ECON 222 Macroeconomic Theory I Fall Term 2011

Transcription

1 ECON 222 Macroeconomic Theory I Fall Term 2011 Assignment 1 ANSWER KEY Due: Drop Box 2nd Floor Dunning Hall by October 5th 2011 No late submissions will be accepted No group submissions will be accepted No Photocopy answers will be accepted Question 1: National Accounting is Fun (15 Marks) Use the following information to derive: 1) Private Savings Rate for both years (5 marks) Note that S govt Budget Deficit and PSR = Spvt PDI S pvt = S S govt S 2009 pvt = 145 and S 2010 pvt = 180 PDI = Y + NFP + TR+ INT T =(GNP NFP)+NFP + TR t(gnp NFP)whereT = t Y PDI 2009 = 545 and PDI 2010 = 625 PSR 2009 = % = 26.6% 545 PSR 2010 = % = 28.8% 625 2) Nominal GDP and all its components using the expenditure approach for both years (10 marks) S govt = T TR INT G G 2009 = 150 and G 2010 = 245 CA = NX + NFP NX 2009 = 125 and NX 2010 = 15 Y +NFP C G = S = I+CA C 2009 = 400 and C 2010 = 445 ; I 2009 = 185 and I 2010 = 125 Y = C + I + G + NX Y 2009 = 600 and Y 2010 = 800 Table 1: Below Variable National Savings TaxRevenue GDP Net Factor Payments Budget Deficit GNP Transfer Payments Current Account Balance Interest Payments 20-10

2 Question 2: Output and Inflation (20 Marks) U.S. Nominal GDP for the year 2010 was $14,658 billion, whereas in 2000 it was $9,951 billion. The GDP deflator for 2000 (2010) was 88 (111) using 2005 as a base year. Using the compound growth formula x t+n = x t (1 + g) n (where x t is the value of the series at time t, n is the number of years, and g is the annual growth rate), find (correct to 2 decimal places, ie. 2.75%): 1) The ten-year (You can call it gn 10 ; the index 10 is not an exponent, just a superscript to keep your solutions clear) as well the average annual growth rate (g n ) of nominal GDP between the years 2000 and 2010? (5 mark) g 10 n = % = 47.30% = 9951(1 + g n ) 10 [or you can use: (1 + g n ) 10 =1+gn 10 g n =(1+g ] g n = % = 3.95% 2) The ten-year (π 10 ) as well as the average annual inflation rate (π) as measured by the GDP deflator? n ) = 88 (1 + π) 10 π = % = 2.35% 111 = 88 (1 + π 10 ) π = % = 26.14% 3) Real GDP for both years, the ten-year (g 10 r ) and the average annual growth rate (g r) between 2000 and 2010? (5 marks) Y 2000 = = ; Y 2010 = = = (1 + g r) 10 g r = % = 1.56% g 10 r =( ) 10 1= % = 16.78% 4) Does gn 10 = π10 + gr 10? Why or why not? Explain. Express gn 10 as function of π10 and gr 10 (Again, the 10 s are superscripts, not exponents). Be precise! (5 marks) No, it doesn t. (1 + gn 10)=(1+π10 )(1 + gr 10) g10 n = π10 + gr 10 + π10 gr 10 Ignoring the last term will not significantly affect the solution if both terms are very small. However, if the growth rates are significant and if the time period is lengthened, ignoring the last term can result in significant error. Here, for instance, the last term accounts for 4.4% of nominal GDP in 2010 (approximately: $ = $643 billion) 2

