Solutions Mannual

Barro, Chapter 2 Questions Craig Burnside Economics 302 University of Virginia 2.2 Total product is f (l) while marginal product is MP L, which is the slope of f (l). If (a) MP L is positive and increasing it means the slope of f (l) is positive (as in Ch. 2), but the slope of f (l) also gets steeper for bigger l (not like Ch.2); (b) MP L is positive and decreasing it means the slope of f (l) is positive and gets ßatter as l increases (as in Ch. 2); (c) MP L is negative it means f (l) actually gets smaller as l increases. 2.3 A utility function is a mathematical function that tells us how much utility or sense of well-being a person gets from consuming a particular basket of goods. Illustrating indifference curves is simple and is described in Ch.2. Indifference curves shift around if people have changing tastes. 2.5 Along an indifference curve 0 = ∆u = MUC × ∆c + MUL × ∆l

(1)

and the question tells us that we are at a point on an indifference curve where ∆c must equal 1 if ∆l equals 1, in order to stay on that indifference curve. Using (1) this implies that in this example 0 = MUC + MUL or MUC = −MUL. The question also tells us that MP L > 1, so if the person works an extra unit of time, i.e. ∆l = 1, his net change of utility will be ∆u = MUC × ∆c + MUL × ∆l = MUC × MP L × ∆l + MU L × ∆l. Using the facts that ∆l = 1, MU C = −MU L and MP L > 1 we have ∆u = −MUL × (MP L − 1) > 0 so that the person whould work the extra unit of time. If MP L < 1, of course, the person should not work the extra unit of time. 2.6 Because the shift in the production function increase f (l) for every l, but also raises the MP L for every l, the effects of the shift will combine income and substitution effects. The upward shift, by itself, induces an income effect that makes the person want to consume more and work less (consume more leisure) as long as consumption and leisure are both normal goods. The increase in the MP L, by itself, induces a substitution effect that makes the person want to consume more and work more (consume less leisure). So c increases, while the effect on l is ambiguous. If consumption were an inferior good, then the income effect would induce consumption to fall, and so the net effect on consumption would also become ambiguous. If leisure were an inferior good, then the income effect would induce work to rise (leisure to fall), so the net effect on l would be that it would increase. 1

2.8 If f (l) = Al1/2 + B, then MP L = Al−1/2 /2. (a) If A increases this shifts f (l) and MP L up, so we get combined income and subsitution effects: c rises, y = c rises, and the effect on l is ambiguous. (b) If B increases this shifts f (l) up leaving MP L unchanged, so we get a pure income effect: c rises, y = c rises, and l falls as long as leisure is normal. 2.11 (a) To answer this part you would just need to show that the slope of f (l) at a point such as lI , always is lower than the slope of the line from the origin (0,0) to (lI , y I = f (lI )). If you drew a line tangent to f (l) at lI it would be obvious that its slope was less than the slope of the line from the origin. (See my Figure 1.) (b) From the text and the class we know that a proportional (not parallel) shift of f (l) induces income and substitution effects: c and y rise, but the effect on l is ambiguous. Barro points out that historical data show that over time (as technology has improved) productivity, y/l, rises. This suggests that whatever happens to l over time, y rises fast enough that y/l rises. (c) Recall, from (1), above, that on an indifference curve 0 = MUC × ∆c + MUL × ∆l. Hence, the slope of the indifference curve, which represents the amount of consumption, ∆c, needed to compensate per additional unit of time spent working, ∆l, is ∆c MUL =− . ∆l MUC The question asks you to consider what happens if the indifference curves get ßatter because −MUL/MUC becomes smaller. From my Figure 2, this implies that l and c both rise. Since f (l) has not changed, it is obvious that productivity, y/l, falls since the line from the origin will become ßatter.

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FIGURE 1

y

slope of f(l) at lI = slope of tangent line f(l) yI

slope of line from origin to (lI,yI) = yI/lI

lI

0

l

FIGURE 2 new indifference curves after shift in tastes

y original indifference curve

f(l)

optimum (l,c) combination after shift in tastes original optimum (l,c) combination

0

l

Barro, Chapter 3 Questions Craig Burnside Economics 302 University of Virginia 3.8 a) P B should be less than 1. Even though Barro calls the face value of the discount bond the principal, there’s a sense in which you can think of $P B as the principal on the discount bond (because the principal is usually the amount borrowed by the borrower). The face value (1$) is paid a period in the future so its as if you are being repaid what I am calling the principal (P B ) plus the interest, 1 − P B . So for the interest to be positive, P B must be less than 1. b) The 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. c) The interest rate is the interest, 1 − P B , expressed as a fraction of the principal, P B , and must equal R: hence 1 − PB R= PB or 1 . PB = 1+R This illustrates a general rule: the price of a discount bond is the present value of its face value. d) The interest rate per period for a two period discount bond should also be R. Using the general principal that the price of a discount bond is the present value of the face value we should have 1 B P(2−period) = (1 + R)2 and B = P(j−period)

1 . (1 + R)j

3.10 (a) If you have b0 > 0 and there is an increase in P , the real value of your bonds will decrease. Hence you suffer a negative wealth effect. (b) This should imply no wealth effect. Notice that you can always write the lifetime budget constraint as y3 − c3 y2 − c2 b0 + + · · · + (1 + R) = 0 y1 − c1 + 2 1+R (1 + R) P so if yt = ct for all t, there is no impact on the y1 − c1 +

y3 − c3 y2 − c2 + + ··· 1+R (1 + R)2 1

part, and if we also assume b0 = 0, then there is a zero wealth effect. (c) In this case we can write y1 − c1 +

y3 − c3 y2 − c2 + + · · · = 0. 1+R (1 + R)2

For the periods up to time T the household is a lender, ct < yt so the increase in R makes (yt − ct )/(1 + R)t−1 terms smaller positive numbers. But for t > T , ct > yt so the increase in R makes (yt − ct )/(1 + R)t−1 terms smaller negative numbers. But the ones that are negative are being discounted more so they shrink more than the early terms, so for the same yt and ct we have y2 − c2 y3 − c3 + · · · > 0. y1 − c1 + + 1+R (1 + R)2 Hence there will be a positive wealth effect, and the household will raise consumption until, once again, y3 − c3 y2 − c2 y1 − c1 + + + · · · = 0. 1+R (1 + R)2 3.11 Someone should be indifferent (either a borrower or a lender) between the two ways of borrowing for two periods: • You borrow $1 in period 1 and repay $(1 + 2R) in period 3. • You borrow $1 in period 1, and repay $(1 + R1 ) in period 2 by borrowing $(1 + R1 ) in period 2 and repaying $(1 + R1 )(1 + R2 ) in period 3. (a) You would only be indifferent as a borrower or lender if 1 + 2R = (1 + R1 )(1 + R2 ) or R=

R1 + R2 + R1 R2 . 2

(b) If R2 > R1 then

R12 2R1 + R12 = R1 + > R1 . R> 2 2 So the long-term interest rate is greater than the short-term interest rate. 3.12 (a) The household might only use horizon j = T2 because it has no bequest motive. In this case, the household would not care about leaving any wealth for future members of the household to use. We can rewrite the budget constraint as P y1 +

P yT1 P cT2 P y2 P c2 bT2 +· · ·+ +· · ·+ +b (1+R) = P c + + . 0 1 (1 + R) (1 + R)T1 −1 1+R (1 + R)T2 −1 (1 + R)T2 −1

2

The household would set bT2 = −∞ if it could, because this would let it borrow huge amount and leave the debt unpaid at death. Assuming that nobody is willing to let someone borrow if they won’t repay at death, similar logic implies that households will set bT2 = 0. (b) The household will have bt −bt−1 < 0 during the retirement period because the household has no income from selling output, so its consumption is Þnanced by dissaving. From the budget constraint, when yt = 0 we will literally have bt − bt−1 = Rbt−1 − P ct . (c) 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). This increase in work effort would tend to offset the decline in consumption a little by expanding output. (d) When people care about children and grandchildren it’s not obvious they want to set bT2 = 0, if T2 is the period of their death. Instead, the planning horizon might include the lifetimes of the children and grandchildren. So it is not obvious how the planning horizon should be set or that it should not be set to ∞.

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Barro, Chapter 4 Questions Craig Burnside Economics 302 University of Virginia 4.2 a) If the person spends P c = $6000 per year and makes monthly (T = 1/12) withdrawals then each withdrawal is P cT = $500. Average money holdings are P cT /2 = $250. See Figure 1. b) Now T = 1/6 so each withdrawal is P cT = $1000. Average money holdings are P cT /2 = $500. See Figure 1. 4.3 If P c = $9000 per year and T = 1/12 then each withdrawal is P cT = $750 and average money holdings are P cT /2 = $375. It may or may not be optimal for the frequency to remain at T = 1/12, if T = 1/12 was optimal when P c = $6000. The formula for the optimal T is: ! 2 γ T = . Rc P If P c rose due to a rise in P , but γ also rose in proportion to P (Barro indicates he wants us to think about the transactions cost as being Þxed in real terms) then there is no change in the optimal T . However, if P c rose due to a rise in c, then T should fall: the person should visit the bank more often. 4.5 The real demand for money is M/P = Φ(R, C, γ/P ). (a) Φ is increasing in R, so a decrease in R raises the real demand, M/P . (b) Φ is increasing in γ/P , so an increase in γ/P raises the real demand, M/P . (c) Φ is increasing in C, so an increase in C raises the real demand, M/P . (d) Φ is invariant to P (holding γ/P Þxed), so an increase in P has no effect on the real demand, M/P . 4.6 Velocity is Y /Φ, so the effects on velocity are all in the opposite direction to the effects on M/P . 4.7 (a) If we were to incorporate transactions costs into the lifetime budget constraint we would have something like PV(y) − PV(

∆m b0 ) − PV(trans. costs) + (1 + R) = PV(c) P P

Up til now we have ignored transactions costs in the lifetime budget constraint. Transactions costs lower wealth. (b) If γ/P rises people will suffer a negative wealth effect, so they will decrease consumption and increase labor supply. The effect on saving is not entirely clear. With labor supply rising, 1

output will increase. With consumption decreasing this suggests that saving will rise. But to know for sure that saving increased we would need to check whether y − c−transactions costs increased, and this is not clear. (c) Yes, we did leave out a substitution effect. In effect, when γ/P rises it raises the effective price of consumption relative to leisure. So people will want to raise leisure and lower consumption. The rise in leisure (decline in work effort) offsets the wealth effect discussed in part (b). 4.11 (a) C rising causes M/P to rise. Barro is exploring a subtlety here. He wants you to realize that since m = φ(R, c, γ/P ) P the aggregate M/P is " M φ(R, c, γ/P ) = P i where the notation

" i

indicates the sum over all individuals (indexed by i). It’s also true that C=

"

c

i

So if the number of people rises (say by 10%) both C and M/P will rise by the same percentage (10%). But if little c rises, say by 10%, we know that M/P rises, but not necessarily by 10%. (b) Velocity is V = Y /Φ(R, C, γ/P ). We might expect that as an economy develops, transactions costs decline, so that γ/P is falling. This makes Φ fall and V rise. Some economic theories predict that R should fall as an economy develops, but there is little empirical evidence to support this hypothesis. A fall in R would make Φ rise and V fall. As an economy develops GDP rises, i.e. Y increases. But it is also true that C increases. These changes would have opposite effects on velocity. However, the evidence that Barro cites from Goldfeld suggests that the effect of rising GDP on real money demand (through its impact on real consumption demand), is the weaker effect. I.e. if Y increased 10%, M/P would only increase, say, 6%. So we might expect this to be another reason V should rise with economic development.

2

FIGURE 1

Money holdings

Withdrawals every 2 months

1000

500

2

4

6

8

10

12 Month

Withdrawals every month

Barro, Chapter 5 Questions Craig Burnside Economics 302 University of Virginia 5.2 a) Aggregate (consumption) demand falls with a rise in the interest rate because the incentive to save is greater. Aggregate supply rises with a rise in the interest rate because labor supply increases. These effects are captured as movements along the aggregate demand and supply curves illustrated in Chapter 5. b) An increase in wealth shifts the aggregate demand curve to the right since it increases consumption demand. It also shifts the aggregate supply curve to the left because an increase in wealth causes an increase in leisure and a reduction in labor supply. c) A shift of the production function (if it does not changes the MP L schedule) increases wealth. Therefore, the effect on aggregate demand is identical to what it was in (b): a shift to the right. On the other hand, even though work effort will decline, the improvement in the production function will overwhelm that effect, so that the aggregate supply curve also shifts to the right (in contrast to what we saw in part (b)). If the shift in the production function is also accompanied by an increase in the MP L, the rightward shift of the supply curve is even bigger since the incentive to work increases. 5.4 For (a) Figure 5 in the class notes would be an appropriate graph. For (b) Figure 6 would be an appropriate graph. 5.5 In part (a) of 5.4 the increase in consumption demand is as large as the increase in output so the marginal propensity to consume is 1. This suggests a permanent shock to the production technology. In part (b) of 5.4 the increase in consumption demand is much smaller than the increase in output so the marginal propensity to consume is less than 1. This suggests a temporary shock to the production technology. 5.9 Aggregate output should fall in proportion to the decline in the population, as should aggregate work effort. Why? At the individual household level, wealth should be the same, as should the marginal product of labor. So, if we considered aggregate demand and supply they should both simply shift to the left by exactly the same amount, leaving the interest rate unchanged. Since P = M/Φ(R, Y, . . .) we might expect the price level to rise since M does not change (by assumption), and Y falls in proportion to the loss of population. 5.10 In this question Barro is asking you to analyze what happens if household preferences ˜ say. In change. This means they go from having a utility function U to a utility function U, the question Barro is basically saying that if we look at any point (c, l), the new marginal

1

rate of subsitution between consumption and leisure will be such that ˜ −M UL −MUL . < ˜ MUC M UC Why? The question says that the household starts to like consumption more than before relative to leisure. This tells you that for a given increase in work effort (decline in leisure), the household needs to be compensated with less consumption than before (because it likes consumption more than it did before). This tells you that indifference curves are ßatter. The slope of an indifference curve is −MUL/MU C so it must be a smaller number for the indifference curves to be ßatter. He also wants you to assume that at any choice of (c1 , c2 ) that ˜ 1 MUC1 M UC = . ˜ 2 MUC2 M UC That is, there is no change in how the household values consumption in one period relative to another. At the old equilibrium MP L = −MUL/MUC. The old equilibrium, (c∗ , l∗ ) cannot still ˜ be the equilibrium because MP L would be the same, and −M UL/M U˜ C would be smaller. Of course, that would not be optimal. Since the household likes consumption more than before, it will raise its labor supply and ˜ ˜ to rise some until they are its consumption causing MP L to fall some, and −M UL/M UC equal again. This means output will also rise. There should be no effect on the interest rate because consumption (and labor) will rise in all periods, and consumption demand should shift out by about the same amount as aggregate supply. For a Þxed money supply, an increase in output shifts the demand for real balances to the right, causing the equilibrium value of P to fall. 5.13 (a) If we considered a rise in the interest rate out of equilibrium, then as we increased the interest rate C would fall as we moved along the demand curve, and Y would rise as we moved along the supply curve. So C/Y would fall. But, of course, this is not what is going on in the example. In the example, the shock to the production technology causes a small decrease in wealth which induces a small decline in C d for any given interest rate. On the other hand, the shock to the production technology causes a large decrease in output supplied, Y s , at any given interest rate. So at the original interest rate C d /Y s would actually rise. However, C/Y must always equal 1 in equilibrium. The only way for this to happen in the example, is to increase the interest rate to make C d even smaller (along the new demand curve) and make Y s larger (along the new supply curve), until they are equal. Aggregate saving must always equal 0 in this version of our model. (b) A study such as the ones Barro is describing looks at time series data on the ratio C/Y and R. That is, it looks at a sequence of observations on Ct /Yt and Rt and tries to see how R affects C/Y using econometric methods. Of course, our theory predicts that C/Y is always 1, and does not respond at all to changes in the interest rate. However, one would not want to conclude from this that the interest rate does not affect household decision making. 2

It does, of course, as reßected in the slope of the agrgegate demand curve. But the kind of study Barro is referring to cannot uncover that slope. (c) Studying movements in aggregate Ct+1 /Ct and Rt could tell me something about individual behavior but I’d have to be very careful in examining the data. Remeber that C changes depending on the interest rate and level of wealth. So within the context of our model, I’d have to have a method for Þguring out what part of any movement in consumption was due to the interest rate changing and what part was due to ongoing changes in perceived wealth. Consider the case illustrated in Figure 5.4. Suppose the shock to the production technology lasts for only one period. Then in period t consumption falls to the amount Ct = Y ∗" and interest rates rise to the level Rt = R∗" . If there are no other shocks in period t + 1, the demand curve should still be the one labeled (C d )" (recall that the shock to the production technology reduces the household’s lifetime wealth—not by much—and the effect on wealth— not output—is permanent). On the other hand the supply curve should be the one labeled Y s because the shock to technology lasts for just one period. Notice that this means in period t + 1 the equilibrium will be at the intersection of (C d )" and Y s so that Ct+1 = Y ∗"" a value slightly less than Y ∗ and Rt+1 = R∗"" a value slightly less than R∗ . What’s interesting about this case, though, is that the comparison between Ct and Ct+1 in my example represents a comparison between consumption levels on the same demand curve. Therefore, examining the data in this way, could, in principle, uncover something about the relationship between C d and R. (d) The answers to this question suggest that we should take the approach described in part (c). But we would have to be careful to Þgure out how to handle the possibility of shocks to the economy at t + 1 that might pollute the relationship between Ct+1 /Ct and Rt . Econometricians have Þgured out methods for doing this. 5.16 (a) In a Robinson Crusoe economy, if f (l) shifts down in a parallel fashion this is the opposite of Figure 2.9 in the textbook. The fall in f (l) would cause c and y to fall, and l to increase. In the market economy the effects are identical. The permanent wealth effect shifts C d by about the same amount as Y s . This means R doesn’t change and C falls. The negative wealth effect causes L to rise. This is the opposite of Figure 5 in the class notes. (b) With the temporary shock, in the market economy the interest rate rises. This induces a bigger increase in labor supply than in part (a), and so the declines in C and Y are not as large as in part (a). This is the opposite of Figure 6 in the class notes. (c) They don’t. (d) The reason the market economy and the Robinson Crusoe case differ (in the case of a temporary shock) is due to the existence of the bond market. This allows people to think about intertemporal substitution and this is what induces the effect on labor supply in the market economy. In the absence of a linkage between periods the Robinson Crusoe case is fundamentally different in this respect.

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Barro, Chapter 6 Questions Craig Burnside Economics 302 University of Virginia 6.3 An increase in the interest rate shifts the supply curve for labor to the right (increases) because households substitute towards less leisure today, and more leisure tomorrow. Since the MP L schedule is unaffected by R, the real wage falls and the equilibrium quantity of labor rises. 6.6 In the example in the text, region A uses technology f A (l) to produce output, while region B uses technology (with a higher MP L schedule), f B (l). If there were separate labor markets in the two regions, they would have separate wage rates given by MP LA (lA ) and MP LB (lB ) where lA and lB are household labor supply in regions A and B. I’m not sure how subtle an answer Barro wants here. A simple answer would run something like the following: suppose lA and lB are roughly equal so that lA = lB = l–this implies that real wages are higher in region B, (w/P )B = MP LB (l), than in region A, (w/P )A = MP LA (l) If the level of technology is similar across the regions, this means that output is similar across the two regions so that y A = f A (lA ) = f A (l) = f B (l) = f B (lB ) = y B = y.1 Since real profits are given by the difference between the value of output and the wage bill it is clear that real profits are higher in region A: π

A

πB









w A A w A = y − l =y− l P P  B  B w w = yB − lA = y − l < πA . P P A

This means that when trade in labor opens up, which will equate real wages across regions, real wages paid by firms in region B will fall and in region A they will rise, because workers from region A will cross to region B to perform work. So you might think of the answer to this question being: workers from region A will like the single market, while workers from region B will not. It turns out that the reverse is true for firms. Because real wages rise in region A firms there also reduce labor input, and both effects tend to reduce profits. The reverse is true in region B, where falling real wages and rising labor input will raise profits. Hence, shareholders in region A are unhappy while shareholders in region B are happy. Now, of course, our average household is both a worker and a shareholder. So the opening of the larger market tends to have both positive and negative effects. The overall effect is positive, however, because the opening of the economy-wide labor market creates an efficiency gain. If aggregate labor supply is unchanged, aggregate production will rise because it is more efficiently allocated. So aggregate consumption will rise. As long as this increase in consumption is shared across the two countries the utility of the average household is likely to increase. 1

If the level of output is not similar in the two regions then everything gets more complicated.

1

Of course, in the real world, where ownership of firms is not allocated equally across the population, the effects of the opening of a single labor market could have an important political impact: if workers are not also shareholders, then workers in region B will be opposed to opening the single market, whereas firms in region B will be opposed. The opposite should be true in region A. Debates on free trade have a similar flavor to this discussion, although the issues aren’t quite the same because they don’t have to do with labor mobility. Debates over immigration also have this flavor. 6.9 (a) In Chapter 5 we saw that a temporary parallel downward shift in the production technology lowers wealth, by a small amount, and therefore shifts C d slightly to the left. We also saw that it shifts Y s substantially to the left (see Figure 5.4). This lowers C and Y and raises R, the equilibrium interest rate. In the labor market diagram, we know that the increase in the interest rate shifts the labor supply curve to the right, and that labor demand is unaffected by the shock. So w/P should fall and L should increase. See Figure 1(a) (b) No. (c) The answer to (a) changes. A downward shift in MP L combined with the downward shift in f (l), i.e. a proportional shift of f (l), will have similar implications for C, Y and R. However, in the labor market diagram, at the same time as the labor supply curve is shifting to the right, the labor demand curve will shift down. We usually assume the effect on labor demand is strong enough to outweigh the effect on labor supply so that both w/P and L should fall. See Figure 1(b). 6.11 (a) Average productivity or AP L, is not the same as MP L. Average productivity tells you about the ratio of total production to total labor input. So in a sense, it tells you about the average contribution of each unit of labor being used to total output. On the other hand, the marginal product of labor tells you something about the contribution of the last unit of labor to the production process to total output. Since the contribution of additional units of labor is always less than the contribution of previous units, MP L is less than AP L. Notice, however, that AP L should also decline as the number of workers used increases. Figure 2(a) illustrates what the MP L and AP L might look like. (b) i. In 6.10a and 6.9a, we imagine business fluctuations driven by a decline in the f (l) function that somehow leaves the MP L schedule unchanged. As a result of this shock Y falls and L rises. So it is unambiguous that AP L = Y/L falls. So AP L falls when Y falls. (b) ii. In 6.10b and 6.9c, instead, we imagine that the f (l) function and the MP L schedule both fall. We saw that this implies Y and L both falling. So offhand, it is not obvious in which way AP L moves. Notice, however–from Figure 1(b)–that the equilibrium MP L falls despite the fall in L (this is because the MP L schedule changed). This means the marginal product of the last worker shifted. If the whole MP L schedule shifted, as a result of a proportional shift in f (l), this should mean that the contribution of each worker (from first to last) to output declined. Hence the average product of labor should have declined as well. So again, AP L should fall.

