Problem Set 3 Mr. Nordhaus and Staff Due: At the beginning of class,
Wednesday, October 3rd
Economics 122a. Fall 2012. Problem Set 3 Please, write your name, the name of your Teaching Fellow and section day and time in each sheet of paper you hand in. Recall that problem sets are graded on a random basis. This problem set will be graded if the last two digits of the winning number of CT Lottery “3Play Day” of October 4 are between 00 and 41. If it is between 42 and 99, it is “ungraded.” Ungraded means that we will look it over to make sure you did the problems adequately, but they will not be corrected. You can check the results of the draw in http://www.ctlottery.org/modules/games/default.aspx?id=3&winners=1. Please read the rules on “Rules on problem set joint work” in the folder “Problems.” Write the names of any students you have worked with at the end of your problem set. 1. Consider the life‐cycle model discussed in class. A 22‐year‐old Yale graduate named Carol has just accepted an analyst position at a solvent investment bank. She knows her income will be $80,000 per year (in 2012 dollars) until she is promoted to managing director, at age 32. As a managing director, she will earn $300,000 per year until she resigns at age 47 to pursue her life‐long dream of being a composer. At this point, she writes music and earns $50,000 per year until she retires at the age of 62. Carol expects to live until her 82th birthday. The real interest and discount rates are zero. a. Assume Carol can borrow and lend without limit but has zero net worth at graduation and death. Compute consumption expenditures for each year. Lifetime Income 10 years at $80,000 $ 800,000 15 years at $300,000 $ 4,500,000 15 years at $50,000 $ 750,000 20 years at $0 $ ‐ Total Income $ 6,050,000 Consumption per year: C = $6,050,000/(82‐22) = $100,833.33 b. Graph income, consumption, and savings as a function of age. Due: At the beginning of class,
Wednesday, October 3rd
Problem Set 3 Mr. Nordhaus and Staff 350000
300000
250000
200000
150000
Income
100000
Consumption
50000
Savings
0
‐50000 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81
‐100000
‐150000
Carol's Age
c. What is Carol’s marginal propensity to consume for the year if her great‐uncle Milton passes away and she receives an unanticipated one‐time windfall of $100,000 when she is 32. MPC = 1/(82‐32) = 1/50 = 0.02 d. Repeat (a) through (c) assuming that Carol cannot borrow. If Carol cannot borrow she will consume all of her income until she is 32, and after that she will smooth consumption. Consumption per year until Carol is 32: C = $80,000 Consumption per year after Carol is 32: C = (Total remaining income)/(82‐32) =($5,250,000)/50 = $105,000 350000
300000
250000
200000
150000
Income
100000
Consumption
50000
Savings
0
‐50000 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81
‐100000
‐150000
Carol's Age
MPC = 1/(82‐32) = 1/50 = 0.02 e. Assume that the population distribution is stationary (equal numbers at each age). Calculate the nation’s wealth‐income ratio and the net savings rate. Age Income Consumption Savings Wealth Problem Set 3 Mr. Nordhaus and Staff 22 80,000 23 80,000 24 80,000 25 80,000 26 80,000 27 80,000 28 80,000 29 80,000 30 80,000 31 80,000 32 300,000 33 300,000 34 300,000 35 300,000 36 300,000 37 300,000 38 300,000 39 300,000 40 300,000 41 300,000 42 300,000 43 300,000 44 300,000 45 300,000 46 300,000 47 50,000 48 50,000 49 50,000 50 50,000 51 50,000 52 50,000 53 50,000 54 50,000 55 50,000 56 50,000 57 50,000 58 50,000 59 50,000 60 50,000 61 50,000 62 ‐ 63 ‐ 64 ‐ Due: At the beginning of class,
Wednesday, October 3rd
100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 100,833 ‐ 20,833 ‐ 20,833 ‐ 20,833 ‐ 20,833 ‐ 20,833 ‐ 20,833 ‐ 20,833 ‐ 20,833 ‐ 20,833 ‐ 20,833 199,167 199,167 199,167 199,167 199,167 199,167 199,167 199,167 199,167 199,167 199,167 199,167 199,167 199,167 199,167 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 50,833 ‐ 100,833 ‐ 100,833 ‐ 100,833 ‐ 20,833 ‐ 41,667 ‐ 62,500 ‐ 83,333 ‐ 104,167 ‐ 125,000 ‐ 145,833 ‐ 166,667 ‐ 187,500 ‐ 208,333 ‐ 9,167 190,000 389,167 588,333 787,500 986,667 1,185,833 1,385,000 1,584,167 1,783,333 1,982,500 2,181,667 2,380,833 2,580,000 2,779,167 2,728,333 2,677,500 2,626,667 2,575,833 2,525,000 2,474,167 2,423,333 2,372,500 2,321,667 2,270,833 2,220,000 2,169,167 2,118,333 2,067,500 2,016,667 1,915,833 1,815,000 1,714,167 Due: At the beginning of class,
Problem Set 3 Wednesday, October 3rd
