Question

A closed economy has the following Cobb-Douglas production function: F(KL) = K2/5 (EL)3/5, where the notation is as in class. The depreciation rate is 1.5% and the saving rate is 20%. The economy is in steady state, where the population decreases at a rate 1% and capital K increases at a rate 1%. (a) Find the growth rates of the following variables (i) labor efficiency, E (ii) the number of workers per machine, L/K (iii) the average productivity of capital, Y/K (iv) the price of capital relative to the price r/w (v) the difference between labour income and capital income wL- rK (b) If the real GDP is 100 billion this year, find the number of machines next year. (c) Bu how many percentage points should the government change the savings rate so that the economy may converge to the golden rule steady state( use + for increase and – for decrease)?. How would the current generation feel about the change

Answer #1

An economy has the following Cobb-Douglas production
function:
Y = Ka(LE)1-a
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. Does the economy have more or less capital than at the Golden
Rule steady state? How do you know? To achieve the Golden Rule
steady...

Assuming the following Cobb-Douglas production
function is given for a closed economy without government.
i. Where returns to capital = 0.5; and rate of
depreciation of physical capital
Determine the steady-state level of capital per worker. What is the
savings rate at which the steady-state level of capital is
achieved?
[6marks]
ii Prove that the steady-state level of output is the
ratio of the saving rate to the rate of
depreciation
[6 marks]
iii. Assuming that , what will be...

Consider the following production function: Y = A ̄K2 L1 , where
Y is production, A ̄ is productivity, K is capital, and L is labor.
Let w denote the wage rate and r denote the rental rate of capital.
21 Suppose you solve the profit maximization problem of the firm:
max A ̄K L wL rK. What is K,L the expression for wL ? Y (a) wL =1↵.
Y (b) wL =↵. Y (c) wL = 1. Y3 (d)...

An economy has a Cobb–Douglas production function:
Y=Kα(LE)1−αY=Kα(LE)1−α
The economy has a capital share of 0.30, a saving rate of 42
percent, a depreciation rate of 5.00 percent, a rate of population
growth of 2.50 percent, and a rate of labor-augmenting
technological change of 4.0 percent. It is in steady state.
Solve for capital per effective worker (k∗)(k∗), output per
effective worker (y∗)(y∗), and the marginal product of capital.
k∗=k∗=
y∗=y∗=
marginal product of capital =

In a solow-type economy with Cobb-Douglas production, assume
that the population growth rate depends on the current level of
output per worker, y, so that n=my, where m is a positive constant.
For simplicity, assume d=0
a) Find an expression for the growth rate of the capital-labor
ratio, k̇ / k
b) Find expressions for the steady states of y and k
c) Find an expression for the growth rate of Y in steady state

Consider the following Cobb-Douglas production function: y(K,L)
= 2K^(0.4)*L^(0.6), where K denotes the amount of capital and L
denotes the amount of labour employed in the production
process.
a) Compute the marginal productivity of capital, the marginal
productivity of labour, and the MRTS (marginal rate of technical
substitution) between capital and labour. Let input prices be r for
capital and w for labour. A representative firm seeks to minimize
its cost of producing 100 units of output.
b) By applying...

Consider two countries: Country A and Country B. Each country
has the following Cobb-Douglas type production function:
Country A: Y = (K0.5)(EL)0.5 Country B: Y =
(K0.7)(EL)0.3
Unfortunately, your knowledge of Country A is a bit limited.
You have pieces of information, but you don’t know the entire
picture.
o Savings rate (s): unknown for Country A and 14.29% for
Country B
o Steady-state value of capital per effective worker: unknown
for both countries, but you have
heard that Country...

Consider the following Cobb-Douglass production
function?≡??(?,?):?=??1/3?2/3
where Y is output, the constant z measures productivity, K is
physical capital, and N is labor. Suppose ?=2, ?=0.16, ?=0.06, and
?=0.02.
a. What are the steady-state (numerical) values of ?, ?, and
??
b. What is the golden-rule (numerical) level of capital per
worker?
c. If the government wants to achieve the golden rule level of
k, should savings increase, decrease or remain unchanged? Solve
for/obtain its (numerical) value. Explain briefly.

6.7 The production function
Q=KaLb where 0≤ a, b≤1 is called a Cobb-Douglas production
function. This function is widely used in economic research. Using
the function, show the following:
a. The production function in Equation 6.7 is a special case of
the Cobb-Douglas.
b. If a+b=1, a doubling of K and L will double q.
c. If a +b < 1, a doubling of K and L will less than double
q.
d. If a +b > 1, a doubling...

QUESTION 1
Suppose an economy can be characterized by a Cobb-Douglas
production function with capital share of 1/3, and A =
200. The investment rate is 0.12 (12%), the annual rate of growth
of the labor force is 0.02 (2%), and the annual depreciation rate
of capital is 0.04 (4%). According to the Solow growth model, this
economy's steady state capital/labor ratio (capital per worker,
k) is
4,000
8,000
10,000
12,000
None of the above.
QUESTION 2
The steady state...

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