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Midterm Exam Economics 514 Macroeconomics Analysis October 11, 2004 Short Questions (10 points each) 1. You measure the real wage rate in an economy as w = $25 per hour. The consumption of an average household as being equal to C = 2400 per month. The household is endowed with TIME = 24*30 = 720 hours in an average month and is observed to spend L = 240 hours in labor. Assume that the household has a utility function of U = ln Ct + ½ ln lst. Assume that the household chooses its optimal, utility-maximizing combination of consumption and leisure. Calculate the marginal rate of substitution between leisure and consumption. The marginal rate of substitution can be derived as the ratio of the marginal utility of 1 1 1 MU ls 2 ls ls C 2400 MRS 12 12 leisure to the marginal utility of consumption = 1 MU C ls 480 C 2.5. Also at utility maximization, the slope of the indifference curves (-MRS) equals the slope of the budget constraint (-w), so the MRS = w = 25. Whoops. Either answer was accepted for full credit. 2. You compare the price levels of several economies by measuring the price level in each country of a common market basket of goods. To save time, you assume that the typical market basket of goods includes only one good: Big Mac sandwiches from McDonalds. Using the table below, convert the GDP per capita of China and Japan into US dollars using the exchange rate method and the PPP method. Calculate the real exchange rate with Japan and the US treating China as the home country. Country USA Japan China GDP per Capita US$49735.14 ¥3899026 3. US$2.9 ¥261.87 RMB10.411 RMB9073 XP USA Japan China Big Mac Price X Rate PPP S DCU per $ 111 8.28 s 2.306407 90.3 35126.36 43178.58 111 1.876293 3.59 1095.773 2527.298 8.28 Assume that money demand in a home and foreign economy is given by M .2 Q i . Assume that at a particular point in time, the two economies have P equal supplies of money (i.e. M = MF) and equal levels of real output (Q = QF) and in each economy the real interest rate is equal to the growth rate of output. In the home economy, the money growth rate is 4% and in the foreign economy the money growth rate is 16%. Calculate the spot/nominal exchange rate if absolute PPP holds. Absolute PPP implies SPF = P S and M = MF and Q = QF . S P PF gM F g M 5 5 M Q i MF M QF MF Q iF . With r = gQ , i=gM i QF iF MF g gM .16 .04 4 2 Note, I also accept .5 as an answer. 4. We measure an economy in which the output per labor hour, Q = qt = $30, and L W = wt = $20. We also measure that the ratio of GDP P to capital in the economy is equal to 1. Calculate TFP in this economy i. GDP is given by Q K (ZL)1 Labor share of income is WL WP 1 Q 2 13 . The level of TFP can be derived as 3 PQ L real wages are measured as TFP Z 5. 1 1 q Q 2 1 2 30 3 1 3 30 3 9.655 K We find that the price level in a foreign economy in 1980 was equal to 1 and equal to 1.75 in year 2000. We find that the price level in the domestic economy was equal to 1 in year 1980 and equal to 1.5 in year 2000. The exchange rate between the domestic economy equal to 1 in year 1980 and was equal to .75 in year 2000. Calculate the average inflation rate in the domestic economy and the foreign economy over the 20 year period 1981-2000. Estimate the average depreciation rate of the domestic currency. Estimate the average real depreciation rate (i.e. the growth rate of the real exchange rate). [Hint: Use the logarithm tables]. The average inflation rate over the period is the average growth of prices between 1981 1 and 2000 which can be derived as 81:00 [ln P2000 ln P1980 ] 0.020273255 . Similarly 20 1 F F F [ln P2000 ln P1980 ] 0.027980789 . The depreciation rate is foreign inflation is 81:00 20 1 S [ln S2000 ln S1980 ] -0.014384104 . The the growth rate of the exchange rate is g81:00 20 s S F g g real depreciation rate is 81:00 81:00 81:00 81:00 -0.00667657 . X 0.25 0.5 0.75 1 1.25 1.5 1.75 2 ln(X) -1.386294 -0.693147 -0.287682 0 0.223144 0.405465 0.559616 0.693147 -0.069315 -0.034657 -0.014384 0 0.011157 0.020273 -0.006677 0.027981 0.027981 0.034657 Long Answer Questions 6. (20 points) Households in an economy get utility as a linear function from consumption and leisure. U Ct 1 lst 2 Households in the economy are either unemployed in which case, they enjoy ls = 672 hours of leisure, or employed in which case they enjoy ls = 472 hours of leisure and must work 200 hours. If they are unemployed they receive a benefit level of C = b = 900. If they are employed, they get a consumption level of C = w ∙ 200. a. Calculate the reservation wage (i.e. the wage offer that the household would have to receive to give them a utility equal to the utility of an unemployed person. The reservation wage that sets utility of an unemployed person equal to the utility of an employed person solves b 1 2 672 900 336 1236 U u U e (200 w) 1 2 472 (200 w) 236 (200 w) 236 1236 w 5 Wage offers are uniformly distributed over a range of [0, 20]. This means that for any reservation wage level, ŵ , the probability that a job applicant would get a 20 w wage offer higher than ŵ is . 20 b. What is the probability that a worker will receive a wage offer better than the reservation wage from 6a. 20 5 .75 The probability of finding a job that offers a wage greater that 5 is 20 c. If the probability that someone in the workforce has some event that would cause them to leave their job if they have one is 5% (i.e. s = .05) and the probability that someone searching for a job will receive a job offer is 100% (i.e. p = 1), what is the steady-state unemployment rate. s .05 The steady state unemployment rate is ur SS .05 .0625 s pH ( w) .05 .75 .80 7. (20 points) In an economy, there is a fixed supply of labor (which we will normalize to 1), Lt = 1, so the growth rate of the labor supply is n = 0. The household can divide the labor supply between producing goods or to doing R & D. The amount of time spent in producing goods is equal to u ∙ L = u. The amount of time spent in R & D is (1-u) ∙ L = (1-u). The amount of goods produced are given by a Cobb-Douglas production function where Kt is the capital stock and Zt is the technology produced. 1 1 1 1 Qt Kt 2 (Zt Lt ) 2 Kt 2 Zt 2 The amount of new technology produced is given by Z 2 Z t (1 u ) Lt 2 Z t (1 u ) If the share of time spent in R & D is 2% (i.e. u = .98) what is the long-run growth rate of labor productivity when the economy is on its balanced growth path. Assume that the investment rate, s = .2. Calculate the capital deepening rate of capital and the growth rate of labor productivity when capital productivity is 2. What is the steady-state capital productivity level when the economy is on its balanced growth path? Z The growth rate of technology is 2 .02 .04 . The growth rate of the capital labor Z ratio is equal to the growth rate of capital K I Q Q K (1 ) K I s .2 .08 . Under Cobb-Douglas, the K K K K growth rate of output, is a weighted average of the growth rate of capital and technology (when labor is fixed). Q gtQ 12 gtK 12 gtz 12 [.2 .08] 12 .04 K When capital productivity is 2, the growth rate of output is .18. The steady state capital productivity sets the growth rate of capital equal to the growth rate of technology so the Q steady state capital productivity is .6 . K (10 points) Assume that the money demand is as described as by the BaumolTobin model. On a graph with P on the vertical axis and Q on the horizontal axis, draw the relationship between P and Q taking M and i as given. Assume that Q is fixed (draw this as vertical level of supply that is not affected by the price level). Where the two curves cross is the equilibrium price level. Show in two graphs, how an increase in M or an increase in I would affect the equilibrium price level. P M↑ PEQ PEQ Q P i↑ PEQ PEQ Q