2

FIGURE 1 (a) Temporary Downward Parallel Shift of f(l)

Ls

w/P

(Ls)'

(r*,L*) (r*,L*)'

Ld

L (b) Temporary Downward Proportional Shift of f(l)

Ls

w/P

(Ls)'

(r*,L*)

(r*,L*)'

Ld (Ld)' L

FIGURE 2 The Average and Marginal Products of Labor

MPL APL

APL MPL

L

Barro, Chapter 7 Questions Craig Burnside Economics 302 University of Virginia 7.5 (a) Yes, this theory of money demand accords with what we learned in Chapter 4. The only difference is that in Chapter 4, Barro spent some time arguing that money demand increases with c, but he also argued that it might not go up one-for-one with c (see page 143) because of economies of scale in cash holding. And, of course, the Baumol-Tobin model predicted cash holdings in proportion to c1/2 not c. (b) Notice that if M/P = Y Ψ(R) then the inßation rate is π t ≈ µt − gt − ψ t where µt = Mt+1 /Mt − 1 is the money growth rate, gt = Yt+1 /Yt − 1 is the growth rate of real GDP, and ψ t = Ψ(Rt+1 )/Ψ(Rt ) − 1 is the growth rate of Ψ(R).1 Assuming that over long periods of time we can ignore the ψ t term (and set it to zero), this means that inßation will equal money growth minus real growth. So the higher real growth is the more money growth an economy can absorb without suffering inßation. (c) If the nominal interest rate increases between t and t + 1 then Ψ(Rt+1 )/Ψ(Rt ) will be less than 1, hence, ψ t will be negative. Hence, inßation will be higher than it otherwise would be, between periods t and t + 1. (d) The answer is much as in part (c). Since R ≈ re + π e , re is constant and π e has increased, this means R has increased, just as in part (c). 7.6 See the spreasheet for the results. (a) Since Mt /Pt = Φ(Rt , Yt ), we have π t ≈ µt − φt where π t = Pt+1 /Pt − 1, µt = Mt+1 /Mt − 1 and φt = Φ(Rt+1 , Yt+1 )/Φ(Rt , Yt ) − 1. The regression estimates a relationship between πt and µt of the form π t = a + bµt . Given our theory we would expect b = 1 as long as φt was roughly constant. The estimate of a would be that constant value of φ. In fact, we get the following results: πt = −3.2 + 1.001 × µt . (b) Notice that µt − π t ≈ φt . If we run a regression of the form µt − π t = d + cgt 1

To see this, notice that since Mt /Pt = Yt Ψ(Rt ), this means Pt+1 /Pt = (Mt+1 /Mt )/{(Yt+1 /Yt )[Ψ(Rt+1 )/Ψ(Rt )]}, so 1 + π t = (1 + µt )/[(1 + gt )(1 + ψ t )]. It is mathematically true that if µt , gt and ψ t are small decimals, π t ≈ µt − gt − ψ t .

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where gt = Yt+1 /Yt − 1 we might think of the gt term as capturing the effect of output growth on the growth for the demand for real balances, φt . So we would expect a positive relationship. We end up with the following estimate µt − π t = 0.2 + 0.8gt . (c) Finally, if we add gt to the regression we have πt = a + bµt + cgt . We end up with parameter estimates π t = −0.14 + µt − 0.8gt . These results seem very consistent with parts (a) and (b) and our theory. 7.7 (a) A borrower (who issued bonds denominated in Þxed units of currency—as opposed to indexed bonds) is hurt if inßation turns out to be less than expected. In this case, it means the borrower will expect his payments to be more costly in real terms than he previously expected. This means that if there is a decline in inßation expectations between when the borrower issued the bonds, and the current period, the borrower may well try to prepay (before the previously unanticipated decline in inßation occurs). So we would expect to see prepayment during a time of falling inßation expectations and interest rates. (b) In the mid to late 1980s interest rates were on the decline so as in (a), borrowers (customers) wanted to prepay. In the mid to late 1970s, interest rates were rising so banks (the lenders) would have tried to encourage customers to prepay, but the customers would not have wanted to. (c) Greater volatility in interest rates means that there is more likelihood the customer will want to exercise the option to prepay in any given year. This makes the option more volatile.

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Barro, Chapter 8 Questions Craig Burnside Economics 302 University of Virginia 8.2 If someone gives up a unit of consumption at date t, they gain Pt dollars, which can be used to purchase bonds maturing at t + 1 that will pay Pt (1 + R) dollars. These additional dollars, in turn, will allow the person to purchase Pt (1 + R)/Pt+1 = (1 + R)/(1 + π) = 1 + r extra units of consumption at time t + 1. 8.4 (a) i. 1 + R = (1 + r)(1 + π) = 1.04 × 1 = 1.04 so that R = 0.04, or 4%. (a) ii. If π = 0.10, or 10%, we can use the approximation, R ≈ r + π = 0.04 + 0.1 = 0.14 (or 14%). On the other hand if we wanted an exact figure we would need to compute 1 + R = (1 + r)(1 + π) = 1.04 × 1.1 = 1. 144, so that R = 0.144 or 14.4%. (b) There would be excess demand. If R didn’t rise in proportion to π, then the real interest rate would be lower than the equilibrium real interest rate. So people want to borrow too much to finance higher consumption, and output would be too low because labor supply would be lower. 8.8 The answer here depends on how you define revenue. If you define it in dollar terms then the answer is yes. The government can always dump as much money into the economy as it wants, and its revenue is Mt − Mt−1 , so the more money it pours in the more revenue it gets. But if you ask the question in terms of the purchasing power of the government’s revenue then the answer is no. Notice that the purchasing power of the government’s money is Mt − Mt−1 . Pt Take the example where Y and r and the money growth rate, µ, are all constant. In that case, we saw that inflation will be constant and equal to µ. Our model predicts Mt /Pt = Φ(r + µ, Y ). Notice that this implies Mt − Mt−1 Mt Mt−1 Pt−1 = − Pt Pt Pt−1 Pt 

= Φ(r + µ, Y ) 1 −

1 1+π



= Φ(r + µ, Y )

µ . 1+µ

An increase in π will make the Φ(r + µ, Y ) term smaller, while making the µ/(1 + µ) term bigger (you should verify the latter statement by checking that µ/(1 + µ) goes up if µ goes up). One effect makes revenue smaller while the other makes it bigger. For small values of µ, the latter effect is probably stronger so that revenue should rise as µ rises. Eventually, however, we might expect this effect to be reversed. For very large values of µ we might see Φ(r + µ, Y ) → 0, whereas µ/(1 + µ) → 1, implying that revenue might actually get close to 0 for very large money growth rates. 1

8.9 (a) If you could hold goods between periods (i.e. they were not perishable) and their real value depreciated at the rate δ between periods, then the “nominal interest rate” to holding them would be given by 1 + RG =

Pt+1 (1 − δ) = (1 + π)(1 − δ). Pt

So RG ≈ π − δ. The interest rate on bonds would still be R = r + π, so it seems unlikely that anyone would actually store the goods as long as R > RG , or r > −δ. The demand for money should still depend on R, and should remain unaffected as long as r > −δ. (b) Since R = r + πe , if π e rises and R stays the same, r must be falling. If r fell so far that it became less than −δ then people would no longer hold bonds and would switch to storing goods as a way of saving. At that point, the demand for money would depend on the spread between the interest rate on storing goods, π − δ, and the interest rate on money, 0. For this reason money demand would probably still fall because π e and π would be rising. 8.10 (a) Real money demand is Φ(R, Y ), so it falls if R rises. (b) People economize on money holdings by going to the bank more often. Therefore, they pay more transactions fees. (c) This implies a negative wealth effect, so C and leisure will fall and L rise. (d) Yes. Leisure also becomes relatively cheap because you don’t have to use money to purchase it. This means C will fall further, and the negative (positive) impact of the rise in R on leisure (labor supply) is mitigated by the substitution effect. 8.13 The House of Lords was right. By issuing the getting the bank note printers (Waterlow Co.) to deliver them fake notes the swindlers were able to use the fake notes to purchase real goods from the Portuguese economy. In a sense they were able to take some of what would otherwise have been the Portuguese government’s “tax” revenue and use it to purchase goods the Portuguese government could otherwise have acquired. Although printed money is not a claim to anything concrete, it has a value determined by people’s willingness to hold it. As a result, the swindlers were able to acquire goods by doing something only the Portuguese government was allowed to do: i.e. printing money.

