Question

A firm produces good X and has a production function X =
2L^0.25K^0.25, where L and K are the inputs.

Assume that the price of L is $6 and the price of capital is $12.
Let the firm have a target output

of X1 units.

a. Find the firm’s conditional demand for labor and capital.

b. Find the firm’s total cost function.

c. What is the firm’s marginal cost?

Answer #1

A firm produces good Q using inputs L & K. The firm’s
production function is X = 20L^0.5 + 11K. The
price of K is $P_K a unit and the price of L is $P_L a unit, and in
the short‐run, the capital input is
fixed at 3 units.
a. If the firm needs an output of X_1 in the short‐run, what is the
firm’s total cost and marginal
cost of production?
b. What is the firm’s fixed cost and...

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...

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 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...

A firm’s production function is Q = min(K , 2L), where Q
is the number of units of output produced using K units of capital
and L units of labor. The factor prices are w = 4 (for labor) and r
= 1 (for capital). On an optimal choice diagram with L on the
horizontal axis and K on the vertical axis, draw the isoquant for Q
= 12, indicate the optimal choices of K and L on that isoquant,...

1. Suppose a short-run production function is described as Q =
2L – (1/800)L^2 where L is the number of labors used each hour. The
firm’s cost of hiring (additional) labor is $20 per hour, which
includes all labor costs. The finished product is sold at a
constant price of $40 per unit of Q.
a. How many labor units (L) should the firm employ per hour
b. Given your answer in a, what is the output (Q) per hour...

A firm has the following production function:
q=5LK^0.5+2L^2K-L^3K
What is its short-run production function if capital is fixed at
K=9?
What are the firm’s marginal product of labour and average
product of labour in the short run?
Show that the firm’s elasticity of output with respect to labour
in the short run is a function of marginal product of labour and
average product of labour. Calculate the short-run elasticity of
output with respect to labour

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...

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 ,...

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...

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