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Homework 2 Answer Key March 18, 2005 1. Compare two countries. Each country has a population growth rate of 2% (i.e. n = .02) and a depreciation rate of 10% (i.e. d = .10). Both countries have the same CobbDouglas production function Yt = Kta(QLt)1-a with constant technology level Qt = 1. Country A has an investment rate of s = .3 while Country B has an investment rate of .18. Calculate the ratio of steady state labor productivity in Country A to steady state labor productivity in Country B, when a = ⅓, ⅔, ½, and ⅞. If capital intensity is strong enough, can differences in capital fundamentals explain differences in labor productivity seen in different areas of the world? Steady state occurs when gross investment per labor unit equals replacement investment needs per labor unit a 1 a K a L1 a s (n d )k SS s q s s k SS k SS L nd a 1 s 1 a s 1 a SS k q nd nd When we want to compare the steady-state productivity across countries, keeping in mind investment rates differ but depreciation and population growth are the same. SS q Ass q ss B sA nd a 1 a s A sB a a 1 a .3 .18 a 1 a 5 3 a 1 a sB 1 a nd This is a function of the level of capital intensity. See the following table. a 0.333333 0.5 0.666667 0.875 a 1 a 0.5 1 2 7 5 3 a 1 a 1.290994 1.666667 2.777778 35.72245 At a realistic capital intensity a = ⅓, the gap in capital fundamentals only explains a 30% difference in productivity levels. But if capital is very important in production, capital fundamentals will have a bigger impact on steady state productivity. In particular, the higher is a, the more slowly capital returns will diminish as the capital stock is added. When a is high, a country with a higher investment rate will be able to support a much higher level of investment and have higher steady state capital stock. When a is high enough, near a = ⅞, differences in capital fundamentals could explain differences in productivity of a factor of more than 35. 2. An economy is closed so that output is either used for household consumption, Ct ; capital investment, It; or government spending, Gt. The government collects a constant share of output for its spending on output, Gt = τYt. The household consumes a constant fraction of its after-tax income, Ct = c(1-τ)Yt, and invests the rest, It = (1-c)(1-τ)Yt. Assume zero population growth, n = 0, and a realistic depreciation rate of 8%, d = .08. The average rate of household consumption as a share of GDP not spent by government over the last 10 years in Hong Kong is 66.4% , so set c = ⅔. The average of government spending as a share of GDP in Hong Kong over the last 10 years is 9% . a. Assume that production in country H is given by the Cobb-Douglas function Yt = Kt⅓ (Lt) ⅔ with a = ⅓. Calculate steady state labor productivity levels when government spending share is τ = .09. Calculate steady state labor productivity levels when government spending share is τ = .18. With fixed technology, steady state is achieved when gross investment equals replacement investment. When n = 0, all replacement investment goes toward replacing depreciation. 1 a s s 1a s 1a so that productivity is dk s k k k SS q SS d d d an increasing function of the investment rate. In this economy the investment rate is a function of the taxes taken by government. I 1 s (1 )(1 c) . We see that Y 3 SS SS a SS a 1 a 1 1 .5 1 1 1 3 1 SS 3 3 q d .08 .24 The greater is the tax rate, the lower is long-term productivity 1 a 3 qSS 0.09 1.94722 0.18 1.848423 b. Now assume a production function with nondiminishing returns to capital, Yt = Kt. Calculate the rate of growth of output when τ = .09 and is τ = .18. The productivity of capital is constant Y 1 . The growth rate of output is equal to K the growth rate of capital. Capital accumulation is done through investment K t 1 (1 d ) K t I t K t 1 K t I t dK t gtK1 K t 1 K t I I Y 1 t d t t d sd .08 Kt Kt Yt K t 3 1 .24 1 ( .24) 3 3 3 The growth rate of output and capital is a negative function of the tax rate. gtK1 gY 0.09 0.223333 0.18 0.193333 3. Examine an economy in which the production function is quadratic so 1 Yt a Lt Lt 2 which implies a linear marginal product of labor curve, 2 Y MPL L a Lt . In this economy, a. Assume that a = 100. What is the demand for labor in this economy if the market real wage W = 10. P The demand for labor is W a LD LD a W 100 W . With P P P W = 10, LD = 90. P b. Calculate the equilibrium real wage in this economy when a = 100 and the