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Barro, Chapter 9 Questions Craig Burnside Economics 302 University of Virginia 9.4 No, because we know that individual households can borrow in order to Þnance investment. Of course, since aggregate borrowing must be zero, at the aggregate level higher investment requires higher saving. Saving always equals investment in equilibrium. If saving were less (greater) than investment, the interest rate would rise (fall), thereby increasing (decreasing) people’s desire to save and decreasing (increasing) investment demand. 9.6 This is the opposite case to the one illustrated in Figure 4 in the class notes. A decline in the MP K reduces investment demand and therefore shifts the aggregate demand curve to the left. This causes a fall in equilibrium r and Y . The fall in r raises consumption demand, so with Y falling and C rising it must be the case that I = Y − C falls. 9.7 (a) According to one way of thinking about this question is that the price of consumables equals the price of capital because there is no real distinction between the investment good and the consumption good. If someone wants a consumption widget or an investment widget they just go to the single widget market and purchase the widget for whatever purpose they want to use it for. If the prices were different then nobody would buy the more expensive widgets. Another way to think about it is that they are distinct goods but the technology for producing them is identical and they use the same inputs. Then they have to have the same price because otherwise producers would only produce the more expensive one. (b) If for some reason investment demand was so low that aggregate I d was actually negative, then the price for investment goods would fall below the price of consumption goods, and production of investment goods would be zero. (c) No, the prices could be different because the marginal rate of transformation between consumption and investment goods would no longer be 1. 9.9 Think about the return to investment. If I buy a unit of investment at time t, the government will give me back a fraction a of its cost, so my net purchase cost will be (1 − a)Pt . The payoff at t + 1 will be Pt+1 MP Kt + Pt+1 (1 − a)(1 − δ) because Barro assumes you have to give back the same percentage tax credit to the government if you resell the capital. Hence, the nominal return to capital is Pt+1 MP Kt + Pt+1 (1 − a)(1 − δ) (1 − a)Pt which must equal 1 + Rt in equilibrium. Hence we get the result MP Kt = (rt + δ)(1 − a) 1

(1)

whereas if the tax credit was not in place we would have MP Kt = rt + δ. Clearly the higher the tax credit, 0 ≤ a < 1, the lower MP Kt will be, and hence the higher desired capital and investment demand will be. 9.11 (a) Separating ownership of the capital from the use of the capital might be useful simply because it might generate returns from specialization. Not all people who are skilled in managing a portfolio of owned capital are also skilled in capital use. So there might be gains from specialization if the owners of capital (shareholders) allowed others to be the users (managers) of the capital. The downside to such an arrangement is that the owners still need to worry about monitoring the users of capital. This is called an agency problem. (b) The price of an ownership certiÞcate will always be the same as the price of capital given the setup. In our model this means shares trade at Pt dollars per share. The ‘real’ price, is of course, 1, since a share is always worth 1 widget. (c) Shares in a company often represent claims to intangible capital, whose value is hard to determine. Also the tangible capital of a company may have many Þrm-speciÞc qualities which make it much harder to value than the capital in our model. 9.12 Once we added capital to Robinson’s world, it would work a lot like the market economy we have described. Robinson would now have a way to save from one period to the next. By investing in capital, rather than consuming all his output, he could save more (make more capital goods) when times were good, and save less (make less capital goods) when times were bad.

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Barro, Chapter 10 Questions Craig Burnside Economics 302 University of Virginia 10.2 On reason to reject an offer of a wage above wu is that job search is costlier if you are are working. So there may be some value in staying unemployment in order to wait for a higher paying job, or a job with better nonwage characteristics. 10.3 Some jobs are strictly temporary to begin with. (Seasonal jobs like lifeguarding.) Firms and workers make matches when they still don’t have full information about each other. As information is revealed, they may begin to realize they are a bad match. 10.4 The natural rate of unemployment is the rate of unemployment at which the number of workers moving into employment (findings) matches the number of workers moving out of employment (separations). The natural rate can change if any of the permanent factors that affect the finding and separation rates change. The unemployment rate itself can change due to a change in the natural rate, but also because of temporary changes in job finding and separation rates. 10.5 (a) By raising the reservation wage, wu , an increase in unemployment benefits reduces the job finding rate and raises the duration of unemployment. (b) Raising the minimum wage also lowers the job finding rate and raises the duration of unemployment. In some cases, potential workers will stay out of the labor force and won’t be counted as unemployed. (c) A technological improvement shifts the distribution of wage offers up more than it shifts the reservation wage. Hence, it increases the job finding rate and decreases the duration of unemployment. Workers with a very wide distribution of wage offers will have a low job finding rate because the option value of remaining unemployed will be higher when a job offer greater than wu comes in. 10.6 Use the formulas Lt+1 = (1 − σ)Lt + φUt Ut+1 = σLt + (1 − φ)Ut .

(1) (2)

with σ = 0.01 and φ = 0.2. In this case the natural unemployment rate is σ/(σ + φ) = 0.01/0.21 = 0.0476, or just under 5 percent. We would expect to see about 4.76 million unemployed in the long-run in this economy since the labor force is 100 million. We can illustrate the paths of Lt and Ut using a table: time 0 1 2 3 · · · Long-run Lt 92.0 92.7 93.2 93.6 95.2 Ut 8.0 7.3 6.8 6.4 4.8 1

Barro, Chapter 11 Questions Craig Burnside Economics 302 University of Virginia 11.1 No. The change in the capital stock is ∆K = sf(K, L) − δK. As can be seen in Figure 11.2, for K > K ∗ , sf (K, L) < δK, hence for K > K ∗ the capital stock is decreasing. 11.2 The steady state capital stock, K ∗ , is the value of K for which sf (K, L) = δK. (a) Using Figure 11.2 it is clear that raising s, which raises the curve sf(K, L), raises K ∗ . (b) The same is true if you raise f (K, L). (c) A rise in δ, on the other hand, raises the line δK, and therefore reduces K ∗ . 11.5 Convergence refers to the fact that in the growth model, the capital stock and the income level (or the capital-labor ratio and output-labor ratio) converge to long-run target values. Absolute convergence refers to the situation where all countries are converging towards the same long-run targets. Relative, or conditional, convergence refers to the situation where different countries may have different long-run target values. 11.6 The association between the real interest rate and the capital stock depends on whether the convergence process is still happening or not. For example, suppose that the capital stock is growing both because of convergence effects (because the actual capital stock is below the long-run target path) and because of technological progress and population growth (the long-run target path is rising). In this case, the convergence effects imply that the MP K is higher than the MP K would be if capital were on the long-run target path, and therefore that r is higher than its long-run value. As K grows and gets closer to the long-run path, the MP K falls and gets closer to its long-run path, and the real interest rate falls. It turns out, however, that once the convergence effects have pretty much worn off, so that K only grows because of technical progress and population growth, that the real interest rate will remain constant. So the theory is not at odds with the data if you think of 1840-1900 as the US period of convergence, and 1900-2000 as a period in which the US had already converged very close to its long-run target path. 11.9 (a) sf(K, L) = δK becomes sAK α L1−α = δK which, if you solve it for K implies that K ∗ = (sA/δ)1/(1−α) L. Of course this means Y ∗ = f (K ∗ , L) = A1/(1−α) (s/δ)α/(1−α) L.

