solutions mannual

Barro, Chapter 2Questions

Craig BurnsideEconomics 302

University of Virginia

2.2 Total product is f(l) while marginal product isMPL, which is the slope of f(l). If (a)MPL is positive and increasing it means the slope of f(l) is positive (as in Ch. 2), but theslope of f(l) also gets steeper for bigger l (not like Ch.2); (b)MPL is positive and decreasingit means the slope of f(l) is positive and gets ßatter as l increases (as in Ch. 2); (c)MPL isnegative 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 ofwell-being a person gets from consuming a particular basket of goods. Illustrating indifferencecurves is simple and is described in Ch.2. Indifference curves shift around if people havechanging 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 equal1 if ∆l equals 1, in order to stay on that indifference curve. Using (1) this implies that inthis example 0 =MUC +MUL or MUC = −MUL.The question also tells us that MPL > 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 ×MPL×∆l +MUL×∆l.

Using the facts that ∆l = 1, MUC = −MUL and MPL > 1 we have∆u = −MUL× (MPL− 1) > 0

so that the person whould work the extra unit of time.If MPL < 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 raisesthe MPL 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 consumemore and work less (consume more leisure) as long as consumption and leisure are bothnormal goods. The increase in the MPL, by itself, induces a substitution effect that makesthe 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 werean inferior good, then the income effect would induce work to rise (leisure to fall), so the neteffect on l would be that it would increase.

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2.8 If f(l) = Al1/2 +B, then MPL = Al−1/2/2.(a) If A increases this shifts f(l) and MPLup, so we get combined income and subsitutioneffects: c rises, y = c rises, and the effect on l is ambiguous.(b) If B increases this shifts f(l)up leavingMPLunchanged, 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 pointsuch as lI , always is lower than the slope of the line from the origin (0,0) to (lI , yI = f(lI)).If you drew a line tangent to f(l) at lI it would be obvious that its slope was less than theslope 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) inducesincome and substitution effects: c and y rise, but the effect on l is ambiguous. Barro pointsout 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

∆l= −MUL

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. Sincef(l) has not changed, it is obvious that productivity, y/l, falls since the line from the originwill become ßatter.

2

FIGURE 1

FIGURE 2

y

l

f(l)

0 lI

slope of f(l) at lI = slope oftangent line

yI

slope of line fromorigin to (lI,yI) = yI/lI

y

l

f(l)

0

original indifference curve

new indifference curves aftershift in tastes

original optimum (l,c)combination

optimum (l,c) combinationafter shift in tastes




3.8 a) PB should be less than 1. Even though Barro calls the face value of the discountbond the principal, there�s a sense in which you can think of $PB as the principal on thediscount bond (because the principal is usually the amount borrowed by the borrower). Theface value (1$) is paid a period in the future so its as if you are being repaid what I amcalling the principal (PB) plus the interest, 1 − PB. So for the interest to be positive, PBmust be less than 1.

b) The interest rate on bonds is R. A discount bond should also have interest rate R as longas the issuer of the discount bond is no riskier than the issuer of the bond.

c) The interest rate is the interest, 1−PB, expressed as a fraction of the principal, PB, andmust equal R: hence

R =1− PBPB

or

PB =1

1 +R.

This illustrates a general rule: the price of a discount bond is the present value of its facevalue.

d) The interest rate per period for a two period discount bond should also be R. Using thegeneral principal that the price of a discount bond is the present value of the face value weshould have

PB(2−period) =1

(1 +R)2

and

PB(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 willdecrease. Hence you suffer a negative wealth effect.

(b) This should imply no wealth effect. Notice that you can always write the lifetime budgetconstraint as

y1 − c1 + y2 − c21 +R

+y3 − c3(1 +R)2

+ · · ·+ (1 +R)b0P= 0

so if yt = ct for all t, there is no impact on the

y1 − c1 + y2 − c21 +R

+y3 − c3(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 + y2 − c21 +R

+y3 − c3(1 +R)2

+ · · · = 0.

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 inR makes (yt−ct)/(1+R)t−1 terms smaller negative numbers. But the ones that are negativeare being discounted more so they shrink more than the early terms, so for the same yt andct we have

y1 − c1 + y2 − c21 +R

+y3 − c3(1 +R)2

+ · · · > 0.

Hence there will be a positive wealth effect, and the household will raise consumption until,once again,

y1 − c1 + y2 − c21 +R

+y3 − c3(1 +R)2

+ · · · = 0.

3.11 Someone should be indifferent (either a borrower or a lender) between the two waysof 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) inperiod 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 +R1R2

2.

(b) If R2 > R1 then

R >2R1 +R

21

2= R1 +

R212> R1.

