Question

- A firm’s production function is given by Q = 5K
^{1/3}+ 10L^{1/3}, where K and L denote quantities of capital and labor, respectively.- Derive expressions (formulas) for the marginal product of each input.
- Does more of each input increase output?
- Does each input exhibit diminishing marginal returns? Prove.
- Derive an expression for the marginal rate of technical substitution (MRTS) of labor for capital.
- Suppose the price of capital, r = 1, and the price of labor, w = 1. The firm wants to produce 200 units of output. What quantities of capital and labor would minimize the costs of producing 200 units of output? Show all your work.

Answer #1

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...

Cobb-Douglas Production Function & Cost of
Production
A firm’s production function is given as –
q =
2K0.4N0.6
What kind of returns to scale does this production technology
exhibit? Justify your answer.
Find out the expression for the marginal product of labor.
Find out the expression for the marginal product of
capital.
Find out the expression for MRTS.

Suppose a competitive firm’s production function is Y= 20
L1/2 K1/3. L is Labor , K is capital and Y is
output.
a) (4) Find the marginal product of labor and capital.
b) (4) What is Marginal Rate of technical Substitution of Labor
for Capital?
c) (2) Does this production function exhibit increasing,
decreasing or constant returns to scale? Show your work.

Suppose a firm’s production function is given by Q = L 1/2 , K
1/2.
a) Suppose the firm has a fixed cost FC=6, the price
of labor is w = 64 and the price of capital is r = 4. Derive the
firm’s total cost function, TC(Q).
b) What is the firm’s marginal cost?
c) Graph the firm’s isoquant for Q = 20 units of
output. On the same graph, sketch the firm’s isocost line
associated with the total...

A firm has the production function:
Q = L 1 2 K 1 2
Find the marginal product of labor (MPL), marginal
product of capital (MPK), and marginal rate of technical
substitution (MRTS).
Note: Finding the MRTS is analogous to finding the
MRS from a utility function:
MRTS=-MPL/MPK. Be sure to simplify your
answer as we did with MRS.
A firm has the production function:
Q = L 1 2 K 3 4
Find the marginal product of labor (MPL),...

A firm’s production function is Q(L,K) = K^1/2 + L. The firm
faces a price of labor, w, and a price of capital services, r.
a. Derive the long-run input demand functions for L and K,
assuming an interior solution. If the firm must produce 100 units
of output, what must be true of the relative price of labor in
terms of capital (i.e. w/r) in order for the firm to use a positive
amount of labor? Graphically depict this...

Consider the following production function q
= K2 + L2.
Does this production function exhibit constant, increasing or
decreasing returns to scale?)
Find an expression for the marginal rate of technical
substitution. Does this production function exhibit diminishing
marginal rate of technical substitution? Explain

Consider the following function:
q = 9LK + 6L^2 - (1/3)L^3
Given the following expression for the marginal productivity of
each input:
MPL = 9K + 12L - L^2 and MPK = 9L
Assuming Capital is plotted on the vertical axis and labor is
plotted on the horizontal axis, determine the value of the marginal
rate of technical substitution when K=20 and L =10. (Round your
answer up to two decimal places and include the proper sign.)
MRTS= ___________

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...

A firm’s production is represented by the following Cobb-Douglas
function: ? = ?^2/3?^1/3. The rental rate, r, of capital is given
by $200 and the price of labor is $100.
a) For a given level of output, what should be the ratio of
capital to labor in order to minimize costs?
b) How much capital and labor should be used to produce those
300 units?
c) What is the minimum cost of producing 300 units?
d) What is the short...

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