Assignment 1 Answers

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1 Fall 204 ECON 302 Intermediate Macroeconomics Professor Ananth Seshadri 2.7 Assignment Answers. A concave function with B equal to the Y-axis intercept. 2. MPL= dy dl = 2 Al if l 0. It s positive and diminishing in labor. 3. An increase in A means MPL increases, which has substitution effect (c, l ) and wealth effect (c, l ). Assume SE > WE: c, l. Please refer to page in the textbook for more details. 4. An increase in B means a parallel shift up of production function. It has a pure wealth effect (c, l ) An increase in A has substitution effect (c, l ) and wealth effect (c, l ). y = c, so y. Overall, y, c, l all increase. 2. An increase in B has a pure wealth effect, and y = c, so c, l, y For temporary change in the current period t =, there is a pure wealth effect only in this period. So c t=, l t=. Consumers smooth the consumptions, so he will save some extra wealth today and consume more in the future, MPC<, all future consumptions c 2, c 3, c 4... increase a little bit. But l 2, l 3,... do not change. 2. For permanent change, there are positive wealth effects in each period. MPC=. c t and l t for all t. 2.. o answer this part you would just need to show that the slope of f(l) at a point such as l I, always is lower than the slope of the line from the origin (0, 0) to (l I, y I = f(l I )).

2 2. A proportional change is like an increase in A in problem 2.8. here are substitution effect and wealth effect. Assume SE dominates WE: y = c, l. Note that y increases because of more labor (l ) and technological progress (A ). Barro points out that historical data show that in the long run (with technology progress), productivity, y/l, rises. his suggests with technological progress, y rises more fast then l, as the result, y/l rises. 3. People are more willing to work (l ), but need more consumption to compensate the decrease in leisure (c ). c = y, the increase in output is due to the increase in labor. his rise in y in graph is moving along the f(l) (no technology change, the production function is the same one ), and it is obvious that productivity, y/l, falls since the line from the origin will become flatter. Additional Problem max U(c, l) = ln(c) l s.t. c = y = Al 0.5 Plug c = Al 0.5 into U = U = ln A ln l l. Set the first order derivative to zero, we get du dl = 2l = 0 So l = /2., and c = A/ P B is less than. 2. he interest rate on bonds is R. A discount bond should also have interest rate R as long as the issuer of the discount bond is no riskier than the issuer of the bond. 3. he principal of the bond is P B, and the interest received is P B, so the return of the bond is P B, and this return should be equal to R. So P B or /P B is called the return of the bond. R = P B P B, P B = + R. 2

3 4. If you save one dollar in a bank for two periods, you will receive ( + R) 2. And the return of the two period bond is /P B. So the interest rate per period of this bond is P = ( + B r)2 r =. P B For j periods bond, we have P B = ( + r)j If you have b 0 > 0 and there is an increase in P, the real value of your bonds will decrease. Hence you suffer a negative wealth effect. 2. here is imply no wealth effect. Notice that you can always write the lifetime budget constraint as y c + y 2 c 2 + R + y 3 c 3 ( + R) + + b 0( + R) 2 P = 0 Since b 0 = 0, effect. c t = y t, all terms above are zero. A change in R does not have any 3. b 0 = 0, consider y c + y 2 c 2 + R + y 3 c 3 ( + R) 2 + = 0 () We can divide the above equation into two parts: t < and t as y c + y 2 c 2 + R + + y c + y + c = 0 ( + R) }{{ ( + R) }}{{ } (a) (b) Note (a)> 0, (b)<0, and (a)+(b)=0 before the increase in R. he increase in R makes both (a) and (b) smaller. But part (b) is discounted more so they shrink more than (a). As the result, after the increase in R, we have (a) + (b) > 0, Eq () becomes y c + y 2 c 2 + R + y 3 c 3 ( + R) 2 + > 0 Hence there is a positive wealth effect, and the household will raise consumption until, y c + y 2 c 2 + R + y 3 c 3 ( + R) 2 + =

