Question

Suppose you own a firm that producing shoes using both capital and labor. The production function is q=f(K, L)=0.5K2 L4 . In long run both capital (K) and labor (L) are variable. Price for each pair of shoes is $50 (p=50), the wage rate is 0.04 (w=0.04) and the rental price for capital is 1 (r=1). Given those output and input prices, what is the profit maximizing input level of K and L (K* & L* )?

Answer #1

Consider a firm using the production technology given by q =
f(K, L) = ln(L^K)
If capital is fixed at K = 2 units in the short run, then what
is the profit maximizing allocation of output if the price of
output and respective input prices of labor and capital are given
by (p, w, r) = (2, 1, 5)?

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...

2. Suppose a firm is producing 200 widgets. The firm’s
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Douglas with decreasing returns to scale. (This means we have
normal, convex
isoquants). The firm uses K’ units of capital and L’ units of
labor. Initially, the input prices
are w’ and r’. However, an exogenous shock in the labor market
causes an increase in
the wage rate, resulting in an increase in input prices from w’
to w’’ where w’<w’’. Using
a graph (of isoquant...

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 (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...

FACTOR PRICES QUESTION
Imagine firm Oracle is producing computers following
production function q(L,K) =L^0.5 K^2. In the short run, capital is
ﬁxed at K¯ = 5. Oracle faces price p = 50 and can hire as many
workers as it would like at a constant wage w = 25.
A. Find equilibrium labor (L∗) and wages.
B. What are Oracle’s proﬁts at this
equilibrium?
C. Prove that this proﬁt level is a global
maximum.

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.

The Pear Corp produces high end consumer electronics using labor
L and capital K according to production function Q = F (L,K) =
(100)(L^1/4)(K^1/4). Let the price of a unit of labor be given by W
and the price of a unit of capital is given by R. The output price
is P = 1 which Pear Corp takes as given for any choice of output
level Q. Both labor and capital are fully adjustable. Set input
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A firm’s production function is Q(L,K) = K^1/2 + L. The firm
faces a price of labor, w, and a price of capital services, r.
a. Derive the long-run input demand functions for L and K,
assuming an interior solution. If the firm must produce 100 units
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Suppose that a firm has production function F(L, K) = L1/4 K3/4
for producing widgets, the wage rate for labor is w = $32, and the
rental rate of capital is r = $6. Suppose the firm has an order to
produce 40 units of output.
a) Carefully write out the firm’s cost minimization problem,
using information specific to this problem.
b) Express two equations—specific to this problem—that the
optimal solution satisfies.
c) Solve these two equations for L* and...

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