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(b) Steady state investment and consumption are α

I ∗ = δK ∗ = δ− 1−α (sA)1/(1−α) L C ∗ = Y ∗ − sY ∗ = (1 − s)A1/(1−α) (s/δ)α/(1−α) L (c) ∆K = sAK α−1 L1−α − δ K To know whether this growth rate rises or falls with the capital stock we need to find the derivative of the growth rate with respect to K: ∂(∆K/K) = (α − 1)sAK α−2 L1−α < 0 ∂K because 0 < α < 1. So the growth rate of the capital stock falls as K rises. (d) If L is constant then ∆Y ∆K =α = α(sAK α−1 L1−α − δ). Y K Obviously the growth rate of Y falls as K rises, because the growth rate of K falls as K rises. Also, the growth rate of Y falls as Y rises, because Y only rises when K rises. 11.10 (a) Yes, but we just have to use the time varying s, not a constant s. (b) If s falls as K rises, this reinforces the effect discussed in the previous question that when K rises, ∆K/K falls. Now ∆K/K will fall for two reasons as K rises. This means that the convergence process will be fast in the early stages of development but it will slow down more as development continues. c) In this case the rise in s as K rises will tend to counter the direct effect of K rising on the growth rate. This implies than in the early stages of development the convergence process will be slower but eventually the convergence process will speed up. 11.12 (a) Now we would have ∆K = sAK − δK.

The sf(K, L) curve becomes a straight line. (b) The growth rates of capital and output are constant: ∆K/K = (sAK − δK)/K = sA − δ ∆Y /Y = ∆K/K = sA − δ. These growth rates are positive if sA > δ. There isn’t really a convergence property in this model because the economy simply grows at a constant rate right from the beginning. (c) Diminishing returns is what makes the sf (K, L) curve cross the δK line, and implies that there is a long-run target, K ∗ , towards which the economy converges. If there was only physical capital non-diminishing returns wouldn’t make much sense, because the more machines you add to the production process with the same amount of labor, you should eventually expect diminishing returns to set in. However, if capital includes human capital, and the same argument doesn’t apply. As long as human and physical capital increase together, it is not as obvious that diminishing returns will kick in. 2

Barro, Chapter 12 Questions Craig Burnside Economics 302 University of Virginia 12.3 The real interest rate rises with a temporary increase in G because the aggregate demand curve shifts to the right more than the aggregate supply curve. In order to make demand equal to supply, the real interest rate has to rise. Demand shifts to the right by (1 − α)∆G, because consumption demand decreases by α∆G. Supply shifts to the right by β∆G because output is directly increased by the increase in G. But the demand shift is greater than the supply shift because α + β < 1. 12.4 Crowding out is the description of what happens to consumption and investment demand as the result of an increase in government purchases. Part of this is the direct substitution effect that happens because households don’t want to change their effective consumption, C E = C + αG, so if G rises they reduce C. The other part of this is the intertemporal substitution effect that when the real interest rate rises, C E and I both fall. There is no direct substitution effect on investment in the model we developed in class, but we could easily write down a model where there was one, if we thought of G as included government investment purchases, not just consumption purchases. 12.8 In this question the permanent increase in government purchases will happen in the future. So people will perceive much the same wealth effects of the future increase in government purchases that they would if the increase happened right away. (They would discount them a little bit because they would be coming in the future.) So the shift of the aggregate consumption demand curve would be by the amount (α + β − 1)∆G + MP L∆L, where ∆L represents the increase in labor supply due to the wealth effect. This means that on net we cannot be sure which way consumption demand is shifting (one effect is negative the other is positive). It seems safe to say (from the condition that MUC ×MP L = −MUL, that the shift of the consumption demand curve has to be negative otherwise the shift in labor supply would not be positive. There is no immediate direct substitution effect because G does not rise right away. So we don’t get the additional “α”-shift to the left of aggregate consumption demand curve. We also don’t get the direct shift to the right of G. So, on net, the demand curve shifts left only due to the wealth effect. The supply curve shifts to the right because of the increase in labor supply because of the wealth effect, by MP L∆L. There is no direct effect on supply because G does not rise right away. See Figure 1. So one thing is certain. Supply shifts to the right relative to demand. (a) The net result is a big decline in the real interest rate and an increase in output. Investment has to rise because of the decrease in the real interest rate. But consumption demand will be affected by offsetting forces (a negative wealth effect, but a positive intertemporal substitution effect). Initially, G is unchanged. Labor supply is hit by offsetting forces (a wealth effect that increases labor supply, but an intertemporal substitution effect that decreases it). However, output can only rise if L rises. 1

(b) The nominal interest rate falls because R = r + π falls when r falls. P = M/Φ(R, Y ), so the decline in R, by raising the demand for real balances, Φ, tends to lower the price level. Since Y also rises we can be sure that Φ rises and, therefore, P falls. (c) Prospective G might rise if people foresaw a future war, or a change in administration towards one with the intention of spending more. 12.9 The real wage is equal to the MP L. We know the MP L schedule is unaffected by a G-shock, so all we have to worry about is the effect on labor supply. (a) For the temporary increase in G we found that r rises, and that the level of wealth is roughly unchanged. Therefore, L rises, so MP L falls and w/P falls. (b) For the permanent increase in G we found that r is unchanged, but wealth declines. Therefore, L rises, so MP L falls and w/P falls.

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FIGURE 1 Short-run Effects of a Prospective Permanent Increase in G

r

( β + α − 1)∆G + MPL∆L C

d

MPL∆L

Ys

Id

G

Yd Y

Barro, Chapter 13 Questions Craig Burnside Economics 302 University of Virginia 13.2 If we didn’t hold tax revenue constant then an increase in the tax rate would mix substitution effects with negative wealth effects. 13.3 A rise in the tax rate discourages consumption and investment demand and shifts aggregate demand more to the left than it shifts the aggregate supply curve. Therefore, the after-tax real interest rate declines. Although the after-tax real interest rate declines, the net effect is a decrease in investment demand which reduces the capital stock in the long-run because the tax change is permanent. Notice that the condition (1 − τ )(MP K − δ) = r˜ implies that it is still true that MP K − δ = r

since r˜ = r(1 − τ ). The long-run decline in K raises MP K, and therefore must be associated with a long-run rise in r. 13.4 Since a rise in τ reduces (1 − τ )MP L it should cause labor-supply to decline. But there is another long-run effect of taxation that we ignored in class. A long-run impact of the tax change comes from the change in the long-run capital stock which will be lower. This raises the long-run MP K as in the previous question. But it should lower the long-run MP L because having less capital to work with should lower the marginal product of labor (this effect is absent in the short-run because it takes time for the reduction in capital to occur). Thus labor supply should decline in the long-run. 13.8 A positive income tax rate (as opposed to a zero income tax rate), with an exemption that leaves total taxes paid equal to 0, causes people to reduce consumption demand and labor supply (as we saw in the chapter). One of these changes lowers utility while the other raises utility. However, the tax change can’t we welfare improving. If it were, the household could have afforded to make the choice it now makes before, but it didn’t before. The opposite is true with a negative income tax rate with a negative exemption that leaves taxes raised equal to 0. So both are welfare reducing. 13.9 (a) We have tt /Pt = τ (ct −et ) so that the government budget constraint is Pt Gt +Vt = Tt + Mt − Mt−1 , with Tt /Pt = τ (Ct − Et ). The household budget constraint becomes bt−1 mt−1 vt bt mt + + − τ (ct − et ) = ct + it + + . Pt Pt Pt Pt Pt

