Question

Find the cost minimizing input combinations in the following problems:

- f(k, ℓ) = 4k^3 ℓ^2 , r = 2, w = 1, q = 100

- f(k, ℓ) = min{2k, 3ℓ}, r = 2, w = 3, q = 10

2. In the above question, you found the cost minimizing input combination that produces 100 units of output. Now find the cost minimizing input combination for any positive quantity of output q to obtain the firm’s conditional factor demand functions at the given input prices. Next, write down the firm’s cost function.

3. Suppose the firm is operating in the short run and cannot change its capital input, which is currently fixed at 64 units. In each case above, how many units of labor does the firm need in order to produce 100 units of output? Given that capital is fixed at 64 units, find the cost minimizing input combination that produces 100 units of output.

(a) Assuming again that capital is fixed at 64 units, find the cost minimizing input combination for any positive quantity of output q to obtain the firm’s short-run conditional factor demand functions at the given input prices. Next, write down the firm’s short-run cost function

Answer #1

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. 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...

The firms production function is: Q=2L^2/3 K^1/3
A) Suppose the firm wants to determine the cost minimizing
combination for L and K for any given values of q, w, and r. Solve
for the the firms factor demand functions for L and K (i.e. express
the optimal quantity of L and K in terms of W, r and Q)
B) Using these factor demand functions, solve for the firm's
long run cost function.

If input prices are w = 4, and r = 1, and q = 4K^0.5L^0.5
a. What is the least-cost input combination required to produce
40 units of output?
b. Suppose instead that capital was fixed at 16 units. What
would be the implications for labor usage and total cost?

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...

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.

1. Acme Inc. produces widgets. Its production function
is ? = ? 1/3 (? − 1) 1/3 , for ? ≥ 0 and ? ≥ 1, where ? denotes
units of capital input, ? denotes units of labor input, and ?
denotes units of output. (Note that ? = 0 for all ? < 1.) The
price of a unit of capital input is ? and the price of a unit of
labor input is ?.
a. Find Acme’s ???...

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)?

1. Consider the production
function q=K2L0.5
a) Find the cost minimizing quantities of K and L for q = 100, r
as the price of K and w as the price of L.
b) Find the cost minimizing quantities of K and L for q = 1000,
r as the price of K and w as the price of L. Explain whether or not
the output expansion [change from part a) to part b)] is labor
saving or capital saving.

There are two inputs labor, L, and capital, K. Their cost
minimizing levels are given by K(y) = 2y and L(y) =y^2.L and K are
respectively priced w=1/2 and r= 3
.a) Find the firm’s cost curve.
b) What is the firm’s exit price
c) Graphically show how the long run supply curve is derived
from cost curves (make sure to label the axes, the curves, the
intercepts, and the slope).
d) A tp= $12, what is the profit-maximizing level...

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