Review consumer theory and the theory of the firm in Varian. Review questions. Answering these questions will hone your optimization skills.

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1 Econ 6808 Introduction to Quantitative Analysis August 26, 1999 review questions -set 1. I. Constrained Max and Min Review consumer theory and the theory of the firm in Varian. Review questions. Answering these questions will hone your optimization skills. You will soon have a quiz on all or some of this material. Solve the following problems using, as you deem appropriate, either direct substitution or the Lagrangian technique. Use Mathematica only as needed to make your life easier. For each question, along with the math, explain all of your steps in words. Also make sure you can do all of the problems in the min%max notes. 1. Discuss in general terms how one would search for the global max or min of a differentiable function of one variable, f(x). What are the potential pitfalls of assuming the max (min) is at a level of x, x 0, where f x (x0) = 0. Explain 2. Assume that you have found an x, x 0, where f x (x 0 ) = 0. Identify an additional condition that is sufficient for f(x) to be globally maximized at x =x 0. Explain. Why isn't your condition necessary? 3. Define directional derivatives and relate them to partial derivatives. As part of your answer give an example of a directional derivative that is not a partial derivative. Explain the usefulness of directional derivatives in the search for maximums and minimums. 4. Define the term upper-level set for the function m = h(y 1, y 2,...,y N ). Now define, both in functional notation and in words using economic terminology, the upper-level sets of the production function x = f(k, l). 5. Assume p = px-c(x) where p = $40 and c(x) = x 3-12x x Determine the profit maximizing level of output. Don't forget to check the second-order conditions. 6. Assume a profit maximizing monopolist whose demand function is x = x(p) = b-ap b>0 and a>0

2 and whose cost function is c(x,w,r) = x 2 g(w,r) where g(w,r)>0 Derive the profit maximizing level of output, x s, as a function of w and r i.e., x s = x s (w,r) Don't forget to check the second order conditions. 7. Assume that the consumer max U = U(X 1, X 2, X 3 ) = ax 1 + bx 2 ½ + cx 3 ½ where a,b,c>0 subject to the budget constraint Y = P 1 X 1 + P 2 X 2 + P 3 X 3 a. Derive the demand functions for X 1, X 2 and X 3. b. What does this theory predict will happen to the quantity demanded of good 1 when the price of good 3 increases. 8. Assume a world of three goods (X 1, X 2, and X 3 ). Assume that prices (p 1, p 2, and p 3 ) and income (Y) are parametric to the consumer and that the consumer's preferences can be represented by the utility function U = U (X 1, X 2, X 3 ) = X 1 X 2 + X 3 Derive the demand functions for the three goods (Don't worry about checking the second order conditions for a max.) 9. Assume that the Snerd Corporation produces product X using L and K where X = f(k,l) = L.5 K.5 Further assume that the firm buys labor and capital at the parametric prices w and r.

3 Using the technique of direct substitution, derive the firm's conditional demand function for labor, i.e., derive L d = L d (X,w,r). Then derive the firm's conditional demand function for capital, K d = K d (X, w, r). Now use, instead, the Langrangian technique to derive those same two conditional demand functions. Interpret your Lagrangian multiplier. Note that the conditional demand function for L identifies the amount of labor the firm will purchase to minimize the total cost of producing X given w and r. You do not have to check the second order conditions for a max 10. Assume a competitive firm produces product X using K and L. Further assume that X is sold at the parametric price p and that r and w are the parametric prices of K and L. a. Define, in words, the firm's long run demand function for capital services, K * = K(P,w,r). b. Derive the firm's long run demand function for labor, L *, and capital services, K *, assuming X = f(k,l) = K.2 L.5 c. Now derive the firm's supply function [Don't worry about checking the second-order conditions for the max.] Explain your steps as you proceed. 11. Assume a competitive firm sells its output x at the parametric price p and that it can purchase labor and capital at the parametric prices w and r. Further assume that the firm's cost function is c=c(x,w,r)=x.5 wr Determine the profit maximizing level of output, x*. Show all your work and explain all your steps in words. 12. Assume that the Gomer Corporation is the sole producer of gomers (i.e., it's a monopoly). Specifically assume that the aggregate demand function for gomers is

4 G= G(P) = P Where G / is the quantity demanded of gomers and P / is the price of one gomer And that the cost function for producing gomers is C = C(G) = G 3-12G Derive the profit maximizing quantity of gomers. Explain each step and show all of your work. Make sure to check the second-order condition for profit max. 13. A. Describe, in words, a simple theory to explain the market behavior of an individual consumer. B. Now describe this theory in general functional notation. C. Describe both in words, and functional notation, the solution to the consumer's choice problem. 14. Assume Wilbur's utility function is U = X 1 X 2 X 3, that Y is Wilbur's income, and that p 1, p 2, and p 3 are the parametric prices of goods one, two and three. Further assume that the law dictates that Wilbur consume two units of x 2 for every unit of x 1. a. Determine Wilbur's demand function for good 1. In this part of the question do not worry about the second-order conditions for utility maximization. For now assume that the critical value of X 1 that you derive maximizes utility. Hint: Start by turning Wilbur' problem into an unconstrained problem in one variable. Explain, in words, all the steps in your derivation of his demand function for good 1. b. Now derive Wilbur's demand functions for goods 2 and 3.. c. Now check the second-order conditions to make sure the demand functions that you derived in part b are, in fact, the demand functions for the three goods.. d. How many units of the three goods will he choose to purchase if his income is $72, p 1 = 1, p 2 =.5 and p 3 = 7.

5 15. (This one is long and has a lot of algebra.) Assume the Snerd Corporation produces product X using L and K. Such that x = f (K,L) = L a K (1-a) Where 0<a<1 Further assume that the firm buys labor and capital at the parametric prices w and r. Derive the firms conditional demand function for L and K, i.e., Derive L D = L D (x,w,r) K D = K D (x,w,r) The conditional demand function for L and K identify the a mount of L and K the firm will purchase to minimize the total cost of producing x given w and r. Don't worry about the second-order conditions Once you have the conditional demand functions, use them to derive the function C = C (x,w,r) 16. Consider OLS estimation of the parameters in the linear regression equation y = a +ßx +e. Identify the OLS estimates of a and ß as the solution to a Minimization problem. (There is a problem that derives OLS estimates in your optimization notes.) 17. Assume Wilbur's preferences can be represented by the utility function u(x 1, x 2 ). Now assume some other utility function U = U(x 1, x 2 ). List a set of necessary and sufficient conditions on U(x 1, x 2 ), in terms of u = u(x 1, x 2 ), such that both functions represent the same preferences. As part of your answer define preferences and discuss what it means for Wilbur to have preferences over bundles of x 1 and x Discuss the role of marginal utility in demand theory. 19. Discuss the distinction between ordinal and cardinal preferences. As part of your answer define both. Discuss the representation of preferences with a utility function.