Question

The firm’s production function is given by q =
4K^{0.5}L^{0.5}, 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 MP_{L} = 2K^{0.5}L^{–0.5}
and the marginal product of capital is given by MP_{K} =
2L^{0.5}K^{–0.5}.

^{1.} Find the cost-minimizing levels of capital (K*)
and labor (L*) required to produce the target level of output of
q_{o} = 480 if the cost of capital is r = $40/sq foot and
the wage rate is w = $10/hour.

2.

Find the cost-minimizing levels of capital (K*) and labor (L*)
required to produce the target level of output of q_{o} =
480 if the cost of capital is r = $40/sq foot and the wage rate is
w = $10/hour. What is the firm’s total cost of producing 480 units
of output?

3.

Find the cost-minimizing levels of capital (K*) and labor (L*)
required to produce the target level of output of q_{o} =
480 if the cost of capital is r = $40/sq foot and the wage rate is
w = $10/hour. What is the firm’s average cost of producing 480
units of output?

Answer #1

**1)**

**q =
4K ^{0.5}L^{0.5} **

**MP _{L}
= 2K^{0.5}L^{–0.5}**

**MP _{K} =
2L^{0.5}K^{–0.5} **

**MRTS
= 2K ^{0.5}L^{–0.5} /
2L^{0.5}K^{–0.5} **

**= K/L**

**cost minimising condition
is**

**MRTS =
w/r**

**K/L =
10/40**

**K/L =
1/4 **

**L = 4K**

**Put L = 4K in production
function**

**q =
4K ^{0.5}L^{0.5} **

**q _{o}
= 4K^{0.5}L^{0.5} **

**q _{o} =
4K^{0.5}(4K)^{0.5}**

**q _{o} =
4K^{0.5}(2K^{0.5}) **

**q _{o} =
8K**

**q _{o}/8 =
K**

**K =
q _{o}/8**

**L = 4K**

**L =
4(q _{o}/8)**

**L =
q _{o}/2**

**q _{o}
= 480 **

**K = 480/8 =
60 **

**L = 480/2 =
240**

**Thus K* =
60**

**L* =
240**

**2) **

**Total cost C = wL +
rK **

**= 10(240) +
40(60) **

**= 2400 +
2400**

**= 4800**

**3)**

**Average cost AC =
C/q **

**=
4800/q **

**=
4800/480 **

**=
10 **

** **

Suppose a firm’s long-run production function is given by
Q=K^0.25 L^0.25 ,where K is measured in machine-hours per year and
L is measured in hours of labor per year. The cost of capital
(rental rate denoted by r) is $1200 per machine-hour and the cost
of labor (wage rate denoted by w) is $12 per hour.
Hint: if you don’t calculate the
exponential terms (or keep all the decimals when you do), you will
end up with nice numbers on...

a. A cost minimizing firm’s production is given by Q=L1/2K1/2.
Suppose the desired output is Q=10. Let w=12 and r=4. What is this
firm’s cost minimizing combination of K & L? What it the total
cost of producing this output?
b. Suppose the firm wishes to increase its output to Q=12. In
the short run, the firm’s K is fixed at the amount found in (a),
but L is variable. How much labor will the firm use? What will the...

a. A cost minimizing firm’s production is given by
Q=L^(1/2)K^(1/2)
. Suppose the desired output is
Q=10. Let w=12 and r=4. What is this firm’s cost minimizing
combination of K & L? What it the
total cost of producing this output?
b. Suppose the firm wishes to increase its output to Q=12. In
the short run, the firm’s K is fixed
at the amount found in (a), but L is variable. How much labor
will the firm use? What will...

A firm’s production technology is given by the production
function q = 0.25 LK where L represents labor hours, K machine
hours and q the amount of output. The market wage and rental rates
are, w= $16 and r = $256. The firm is operating in the long run
where it can adjust both inputs.
(b) Suppose that the firm wants to produce 100 units of output.
Determine the cost minimizing combination of L and K. Calculate the
resulting long...

A firm’s production function is q = 10KL with per unit input
prices for labor w = 3 and capital r = 2. Support your answers with
a graph of isoquant-isocosts.
a. Calculate the least-cost input combination of L and K to
produce 60 units of output.
b. Suppose the wage decreases to $2. How does this affect input
use holding constant output at 60?
c. What are the total costs of producing the two output levels
in parts (a)...

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...

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.

Suppose a firm’s production function is given by Q = L 1/2 , K
1/2.
a) Suppose the firm has a fixed cost FC=6, the price
of labor is w = 64 and the price of capital is r = 4. Derive the
firm’s total cost function, TC(Q).
b) What is the firm’s marginal cost?
c) Graph the firm’s isoquant for Q = 20 units of
output. On the same graph, sketch the firm’s isocost line
associated with the total...

Suppose the firm's production function is Q = K 1/3L 2/3 . a. If
the rental rate of capital R = $30 and the wage rate W = $40, what
is the cost-minimizing capital-to-labor ratio? b. If the rental
rate of capital R is $35 and the wage rate W is $70, how many units
of labor and capital should the firm use to produce 12 units of
output?

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.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 5 minutes ago

asked 10 minutes ago

asked 16 minutes ago

asked 41 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago