Harvard Economics 2020a Problem Set 4

API 111 / Econ 2020a / HBS 4010 Fall 2007 -- Problem Set #4 Due: Wednesday, October 24, 2007

1. Consumption over time (Adapted from Silberberg, 1990) The world lasts T > 2 periods. A consumer has income wt in any period t. The real interest rate is r ≥ 0, and the consumer can borrow and lend as much as he/she wants at this rate. The consumer consumes ct in each period and has utility function U(c). 1a. Write down the consumer’s budget constraint. 1b. Write down the consumer’s utility maximization problem. Consider the following two utility functions: (i)

t

 1   log ct U (c) = ∑  t =0  1 + ρ  T

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 1  a  ct for 0 < a < 1. (ii) U (c) = ∑  t =0  1 + ρ  1c. Show that for the optimal consumption stream, consumption in any time period after period 0 can be expressed as a constant times the consumption in the previous period. T

1d. Under what conditions on the exogenous variables will consumption be increasing over time? Under what conditions will consumption be decreasing over time?

(Problem Set Continues)

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Production 2. From MWG, Exercise 5.C.13. Hint: What is the firm’s revenue maximization problem? Given that, what is the function that the econometrician has estimated? Apply the envelope theorem (or a version of Roy’s Identity). [See Simon and Blume p. 453-457 or MWG p. 964966 for the envelope theorem.] 3. Cobb-Douglas Production Function Consider the following production function:

y =f (z) = z1 4 z2 2 . 1

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Assuming that the price of the output is p and the prices of inputs are w1 and w2 respectively: Part I: Profit Maximization: (a) State the firm’s profit maximization problem. (b) Derive the firm’s factor demand functions for z1 and z2. (c) Derive the firm’s supply function. (d) Derive the firm’s profit function. (e) Verify Hotelling’s lemma for q(w,p), z1(w,p), and z2(w,p). Part II: Cost Minimization: (f) (g) (h) (i)

State the firm’s cost minimization problem Derive the firm’s conditional factor demand functions. Derive the firm’s cost function. Verify Shepard’s Lemma.

Part III: PMP and CMP: (j) Suppose the firm chooses quantity q to maximize p q – c(w,q). Show that the supply function determined from this problem is the same as the supply function determined by solving the profit maximization problem in Part I. (Problem Set Continues)

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4. The Le Châtelier Principle Part I: Comparative statics when both inputs are variable. A profit-maximizing firm produces a single output q using two inputs, z 1 and z2. The price of the single output is p and the cost of the inputs are w1 and w2 respectively, and these prices are taken as given by the firm. The firm’s production function q = f(z1,z2) is strictly concave. The fact that f(z) is concave implies that its second derivative matrix is negative definite, which in turn implies that: f11 < 0 f22 < 0 f11f22 – (f12)2 > 0

a) State the firm’s profit maximization problem in terms of inputs z1 and z2. b) Derive the firm’s first-order conditions. Denote the firm’s factor demand functions by z1(w,p) and z2(w,p). c) Use the factor demand functions and the first order conditions derived in b) to produce two identities (i.e., substitute the solution into the first-order conditions). We are interested in determining the effect of an increase in w1 on the firm’s utilization of inputs z1 and z2. These effects are captured by the derivatives dz1(w,p)/dw1 and dz2(w,p)/dw1.

d) Totally differentiate the two identities in c with respect to w1. This yields two equations that are linear in dz1(w,p)/dw1 and dz2(w,p)/dw1.

e) Solve the equations in part d) for dz1(w,p)/dw1 and dz2(w,p)/dw1. Your answer should be in terms of the price of output and the second derivatives of the production function. f) Use the fact that f(z) is concave to determine the sign of the dz1(p,w)/dw1. What is that sign? Under what conditions will dz2(p,w)/dw1 be positive? Negative? What is the economic meaning of these conditions? Part II: Comparative statics when one of the inputs is not variable. Now assume that the level of z2 is fixed at z 2* . This would be true, for example, if input 2 is the size of the firm’s plant, and it is unable to vary its plant size in the short run.

g) State the firm’s profit maximization problem when the level of input 2 is fixed at z 2* . h) State the firm’s first order condition. State the firm’s second order condition. Denote the solution to the firm’s short-run profit maximization problem as z1S(w1,p, z 2* )

i) Substitute z1S(w1, p, z 2* ) into the first order condition in h) in order to obtain an identity. Totally differentiate this identity with respect to w1 and solve for dz1S(w1,p, z 2* )/dw1. What is the sign of dz1S(w1,p, z 2* )/dw1? j) Compare dz1(w,p)/dw1 to dz1S(w1,p,z2*). Is the firm more responsive to an increase in factor prices in the short run (when input 2 is fixed) or in the long run? k) Explain you answer to part j) intuitively using the example of input 2 being plant size. This is an example of the Le Châtelier Principle, a quite general phenomenon relating

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changes in variables when some other factors are fixed to changes when all factors are variable.

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