Question

The production function for a firm is Q = −0.6L 3 + 18L 2K + 10L where Q is the amount of output, L is the number of labor hours per week, and K is the amount of capital.

(a)Use Excel to calculate the total short run output Q(L) for L = 0, 1, 2...20, given that capital is fixed in the short run at K = 1.

(b) Use Excel to calculate the total long run output Q(L) for L = 0, 1, 2...20 and K = 0, 1, 2.......20. (Note: You can use one formula to perform all calculations in this question).

(c) Suppose firm currently employs one unit of labor and one unit of capital. Use the matrix generated in part (b) to determine the returns to scale for this production function. Suppose that the wage is $100, the rental rate is $800 per time period and the capital is fixed at K = 1

(d) For each quantity of labor in part (a), calculate the average product of labor (APL), marginal product of labor (MPL), total variable cost (T V C), total cost (T C), average variable cost (AV C), average total cost (AT C) and marginal cost (MC).

(e) For each quantity of labor in part (a), calculate w/APL and w/MPL, and confirm that they equal AV C and MC, respectively. Explain why these relationships hold.

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. The following table show the relationship between labor and
output for a firm that is using 100 units of capital (K).
L 40 41 42 43 44
Q 1,000 1,018 1,035 1,051 1,066
a) Calculate the MPL and APL for labor units 41 through 44.
b) The cost of one unit of K is $45 (r = 45) and the cost of one
unit of labor is $30 (w = 30). Use these figures, the amount of
capital you...

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...

Ed’s building company has the following production function q =
10KL − 1/2KL^2 where q is the number of houses built, L is the
quantity of labor Ed employs and K is the quantity of capital Ed
uses. In the short run, K is fixed at K¯ = 2
a. Derive MPL and APL.
b. For what values of L is the MPL > 0?
c. For what values of L is the MPL diminishing?
In the long run, both...

Suppose that a firm's fixed proportion production function is
given by q = min(2k, 4L), and that the rental rates for capital and
labor are given by v = 1, w = 3.
A) Calculate the firm's long-run total, average, and marginal
cost curves.
B) Graph these curves.
C) Suppose that k is fixed at 10 in the short run. Calculate the
firm's short-run total, average, and marginal cost curves and graph
them.
D) Now suppose in the long run...

2. Assume that a manufacturer faces a Cobb-Douglas production
function, q=40K^0.5L^0.5
where q is output per period, L is labor, K is capital. The
market price of labor (w) is $50 per unit and the price of capital
(r) is $200 per unit.
a. Specify and illustrate graphically the short-run MPl and APl
for L = 5 to 30 units (assume that the level of capital is 25; use
increments of 5 units of labor). Is this firm operating in...

Given the short run production function, Q = 3L2 –
0.1L3
( a) Write down the equations for,
(i) the marginal product of labor, MPL
(ii) the average product of labor, APL.
(b) Find the value of Q for which the MPL and APL are
maximized.
(c) Show that the MPL= APL when the APL is at a maximum

Suppose that a firm has the following production
function:
Q = 12KL + .7KL2 – 1/30 KL3
Assume the firm is operating in the long run show the
expression for 100 units of output
Now the firm is operating in the short run and capital
(K) is fixed at K = 5, to determine the following:
a.
The maximum output the firm can produce when K =
5.
b.
The level of use of input L where APL...

The firm’s production function is given by q =
4K0.5L0.5, where q denotes the output
(measured as the number of research reports per month). The firm
hires people (“labor”, measured in hours of work) and rents office
space (“capital”, measured in sq. feet). The marginal product of
labor is given by MPL = 2K0.5L–0.5
and the marginal product of capital is given by MPK =
2L0.5K–0.5.
1. Find the cost-minimizing levels of capital (K*)
and labor (L*) required to produce...

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...

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