第12章作業(1,3,6,7題)之補充說明與提示(Hints)

本週交第 12 章 Problems 第 1、3、6、7 題。

注意：本章作業 所有計算都算到小數點以下第 3 位。

提示(Hints)

有關第 1 題

本題皆是用第 11 章 (11.10) 式把 n 改成 h 後 (即 12.2 式 ) 很簡單而直接的計算。問題是要您觀察某個變數 (例如 y，或 y － h ) 受另一個變數 (例如 a，或 k，或 h ) 改變而影響的變動趨勢。

" Are these the results you would expect？ Why or why not？" 是指從 Solow growth model 的公式觀點，是否這些 results 是您所預期得到的。

有關第 3 題

本題計算皆是有關課本 p. 415 (12.4) 與 (12.5) 式的應用。

有關第 6 題

令 f_d 為 " the fraction of the per person capital of each type that is being diverted "，即所謂的 diversion rate。以 f_d 代入公式求解。

本題假設 country 1 與 country 2 生產函數完全相同，來算出 diversion rate (貪污率或說回扣率)要多高才會導致所得水準相差 10 倍。如此所算出的 diversion rate 會相當高。實際世界窮國的生產函數比富國落後，diversion rate 就不須要這麼高而能觀察到所得水準低於美國 10 分之 1。

有關第 7 題

本題要分別 growth rate of labor productivity ( y － h ) 與 growth rate of output per capita ( y － n ) 的不同。參考課本 pp. 413-414 的討論。

Chapter 12: 解答 ( 1, 3, 6, 7 題 )

1.  (a)    y changes from 3.75 to 3.25 to 2.75 to 2.25 and y - h changes from 2.75 to 2.25 to 1.75 to 1.25 as a declines.

(b)  y changes from 3.75 to 3.625 to 3.5 to 3.375 and y - h changes from 2.75 to 2.625 to 2.5 to 2.375 as k declines.

(c)  y changes from 3.75 to 4.125 to 4.5 to 4.875 and y - h changes from 2.75 to 2.625 to 2.5 to 2.375 as h increases.

These calculations show that output growth and labor productivity growth both decline as MFP growth and capital input growth declines. In (a), the declines in output and labor productivity growth match the declines in MFP growth. In (b), the declines in output and labor productivity growth are less than the declines in capital input growth because the elasticity of output to capital (b) is less than 1. As labor input growth increases, output growth also increase but labor productivity growth declines because both MFP and capital input growth are held constant. These results should be expected based on the Solow growth formula.

3.  (a)  For labor’s share of income to remain constant, (W/P) must increase at the same rate that (Y/N) does (see equations (12.4), (12.5) and (12.6) on p. 415). Since (Y/N) is rising at 2% per year, the nominal wage (W) must rise at 4% per year if (W/P) is to increase at 2% per year, given that the GDP deflator (P) rises by 2% per year.

(b)  Since labor productivity growth (y - n) has declined to 1.2% per year, labor share’s of income rises since the real wage is rising (w - p = 2%) faster than labor productivity.

      For labor’s share of income to remain constant, (W/P) must increase by 1.2% per year, the same rate that labor productivity does. Therefore W must rise at 3.2% per year if (W/P) is to increase at 1.2% per year, given that P continues to rise by 2% per year.

(c)  Since labor productivity growth has increased to 3% per year, labor share’s of income falls since labor productivity is rising faster than the real wage.

      For labor’s share of income to remain constant, the real wage must increase by 3% per year, the same rate that labor productivity does. Therefore W must rise at 5% per year if the real wage is to increase at 3% per year, given that P continues to rise by 2% per year.

6.  Let Y₁ be the level of output in country 1 and let Y₂ be the level of output in country 2. Furthermore, let f_d be the fraction of capital (of all types) in Country 2 that is diverted. Then,

 and 

        Hence,

        Therefore, , from which it follows that  and .

7.  (a)  The labor productivity growth rate in Country 1 equals 3 + 0.25(3.4 – 1) = 3.6% per year. The growth rate of output in Country 1 equals 3.6 + 1 = 4.6% per year. Therefore the growth rate of output per capita in Country 1 equals 4.6 – 1 = 3.6% per year.

(b)  To have the same rate of growth of output per capita as Country 1, 3.6% per year, Country 2’s output must grow by 5.6% per year, since its population growth rate equals 2% per year. For output to grow by 5.6% per year, labor productivity must grow by 4.6% per year, given that labor is growing by 1% per year. Finally, for labor productivity to grow by 4.6% per year, multifactor productivity must grow by 4.6 – 0.25(0.38 – 1) = 3.9% per year.