3 Question 3 : Production and Technological Growth (25 marks) Assume that production (Real GDP) in the economy follows a Cobb-Douglas production function Y = µk α N 1 α where α =.25. Suppose the parameter µ measures TFP in the economy. Real GDP in 2006 totalled 900, the Capital-Labour K ratio was 16 and the real wage was 9. N a) Find the values of TFP (µ), the level of employment (N), and the capital stock (K). (10 marks) K N = 16 K = 16N w = MPN = Y N =(1 α)µkα N α =(1 α)µ(16n) α N α µ =6 Y = µk α N 1 α = µ16 α N 900 = 6 2N N = 75 K = 1200 b) In 2010, Immigration and capital accumulation resulted in an 8% growth in the capital stock over 2006 levels with no change in the capital-labour ratio. What was the average annual growth rate of real GDP during that time period? What happened to the real wage, w? Are workers better off? Explain. (5 marks) Y 2010 = 6( ).25 ( ).75 = 6( )(75.75 )1.08 = 1.08Y 2006 = = 900(1 + g avg ) 4 g avg = = % = 1.943% w 2010 =(1 α)µ(16n 1.08) α (N 1.08) α = =9=w 2006 Workers are not better off since their real wage, w, has not changed. A worker can t afford to purchase any more goods than she could afford in c) Now assume that technological progress has allowed the TFP of the economy to double over that period. What is the average annual growth rate of real GDP (g dbl ) during the years 2006 and 2010? Use the exact growth rate formula (hint: Q#2 part 4). What is the average annual growth rate of µ, g µ (correct to 2 decimal places)? What is the real wage in 2010, w 2010?Expressg dbl as a function of g partb and g µ. (10 marks) µ 2010 = 12 ; Y 2010 = 12(16N ).25 (1.08N 2006 ).75 = 2(1.08)Y 2006 = = 900(1 + gy dbl)4 gy dbl = % = 21.23% µ 2010 = µ 2006 (1 + g µ ) 4 g µ = 18.92% w 2010 =(1 α)µ 2010 ( K N )α =.75(12)16.25 = 18 = 2 w 2006 Workers can now afford twice as many goods as in 2006 given their new wage. g dbl = g partb + g µ + g partb g µ 3

4 Question 4: Mikey Needs to Borrow Money (20 Marks) Mikey cuts hair to pay for his tuition at Queen s. Unfortunately, the cost of a haircut in Kingston hasn t risen in years, and neither has his hourly wage. To make matters worse, the cover charge at his favourite club, Stages, keeps rising. He asks you to lend him $600, promising to repay you with interest at the end of one year. You try to convince him to stay home and do his work, but he doesn t waiver. You decide to charge Mikey 5% interest, the same as the fixed rate on your student loan. At the end of the year, he repays you principal plus interest in full. For the following, only consider the loan between yourself and Mikey. You may use the approximated formula from Appendix A7 (math tutorial) for parts a and b only. a) On the day the loan is repaid, you happen to notice in the newspaper that the CPI grew to 126 from 120 in the previous year. What was the real rate of return on your loan? R = i π =.05 ( 126 1) = 0% (Identical with exact formula) 120 b) Redo part (a) but now assume that the CPI grew instead to 138. R =.05 ( 138 1) =.1 100% = 10% (-8.74% with exact formula; Look at part d) 120 c) Now use the exact growth rate formula to derive an expression for the expected real interest rate as a function of only the nominal rate, i, and the expected inflation rate π e : r = r(i, π e ) 1+i =(1+π e )(1 + r) i π e = r(1 + π e ) r(i, π e )= i πe 1+π e d) Find the precise expected real rate using the CPI figures from parts (a) and (b) (correct to 2 decimal places ie. 4.75%). What would be the expected real interest rate if you never considered that prices could change during the year? Answer: r a =.05 ( ) = 0% r b =.05 ( ) 1+( ) = % = 8.7% r πe =0 = = % = 5% 4

5 Question 5: The Aggregate Production Function (20 Marks) This question focuses on labour productivity, labour demand and generally the production function. Assume that the Aggregate Production Function is represented by the following equation Y t = µk α t (H)1 α t where K is capital and H is human capital, and α is between 0 and 1. a) Derive an analytical expression for: i) the marginal product of capital. Answer: MPK = Y K = αµkα 1 H 1 α = α Y K ii) the marginal product of human capital. Answer: MPH = Y H =(1 α)µkα H α =(1 α)µ Kα H 1 α H =(1 α) Y H b) i) Find an expression for the wage for human capital, p h and return on capital, r. (Hint, wage=mpl when L is labour.) Answer: r = MPK = α Y K or αµ( H K )1 α Answer: p H = MPH =(1 α) Y H or (1 α)µ( K H )α ii) Derive an expression for total capital income and total labour income. Answer: Total Capital Income= Total Capital Rate of Return rk = α Y K K = αy Answer: Total Labour Income = Total Human Capital Wage p H K =(1 α) Y H =(1 α)y H iii) Now divide the expressions you derived in part ii) by total output. What is α in this context? Answer: rk Y = αy Y = α. p HH Answer: =1 α Y Y α is capital s share of total income in the economy while (1 α) is human capital s share of total income in the economy. Owner s of capital get αy while owner s of human capital get (1 α)y. = (1 α)y 5