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 ofthe household to use. We can rewrite the budget constraint as

Py1+Py2

(1 +R)+· · ·+ PyT1

(1 +R)T1−1+b0(1+R) = Pc1+

Pc21 +R

+· · ·+ PcT2(1 +R)T2−1

+bT2

(1 +R)T2−1.

2

The household would set bT2 = −∞ if it could, because this would let it borrow huge amountand leave the debt unpaid at death. Assuming that nobody is willing to let someone borrowif 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 householdhas no income from selling output, so its consumption is Þnanced by dissaving. From thebudget constraint, when yt = 0 we will literally have

bt − bt−1 = Rbt−1 − Pct.

(c) If people were forced to retire earlier, then, other things equal, they would understandthat they would need to have a bigger stock of assets to run down from their retirementuntil their death. So early in life they would save more (by consuming less) and, becauseconsumption falls, they would also decrease leisure (increase work effort). This increase inwork 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 setbT2 = 0, if T2 is the period of their death. Instead, the planning horizon might include thelifetimes of the children and grandchildren. So it is not obvious how the planning horizonshould be set or that it should not be set to ∞.

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4.2 a) If the person spends Pc = $6000 per year and makes monthly (T = 1/12) with-drawals then each withdrawal is PcT = $500. Average money holdings are PcT/2 = $250.See Figure 1.

b) Now T = 1/6 so each withdrawal is PcT = $1000. Average money holdings are PcT/2 =$500. See Figure 1.

4.3 If Pc = $9000 per year and T = 1/12 then each withdrawal is PcT = $750 and averagemoney holdings are PcT/2 = $375. It may or may not be optimal for the frequency toremain at T = 1/12, if T = 1/12 was optimal when Pc = $6000. The formula for theoptimal T is:

T =

!2

Rc

γ

P.

If Pc rose due to a rise in P , but γ also rose in proportion to P (Barro indicates he wants usto think about the transactions cost as being Þxed in real terms) then there is no change inthe optimal T . However, if Pc rose due to a rise in c, then T should fall: the person shouldvisit 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 realdemand, M/P .

4.6 Velocity is Y/Φ, so the effects on velocity are all in the opposite direction to the effectson M/P .

4.7 (a) If we were to incorporate transactions costs into the lifetime budget constraint wewould have something like

PV(y)− PV(∆mP)− PV(trans. costs) + (1 +R)b0

P= PV(c)

Up til now we have ignored transactions costs in the lifetime budget constraint. Transactionscosts lower wealth.

(b) If γ/P rises people will suffer a negative wealth effect, so they will decrease consumptionand 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. Butto know for sure that saving increased we would need to check whether y − c−transactionscosts increased, and this is not clear.

(c) Yes, we did leave out a substitution effect. In effect, when γ/P rises it raises the effectiveprice of consumption relative to leisure. So people will want to raise leisure and lowerconsumption. The rise in leisure (decline in work effort) offsets the wealth effect discussedin part (b).

4.11 (a) C rising causes M/P to rise. Barro is exploring a subtlety here. He wants you torealize that since

m

P= φ(R, c, γ/P )

the aggregate M/P isM

P="i

φ(R, c, γ/P )

where the notation "i

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

C ="i

c

So if the number of people rises (say by 10%) both C and M/P will rise by the same per-centage (10%). But if little c rises, say by 10%, we know thatM/P rises, but not necessarilyby 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 Rwould make Φ rise and Vfall.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 Barrocites from Goldfeld suggests that the effect of rising GDP on real money demand (throughits impact on real consumption demand), is the weaker effect. I.e. if Y increased 10%, M/Pwould only increase, say, 6%. So we might expect this to be another reason V should risewith economic development.

2

Month

2 4 6 8 10 12

500

Money holdings

1000

Withdrawals every 2 months

Withdrawals every month

FIGURE 1




5.2 a) Aggregate (consumption) demand falls with a rise in the interest rate because theincentive to save is greater. Aggregate supply rises with a rise in the interest rate becauselabor supply increases. These effects are captured as movements along the aggregate demandand 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 increasein wealth causes an increase in leisure and a reduction in labor supply.c) A shift of the production function (if it does not changes theMPL schedule) increases

wealth. Therefore, the effect on aggregate demand is identical to what it was in (b): a shiftto the right. On the other hand, even though work effort will decline, the improvement inthe production function will overwhelm that effect, so that the aggregate supply curve alsoshifts 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 MPL,

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 6would be an appropriate graph.

5.5 In part (a) of 5.4 the increase in consumption demand is as large as the increase inoutput so the marginal propensity to consume is 1. This suggests a permanent shock to theproduction 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 temporaryshock to the production technology.

