Question

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 when r=9 and
w=1.

Answer #1

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...

(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...

(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.

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 produces a product with labor and capital as inputs. The
production function is described by Q=LK. the marginal products
associated with this production function are MPL=K and MPK=L. let
w=1 and r=1 be the prices of labor and capital, respectively
a) find the equation for the firms long-run total cost curve
curve as a function of quantity Q
b) solve the firms short-run cost-minimization problem when
capital is fixed at a quantity of 5 units (ie.,K=5). derive the...

Answer the following questions about the producer’s production
function: Q = 2K1/2L2
Does the production function display increasing, constant, or
decreasing returns to scale? [Prove your answer by increasing all
inputs by a factor of c in your analysis.]
Find MPL if capital is fixed at K0=9 and
determine whether the production process follows the law of
diminishing returns (LDR) to labor.
If input prices are r=5 and w=4 for capital and labor,
respectively, and suppose MPK=40 and the firm...

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),...

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.

Consider the following production function: x = f(l,k) =
Albkbwhere 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...

Suppose a firm’s production function is given by Q = 2K^1/2 *
L^1/2 , where K is capital used and L is labour used in the
production.
(a) Does this production function exhibit increasing returns to
scale, constant returns to scale or decreasing returns to
scale?
(b) Suppose the price of capital is r = 1 and the price of
labour is w = 4. If a firm wants to produce 16 chairs, what
combination of capital and labor will...

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