1. Assume the following smooth production function: Q = Q(K,L) with positive marginal productivities. Let w and r the prices of labor and capital, respectively. a. Formulate the problem of minimizing costs subject to the technology. b. Explain under what conditions you might have to consider the case of corner solutions (optimal labor or capital equal to zero). Provide an example. Assuming interior solution, present the first order conditions.
1. Assume the following smooth production function: Q = Q(K,L) with positive marginal productivities. Let w and r the prices of labor and capital, respectively. a. Formulate the problem of minimizing costs subject to the technology. b. Explain under what conditions you might have to consider the case of corner solutions (optimal labor or capital equal to zero). Provide an example. Assuming interior solution, present the first order conditions.
Chapter1: Making Economics Decisions
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Transcribed Image Text:1. Assume the following smooth production function:
Q = Q(K,L)
with positive marginal productivities. Let w and r the prices of labor and capital, respectively.
a. Formulate the problem of minimizing costs subject to the technology.
b.
Explain under what conditions you might have to consider the case of corner solutions
(optimal labor or capital equal to zero). Provide an example.
c. Assuming interior solution, present the first order conditions.
d.
Provide an economic interpretation to the optimality condition. In your answer, refer to the
Lagrange multiplier.
e.
Provide a graphical representation of the resulting optimal input combination.
f. Present the second order condition.
g. Explain how the strict convexity of the isoquants would ensure a minimum cost.
h.
Explain how quasi-concave production function can generate everywhere strictly convex,
downward-sloping isoquants.
i. Now, assume Q = AL" KB. Show that the expansion path (optimal combinations of capital
and labor for different isocosts) is characterized by a linear function.
j.
Show the previous result holds for all homogeneous production functions.
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