Mr. Nordhaus and Staff 65 ‐ 100,833 ‐ 100,833 1,613,333 66 ‐ 100,833 ‐ 100,833 1,512,500 67 ‐ 100,833 ‐ 100,833 1,411,667 68 ‐ 100,833 ‐ 100,833 1,310,833 69 ‐ 100,833 ‐ 100,833 1,210,000 70 ‐ 100,833 ‐ 100,833 1,109,167 71 ‐ 100,833 ‐ 100,833 1,008,333 72 ‐ 100,833 ‐ 100,833 907,500 73 ‐ 100,833 ‐ 100,833 806,667 74 ‐ 100,833 ‐ 100,833 705,833 75 ‐ 100,833 ‐ 100,833 605,000 76 ‐ 100,833 ‐ 100,833 504,167 77 ‐ 100,833 ‐ 100,833 403,333 78 ‐ 100,833 ‐ 100,833 302,500 79 ‐ 100,833 ‐ 100,833 201,667 80 ‐ 100,833 ‐ 100,833 100,833 81 ‐ 100,833 ‐ 100,833 0 Total 6,050,000 6,050,000 0 74,375,000 Net Savings Rate = Total Savings / Total Income. Since the population is stationary, we’ll say there are N individuals alive at each age. Total Savings = Sum of (number of people alive at each age * savings at each age) = Sum of (N*savings at each age) = N*Sum of (savings at each age) = N*0 = 0 Thus, Net Savings Rate = 0. Similarly, Wealth‐Income Ratio = Total Wealth / Total Income. Total Wealth = Sum of (number of people alive at each age * wealth at each age) = Sum of (N*wealth at each age) = N*Sum of (wealth at each age) = N*74,375,000 Total Income = Sum of (number of people alive at each age * income at each age) = Sum of (N*income at each age) = N*Sum of (income at each age) = N*6,050,000 Thus, Wealth‐Income Ratio = N*74,375,000/(N*6,050,000) = 12.30 Due: At the beginning of class,
Problem Set 3 Wednesday, October 3rd
Mr. Nordhaus and Staff 2. Assume that Franco “Irv” Fisher lives two periods. Irv can borrow and lend money at the interest rate r = 10 % per period. His income is $500 in the first period and $1,100 in the second. a. Graph the budget constraint. Be sure you indicate the income level in each period and the slope of the budget constraint. What is the present value of Irv’s income. Present value of income is Y1 + Y2/(1+r) = 500 + 1100/1.1 = 1500 Future value of income is Y1*(1+r) + Y2 = 500*1.1 +1100 = 1650 b. You observe that Irv consumes $600 in the first period. Graph an indifference curve consistent with this consumption. How much will he consume in the second period? How much does he save in the first period? Consuming $600 in the first period means that Irv must borrow $100. He thus has savings of ‐$100 in the first period. In the second period he has income of $1,100, but must repay the loan of $100, and interest of $10. After repaying the loan, he has $990, which he uses to consume in the second period. Problem Set 3 Mr. Nordhaus and Staff Due: At the beginning of class,
Wednesday, October 3rd
c. Assume that President G.W. Shrub enacts tax cuts that will be applied in the first period, but the Shrub‐Era tax cuts will expire in the second period. The tax cuts will transfer $100 to each citizen in the first period, but in the second period (after the tax cuts expire) the government will impose a tax of $110. Graph the budget constraint. Again indicate the income level in each period and the slope of the budget constraint. Present value of income is Y1 + Y2/(1+r) = 600 + 990/1.1 = 1500 Future value of income is Y1*(1+r) + Y2 = 600*1.1 +990 = 1650 Here we see that the budget constraint does not change. The income levels have changed from (500, 1100) to (600, 990), but these two income distributions both have the same present value. The slope also remains the same. Problem Set 3 Mr. Nordhaus and Staff Due: At the beginning of class,
Wednesday, October 3rd
d. What is the impact of the tax cut and later tax on consumption and saving in each period? The tax has no impact on the budget constraint or the utility function, so consumption remains the same at (600, 990). There is no longer a need to borrow, so savings is (0, 0) over each period. e. Redo parts c) and d) assuming that Irv cannot incur any debt. If Irv cannot incur any debt, then he is unable to consume $600 in period one. He can at most consume $500. The budget constraint is the solid line segment from (0, 1650) to (500, 1100). We know that there was an indifference curve tangent to the point (600, 990), so Irv is now at a lower indifference curve tangent to the point (500, 1100). That is, Irv is receiving a lower utility (the solid indifference curve lies to the left of the original dashed indifference curve). Problem Set 3 Mr. Nordhaus and Staff Due: At the beginning of class,
Wednesday, October 3rd
If the (100, ‐110) tax is introduced, then the budget constraint line segment is extended to (600, 990) and the original indifference curve is achieved. That is, consumption changes from (500, 1100) to (600, 990), and savings remains at zero over all periods. Due: At the beginning of class,
Wednesday, October 3rd
Problem Set 3 Mr. Nordhaus and Staff 3. Do exercise number 5 on page 422 of Jones Chapter 15. Show your work. Given: Solve these two equations for c1 and c2 when beta is not 1.
Implies The 1+R terms cancel and we can rearrange to get Due: At the beginning of class,
Wednesday, October 3rd
Problem Set 3 Mr. Nordhaus and Staff Thus, we can plug this into the equation for c2
When beta = 1, this gives
which matches the equations on page 409. When beta < 1, then Taking the inverse of both sides gives us Thus, So, when beta < 1, period 1 consumption is higher than period 1 consumption when beta = 1. This says that when people dislike waiting, they will choose to consume more today than tomorrow.