(1)

bt−1 mt−1 vt bt mt + + + τ et = (1 + τ )ct + it + + . Pt Pt Pt Pt Pt

(2)

yt + (1 + Rt−1 ) which we can rewrite as yt + (1 + Rt−1 )

1

(b) The after-tax real interest rate is just rt because interest is no longer taxed. (c) Suppose the household decreases ct by 1 widget: this leads to a utility loss = MUCt . We know from (1) that this allows the household to purchase (1 + τ )Pt dollars of bonds (because you save on the purchase and on tax payments) and that this means extra interest and principal at time t + 1 equal to (1 + Rt )(1 + τ )Pt dollars, or (1 + rt )(1 + τ ) = (1 + Rt )(1 + τ )Pt /Pt+1 in widget terms. But the household has to pay consumption taxes on widget purchases, so the household ends up being able to purchase 1 + rt extra widgets at time t. So we get our old condition (3)

MUCt = (1 + rt )MUCt+1

Suppose the household works harder by raising lt by 1 unit of time: this leads to a utility loss = −MULt . From (2) that the household’s extra unit of work translates into MP Lt extra units of output. These extra widgets worth of output allow the household to buy MP Lt /(1 + τ ) extra widgets for consumption.. If the household consumes these it gets a utility gain = MU Ct × MP Lt /(1 + τ ). So MP Lt × MUCt /(1 + τ ) = −MU Lt . Investment decisions aren’t distorted so we get the old condition MP Kt − δ = rt . The net implication of the changes in three of our key equations is that we should now write C d (r , wealth, permanent changes in MP L/(1 + τ ) schedule) −

+

+

Ls (r , wealth, MP L/(1 + τ ) schedule) +

−

+

d

I (r , MP K schedule). −

+

and Y s (r , wealth, MP L/(1 + τ ) schedule, technology level). +

−

+

+

(d) Now a permanent increase in the tax rate τ should shift leftward consumption demand and output supply by roughly the same amount in a diagram increase with the real interest rate on the vertical axis. Investment demand should be unchanged. Thus Y will fall, r will stay about the same, and C and L will both fall. Of course, I will be unchanged. The big difference versus an income tax, is that it causes investment demand to shift to the left as well, so that the after-tax real interest rate would be lower and I would be lower in equilibrium.

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Barro, Chapter 14 Questions Craig Burnside Economics 302 University of Virginia 14.2 If you hold government purchases constant there is no effect on private consumption. The increase in lump-sum taxes allows the government to reduce its debt stock by the amount of the increase in taxes. This means people are lending less to the government. The amount that their lending to the government is reduced by is the same as the amount their disposable income is reduced by so their consumption and investment decisions need not change. 14.3 If the government runs deficits and pays for these by printing money then inflation is the consequence. This will cause the nominal interest rate to rise but the real interest rate will be unaffected. If the government runs a deficit, borrows to finance it, and runs surpluses in the future to pay for the debt it created, then inflation will not result from running the deficits. 14.5 Since taxes will be lower in the future, for a given value of the after tax real interest rate, consumption demand today will be unaffected, but labor supply will decrease now and increase later through a simple intertemporal substitution effect. This is intuitively clear since the disincentive to work now is stronger than the disincentive to work in the future. You can easily see how this effect works by taking an example where the tax rate at time t is τ t and the tax rate at time t + 1 is τ t+1 < τ t . The usual intertemporal condition for consumption is MUCt = (1 + r˜t )MUCt+1 . (1) This condition is not affected by the future tax cut, so as long as there is no wealth effect the consumption demand schedule won’t shift. But notice that the intratemporal conditions for periods t and t + 1 are (1 − τ t )MP Lt × MUCt = −MULt . (1 − τ t+1 )MP Lt+1 × MUCt+1 = −MU Lt+1 . These conditions together with (1) imply: −MULt+1 −MULt = (1 + r˜t ) . (1 − τ t )MP Lt (1 − τ t+1 )MP Lt+1 Since τ t > τ t+1 , for a given MP L schedule and a given real interest rate, the fact that 1 − τ t is smaller than 1 − τ t+1 implies that −MU Lt must be smaller than −MU Lt+1 , so Lt must fall and Lt+1 rise. The net effect of this is that the aggregate supply curve today will shift to the left while the aggregate demand curve remains the same. The result is an increase in r˜t , a decline in Yt , declines in Ct and It , a decline in Lt (implied by the decline in Yt ) and a rise in the real wage rate. 1

14.7 There is no aggregate wealth effect from the tax cut, because the lifetime income of an infinitely lived household is unaffected. The tax cut now implies a tax increase in the future. (a) If people have finite lives and don’t care about descendants they will feel wealthier because they will perceive that the taxes will be borne by others they don’t care about. If they do care about their descendants then they won’t feel wealthier. (b) Wealth increases for people with no children, but decreases for people with more than the average number of children. So the net effect could be zero. (c) If people are risk averse, uncertainty will imply a reduction in wealth. (d) This raises people’s perceptions of future transactions costs, but as long as these are small people will not significantly less wealthy. (e) The net effect could be a rise in wealth if some people cannot borrow due to market imperfections. But it implies that the government has better skill at collecting taxes from people who are bad credit risks that creditors would have at collecting debt payments from them. 14.8 (a) This implies that lump-sum taxes will rise in the future. None of the variables mentioned is affected. (b) This implies no change in the real interest rate, output, and investment. However, the inflation rate and, therefore, the nominal interest rate will rise. The demand for real balances will fall, so the price level will rise. (c) Same as part (a). A decline in transfers is equivalent to a lump-sum tax increase. (d) See the answer to question 12.8, which discusses the effect of a future increase in G, and reverse the effects. 14.9 The two plans are identical with respect to the paths of government spending, so they would have the same implications, except for the fact that the timing of taxes would be different. Under the Reagan plan, the fact that taxes were higher in the early part of the 3-year phase in plan and lower later, implies that the tax cut would induce intertemporal substitution effects similar to what you saw in question 14.5. Relative to the end of the phase-in period, the earlier period would be characterized by lower employment, output and investment and a higher real interest rate. An immediate tax-cut to the long-run tax rates would have induced no intertemporal effects of that kind.

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Barro, Chapter 17 Questions Craig Burnside Economics 302 University of Virginia 17.2 A bank holds checkable deposits as a liability, and things like loans and bonds as earning assets. The spread between the rate of interest on the earning assets and the deposits reflects the bank’s cost of doing business: (i) reserve requirements, (ii) administrative costs of handling deposits and loans, (iii) and normal profit. An increase in the spread could reflect any change in the above items. One of those could be a change in administrative costs, which would presumably lead to a decline in deposits. 17.4 People have an incentive to withdraw their funds if they think a bank failure is imminent because banks only keep a fraction of the deposits on reserve. So if a failure takes place there is a depositor will find himself unable to withdraw his funds when he wants to when he wants to (or, as in the days before deposit insurance, he may find himself losing his money). So there is an incentive to withdraw the funds right away. If one person feels this way, all people may feel this way, and so everyone may run on the bank. Given that the bank only keeps a fraction of deposits on reserve it will fail. Insurance eliminates the risk of losing your money so it reduces the risk of a bank run occurring. 17.5 A shift in money demand away from currency, in favor of checkable deposits, reduces the demand for the real monetary base. Therefore the price level will rise to reduce the real supply of money. 17.7 (a) Raising the reserve requirement raises the demand for the real demand for base money. (b) Thus, it lowers the price level. (c) It lowers the nominal quantity of M1. (d) Raising reserve requirements causes disintermediation, which will lower output and investment.

