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Name________________ Student ID ________________ Economics 514 Macroeconomic Analysis Midterm Exam 2 November 15, 2007 Write all answers on the white exam paper. Do not turn in the blue books. 20 points each. 1. Interest Rates and Saving There are three households each of who live for two periods (period 0 and period 1). Each household begins the period with zero financial wealth and has a utility of the form ln(C0 ) ln(C1 ) . Note that this utility function implies the household is patient and has a subjective discount factor β = 1. Each household can save or borrow across time to maximize utility and faces a real interest rate of zero, r = 0, 1+r = 1. The only difference comes in terms of the time path of income Y0 and Y1 which is given in the following table Household Y0 Y1 i) 0 100 ii) 50 50 iii) 100 0 a. Calculate the present value of lifetime income for each household. Present value of lifetime income is hw W Y0 Y1 100 for all households 1 r b. Calculate consumption, C0, and savings, S0 = Y0 – C0 for each household. Maximize utility, ln(C0 ) ln(C1 ) subject to hw C0 u '(C0 ) (1 r ) u '(C1 ) C1 the first order condition is 1 r 1 1 (1 r ) (1 r ) 1 C0 C1 C0 C1 Household C0 S0 i) 50 -50 ii) 50 0 iii) 50 50 1 c. Assume that the real interest rate goes up by 1% to (1+r) = 1.01. By what % will C the ratio 0 change (i.e. what is the intertemporal elasticity of substitution) for C1 each household? C ln 0 C C 1 C1 0 ln 0 ln(1 r ) 1 ln(1 r ) (1 r ) C1 C1 d. What will be consumption and saving after the interest rate rise for each household If r = .01, then the first order conditions suggests C1 = (1+r)C0 so (1 r )C0 hw C0 C0 C0 . This means that regardless of the interest rate, 1 r households consume half of human wealth. Calculate the human wealth under the different interest rates. Household Hw C0 S0 i) 99.0099 49.50495 -49.505 ii) 99.50495 49.75248 0.247525 iii) 100 50 50 2 e. Explain how the response of saving by household iii) to the rise in the interest rate is qualitatively different than the response of households i) and ii). Explain why. An interest rate rise has two effects on consumption. The higher relative price of current consumption causes a shift of the consumption pattern toward the future. This substitution effect tends to reduce current consumption. Second, the higher interest rate, will change income that a household earns on its savings or must pay on its borrowing. This income effect will give more funds to consume for savers and fewer funds to consume for borrowers. For household 1, the income effect and the substitution effect both work to reduce consumption and increase saving. For household 2, there is no income effect and the substitution effect reduces consumption and increases saving. For household 3, the substitution effect cancels out the substitution effect. 3 2. Retirement and Savings In our economy, households will begin period 0 of their life with no financial wealth and live for 61 years until time T = 60. The household will work for 41 years and then retire. The household has an income stream equal to Yt = 100 from period 0 to period N = 40 and Yt = 0 from period 41 to period 60. Assume the real interest rate is constant and the Permanent Income Hypothesis is true so the household borrows or saves to choose a constant consumption level C from time 0 to time 60. a. Calculate the (human) wealth of the household at time 0. Calculate the consumption, C0, and savings, S0 = Y0 – C0, of the household in period 0. The human wealth of the household is Y3 Y Y2 Y41 hw Y0 1 ... 2 3 1 r (1 r ) (1 r ) (1 r ) 41 1 100 100 100 100 (1 r ) 41 100 ... 100 1 1 r (1 r ) 2 (1 r )3 (1 r ) 41 1 1 r The present value of consumption is equal to human wealth. C3 C61 C C2 hw C0 1 ... 2 3 1 r (1 r ) (1 r ) (1 r )61 1 1 C C C C (1 r )61 C ... C 1 1 r (1 r ) 2 (1 r )3 (1 r )61 1 1 r 1 1 1 1 1 1 41 41 (1 r ) (1 r ) (1 r )61 1 r hw C 100 S 100 C 1 1 1 1 1 1 61 61 (1 r ) (1 r ) (1 r )61 1 4 b. Assume that there are a number of households equal to POP so that consumption is C = C *POP. The GDP per capita of the economy is equal to the output of a household that is working (i.e. Y = 100) multiplied by the fraction of the GDP C population that is not retired. Calculate the savings rate of this GDP economy if 75% of the households are employed and 25% are retired. Calculate the savings rate of the household if 50% of the households are employed and 50% are retired. Define f as the share of households who are working. 