Question

Suppose an agricultural firm has the production function:

f(l; k; a) = l^(1/4) * k^(1/4) * a^(1/4)

where the price of labor is w, the price of capital is r and
acreage (a) has price s.

(a) Verify that this is a valid production function.

(b) Solve the rm's cost minimization problem for the conditional
input demands,

cost function, average cost function, and marginal cost
function.

(c) Suppose that there was a tax on one or more inputs. For each of
the following

cases explain how the tax would a ect the eciency of taxation.
Hint: think

about both the cost minimization problem, which you have solved,
and the pro t

maximization problem, which you have not.

i. Only on labor

ii. On both labor and acreage

iii. On all three inputs

Answer #1

A. The production function is valid beacause the production function states the relation between inputs and outputs and the value of these inputs and outputs can be varying.

B.

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...

4. Suppose a small oil drill has the following production
function F(K,L) =
min(4K,L)
where every drill (captial unit) takes 4 people to operate.
Output is measured in barrels.
(a) Suppose there are 10 drills in the oil field. How many
workers are needed to produce 40 barrels of oil (q=40)?
(b) Graph the isoquant curves that represent q=20, q=40, and
q=60.
(c) Setup the cost minimization problem where labor and capital are
flexible. Then find the cost function if...

Suppose one firm has production function f(K, L) =√K+√L, and
another firm has the production function f(K, L) = (√K+√L)^(.3).
Will these firms have the same supply functions?
Show your work

Consider a firm whose production technology can be represented
by a production function of the form q = f(x1, x2) = x α 1 x 1−α 2
. Suppose that this firm is a price taker in both input markets,
with the price of input one being w1 per unit and the price of
input two being w2 per unit. 1. Does this production technology
display increasing returns to scale, constant returns to scale,
decreasing returns to scale, or variable...

(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...

Firm A’s production function and cost line are given by:Q=Q(L,K)=2 L^(1/2) K^(1/2) (The production function)?L+?=30000 (The cost line of firm A):?L is the amount of labor hired.?K is the amount of capital hired.??p_L or the price of labor is 1 dollar per
unit.??p_K or the price of capital is 1 dollars per
unit.C or cost (think of it as the firm’s budget) is 30000
dollars.How much labor and capital should this firm optimally hire?

A firm produces good X and has a production function X =
2L^0.25K^0.25, where L and K are the inputs.
Assume that the price of L is $6 and the price of capital is $12.
Let the firm have a target output
of X1 units.
a. Find the firm’s conditional demand for labor and capital.
b. Find the firm’s total cost function.
c. What is the firm’s marginal cost?

Suppose a firm has a production function given by q = 3L +
K.
The firm can purchase labor, L at a price w = 24, and capital, K
at a price of r = 5.
What is the firm’s total cost function?

Consider a firm with the production function
f(K,L)=(K1/4)(L2/4).
In the short run, the firm has rented 25 units of capital. The
firm is a price taker in both the output market and labor market,
facing an output price of 11 and a market wage of 5 per hour.
Rounded to the nearest tenth, how many hours of labor will the firm
hire? (Make sure your answer only has 1 decimal before
submitting!)

firm can manufacture a product according to the production
function
Q = F (K, L) = K0.75 L 0.25 a. What is this type of function
called? Are the inputs perfect substitutes or should they be used
in a fixed proportion instead? © (3pts) b. Suppose capital is fixed
at 81 units. If the firm can sell its output at a price of $200 per
unit and wage is $50, how many units of labor should the firm hire
in...

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