5.9 Aggregate output should fall in proportion to the decline in the population, as shouldaggregate 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 supplythey should both simply shift to the left by exactly the same amount, leaving the interestrate unchanged.Since P =M/Φ(R, Y, . . .) we might expect the price level to rise sinceM 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 preferenceschange. This means they go from having a utility function U to a utility function �U , say. Inthe 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

M �UC<−MULMUC

.

Why? The question says that the household starts to like consumption more than beforerelative 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 likesconsumption more than it did before). This tells you that indifference curves are ßatter.The slope of an indifference curve is −MUL/MUC so it must be a smaller number for theindifference curves to be ßatter.He also wants you to assume that at any choice of (c1, c2) that

M �UC1

M �UC2=MUC1MUC2

.

That is, there is no change in how the household values consumption in one period relativeto another.At the old equilibrium MPL = −MUL/MUC. The old equilibrium, (c∗, l∗) cannot still

be the equilibrium because MPL would be the same, and −M �UL/M �UC 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

its consumption causing MPL to fall some, and −M �UL/M �UC to rise some until they areequal again. This means output will also rise. There should be no effect on the interest ratebecause consumption (and labor) will rise in all periods, and consumption demand shouldshift out by about the same amount as aggregate supply. For a Þxed money supply, anincrease in output shifts the demand for real balances to the right, causing the equilibriumvalue of P to fall.

5.13 (a) If we considered a rise in the interest rate out of equilibrium, then as we increasedthe interest rate C would fall as we moved along the demand curve, and Y would rise aswe moved along the supply curve. So C/Y would fall. But, of course, this is not what isgoing on in the example. In the example, the shock to the production technology causes asmall decrease in wealth which induces a small decline in Cd for any given interest rate. Onthe other hand, the shock to the production technology causes a large decrease in outputsupplied, Y s, at any given interest rate. So at the original interest rate Cd/Y s would actuallyrise. However, C/Y must always equal 1 in equilibrium. The only way for this to happen inthe example, is to increase the interest rate to make Cd even smaller (along the new demandcurve) and make Y s larger (along the new supply curve), until they are equal. Aggregatesaving 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 seehow R affects C/Y using econometric methods. Of course, our theory predicts that C/Y isalways 1, and does not respond at all to changes in the interest rate. However, one would notwant 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 ofstudy 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 thatC changes depending on the interest rate and level of wealth. So within the context of ourmodel, I�d have to have a method for Þguring out what part of any movement in consumptionwas due to the interest rate changing and what part was due to ongoing changes in perceivedwealth.Consider the case illustrated in Figure 5.4. Suppose the shock to the production technol-

ogy lasts for only one period. Then in period t consumption falls to the amount Ct = Y ∗" andinterest rates rise to the level Rt = R∗". If there are no other shocks in period t + 1, thedemand curve should still be the one labeled (Cd)" (recall that the shock to the productiontechnology 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 periodt+ 1 the equilibrium will be at the intersection of (Cd)" and Y s so that Ct+1 = Y ∗"" a valueslightly 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 demandcurve. Therefore, examining the data in this way, could, in principle, uncover somethingabout the relationship between Cd 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 ofshocks 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 theopposite of Figure 2.9 in the textbook. The fall in f(l) would cause c and y to fall, and lto increase. In the market economy the effects are identical. The permanent wealth effectshifts Cd by about the same amount as Y s. This means R doesn�t change and C falls. Thenegative 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 aslarge 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 thinkabout intertemporal substitution and this is what induces the effect on labor supply in themarket economy. In the absence of a linkage between periods the Robinson Crusoe case isfundamentally different in this respect.

3




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. Sincethe MPL schedule is unaffected by R, the real wage falls and the equilibrium quantity oflabor rises.

6.6 In the example in the text, region Auses technology fA(l) to produce output, whileregion B uses technology (with a higher MPL schedule), fB(l). If there were separate labormarkets in the two regions, they would have separate wage rates given by MPLA(lA) andMPLB(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 some-

thing like the following: suppose lA and lB are roughly equal so that lA = lB = l–this impliesthat real wages are higher in region B, (w/P )B = MPLB(l), than in region A, (w/P )A =MPLA(l) If the level of technology is similar across the regions, this means that output issimilar across the two regions so that yA = fA(lA) = fA(l) = fB(l) = fB(lB) = yB = y.1

Since real profits are given by the difference between the value of output and the wage billit is clear that real profits are higher in region A:

πA = yA − w

P

A

lA = y − w

P

A

l

πB = yB − w

P

B

lA = y − w

P

B

l < πA.

