Question

**1****.** Consider the production
function q=K^{2}L^{0.5}

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.

Answer #1

Without using a Lagrangian, find the cost minimizing levels of L
and K for the production function q = L^.6K^.4 if
the price of labor =10, the price of capital = 15, and desired
output = 100. What is the total cost to produce that output?

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
of output, what must be true of the relative price of labor in
terms of capital (i.e. w/r) in order for the firm to use a positive
amount of labor? Graphically depict this...

Consider the production function q=aK + bL.
a. Show that the cost-minimizing choice of K and L may not be
unique. (The cost-minimizing K and L levels are those used at a
firm’s cost-minimizing point; the levels are not unique if there is
more than one optimal combination of K and L for any one
isoquant.)
b. Show on a diagram that, if the cost-minimizing choice of
inputs is unique, it will generally entail the use of only K or...

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...

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. Consider the Cobb-Douglas production function Q = 6 L^½ K^½
and cost function C = 3L + 12K. (For some reason variable "w" is
not provided)
a. Optimize labor usage in the short run if the firm has 9 units
of capital and the product price is $3.
b. Show how you can calculate the short run average total cost
for this level of labor usage?
c. Determine “MP per dollar” for each input and explain what the
comparative...

The production function is q = (10KL)/(K+L)
where L = labor, K= capital
The cost function is C = wL + vK where w = wages and v = cost of
capital
Assume K is fixed in the short run at K = 20
a.) Find the short run cost function. Find also the short run
average and marginal costs.
b.) The shut-down price is defined as the minimum of average
variable cost. For this cost function, what is 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...

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....

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