Download a. Wide and steady income disparity across countries b. Evidence of

UNIVERSITY OF VICTORIA
DEPARTMENT OF ECONOMICS
ECON 204 (A02): Intermediate Macroeconomics
MIDTERM II
Total Marks: 50



Time: 75 Minutes
Thursday, November 06, 1:00 – 2:15PM, DSB118
Provide necessary diagrams, reasons, and references, where applicable, to complete your answer
For mathematical problems, please show your all calculations to get full credit
Marks are in parentheses
1.
(5) Briefly outline the stylized facts in economic growth.
a. Wide and steady income disparity across countries
b. Evidence of increasing disparity across countries
c. Almost all countries exhibit some growth; that is, as wealthy countries have
grown so too have poorer countries; Parente and Prescott describe this by saying
there is no absolute poverty trap.
d. Wide distribution of growth rates
2. (5) Define, derive mathematically, and interpret the Solow Residual.
Or,
The production function with effects of the changing technology is: 𝑌 = 𝐴𝐹(𝐾, 𝐿)
Where, A is the current level of technology, called total factor productivity.
Taking log and doing the first order total derivatives:
log 𝑌 = 𝑙𝑜𝑔𝐴 + 𝑙𝑜𝑔𝐹(𝑘, 𝐿)
𝑑𝑌 𝑑𝐴
1
=
+
𝐹 ′ 𝐾, 𝐿 𝑑𝐾 − 𝐹 ′ 𝐿 𝐾, 𝐿 𝑑𝐿
𝑌
𝐴 𝐹(𝐾, 𝐿) 𝐾
𝑑𝐴
𝑀𝑃𝐾
𝑀𝑃𝐿
=
+
𝑑𝐾 +
𝑑𝐿
𝐴 𝐹(𝐾, 𝐿)
𝐹(𝐾, 𝐿)
𝑑𝐴 𝑀𝑃𝐾 × 𝐾 𝑑𝐾 𝑀𝑃𝐿 × 𝐿 𝑑𝐿
=
+
.
+
.
𝐴
𝐹(𝐾, 𝐿) 𝐾
𝐹(𝐾, 𝐿) 𝐿
𝑑𝐾
𝑑𝐿
= 𝛼.
+ 1−𝛼 .
+𝑅
𝐾
𝐿
∆𝑌
∆𝐾
∆𝐿
= 𝛼.
+ 1−𝛼 .
+𝑅
𝑌
𝐾
𝐿



