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3. Find the profit function for the Cobb-Douglas production function f (¤1, T2) = Ax¡'x with A > 0, a1, a2 > 0 and a1 +a2 < 1. %3D

3. Find the profit function for the Cobb-Douglas production function f (¤1, T2) = Ax¡'x with A > 0, a1, a2 > 0 and a1 +a2 < 1. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Juan Valdez owns a coffee farm in Colombia. His production function is: f (x1, x2) = (x1 – 1)0.25 x9-5. Assume the price of input 1 is r and the price of input 2 is w.. (a) Write down an expression for the technical rate of substitution. (b) Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c) Find Juan's cost function. (d) What is the supply function of Juan's firm? 2. Show that the profit function is convex in (p, w). 3. Find the profit function for the Cobb-Douglas production function f(¤1, 12) = Ax†' x" with A > 0, a1, ¤2 > 0 and a1 + a2 < 1. 4. Verify Hotelling's Lemma for the Cobb-Douglas production function with a1+a2 < 1. 5. Find the supply function and the input demand functions for the CES production function: f(r1, F2) = A(ax{ + (1 – a)x2)8/e, %3D where A > 0, B > 0, 0 < a < 1, and 0 + p< 1. 6. Find the profit function for the CES production function. 7. Verify Hotelling's Lemma for the CES production function with B < 1.
Transcribed Image Text:1. Juan Valdez owns a coffee farm in Colombia. His production function is: f (x1, x2) = (x1 – 1)0.25 x9-5. Assume the price of input 1 is r and the price of input 2 is w.. (a) Write down an expression for the technical rate of substitution. (b) Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c) Find Juan's cost function. (d) What is the supply function of Juan's firm? 2. Show that the profit function is convex in (p, w). 3. Find the profit function for the Cobb-Douglas production function f(¤1, 12) = Ax†' x" with A > 0, a1, ¤2 > 0 and a1 + a2 < 1. 4. Verify Hotelling's Lemma for the Cobb-Douglas production function with a1+a2 < 1. 5. Find the supply function and the input demand functions for the CES production function: f(r1, F2) = A(ax{ + (1 – a)x2)8/e, %3D where A > 0, B > 0, 0 < a < 1, and 0 + p< 1. 6. Find the profit function for the CES production function. 7. Verify Hotelling's Lemma for the CES production function with B < 1.
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