labor supply is unaffected by the real wage and is fixed at LS = 100.if the labor market is perfectly competitive and firms have no labor management costs. Labor supply can only equal labor demand when W = 0. P S D 100 L L 100 W 0 W W 0 P P P Now assume that firms must pay the labor management costs per labor unit so E2 W that are a linear function of the real wage c( ) P W P c. The marginal cost of hiring an extra unit of labor is now, 2 W c(W ) W E . Calculate the demand for labor when E = 10, P P P W P a = 100 and the real wage rate is W = 10. Calculate the demand for P labor when E = 10 and the real wage rate is W = 20. P 102 100 100 = 10, the management costs are c(W ) 10 . P P W W 10 P P Total labor costs per labor unit W c(W ) 20 . The firm will set the P P marginal product of labor equal to the extra cost of labor LD 100 W c(W ) 100 10 10 80 . When the wage rate is W = P P P 20, wage costs are higher, but management costs are lower. The marginal 100 cost of labor is W c(W ) W 25 . The demand for labor is P P P W P D L 100 W c(W ) 100 20 5 75 . P P When W Given the cost function, so that the marginal benefit of increasing real wages is c E2 per labor unit while the marginal cost per labor unit of raising 2 W W P P the real wage is 1. d. If the efficiency wage minimizes total costs of hiring a worker by equalizing the marginal benefit of increasing the wage level with the extra cost, solve for the efficiency wage if E = 10. Calculate the demand for labor at the efficiency wage. If the labor supply is LS = 100, what is the level of unemployment (i.e. the difference between labor demand at the efficiency wage and labor supply). What is the unemployment rate (i.e. the level of unemployment divided by the labor supply)? The cost minimizing wage sets the marginal benefit of raising wages to the margical cost, E2 1. 1 we E 10 . We know the management costs when W =10 is 20 and the 2 P e w demand for labor is 80. When the employment is 80 and the labor supply is 100, the unemployment is 20. The unemployment rate is 20%; ur = 20/100 = .2. e. Now assume that a negative technology shock reduces the productivity of labor and decreases a from a = 100 to a = 90. If the efficiency wage rate is unchanged, what is the new demand for labor and what is the new unemployment rate? The efficiency wage is unchanged at we = 10 implying total marginal labor costs of 20. The demand for labor is LD 90 W c(W ) 90 10 10 70 . Labor demand is 70, P P unemployment is 30, and the unemployment rate is 30%. 4. Internet Question Reported below are series for Current Dollar Gross Domestic Product and Population for 10 East Asian Economies. Use the PPP and X-Rate conversion factors for 2002 calculated by the World Bank (to be obtained from http://siteresources.worldbank.org/ICPINT/Resources/Table5_7.pdf) to convert the GDP of one of these into US dollars. Choose the country whose number corresponds with the last digit of your student ID. For comparison, divide by the population to get US dollar GDP per capita for both PPP converted GDP and XRate converted GDP. As an example, I have already completed the results for China. Last Digit Student Country ID Number Name Dave China 0 Bangladesh 1 Hong Kong 2 India 3 Indonesia 4 Japan 5 Korea 6 Malaysia 7 Philippines 8 Singapore 9 Thailand Currency Unit Millions of Renminbi Taka HK Dollar Rupee Rupiah Yen Won Ringgit Peso Sing Dollar Baht Current Price GDP 10517230 3005801 1247381 24633240 1863274700 497897000 684263500 361624 3959648 158064 5446043 Population (Millions) 1284.53 133.4 6.787 1055 211.06 127.377 47.615 24.526 79.503 4.185 63.437 Example Dave China xp 1.8 e 8.28 US Dollars PPP GDP XRATE GDP $5,842,905.56 $1,270,196.86 US Dollars PPP GDP XRATE GDP per Capita per Capita $4,548.67 $988.84 US Dollar GDP per US Dollar GDP Capita e xp xrate PPP xrate PPP 0 Bangladesh 57.89 11.9 $51,922.63 $252,588.32 $389.23 $1,893.47 1 Hong Kong 7.8 6.9 $159,920.64 $180,779.86 $23,562.79 $26,636.19 2 India 48.61 8.8 $506,752.52 $2,799,231.82 $480.33 $2,653.30 3 Indonesia 9311.19 2357.7 $200,111.34 $790,293.38 $948.13 $3,744.40 4 Japan 125.39 146.2 $3,970,787.14 $3,405,588.24 $31,173.50 $26,736.29 5 Korea 738.7 1251.09 $926,307.70 $546,933.87 $19,454.12 $11,486.59 6 Malaysia 3.8 1.6 $95,164.21 $226,015.00 $3,880.14 $9,215.32 7 Philippines 51.6 12.1 $76,737.36 $327,243.64 $965.21 $4,116.12 8 Singapore 1.79 1.6 $88,303.91 $98,790.00 $21,100.10 $23,605.73 9 Thailand 42.96 12.6 $126,770.09 $432,225.63 $1,998.36 $6,813.46