R is the growth in total factor productivity, which is not observable.
R is the change in output that cannot be explained by changes in inputs
o called the Solow residual
Total factor productivity captures anything that changes the relation between measured
inputs and measured output
1|Page
3. Assume an economy has the following production function:
𝒀 = 𝑭 𝑲, 𝑳 = 𝑲𝟎.𝟒 . 𝑳𝟎.𝟔
a. (1) State the per-worker production function.
𝑌
𝐿
𝐾 0.4
=
𝐿
=>𝑦 = 𝑘 0.4
b. (4) If the savings rate is 0.2 and the depreciation rate is 0.05, calculate the
steady-state capital stock per worker, output per worker, and consumption
per worker.
In the steady state: ∆𝑘 = 𝑠. 𝑓 𝑘 ∗ − 𝛿𝑘 ∗ = 0
 0.2 × (𝑘 ∗ )0.4 = 0.05𝑘 ∗
 𝑘 ∗ = 41.66
𝑘 ∗ = 10.08
𝑦 ∗ = 10.080.4 =2.52
𝑐 ∗ = 𝑦 ∗ − 𝑠. 𝑓 𝑘 ∗ = 2.52 − 0.2 × 2.52 = 2.02
4. (3) What does the labor market steady state mean? Find the conditions for the
steady state in the labor market.
-
If U and E are constant
-
Conditions: a) s.E=f.U ; b) L=E+U
5. (2) “Poor countries can catch up rich countries” – when it is possible in the Solow
model?
-
When in the Solow model both poor and rich countries have same level of savings,
population growth rate and technology.
o
Because with low capital poor country will growth faster than rich country
6. (3) What is the Golden Rule level of capital condition? Why might an economic
policymaker choose the Golden Rule level of capital?
-
The golden rule level is derived from consumption function maximization.
When c* is maximized: 𝑀𝑃𝐾 = 𝛿
∗
𝑓 ′ 𝑘𝐺𝑂𝐿𝐷
=𝛿
or,
-
Because policymakers try to maximize society’s welfare, and higher
consumption means better satisfaction to the society
2|Page
7. Consider an economy with the following Cobb-Douglas production function:
𝑌 = 𝐾 1/3 . 𝐿2/3
The economy has 1,000 units of capital and a labour force of 1,000 workers
a) (3) Derive the equation describing labour demand in this economy as a
function of the real wage and the capital stock.
𝜋 = 𝑇𝑅 − 𝑇𝐶 = 𝑃. 𝑌 − 𝑤𝐿 − 𝑟𝐾
𝑃. 𝐾 1/3 𝐿2/3 − 𝑤𝐿 − 𝑟𝐾
1
𝜕𝜋
2
FOC 𝜕𝐿 = 𝑃. 𝐾 3 . 3 . 𝐿−1/3 − 𝑤 = 0
𝑊 −3 2 3
(3 ) 𝐾
𝑃
Or, 𝐿 =
b) (5) If the real wage can adjust to equilibriate labour supply and labor demand,
what is the real wage? In this equilibrium, what are output, and the total
amount earned by workers?
𝑊 −3 2 3
( ) 1000
𝑃
3
𝑊
2
Real wage, =
1000 =
𝑃
3
Output, Y = (1000)1/3(1000)2/3=1000
Labor earn 2/3 of the output. So, MPL.L = 2/3.1000=666.66
c) (3) If the mandated real wage is 1, illustrate the labor market in a diagram.
-
Classical labor market – labor supply is fixed at 1000
W/P= 1
W/P= 2/3
Mandated wage 1 creates unemployment
3|Page
8. (5) “Economics is also able interpret the secretes to happiness”. Do you agree?
Why or why not? Explain.
Yes, according to the Case Study in Chapter 6: People in an economy are happier if both
unemployment and inflation rates are low
9. In the United States, the capital share of GDP is about 30 percent; the average
growth in output is about 3 percent per year; the depreciation rate is about 4
percent per year; and the capital-output ratio is about 2.5. Suppose that the
production function is Cobb-Douglas, so that the capital share in output is
constant, and that the United States has been in a steady state.
Cobb-Douglas production function: 𝑦 = 𝑘 1/3
The capital stock is about 2.5 times one year’s GDP:
k = 2.5 y
(1)
About 4% of GDP is used to replace depreciating capital:  k = 0.04 y
(2)
MPK  k = 1/3 y
Capital income is about 1/3 of GDP:
(3)
a. (3) What must the saving rate be in the initial steady state?
In the steady state:

s y= k=0.04y
s = 0.04
b. (3)What is the marginal product of capital in the initial steady state?
From 1 & 2   .2.5y = 0.04 y
 = 0.016
In the steady state:
s y= k
0.04. (k*)1/3=0.016k
 (k*)2/3 =2.5
From production function:
 k* = (2.5)3/2
MPK= 1/3. (k*)-2/3
MPK= 1/3 (2.5)(3/2)X(-2/3)=1/3 (2.5)-1=1/7.5=0.133
4|Page
c. (5) If this is not the Golden Rule steady state, what will the marginal
product of capital be at the Golden Rule steady state? Compare the
marginal product at the Golden Rule Steady state to the marginal product
in the initial steady state. Explain.
∗
 In Golden rule level: 𝑓 ′ 𝑘𝐺𝑂𝐿𝐷
=  = 0.016
So the Golden rule level MPK =0.016, which is less than the current steady state
MPK=0.133
So, the economy should invest more
Bonus Question
10. (2) What are the main predictions in the basic Solow model? Which variables are
used to describe these predictions? Explain.
Main predictions:
a. Differences in the level of per capita income (output per person);
These are explained in our model by: differences in saving rates, population growth,
and technology growth rates
b. Differences in the growth rates of per capita income which is explained by
technology growth rates
a)
5|Page