Question

Question 1: A firm produces one good with a technology given by the production function y = f (x) = x1/3. The factor price w and the price p for the good are fixed.

a) Explore whether the production function exhibits increasing returns to scale.

b) Determine the cost function

c) Determine the demand function for the input factor.

d) How much will the firm produce?

Answer #1

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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...

Consider the technology of production f(K,L) = 0.3log(x) +
0.3log(y)
a) Check whether the production function exhibits constant,
decreasing or increasing returns to scale. Explain
b) Find the conditional demand functions. Use (p1, w1, w2) to
denote the exogenous prices of output x1 and x2 respectively
c) Find the cost function and verify Shephard's lemma
d) Find the profit function

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...

1. A ﬁrm has two variable factors of production, and its
production function is f(x1,x2) = x1/2 1 x1/4 2 . The price of the
output is 6. Factor 1 receives the wage $2, and factor 2 receives
the wage $2. a. How many units of each factor will the ﬁrm demand?
b. How much output will it produce?
2. Beth produces software. Her production function is f(x1,x2) =
3x1 + 2x2, where x1 is the amount of unskilled labor...

Answer the following questions about the producer’s production
function: Q = 2K1/2L2
Does the production function display increasing, constant, or
decreasing returns to scale? [Prove your answer by increasing all
inputs by a factor of c in your analysis.]
Find MPL if capital is fixed at K0=9 and
determine whether the production process follows the law of
diminishing returns (LDR) to labor.
If input prices are r=5 and w=4 for capital and labor,
respectively, and suppose MPK=40 and the firm...

A firm’s production function is given as y=(x1)^(1/2) *
(x2-1)^(1/2) where y≥0 for the output, x1≥0 for the input 1 and
x2≥0 for the input 2. The prices of input 1 and input 2 are given
as w1>0 and w2>0,
respectively. Answer the following questions.
Which returns to scale does the production function
exhibit?
Derive the long-run conditional input demand functions and the
long-run cost function.

Suppose the aggregate production function is given by Y =
K0.5L0.5. Does it have increasing, decreasing or constant returns
to scale? Show that the marginal products of capital and labour are
declining. Show that they are increasing in the input of the other
factor.

Suppose that the production function
y=f(x_1,x_2) (where: y is output level, x_1 is a
variable input and x_2 is a fixed input), is plotted in the (y,
x_1) space. According to economic theory, we would expect:
a. y to increase with x_1 at a decreasing rate,
due to increasing returns to scale.
b. y to increase with x_1 at an increasing
rate, due to diminishing returns to scale.
c. y to increase with x_1 at a decreasing rate,
due to...

4.. Suppose the aggregate production function is given by Y =
K0.5L0.5. Does it have increasing, decreasing or constant returns
to scale? Show that the marginal products of capital and labour are
declining. Show that they are increasing in the input of the other
factor.

Suppose a firm’s production function is given by Q = 2K^1/2 *
L^1/2 , where K is capital used and L is labour used in the
production.
(a) Does this production function exhibit increasing returns to
scale, constant returns to scale or decreasing returns to
scale?
(b) Suppose the price of capital is r = 1 and the price of
labour is w = 4. If a firm wants to produce 16 chairs, what
combination of capital and labor will...

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