) Write the firm s profit maimization problem and profit function. Let π(w,w) be the profit function. b.) Is the partial derivative of π(w,w) with respect to wi equal to ( ) times the firm s input demand function for input i? Eplain. c.) Is π(w,w) a conve function of (w,w)? Eplain. d.) Now suppose that f, p y y, where > β > 0 and > α > 0 a.) and that Find the optimal input demands and output supply. b.) 3

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5 7. Consider a competitive firm with a well-behaved production function f() that converts an input into a product q. The market price of the product is p and the price of the input is w. Derive the relationship between the curvature of the production function, i.e., f and the elasticity of the product supply curve. A competitive cost-minimizing food producer solves min C w, s. t. f q (A.) 5

6 The properties of f, specified in the tet, guarantee that it has an inverse, h, such that f q h q. The cost of production can thus be written asc whq. The firm equalizes the marginal cost to the output market price MC dc dq wh p (A.) Totally differentiating equation (A.) and rearranging, we obtain dq dp wh (A.3) q qq By Inverse Function Theorem, we have h q f h q or more succinctly q h q f (A.4) Differentiating both sides of (A.4) with respect to q and rearranging yields f h qq f hq f (A.5) 3 f f f f The supply elasticity of a product is defined as S q dq dp p q (A.6) Combining the relationships (A.) to (A.6), we obtain S f f f (A.7) q 8. Given the production function f ln ln, calculate the profit-maimizing,, demand and supply functions, and the profit function. For simplicity assume an interior solution. Assume that i 0. Solved in class 9. Corn (C) is produced from labor (L) using a decreasing returns to scale technology of the formc AL 0,. How is the parameter related to 0., where A is a scale parameter and the price elasticity of the corn supply curve? Solved in class 6

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