In: Economics

Question 1: Show that the Cobb-Douglas Production function Y = zKαN1−α, where 0 < α < 1, satisfies all assumptions made in lecture 5.

Assumptions:

1) Output increases when either the capital stock or the number of workers increase

2) Both the marginal product of capital and the marginal product of labor are decreasing

3) The marginal product of labor increases when capital increases, and the marginal product of capital increases when labor increases

4) For any constant x > 0, F(xK,xN) = xF(K,N)

The total production of a particular commodity is given by the
Cobb-Douglas function: f(x,y)=bx^α*y^(1−α), b is a given positive
constant and 0<α<1 . Assume that we want to maximize
production with a given cost constraint mx+ny−p=0 where m and n are
the cost of a unit of labour and a unit of capital, respectively,
and p the total cost.
Show that each term will always be negative guaranteeing a
maximum (in the sufficiency H test) using the values obtained of...

Cobb-Douglas...again Consider the Cobb-Douglas production
function function of the form, q(k, l) = k α l 1−α
(a) Determine the relation between α and the marginal product of
k and l. For what values of α is the marginal product for each
input: (i) increasing, (ii) constant, and, (iii) decreasing.
(b) Show that the marginal rate of technical substitution (MRTS)
is equal to α 1 − α l k . For what values of α is MRTS decreasing
in k?...

Consider the Cobb-Douglas production function ?=??^??^??^? where
?, ?, ?, ? are positive constants and ?+?+?<1. Let ? be the
amount of labor, ? the amount of capital, and ? be the amount of
other materials used. Let the profit function be defined by
?=?−(??+??+??) where the costs of labor, capital, and other
materials are, respectively, ?, ?, and ?.
Determine whether second order conditions for profit
maximization hold, when the profit function is defined by
?=?−(30?+20?+10?) with ?=5?^0.3?^0.4?^0.2.

An economy has a Cobb-Douglas production function:
Y=K^α(LE)^1-α
The economy has a capital share of 1/3, a saving rate of 24
percent, a depreciation rate of 3 percent, a rate of population
growth of 2 percent, and a rate of labor-augmenting technological
change of 1 percent. It is in steady state.
A. At what rates do total output, output per worker, and output
per effective worker grow?
B. Solve for capital per effective worker, output per effective
worker, and the...

Assume a Solow model that uses Cobb-Douglas production function
with and α = 0.5 for a society that saves 40% of their income but
sees their capital depreciate at a rate of 5%, population grows at
a rate or 2% and technology grows at a rate of 3%.
Determine the steady state level of capital per worker.
Now assume that the economy experiences population growth of
5%, what would their savings rate need to be to make sure that the...

Stanford airline's production function is given by a
Cobb-Douglas form:
where:
Y = number of passengers carried per year
L = number of pilots (labor)
K = number of aircraft (capital)
a) show that the product elasticity for labor is given by
EYL = a,
and that the product elasticity for capital is given by /•'.
,.=/?.
b) show that MPL > 0, MPK >0, and
that tfY / âl: < 0 , &Y / cK2 <
0.
c) show...

The production of a manufacturer is given by the Cobb-Douglas
production function
f(x,y)=30x^(4/5)y^(1/5)
where x represents the number of units of labor (in hours) and y
represents the number of units of capital (in dollars) invested.
Labor costs $10 per hour and there are 8 hours in a working day,
and 250 working days in a year. The manufacturer has allocated
$4,000,000 this year for labor and capital. How should the money be
allocated to labor and capital to maximize...

1.A production function such as Y=AKaLb is called the
Cobb-Douglas (C-D) production function. Only two factors of
production are assumed here: capital (K) and labor (L), with A
interpreted as the level of technology. Later, it was discovered
that human capital (H) is also a "neglected"factor of production,
assuming that the function satisfies constant return to scales for
all the factors of production that it should include (except for
technology). In the correct production function, there should
be:
A. a+b=1。...

QUESTION
Consider the following Cobb Douglas production function: Y=
K2/5L3/5. The rate of depreciation in the
economy is 2% and the marginal propensity to save (mps) is 30%. Any
output that is not saved is consumed and this is a closed economy.
Population growth rate is zero.
Continue with the same data with the exception that mps is
unknown. Solve for the rate of
investment which will ensure golden rule of
consumption per capita . Show
all the steps covered...

Suppose a firm has a Cobb-Douglas production function. Show
graphically that an increase in the rental rate of capital will
increase the amount of labor hired if production remains at the
same amount(10pts).

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