Question

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 shutdown price?

c.) Find the short run supply function. Don't forget to use your answer to part b.

Answer #1

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 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...

Consider the production function Q = f(L,K) = 10KL / K+L. The
marginal products of labor and capital for this function are given
by
MPL = 10K^2 / (K +L)^2, MPK = 10L^2 / (K +L)^2.
(a) In the short run, assume that capital is fixed at K = 4.
What is the production function for the firm (quantity as a
function of labor only)? What are the average and marginal products
of labor? Draw APL and MPL on one...

Suppose a firm’s production function is q = f(K,L) = (K)1/3
(L)1/3
(a) Set up the firm’s problem and solve for K∗ and L∗ here. Show
your work to derive the value of K∗ and L∗ otherwise no marks will
be awarded. Note: your solution
11
should be:
∗ K = P^3/27r2w L = P^3/27w2r
How much does the firm produce (i.e. what is q∗)? What is the
profit earned by this firm (i.e. what is π∗)?
(b) The firm...

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 cost-minimizing firm has the following production function:
Q=LK+2M. Where L denotes Labor, K denotes Capital, and M denotes
Materials. The prices for the inputs are as follows: w=$4, r=$8,
and m=$1. The firm is operating in the long run. Answer the
following questions as you solve for the total cost of producing
400 units of output. Assume an interior solution (i.e. positive
values of all inputs).
a) Set up constrained optimization problem of the firm:
b) Write out the...

A firm has a production function of Q = 10L0.3K 0.6 . The price
of L is w = 9 and the price of K is r = 18
. a. What is its short-run marginal cost curve?
b. What is its average variable cost curve?

(2) Consider the production function f(L, K) = 2K √ L. The
marginal products of labor and capital for this function are given
by MPL = K √ L , MPK = 2√ L. Prices of inputs are w = 1 per hour of
labor and r = 4 per machine hour. For the following questions
suppose that the firm currently uses K = 2 machine hours, and that
this can’t be changed in the short–run.
(e) What is the...

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.

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.

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