4 . After 2, there s no consumption. We can think this household die after 2. If the household does not have incentive to leave bequest, and has to repay all the debts he owns, b j = 0. If there is no constraint on the credit market, the household can borrow and does not repay his debt at 2. hat is, he can consume more by borrowing up to b 2 =. he budget constraint is P y + P y 2 + R + + P y ( + R) +b 0(+R) = P c + P c 2 + R + + P c 2 ( + R) 2 + b 2 ( + R) For t <, the household save, so b t b t > 0. After, the household retire and consume what he saves earlier, so b t b t < If people were forced to retire earlier, then, other things equal, they would understand that they would need to have a bigger stock of assets to run down from their retirement until their death. So early in life they would save more (by consuming less) and, because consumption falls, they would also decrease leisure (increase work effort). his increase in work effort would tend to offset the decline in consumption a little by expanding output. 4. When people care about children and grandchildren (b 2 > 0), the planning horizon might include the lifetimes of their children and grandchildren. And it becomes the infinite horizon, like a dynasty. Without the household s preference for his own consumptions and his offsprings consumptions (how much he cares about his offsprings), we don t know the value of b m = 2 P c = = Now = 24. So m = = From the equation m = P c, we see when, m Instead of continuous spending/shopping in the textbook, this question become discrete spending. So the graph is not triangle (not like Fig4. in the book), and m 2 P c. Note that in Figure, the big letter refers to the wage interval, and the small letter t means the periodic shopping trips interval.wage interval is = /2 (monthly wage payment.) 4

5 m pc t time t t time Figure : Graph for Problem 4.0 (a). t = 4 ( )t m = = $375. In addition, the average goods holding is $25, and $375+$25=$500, the same as answer in 4.9(a). 2. t = 2 m = (500)t = $250 Average inventory of goods is $250. he sum of these two is still equal to $ Compare (a) and (b). Money holding decreases ( ) as t increases. In 4.9 (c), time interval between wage payment () has positive relationship with average money balance (if, then m ). Here, time interval between shopping trips (t) has negative relationship with average money balance (if t, then m ). 4. When the cost of shopping trips increases, the frequency of shopping trips will fall (this means t ). he fewer trips, the smaller the money demand. In an extreme case, someone could do all their shopping on the day of their pay check, which would imply a zero average money balance for the month. herefore there is negative relationship between the cost of shopping trips and money holding. he higher the financial transaction cost, the higher the money demand. here is positive relationship between transaction cost and money demand. Note that this trip cost lengthens t, but has no effect on ; while the financial transaction cost affects, not t he answer is in the textbook, page 7, from Equation (5.), (5.2) to (5.3). 2. From Eq(5.3) in the book, (C d Y s ) + (B d /P ) + (M d /P M 0 /P ) = 0, (5.3) 5

6 If any two markets clear, say, C d = Y s, and B/P d = 0, we know M d /P must be equal to M 0 /P from Eq(5.3). Because of this budget constraint, there are only two independent conditions even though there are 3 markets. 3. he number of independent conditions is equal to the number of market prices. 4. If there is excessive demand in one market, say, C d Y s > 0, then there must be excessive supply at least one other market, e.g., B d /P > 0 or M d /P > M 0 /P, or both. his example shows that consuming more requires the economy to borrow more, or hold more money. And the markets will not clear if any one of the market has excessive demand or supply. Additional Problem: Permanent Income =y + y.07 + y.07 + y.07 2 = y = His annual consumption is equal to the permanent income. c = At the third year, he saves b 3. And at the fourth year c y 4 = ( + R 3 )b 3, b 3 = = hat is, he borrows between the third and fourth year. 2. He can not save or borrow at the first two years. c = y = 5000, c 2 = y 2 = c 3 = c 4 = c c + c = c = Additional Problem: Market clearing. he graphs are exactly Figure 5.3, 5.4 and 5.5 in the textbook. his problem is the same as the example used in the textbook, from page A temporary parallel ship down on f(l) induces a negative wealth effect this period, and people know their income will go back to the original level tomorrow, so they can borrow today and repay tomorrow to smooth the consumption. When people want to borrow more, the interest rate will rise. Labor will increase due to the negative wealth effect, and we know MPL is decreasing with labor, and wage is equal to MPL (note carefully that the MPL decrease because labor increases, we are moving along the f(l).). Over all, c, y, l, MP L, R, P, M, and the real wage, MP L/P. 6

7 2. A permanent shock has no effect on R. Since now MPL also changes, C d and Y s shift to the left side more than in the previous question. his has the same effect as Figure 5.7 in the book. 7