Econ 311 – Assignment 1 Fall 2015 The assignment is due in class on Tuesday, October 6th . Answer each of the following questions. You must provide justification to receive marks. Students may collaborate in solving the problems but must individually write up their answers. Question 1. Suppose that a firm uses two inputs 𝑥1, 𝑥2, to produce a single product y according to the production technology,

𝑦 = 𝐹(𝑥1, 𝑥2) = 𝐴𝑥1𝛼𝑥2

𝛽 𝑎𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝛼, 𝛽 𝑎𝑛𝑑 𝐴 > 0

i) Use the second order test to define conditions on the parameters α and β to ensure that the production function is concave.

For the rest of the question take α =1/4 = β, and A = 1.

ii) Assume the firm is a price-taker in product and input markets. Let (p, w1, w2) be the vector of prices. Define the profit expression for the firm,

𝑝𝐹(𝑥1, 𝑥2) − 𝑤1𝑥1 − 𝑤2𝑥2 and solve the problem of maximizing this function (assuming the solution has x1>0 and x2>0) by finding expressions for the optimal choice of each input x1(p, w1, w2) and x2(p, w1, w2) (these are input demand functions).

iii) If the prices are (p, w1, w2) = (80, 16, 4), what quantity of each of the inputs does the firm use? What are the firm’s profits?

Question 2. Consider the consumer utility maximization problem with given prices and exogenous income (M):

Max 𝑈(𝑥1, 𝑥2, 𝑥3) = 𝑙𝑛(𝑥1) + 2𝑙𝑛(𝑥2) + 𝑙𝑛(𝑥3) 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑝1𝑥1 + 𝑝2𝑥2 + 𝑝3𝑥3 = 𝑀

Use the technique of Lagrange multipliers to derive the consumer’s ordinary demand functions ( xi(p1, p2, p3,M) for i =1,2,3) and then use the bordered Hessian test to check that your stationary point does solve the problem. Question 3. Use the Kuhn-Tucker conditions to find non-negative pairs (x1 ≥ 0 and x2 ≥ 0) to solve the following problems. Provide a sketch of the constraint set and the level sets of the utility function close to your solution (in (𝑥1, 𝑥2) − 𝑠𝑝𝑎𝑐𝑒 ).

i) Max 𝑓(𝑥1, 𝑥2) = −2(𝑥1 − 4)2 − (𝑥2 − 3)2 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 5𝑥1 + 4𝑥2 ≤ 40

ii) Max 𝑓(𝑥1, 𝑥2) = 3𝑥1 + 15(𝑥2)1/2 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 20𝑥1 + 6𝑥2 ≤ 45