Question

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.

Answer #1

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,...

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...

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...

Suppose a firm has a production function q = f(L, K) =2L + 4K,
and the factor prices are w = $2 and r = $2. What is the minimum
cost at which the firm is able to produce 20 units of output?
a. $10 b. $30 c. $45 d. $100 e. $50
please explain why

2. A firm has production function Q = k^1/2L^1/2 and faces a
wage for the labor input w = 1 and a rental price of capital r =
9
a. The policy of the Federal Reserve brings the rental price of
capital to r = 4 Graph the change of the cost minimizing equlibrium
explaining the type of substitution that is happening.
b. Compute the new cost function. Suppose a monopoly and show
graphically if after this change in the...

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...

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...

Consider the following firm with its demand, production and cost
of production functions:
(1) Demand: Q = 230 – 2.5P + 4*Ps + .5*I, where Ps = 2.5, I =
20.
(2) Inverse demand function [P=f(Q)], holding other factors (Ps
= 2.5 and I =20) constant, is, P=100-.4*Q.
(3) Production: Q = 1.2*L - .004L2 + 4*K - .002K2;
(4) Long Run Total Cost: LRTC = 2.46*Q + .00025*Q2 (Note: there
are no Fixed Costs);
(5) Total Cost: TC =...

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...

Suppose the firm's production function is Q = K 1/3L 2/3 . a. If
the rental rate of capital R = $30 and the wage rate W = $40, what
is the cost-minimizing capital-to-labor ratio? b. If the rental
rate of capital R is $35 and the wage rate W is $70, how many units
of labor and capital should the firm use to produce 12 units of
output?

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