Harvard Economics 2020a Problem Set 1

API 111 – Econ 2020a – HBS 4010 Fall 2007 Problem Set #1 Due: Wednesday, September 26, 2007 Please turn in your problem set by noon on the date the assignment is due. Assignments not handed in in class should be put in the assignment drop box on the second floor of KSG, around the corner from room L239 (near the elevator on the second floor). Although you may work in groups on you assignment, each person should write up and hand in their own assignment. Also, note the names of your group members on your problem set. Problem sets submitted late will not be given credit unless prior arrangements are made. Consumer Theory Basics and Comparative Statics Analysis Part 1 (MWG Questions): Please answer the following question from MWG: 2.D.4, 2.E.1 (p. 23), and 2.E.2 (p. 28). Part 2 (Additional Questions): Please answer the following questions. 1. Based on MWG 2.E.4. Suppose that Walrasian Demand x(p,w) is homogeneous of degree one with respect to w, i.e., for each commodity i,

xi ( p, α w ) = α xi ( p, w )

(0.1)

∂xi w , is equal to 1. ∂w xi Hint, begin by differentiating (1.1) with respect to α and evaluating the result at α = 1 . for all α > 0 . Show that the wealth elasticity of demand for good i, ε iw =

Interpret your result. What does it imply for the shape of the consumer’s Engel curves? 2. A Comparative Statics Exercise. A firm produces output q using two inputs, labor, and raw materials. The labor cost of producing q units of output is c(q), where c(q) is a thrice differentiable, non-negative, strictly increasing, and strictly convex function with c(0) = 0. Each unit of output requires m units of material. The price of output is pq , and the cost of each unit of material is pm. Both prices are strictly positive. 2a. Write down an expression for the firm’s profit. What are the exogenous and endogenous variables in this problem? 2b. Assuming the firm maximizes its profit, derive the first-order condition describing the firm’s optimal choice of q. Let q* denote the solution to this first-order condition. You may assume that q* exists and is strictly positive. 2c. Are the second-order conditions for a maximum satisfied at q*? Explain why or why not? 2d. Noting the dependence of q* on the exogenous variables, rewrite the first-order condition in part b as an identity. 2e. Derive an expression that shows how the firm responds to an increase in pm. Can you determine whether this expression is positive or negative? How does the firm’s response to a change in pm depend on m? Interpret your answers.