Question

production function Consider a firm that produces a single output good Y with two input goods: labor (L) and capital (K). The firm has a technology described by the production function f : R 2 + → R+ defined by f(l, k) = √ l + √ k, where l is the quantity of labor and k is the quantity of capital. (a) In an appropriate diagram, illustrate the map of isoquants for the firm’s production function. (b) Does the firm’s technology satisfy the following properties: (i) monotonicity (i.e., are the marginal product of labor and the marginal produc of capital strictly positive), (ii) the no-free-lunch property, (iii) constant/decreasing/increasing returns to scale? (c) What is the firm’s marginal rate of technical substitution?

Answer #1

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...

Consider a firm that produces a single output with a single
input, labor, using production function F(L)=100L−4(L^2), for
L∈[0,12.5]. The input price is W=2.
1. Determine the firm’s cost function C(Q), that is, the lowest
cost of producing Q units of output. State the associated labor
choice for a given required output, L(Q). Consider only the range Q
∈ [0, 12.5] .
2. What is the firm’s marginal cost curve, MC(Q)?

Suppose a competitive firm’s production function is Y= 20
L1/2 K1/3. L is Labor , K is capital and Y is
output.
a) (4) Find the marginal product of labor and capital.
b) (4) What is Marginal Rate of technical Substitution of Labor
for Capital?
c) (2) Does this production function exhibit increasing,
decreasing or constant returns to scale? Show your work.

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...

A firm has the production function:
Q = L 1 2 K 1 2
Find the marginal product of labor (MPL), marginal
product of capital (MPK), and marginal rate of technical
substitution (MRTS).
Note: Finding the MRTS is analogous to finding the
MRS from a utility function:
MRTS=-MPL/MPK. Be sure to simplify your
answer as we did with MRS.
A firm has the production function:
Q = L 1 2 K 3 4
Find the marginal product of labor (MPL),...

4. output Q according to the production function
Q = 6K1/3L1/2, where K = capital and L
=labor.
A. Calculate the marginal product of capital.
Calculate the marginal product of labor.
Calculate the marginal rate of technical substitution
A technological advance occurs which changes the production
function to Q = 2KL.
D. Calculate the new marginal product of capital.
E. Calculate the new marginal product of labor.
F. Calculate the new marginal rate of technical substitution for
Lazy J Enterprises....

A firm produces output (y), using capital (K) and labor (L). The
per-unit price of capital is r, and the per-unit price of labor is
w. The firm’s production function is given by, y=Af(L,K), where A
> 0 is a parameter reflecting the firm’s efficiency.
(a) Let p denote the price of output. In the short run, the
level of capital is fixed at K. Assume that the marginal product of
labor is diminishing. Using comparative statics analysis, show that...

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?

the production function is given as y =f(k,l). find the marginal
rate of technical substitution in terms of marginal products of
capital and labour

Consider a firm that produces a single output with a single
input, labor, using 2 different plants. Denote by L1 the assignment
of labor input into plant 1 and by L2 the assignment of labor
input
into plant 2. Plant 1’s production function is F1 (L1) = 4√L1, for
L1 ≥ 0. Plant 2’s production
function is F2(L2) = 8√L2, for L2 ≥ 0.
1. State the average product function of each plant as a function
of the labor assignment....

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