Question

- A firm has the production function:

Q = L 1 2 K 1 2

Find the marginal product of labor (MP_{L}), marginal
product of capital (MP_{K}), and marginal rate of technical
substitution (MRTS).

Note: Finding the *MRTS* is analogous to finding the
*MRS* from a utility function:
*MRTS=-MP _{L}/MP_{K}. 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 (MP_{L}), marginal
product of capital (MP_{K}), and marginal rate of technical
substitution (MRTS).

Answer #1

Ans.

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...

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,...

2. Consider the following production functions, to be used in
this week’s assignment:
(A) F(L, K) = 20L^2 + 20K^2
(B) F(L, K) = [L^1/2 + K^1/2]^2
a (i) Neatly draw the Q = 2,000 isoquant for a firm with
production function (A) given above, putting L on the horizontal
axis and K on the vertical axis. As part of your answer, calculate
three input bundles on this isoquant. (ii) Neatly draw the Q = 10
isoquant for a firm...

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= ___________

4. output Q according to the production function
Q = 6K1/3L1/2, where K = capital and L
=labor.
A. Calculate the marginal product of capital.
Calculate the marginal product of labor.
Calculate the marginal rate of technical substitution
A technological advance occurs which changes the production
function to Q = 2KL.
D. Calculate the new marginal product of capital.
E. Calculate the new marginal product of labor.
F. Calculate the new marginal rate of technical substitution for
Lazy J Enterprises....

A firm produces output according to the production function.
Q=sqrt(L*K) The
associated marginal products are MPL = .5*sqrt(K/L) and MPK =
.5*sqrt(L/K)
(a) Does this production function have increasing, decreasing, or
constant marginal
returns to labor?
(b) Does this production function have increasing, decreasing or
constant returns to
scale?
(c) Find the firm's short-run total cost function when K=16. The
price of labor is w and
the price of capital is r.
(d) Find the firm's long-run total cost function...

A firm’s production function is given by Q = 5K1/3 +
10L1/3, 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...

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.

A firm’s production function is ? = ?Lα
?β where A, α, and β are positive constants. The firm
currently uses 500 units of labor and 40 units of capital. If the
firm adds 1 more unit of labor, what happens to productivity of
capital? Explain.
b. Given a production function Q
= f(L, K), if marginal product of labor and marginal product of
capital are both positive, then this function displays diminishing
MRTS. Explain if this statement is true...

If a firm has the production function f(L,K) =(L+1)K, and
currently uses no units of labor, but K=3 units of capital, what is
its marginal rate of
substitution?Aretherevaluesofwandrsuchthatthischoiceoffactorinputs
is optimal?

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