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Barro, Chapter 18 Questions Craig Burnside Economics 302 University of Virginia 18.2 (a) Both our market clearing model, and the expectational Phillips curve both suggest that there is no systematic relationship between unemployment and inflation. (b) Empirical findings on the Phillips Curve are unclear as to the trade-off between inflation and unemployment. 18.3 If expected inflation were zero, then all the actual inflation was unexpected. So then the relationship between actual inflation and unemployment would coincide with the relationship between unexpected inflation and unemployment predicted by the expectational Phillips curve. The post-WWI data could be explained by shifts in expectations of inflation. 18.5 Changes in the money supply are the result of government actions. Changes in money demand are the result of private decisions. In the market clearing model a rise in supply has a similar effect to a decline in demand: the price level rises. But if supply and demand rose at the same time it is unclear what the effects would be on the price level. Of course, we expect neither to have an effect on output. 18.6 Endogenous money refers to the fact that some changes in the money supply are driven by changes in real activity (not exogenous government behavior). If the government responds to a real event that raises money demand by raising money supply then the money supply will appear to have a causal impact on the economy, even though it is the other way around.

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Barro, Chapter 19 Questions Craig Burnside Economics 302 University of Virginia 19.2 A relative price is the price of one good or service in terms of another good or service. The real wage is an example of a relative price because it is the cost of a worker in terms of goods (widgets). Buyers and sellers of good z both care about Pt (z)/Pt . If Pt (z) and Pt both change by the sane proportion, this leaves Pt (z)/Pt unchanged. Neither the buyer nor seller is affected. 19.4 Yes, there can be unexpected changes in the money supply with rational expectations. Rational expectations doesn’t mean your forecasts will be perfect, it just means that on average you will be right, and you will use all information available to you in making your forecast. The government can’t use the money supply systematically to counteract business cycles. Suppose the government did this by systematically raising the money supply during recessions. Then in a recession people would be able to anticipate a rise in the money supply, so it wouldn’t take them by surprise. As a result the price level would rise in proportion to the money supply increase. 19.5 Persistent deviations of output from trend (due to monetary shocks) could be explained by the fact that investment might change as the result of a monetary shock. Changes in investment have persistent effects because they affect the stock of capital for years to come. 19.6 (a) The gross nominal return on a unit of investment in the first example is Pt+1 (z)MP Kt + Pt+1 (1 − δ) Pt which must equal 1 + Rt = (1 + π t )(1 + rt ). Hence we have %

&

Pt+1 (z) (1 + π t ) MP Kt + 1 − δ = (1 + π t )(1 + rt ) Pt+1 or

Pt+1 (z) MP Kt − δ = rt . Pt+1 (b) The gross nominal return on a unit of investment in the second example is Pt+1 (z)MP Kt + Pt+1 (z)(1 − δ) Pt (z)

which must equal 1 + Rt = (1 + π t )(1 + rt ). Hence we have Pt+1 (z) (MP Kt + 1 − δ) = (1 + π t )(1 + rt ) Pt (z) 1

or MP Kt + 1 − δ =

(1 + πt )(1 + rt ) . 1 + π t (z)

The gross nominal return on a unit of investment in the third example is Pt+1 (z)MP Kt + Pt+1 (1 − δ) Pt (z) which must equal 1 + Rt = (1 + π t )(1 + rt ). Hence we have Pt+1 (z) (1 + π t )(1 + rt ) . MP Kt + (1 − δ) = Pt+1 Pt+1 /Pt (z) 19.7 If money shocks become less predictable from year to year then: (a) The responsiveness of the perceived relative price, Pt (z)/Pte , to the observed local price will fall; i.e. Pte will tend to move with Pt (z). (b) The effect of a given monetary disturbance on output will be smaller because perceived relative prices won’t change much when the monetary shock occurs. (c) The allocation of resources worsens because when real shocks occur that change relative prices, the economy responds less appropriately to those real shocks. 19.12 (a) The irrelevance is that systematic changes in the money supply will be neutral. (b) No, the unpredictable changes in money cause changes in real variables. (c) No. We saw that (i) u.i. programs matter in chapter 10, (ii) changes in government purchases matter in chapter 12, (iii) cutting income tax rates matters as we saw in chapter 13. 19.13 (a) If the Fed has this policy then people understand that it has this policy. Therefore they expect faster money growth when they see a recession happening. Therefore the changes in money that occur are anticipated and have no real effect. (b) No, in this case, the policy will matter because the recession itself is a surprise from the point of view of the public. Therefore, so is the Fed’s response to it. (c) This gets in a logical loop. If people anticipate that the Fed will try to fool them, then they will adjust up their expectations, so the Fed will have to increase the money supply even more, etc. etc. In this case, there is no sensible rational expectations equilibrium.

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Barro, Chapter 20 Questions Craig Burnside Economics 302 University of Virginia 20.1 (a) The big difference between the theories is that the Keynesian theory puts less emphasis on lifetime income (wealth) and more emphasis on current income as a determinant of consumption demand. In the market-clearing model current income doesn’t matter because a household can always borrow or lend to smooth its consumption. (b) The big difference between the models in this case is that the MP K schedule is an important determinant of investment in the market-clearing model, but in the Keynesian model the producer’s forecast of what his additional sales will be in the future is more important. 20.4 In the Keynesian model an increase in the money supply increases the quantity of real balances because prices are sticky. So, for a given level of real activity, the real demand for money must rise to make supply equal demand in the money market. This happens if the real interest rate (and, therefore, the nominal interest rate) falls. 20.7 What is being described here is a decrease in autonomous consumption. (a) From the Keynesian cross diagram we know this will make output fall for a given level of the interest rate. Labor input also falls for a given interest rate. Investment will tend to fall because Y falls. So it’s bit clear which way S = I tends to fall, even though the question began with an increase in desired saving (a decrease in consumption). (b) If we allow r to adjust then we have to think about the answers from (a) as a shift to the left of the IS curve. So the real interest rate and output will fall. The overall effect on saving is unclear because the decline in Y causes I to fall, but the decline in r causes it to rise. Labor input falls because Y falls. (c) In the market-clearing model we did not really have the analog to changes in autonomous consumption so it is hard to answer this question. 20.8 (a) The multiplicative effect on output comes from the fact that the increase in investment leads to an increase in people’s current income, which leads to an increase in forecast future sales, which leads to more investment, etc. Also, the increase in income leads to increased consumption, which increases current income, which increases consumption and investment, etc. etc. (c) The multiplier tells you how much the IS curve shifts. The increase in Y will be smaller than the shift in IS because LM does not shift. How big the increase is depends on the slopes of IS and LM. If LM is very steep then the multiplier will be small. If IS is very steep then the multiplier will be big. (i) If aggregate demand is sensitive to r then the IS curve is flatter (a small change in r leads to a big change in Y ). Hence, the multiplier will be smaller. (ii) If money demand is sensitive to output then the LM curve is flatter (a given change in Y leads to a bigger change in the real interest rate), so the multiplier is bigger. 1

(iii) If money demand is sensitive to the interest rate, then the LM curve is flatter (a given change in Y requires a smaller change in the real interest rate), so the multiplier is bigger. 20.11 (a) If money demand is insensitive to the interest rate, the LM curve is vertical. Shifts in the IS curve will not have any effect on output but will simply move the real interest rate up and down. (b) In this case, the LM curve his horizontal. In this case, shifts in the IS curve have very large effects on output and no effect on real interest rates. The full multiplier effect is felt. (c) In this case, the IS curve will be vertical. Shifts in the LM curve have no effect on output only on real interest rates. (d) In this case, the IS curve is flat. Shifts in the LM curve have big effects on output and no effect on the real interest rate. 20.12 (a) A decline in autonomous demand that leads to a recession will imply that prices will be too high in the Keynesian model. As a result, inflation will actually slow down (below the initial rate π ∗ ) until the economy recovers. (b) You would have to have an independent rise in π ∗ to get higher inflation during the downturn. This would require an acceleration of money growth.

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