Question

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 and interpret the elasticity of substitution.

d. Set up the optimization problem of the firm and, for arbitrary values of w, r, and Q, solve for the (long run) input demand curves.

e. Derive the own price elasticity of demand for labor and capital.

f. For arbitrary values of w, r, and Q, solve for the long run cost function. Graph this cost function when w=$16 and r=$64.

g. Derive the long run average cost function when w=$16 and r=$64. Does this cost function exhibit economies of scale, diseconomies of scale, or constant returns to scale? Why?

h. Suppose that the firm's capital is fixed in the short run at K=9. For arbitrary values of w, r, and Q, find the firm's short run demand for labor and the corresponding cost function.

Answer #1

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...

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 produces output (y), using capital (K) and labor (L). The
per-unit price of capital is r, and the per-unit price of labor is
w. The firm’s production function is given by, y=Af(L,K), where A
> 0 is a parameter reflecting the firm’s efficiency.
(a) Let p denote the price of output. In the short run, the
level of capital is fixed at K. Assume that the marginal product of
labor is diminishing. Using comparative statics analysis, show that...

Assume that a profit maximizer firm uses only two inputs, labor
(L) and capital (K), and its production function is f(K,L) = K2 x
L. Its MRTS of capital for labor (i.e., how many units of capital
does he want to give up one unit of labor) is given by MRTS = MPL /
MPK = K / (2L) a) Assume that this firm wants to spend $300 for the
inputs (total cost of factors of production). The wage per...

A firm produces a product with labor and capital. Its production
function is described by Q = min(L, K). Let w and r be the prices
of labor and capital, respectively.
a) Find the equation for the firm’s long-run total cost curve as
a function of quantity Q and input prices, w and r.
b) Find the solution to the firm’s short-run cost minimization
problem when capital is fixed at a quantity of 5 units (i.e., K =
5). Derive...

A firm discovers that when it uses K units of capital
and L units of labor it is able to
produce q=4K^1/4
L^3/4 units of output.
a) Calculate the MPL, MPK and MRTS
b) Does the production function (q=4K^1/4 L^3/4) exhibit
constant, increasing or decreasing returns to scale and
why?
c) Suppose that capital costs $10 per unit and labor can
each be hired at $40 per unit and the firm uses 225 units of
capital in the short run....

A firm uses two inputs, capital K and labor L, to produce output
Q that can be sold at a price of $10. The production function is
given by Q = F(K, L) = K1/2L1/2 In the short run, capital is fixed
at 4 units and the wage rate is $5, 1. What type of production
function is F(K, L) = K1/2L1/2 ? 2. Determine the marginal product
of labor MPL as a function of labor L. 3. Determine 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.

A firm’s production function is Q! = min(4L ,5K ). The price of
labor is w and the price of capital is r.
a) Derive the demand function of labor and capital respectively.
How does the demand of capital change with the price of
capital?
b) Derive the long-run total cost function. Write down the
equation of the long-run expansion path.
c) Suppose capital is fixed at K = 8 in the short run. Derive
the short-run total cost function....

To produce traps for capturing humans, the Coyote Cooperative
requires both capital, K, and labour, L. Suppose that the
production technology is given by by the production function
q=20L^0.5K^0.5, where q is the number of traps, MPL =10L^-0.5K^0.5,
and MPK=10L^0.5K^-0.5.
a) What are the returns to scale for this production
function?
b) What is the equation of the Cooperative’s isoquant?
c) What is the equation for a slope of its isoquant?
d) What is this called?
e) What does it...

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