Question

A firm uses two inputs, capital K and labor L, to produce output Q that can be sold at a price of $10. The production function is given by Q = F(K, L) = K1/2L1/2 In the short run, capital is fixed at 4 units and the wage rate is $5, 1. What type of production function is F(K, L) = K1/2L1/2 ? 2. Determine the marginal product of labor MPL as a function of labor L. 3. Determine the value of the marginal product of labor as a function of labor L. 4. How much labor should the firm employ to maximize profits? 5. Calculate the average product of labor, APL, when the level of capital is fixed at 16 units and the firm uses 9 units of labor. How does the average product of labor change when the firm uses 25 units of labor?

Answer #1

Suppose a firm can manufacture a product using the following
production function:
Q = f (K,L) =
"K0.60
"L0.40
(a) What is the APL when the firm uses 10 units of fixed capital
and 30units of labor?
(b) How does the APL change when the firm uses 100 units of
labor?
(c) Write out an expression for the MPL when capital is fixed
(d) Show that the MPL depends on the amount of labor that is
employed by calculating the...

Consider the production function Q = f(L,K) = 10KL / K+L. The
marginal products of labor and capital for this function are given
by
MPL = 10K^2 / (K +L)^2, MPK = 10L^2 / (K +L)^2.
(a) In the short run, assume that capital is fixed at K = 4.
What is the production function for the firm (quantity as a
function of labor only)? What are the average and marginal products
of labor? Draw APL and MPL on one...

1. Suppose a firm can manufacture a product using the following
production function:
Q = f (K,L) = K^0.60 L^ 0.40)
(a) What is the APL when the firm uses 10 units of fixed capital
and 30units of labor?
(b) How does the APL change when the firm uses 100 units of
labor?
(c) Write out an expression for the MPL when capital is
fixed
(d) Show that the MPL depends on the amount of labor that is
employed by...

Assume that a profit maximizer firm uses only two inputs, labor
(L) and capital (K), and its production function is f(K,L) = K2 x
L. Its MRTS of capital for labor (i.e., how many units of capital
does he want to give up one unit of labor) is given by MRTS = MPL /
MPK = K / (2L) a) Assume that this firm wants to spend $300 for the
inputs (total cost of factors of production). The wage per...

Bonus Question. Suppose the production function for a firrm is
Q(K,L) = K1/2L1/2, so the marginal product of labor is MPL = 1 2
K1/2L−1/2 and the marginal product of capital is MPK = 1 2
K−1/2L1/2.
a) Find the equation of the isoquant for Q = 1. That is, when Q
= 1, find L as a function of K or K as a function of L to obtain an
equation for the isoquant.
b) Find K1, K2, L3,...

Consider a firm that used only two inputs, capital (K) and labor
(L), to produce output. The production function is given by: Q =
60L^(2/3)K^(1/3) .
a.Find the returns to scale of this production function.
b. Derive the Marginal Rate of Technical Substitutions (MRTS)
between capital and labor. Does the law of diminishing MRTS hold?
Why? Derive the equation for a sample isoquant (Q=120) and draw the
isoquant. Be sure to label as many points as you can.
c. Compute...

A firm uses capital and labor to produce output according to
the production ? = 4√??
(a) Find the marginal product of labor (MPL) and marginal
product of capital (MPK).
(b) If the wage w=$1/labor-hr. and the rental rate of capital
r=$4/machine-hr., what is the
least expensive way to produce 16 units of output?
(c) What is the minimum cost of producing 16 units?
(d) Show that for any level of output, q, the minimum cost of
producing q is...

An electronics plant’s production function is Q = L 2K, where Q
is its output rate, L is the amount of labour it uses per period,
and K is the amount of capital it uses per period.
(a) Calculate the marginal product of labour (MPL) and the
marginal product of capital (MPK) for this production function.
Hint: MPK = dQ/dK. When taking the derivative with respect to K,
treat L as constant. For example when Q = L 3K2 ,...

firm can manufacture a product according to the production
function
Q = F (K, L) = K0.75 L 0.25 a. What is this type of function
called? Are the inputs perfect substitutes or should they be used
in a fixed proportion instead? © (3pts) b. Suppose capital is fixed
at 81 units. If the firm can sell its output at a price of $200 per
unit and wage is $50, how many units of labor should the firm hire
in...

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),...

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