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(x1, 12) = Ax¶' x with A > 0, a1, a2 > 0 and a1 + a2 < 1. a2 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(T1, F2) = A(ax{ +(1 – a)¤)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 3 < 1.
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(x1, 12) = Ax¶' x with A > 0, a1, a2 > 0 and a1 + a2 < 1. a2 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(T1, F2) = A(ax{ +(1 – a)¤)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 3 < 1.
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.3P
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Transcribed Image Text:1. Juan Valdez owns a coffee farm in Colombia. His production function is:
f(x1, x2) = (x1 – 1)0.25 9-5.
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(x1, 12) = Axa
with A > 0, a1, a2 > 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(T1, #2) = A(ax{ +(1 – a)a%)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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