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Macroeconomics ^宿題 4

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解答は答えだけでなく、理由や計算の過程などを，読み手にわかりやすく丁寧に書くこと。答え だけのものは採点の対象としない．他人のコピーを利用したものなどは，不正行為とみなし厳重

に対処します．A4 の用紙を使い、ホチキス等で綴じること。

1 Assume that the production function is Y

= K

(E

L

)

. In a de-

veloped country (A), the saving rate is 28 percent, and the popu-

lation growth rate is 1 percent per year. In a developing country

(B), the saving rate is 10 percent, and the population growth rate

is 4 percent per year. Both countries’ technological growth rate is

2 percent per year, and the depreciation rate is 4 percent per year.

Answer the following questions.

1.1 Derive the production function per effective worker. (1 point) 1.2 What is the steady state value of output in country A? (1 point) 1.3 What is the steady state value of output in country B? (1 point)

——————————————————————-

2 Prove each of the following statements about the steady state of the

Solow model with population growth and technological progress.

2.1 The capital-output ratio is constant. (2 points)

2.2 Capital and labor each earn a constant share of an economy’s income. (2 points) 2.3 Total capital income grows at the rate of population growth plus the rate of

technological progress, n + g. (2 points)

2.4 total labor income grows at the rate of population growth plus the rate of technological progress, n + g. (2 points)

2.5 The real rental price of capital is constant. (2 points)

2.6 The real wage grows at the rate of technological progress g. (2 points)

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3 In this question, let’s think about the technology. What is “poly-

merase chain reaction”? Explain the detail. Why is this technique

important? Explain over 5 lines. (3 points)

(Hint: see “http://en.wikipedia.org/wiki/Polymerase_chain_reaction” or “^{「生物と無生物の} 間」福岡 伸一 講談社現代新書_”)

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4 Asuume that x(t) = e

, z(t) = e

, and w(t) = e

. Calculate the

growth rate of y(t) for the following cases. (1 point each)

4.1 y = x 4.2 y = z 4.3 y = w 4.4 y = xz 4.5 y = wxz 4.6 y = ^x_z 4.7 y = ^w_x × z

4.8 y = x^βw^1−β, where β = 0.3

4.9 y = x^αw^βz^{１−α−β}, where α = 0.3 and β = 0.6 4.10 y = (x/z)^α, where α = 0.9

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5 Suppose production function in each economy in the world is Y

=

A

K

(E

L

)

where 0 < α < 1. K is capital. L is the labor force.

A > 0 is the level of technology. The national saving rate is s > 0.

The labor force grows at rate n > 0. The capital depreciates at

δ > 0. E is a variable called the efficiency of labor and grows at some

constant rate g > 0. Let k

=

t_×Et)

stand for capital per effective

worker and y

=

stand for output per effective worker.

5.1 Express output per effective worker as a fuction of capital per effective worker.(1 point)

5.2 Write down an expression of the law of motion of k, involving only kt and the exogenous parameters s, n, g, and δ.(1 point)

5.3 Derive the steady state value of k using parameters: s, n, g, and δ.(1 point) 5.4 Derive the steady state value of y using parameters: s, n, g, and δ.(1 point) 5.5 Derive the steady state value of c using parameters: s, n, g, and δ.(1 point) 5.6 What does explain sustained growth in the standard of living (which is output

per worker)?  (1 point)

5.7 Proof that countries with higher saving rate will have higher levels of output per effective worker.   (1 point)

5.8 Proof that countries with higher population growth will have lower levels of output per effective worker.   (1 point)

5.9 Solve for the golden rule level of saving rate which maximizes consumption per effective worker c at the steady state.(1 point)

5.10 Deriving k(t) which is the function of time t by solving the differential equation (Bernoulli’s equation) with the initial value condition k(0) = k0, then we have k(t) =

{(

k₀^1−α₋ ^sA δ + n + g

)

e−(1−α)(δ+n+g)t₊ ^sA

δ + n + g }_1−α¹

as a function of time. If k0 is smaller than the steady state value k^∗, draw the graph of the movement of kt over time. Explain why do we draw such graph from the above equation.(2 point)

5.11 If k0 is larger than the steady state value k^∗, draw the graph of the movement of kt over time. Explain why do we draw such graph from the above equation.(2 point)

6 Growth Accounting: We are going to do a little quantitative re-

search on economic growth. Assume that the production function

is Y

= A

K

L

.

6.1 By using the methodology of growth accounting (see page 64-74 in a Japanese version text), decompose the growth rate into three part: the contribution of total factor productivity (TFP) growth; the contribution of capital stock growth; and the contribution of labor input growth. And show the table as follows. (each series 1 point)

表1: Example of table

Growth rate of real GDP contribution of TFP contribution of K contribution of L 1980–81

1981–82 1982–83 1984–85

... ^... ^... ^... ^...

2007–08

Note: In the above table, some period is omitted. Don’t omit the period (1986–2006) in your answer. (Hint:

Use this web-site to find the data set for calculaton.

http://www.esri.cao.go.jp/en/sna/h20-kaku/22annual-report-e.html For a series of real GDP, you will find in

4. Main Time Series,

(1)Gross Domestic Product(Expenditure approach) (fixed-based method). At constant prices, Calendar Year

For a series of labor input, you will find in Part 1 FLOW 5. Supporting Tables,

(3)Employed Persons, Employees and Hours Worked classified by Economic Activities, Calendar Year, Total number

For a series of real capital stock, you will find in Part 2 Stock 4. Supplementary Tables,

(1)Closing Stocks of Net Fixed Assets,

Real: Fixed-based (which is named ”Constant” in Excel sheet.), Use ”Total”.

)

6.2 According to the result of growth accounting, what do you find in the 1990’s Japan? (2 points)

6.3 According to the result of growth accounting, what do you find in the 2000’s Japan? (2 points)