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Name____________________ Student ID____________________ Mid-Term Exam # 1 Economics 514 Macroeconomic Analysis Thursday, October 12th , 2006 Write your answers on this white exam. Do not write your answers on the blue book! Do not hand in your blue book! 1. Marginal Rate of Substitution A worker has no capital income and, thus, has a budget line given by Ct wt Lt where w is real wages, C is consumption, and L is labor. The worker can use their time for labor or leisure, ls: TIME lst Lt . The workers preferences are given by the utility function of 1 1 U t 2 Ct 2 2lst 2 Normalize the time period to be TIME = 1 and set the real wage w = 25. Calculate the marginal rate of substitution between consumption and leisure when the worker chooses his optimal consumption and labor. [Hint marginal utility of 1 1 consumption is MUC = Ct 2 and marginal utility of leisure, MUls = 2lst 2 ]. MRS = w = 25 2. Forecasts of economic growth. Country A and Country B have the same level of Y labor productivity, y0 = 36, and capital productivity, 0 = 1. ), a population K0 growth rate of 2% (n = .02), and a depreciation rate of 6% (δ = .06). Country A has an investment rate of 20% (s = .2). Country B has an investment rate of 10% (s = .1). Use the following table to answer your questions: x x^20 x x^20 1.01 1.22 1.07 3.87 1.02 1.49 1.08 4.66 1.03 1.81 1.09 5.60 1 1.04 2.19 1.1 6.73 1.05 2.65 1.11 8.06 1.06 3.21 1.12 9.65 a. Country A and B have Cobb-Douglas production functions of the form. Yt K t ( At Lt )1 1 3 where both share the same technology level, At. Assume a technology growth of 2% (i.e. gA = .02). Solve for the level of technology at time 0, A0. Assume that 20 years from now, both countries are on their balanced growth path. Calculate the level of labor productivity in both countries. What is the ratio of labor productivity in country A relative to country B? 1 Y Y Y 1 Y At1 t t At t t A0 36 Kt Lt Kt Lt Technology level is A20 1.49*36 53.64 1 Yt Yt 1 Yt 2 yt At At Lt Kt Kt Steady state capital productivity is Y K SS n g A .1 s s For country A, this is ½ so labor productivity at the balanced growth path is Y ytBGP t Kt 1 2 At 2 At For country B, steady state capital productivity is 1, so y20 2 53.64 75.86 on the balanced growth path, the level of productivity is 53.64. The ratio is the square root of 2. 2 Assume Country A and B both have an AK production function: Yt AK t where both share the same constant technology level, A. Calculate the level of technology, A. Calculate the rate of growth of labor productivity for country A and country B. What is the level of labor productivity in these two economies in 20 years? What is the ratio of labor productivity in country A relative to country B? Yt AKt A Yt 1, yt kt Kt gtk gty s A (n ) s .08 For country A, the growth rate of labor productivity is constant at 12%, after 20 years it will increase to 9.65*36 = 347.4. . For country B, the growth rate is 2%, and so labor productivity is 53.64. The ratio is 6.48. 3 Changing Demographics Country F has a production function of the form: Yt K t ( At Lt )1 1 3 And accumulates capital according to the following function Kt+1 = (1-δ) Kt + It , the I investment rate is constant s t while the labor force growth rate and technology Yt growth rate are constant n0 and gA.. The economy reaches a steady state capital productivity level. Then, at time T, due to a change in fertility rates, population growth drops permanently to a new level n1 < n0. a. Draw a picture that shows labor productivity growth as a function of capital productivity with the old and the new population growth rate. Draw arrows on the graph to indicate the transition to the new steady state NEW Y g y s (n ) (1 ) g A K OLD gA [Y/K]SS Y/K [Y/K]SS b. Explain, in one paragraph or less what happens to labor productivity growth after population growth changes. As population and labor force grows, less investment needs to be devoted to equipping new workers and more can be devoted toward giving each worker more capital to work with. Capital will now grow more quickly than the previous rate which was the growth rate of technology. As capital deepens, diminishing returns will insure that output growth is slower than capital growth, reducing capital productivity over time and decelerating output growth. Soon, the level of capital productivity will reach a new lower steady state level where capital growth and technology growth are equal. 4 Draw a picture of the path of capital productivity over time. Y/K time c. Draw a picture of the path of labor productivity over time y time 5 3. Use the first order conditions of the profit maximization problems of firms to derive a labor demand curve and the first order conditions of the utility maximization problems to derive a labor supply curve. You will then put these together to get an equilibrium. a. The production function of an economy is Cobb-Douglas. 1 1 Yt Kt 2 Lt 2 Real profits are given by 1 1 2 2 PK t t Lt Wt Lt Rt Kt W R Taking the real wage, w = , t and the real capital rental price, rt, t as Pt Pt given solve for the profit maximizing demand for capital and labor. If the supply of capital is elastic at a rate of r = .15, what is the profit-maximizing capital labor ratio. According to the first order condition for optimal labor demand, what would be the wage rate (defined as w*) at that capital-labor ratio. The marginal product of capital is 1 2 MPKt 1 2 kt 2 .15 kt 1/ .3 100 9 MPLt 1 2 kt 2 5 / 3 w * 1 6 b. Workers preferences are given by a log/log utility function. ln Ct 2ln lst where Ct is consumption and lst is leisure. Assume that worker’s have 1 day available for work or leisure, Lt + lst = 1. Assume that non-labor income is equal to 1, so the workers budget constraint is Ct wt (1 lst ) 1 3 Solve for the first order condition that shows optimal labor supply as a function of the real wage. Solve for the optimal amount of labor (defined as L*) that you would have at the real wage that you solved for in part a, w*. How much consumption could be purchased when workers work L* hours at wage rate w*? What would be the optimal amount of capital hired by firms if labor was L*? Ct 5 3 (1 lst ) 1 3 5 3 1 Ct 2 lst 5 Ct lst 6 5 lst 5 3 (1 lst ) 1 2 5 3 lst 6 12 5 10 lst 2 lst .8 6 6 6 L .2 k K 100 K 100 * L 100 *.2 20 L 9 9 9 9 7