ECON 5020 Microeconomic Theory Problem Set # 8
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ECON 5020
Microeconomic Theory
Problem Set # 8
1. The production function of a firm is given by
f (K, L) = 2K1/2L1/2
and factor prices for the inputs capital K and labour L are r and w, respectively. For the exercises it may be useful to draw sketches to support your intuition.
(a) Find the conditional factor demand functions.
(b) What is the cost function?
(c) In the short run, the capital stock is fixed, K = 
. What is the short-run cost function?
(d) Find short-run average and marginal cost, variable and fixed cost, and average variable and average fixed cost.
(e) Show that short-run marginal cost intersects average cost at the minimum of the average cost function.
(f) For the remainder, assume that input prices are w = 4 and r = 1, while desired output is q = 100.
i. What is the long-run total, marginal and average cost, i.e. when capital is not fixed? What is the optimal level of capital at these input prices and this output?
ii. What is the short-run total, variable, fixed, marginal and average cost, when capital is fixed at 
= 50?
iii. What is the short-run total, variable, fixed, marginal and average cost, when capital is fixed at 
= 200?
iv. What is the short-run total, variable, fixed, marginal and average cost, when capital is fixed at 
= 100?
2. The production function of a firm is given by
f (K, L) = AKa L8
and factor prices for the inputs capital K and labour L are r and w, respectively. The price of output is p. We want to derive the cost function from a slightly different approach by using the profit maximization problem.
(a) First, we write down the profit-maximization problem as:
max p . q - r . K - w . L
q,r,w
subject to:
AKa L8 2 q .
Now, denote the Lagrangian multiplier by µ . Form the Lagrangian, write first-order conditions, and find the capital-labor ratio.
(b) Plugging the capital-labor ratio into the constraint, find the conditional factor demands, i.e., K(w, r, q) and L(w, r, q).
(c) Show that the firm’s cost function is
C(w, r, q) = B ╱qra w8、1/(a+8) ,
where
B = (α + β)A − 1/(a+8)α −a/(a+8)β −8/(a+8) .
(d) Discuss whether the optimal quantity of production exists under the following conditions:
i. α + β > 1?
ii. α + β = 1?
iii. α + β < 1?
For what follows, assume that α + β < 1. Get help from the support note whenever needed.
(e) What is the profit-maximizing output as a function of prices?
(f) Show that the supply function is homogenous of degree zero in (p, w, r).
(g) Show that the profit function is
士 α β
π(p, w, r) = Dp 士 − α − β r − 士 − α − β w − 士 − α − β ,
where D = (1 - α - β) ╱Aαa β 8、1/(1 −a −8) .
(h) Show that the profit function is non-decreasing in p and non-increasing in w and r .
(i) Show that the profit function is homogenous of degree one in (p, w, r).
(j) Confirm that q(p, w, r) = ∂π(p, w, r)/∂p.
2023-03-31