1 GDP C 100 f POP C POP C 1 (1 r ) 41 1 1 GDP 100 f POP 100 f f 1 1 (1 r )61 The savings rate is lower when retirees are a larger share of the population 1 5 c. If the household is to continue consuming this level, C after retirement they will need to have acquired financial wealth. From the standpoint of period 41, how much wealth will they need to continue to consume C for the remaining 20 years of their life, if they have no income during those years. W C41 C43 C60 C42 C44 ... 2 3 1 r (1 r ) (1 r ) (1 r )19 1 1 1 20 C C C C (1 r ) (1 r ) 41 C ... 100 1 1 1 r (1 r ) 2 (1 r )3 (1 r )19 1 1 1 r (1 r )61 1 6 3. Investment Prices and the Real Wage Rate According to the Penn-World tables, in China, the average price of investment goods relative to the price of output goods in year 2000 was pI = 1.53. The average price of investment goods relative to in the USA in the same year was pI = .85. Assume that the real interest rate was 4% and the depreciation rate was 12% and expected inflation in output goods prices equaled the expected inflation in investment goods prices. What would be the real capital rental rate for capital invested in year 2000? What would be the marginal product of capital in year 2001 if firms chose a profit maximizing level of investment in 2000? If both countries had a Cobb-Douglas production function with equal technology levels, A = 1, and equal capital intensity functions α = 1/3, what would be the capital labor ratio in each country in 2001? What would be the real wage rate in each country? Rt 1 1 tI1 I 1 rt (1 ) pt . If we have r =.04 and δ Pt 1 1 t 1 R =1, then we have t 1 rt ptI .16 ptI . The marginal product Pt 1 The capital rental price is =.12 and 1 tI1 1 t 1 1 L Y .16 p MPK t 1 t 1 t 1 K t 1 K t 1 I t of capital should be set so that 2 L 3 t 1 K t 1 1 3 32 K t 1 .48 ptI Lt 1 . The real wage rate equals the marginal product of labor which is proportional to labor productivity K Wt 1 Y MPLt 1 (1 ) t 1 1 t 1 Pt 1 Lt 1 Lt 1 2 3 .48 p I t pI USA China 1 2 3 .48 p I t 32 2 3 1 .72 ptI Rt 1 Pt 1 K t 1 Lt 1 0.85 1.53 0.136 0.2448 7 Wt 1 Pt 1 3.837159 1.588916 1.043707 0.777933 4. Structural Employment In an economy with a labor force of 1 million people, the average fraction of unemployed people who find jobs is 50%. The average fraction of employed people who lose their jobs is 2%. Solve for the number of unemployed people in steady state. s .02 1 ur SS s f .5 .02 26 38461.54 8 5. Uncertainty and Saving A household lives for two periods, 0 and 1, has zero financial wealth, and earns income Y0 = 100 in the first period. The household maximizes a utility function U(C0, C1) = ln(C0)+ln(C1). The real interest rate is zero, r = 0. a. If output is, with certainty, equal to Y1 = 100, what would be consumption and savings in period 0. C0 = 100, S0 = 0 9 b. Assume that income in period 1 is uncertain and will be Y1,GOOD = 150 with a 50% probability or Y1,BAD = 50 with a 50% probability. Calculate the expected marginal utility of consumption in period 1, if the savings that the household does is equal to that solved for in part a. Is this greater than or less than the marginal utility in period 0, if the savings were equal to that solved for in part a. If output in the second period were uncertain, would this household have positive savings? Explain your answer. Marginal utility of consumption in time 0 is u '(C1 ) utility is E[u '(C1 )] .5 1 .5 1 1 Expected marginal 100 . IF S = 0, then this is Y1,GOOD S Y1, BAD S 1 1 1 E[u '(C1 )] .5 .5 . The expected marginal utility of consumption 150 50 75 is higher in the future. The household should maximize expected utility by shifting consumption through the future by saving. 10 c. [Warning: Save this for last] Solve for a level of savings which would set the marginal utility of consumption in period 0 equal to the expected marginal utility of consumption in period 1. Reminder: If ax2 + bx + c = 0 then E[u '(C1 )] .5 Y Y 1, BAD 1 Y1,GOOD S .5 S Y1,GOOD S 1,GOOD S Y1, BAD S 1 Y1, BAD S 1 u '(C0 ) Y0 S 2 Y0 S Y1, BAD Y1,GOOD S 2 Y0 S 1 Y0 S Y1,GOOD S Y1, BAD S Y1,GOOD S Y1, BAD S Y1,GOOD S Y1, BAD S Y0 S Y0 S 2 S 2 Y1,GOOD Y1, BAD S (Y1,GOOD Y1, BAD Y0 2 ) 0 2 S 2 200 S 2500 S 100 S 1250 0 S 100 10000 5000 50 25 6 2 S=11.23724 11