Question

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 graph.

(b) What is the marginal rate of technical substitution for this technology?

(c) Are the returns to scale of this production function increasing, decreasing or

constant? Explain.

Answer #1

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 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) Show that the following Cobb-Douglas production function,
f(K,L) = KαL1−α, has constant returns to scale.
(b) Derive the marginal products of labor and capital. Show
that you the MPL is decreasing on L and that the MPK is decreasing
in K.

(2) Consider the production function f(L, K) = 2K √ L. The
marginal products of labor and capital for this function are given
by MPL = K √ L , MPK = 2√ L. Prices of inputs are w = 1 per hour of
labor and r = 4 per machine hour. For the following questions
suppose that the firm currently uses K = 2 machine hours, and that
this can’t be changed in the short–run.
(e) What is the...

3. Consider the production function, Q = [L0.5 +
K0.5] 2 . The marginal products are given as
follows: MPL = [L0.5 + K0.5] L-0.5
and MPK = [L0.5 + K0.5] K-0.5 and
w = 2, r = 1.
A). what is the value of lambda
B). Does this production function exhibit increasing, decreasing
or constant returns to scale?
C).Determine the cost minimizing value of L
D).Determine the cost minimizing value of K
E).Determine the total cost function
F).Determine the...

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...

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.

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,...

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 ,...

production function Consider a firm that produces a single
output good Y with two input goods: labor (L) and capital (K). The
firm has a technology described by the production function f : R 2
+ → R+ defined by f(l, k) = √ l + √ k, where l is the quantity of
labor and k is the quantity of capital. (a) In an appropriate
diagram, illustrate the map of isoquants for the firm’s production
function. (b) Does the...

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