Question

Consider the following production function: x = f(l,k) =
Al^{b}k^{b}where x is the output, l is the labour
input, k is the capital input, and A, b are positive constants.

(a) Set up the cost minimization problem and solve for the first order conditions using the Lagrange Method. Let w be the wage rate and r the rental rate of capital.

(b) Using your answer in (a), find how much labour and capital would the firm use to produce x outputs, given wage rate w and rental rate of capital r? What are these functions called?

(c) Using your answer in (b), find the minimum cost it takes to produce x outputs. What is this function called?

(d) Use your answer in (c) to derive the marginal cost function.

(e) Under what conditions does the production function exhibit increasing, constant, and decreasing returns to scale respectively?

Answer #1

Suppose that a firm has production function F(L, K) = L1/4 K3/4
for producing widgets, the wage rate for labor is w = $32, and the
rental rate of capital is r = $6. Suppose the firm has an order to
produce 40 units of output.
a) Carefully write out the firm’s cost minimization problem,
using information specific to this problem.
b) Express two equations—specific to this problem—that the
optimal solution satisfies.
c) Solve these two equations for L* and...

Consider the following Cobb-Douglas production function: y(K,L)
= 2K^(0.4)*L^(0.6), where K denotes the amount of capital and L
denotes the amount of labour employed in the production
process.
a) Compute the marginal productivity of capital, the marginal
productivity of labour, and the MRTS (marginal rate of technical
substitution) between capital and labour. Let input prices be r for
capital and w for labour. A representative firm seeks to minimize
its cost of producing 100 units of output.
b) By applying...

Consider a firm which has the following production function
Q=f(L,K)=4?LK
(MPL=2?(K/L) and MPK=2?(L/K).
(a) If the wage w= $4 and the rent of capital r=$1, what is the
least expensive way to produce 16 units of output? (That is, what
is the cost-minimizing input bundle (combination) given that
Q=16?)
(b) What is the minimum cost of producing 16 units?
(c) Show that for any level of output Q, the minimum cost of
producing Q is $Q.

Given production function: Q=L3/5K1/5.
Where L is labor, K is capital, w is wage rate, and r is rental
rate.
What kinds of returns to scale does your firm face?
Find cost minimizing level of L and K, and long run cost
function.

Suppose an agricultural firm has the production function:
f(l; k; a) = l^(1/4) * k^(1/4) * a^(1/4)
where the price of labor is w, the price of capital is r and
acreage (a) has price s.
(a) Verify that this is a valid production function.
(b) Solve the rm's cost minimization problem for the conditional
input demands,
cost function, average cost function, and marginal cost
function.
(c) Suppose that there was a tax on one or more inputs. For each...

Suppose the firm's production function is Q = K 1/3L 2/3 . a. If
the rental rate of capital R = $30 and the wage rate W = $40, what
is the cost-minimizing capital-to-labor ratio? b. If the rental
rate of capital R is $35 and the wage rate W is $70, how many units
of labor and capital should the firm use to produce 12 units of
output?

4. Suppose a small oil drill has the following production
function F(K,L) =
min(4K,L)
where every drill (captial unit) takes 4 people to operate.
Output is measured in barrels.
(a) Suppose there are 10 drills in the oil field. How many
workers are needed to produce 40 barrels of oil (q=40)?
(b) Graph the isoquant curves that represent q=20, q=40, and
q=60.
(c) Setup the cost minimization problem where labor and capital are
flexible. Then find the cost function if...

By using the Lagrangean method, construct the total cost
function of a firm with the production function q=K^ 0.3 L ^0.6
facing a wage rate of w and rental rate of capital r.

Given the Cobb-Douglas production function q = 2K 1 4 L 3 4 ,
the marginal product of labor is: 3 2K 1 4 L 1 4 and the marginal
product of capital is: 1 2K 3 4 L 3 4 .
A) What is the marginal rate of technical substitution
(RTS)?
B) If the rental rate of capital (v) is $10 and the wage rate
(w) is $30 what is the necessary condition for cost-minimization?
(Your answer should be...

Suppose a firm's production function is given by LaTeX:
Q\left(L,K\right)=4L^{0.65}K^{0.35}Q ( L , K ) = 4 L^0.65 K^0.35.
The wage is w= 25/ hour and the rent for capital is r= 25/hour. To
produce 350 units per hour, what is the minimum hourly cost of
production? Enter to the nearest $0.1. [number only, no $ sign]

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