400ps3-11

# 400ps3-11 - ECON 400 Winter 2011 Problem Set III(for Friday...

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ECON 400 Hartman Winter 2011 Problem Set III (for Friday, January 21) A firm uses two inputs, capital and labor, to produce its output. The production function is a b Q K L where Q , K , and L denote the firms output and its inputs of capital and labor, respectively, and 0 1 a and 0 1 b are constants. Assume the firm is a price taker in all of its markets, and let p , r , and w denote the output price, and the prices of the capital and labor inputs, respectively. 1. Show that this production function is homogeneous of degree a b . 2. Verify that both marginal products are decreasing, i . e , show that 2 2 / 0 Q K and 2 2 / 0 Q L . 3. What are the first and second order necessary conditions for choosing K and L to maximize profit? Show that the second order necessary conditions for a maximum of profit cannot be satisfied if 1 a b . 4. Show that the second order sufficient conditions for a maximum of profit are satisfied if 1 a b . For problems 5-11 below assume that 1 a b .

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