This means that when trade in labor opens up, which will equate real wages acrossregions, real wages paid by firms in region B will fall and in region A they will rise, becauseworkers from region Awill cross to region B to perform work. So you might think of theanswer to this question being: workers from region Awill like the single market, while workersfrom 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 istrue 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 ispositive, however, because the opening of the economy-wide labor market creates an efficiencygain. If aggregate labor supply is unchanged, aggregate production will rise because it ismore efficiently allocated. So aggregate consumption will rise. As long as this increase inconsumption is shared across the two countries the utility of the average household is likelyto increase.

1If 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 acrossthe population, the effects of the opening of a single labor market could have an importantpolitical impact: if workers are not also shareholders, then workers in region B will beopposed to opening the single market, whereas firms in region Bwill be opposed. Theopposite 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 immigrationalso have this flavor.

6.9 (a) In Chapter 5 we saw that a temporary parallel downward shift in the productiontechnology lowers wealth, by a small amount, and therefore shifts Cd slightly to the left. Wealso saw that it shifts Y s substantially to the left (see Figure 5.4). This lowers C and Y andraises 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/Pshould fall and L should increase. See Figure 1(a)(b) No.(c) The answer to (a) changes. A downward shift inMPL 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 shiftingto the right, the labor demand curve will shift down. We usually assume the effect on labordemand is strong enough to outweigh the effect on labor supply so that both w/P and Lshould fall. See Figure 1(b).

6.11 (a) Average productivity or APL, is not the same as MPL. Average productivitytells you about the ratio of total production to total labor input. So in a sense, it tells youabout the average contribution of each unit of labor being used to total output. On the otherhand, the marginal product of labor tells you something about the contribution of the lastunit of labor to the production process to total output. Since the contribution of additionalunits of labor is always less than the contribution of previous units, MPL is less than APL.Notice, however, that APL should also decline as the number of workers used increases.Figure 2(a) illustrates what the MPL and APLmight 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 MPL schedule unchanged. As a result of this shock Yfalls and L rises. So it is unambiguous that APL = Y/L falls. So APL falls when Y falls.(b) ii. In 6.10b and 6.9c, instead, we imagine that the f(l) function and the MPL

schedule both fall. We saw that this implies Y and L both falling. So offhand, it is notobvious in which way APLmoves. Notice, however–from Figure 1(b)–that the equilibriumMPL falls despite the fall in L (this is because theMPL schedule changed). This means themarginal product of the last worker shifted. If the whole MPL schedule shifted, as a resultof a proportional shift in f(l), this should mean that the contribution of each worker (fromfirst to last) to output declined. Hence the average product of labor should have declined aswell. So again, APL should fall.

2

FIGURE 1

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

Ld

Ls (Ls)'w/P

L

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

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

Ld

Ls (Ls)'w/P

L

(Ld)'

(r*,L*)

(r*,L*)'

FIGURE 2

The Average and Marginal Products of Labor

MPLAPL

L

MPL

APL




7.5 (a) Yes, this theory of money demand accords with what we learned in Chapter 4. Theonly difference is that in Chapter 4, Barro spent some time arguing that money demandincreases with c, but he also argued that it might not go up one-for-one with c (see page143) because of economies of scale in cash holding. And, of course, the Baumol-Tobin modelpredicted cash holdings in proportion to c1/2 not c.(b) Notice that if M/P = YΨ(R) then the inßation rate is

πt ≈ µt − gt − ψtwhere µt = Mt+1/Mt − 1 is the money growth rate, gt = Yt+1/Yt − 1 is the growth rate ofreal GDP, and ψt = Ψ(Rt+1)/Ψ(Rt) − 1 is the growth rate of Ψ(R).1 Assuming that overlong periods of time we can ignore the ψt term (and set it to zero), this means that inßationwill equal money growth minus real growth. So the higher real growth is the more moneygrowth 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, ψt will be negative. Hence, inßation will be higher than it otherwisewould be, between periods t and t+ 1.(d) The answer is much as in part (c). Since R ≈ re + πe, re is constant and πe 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

πt ≈ µt − φtwhere πt = Pt+1/Pt − 1, µt = Mt+1/Mt − 1 and φt = Φ(Rt+1, Yt+1)/Φ(Rt, Yt) − 1. Theregression estimates a relationship between πt and µt of the form

πt = a+ bµt.

Given our theory we would expect b = 1 as long as φt was roughly constant. The estimateof 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 − πt ≈ φt. If we run a regression of the form

µt − πt = d+ cgt1To see this, notice that since Mt/Pt = YtΨ(Rt), this means Pt+1/Pt =

(Mt+1/Mt)/{(Yt+1/Yt)[Ψ(Rt+1)/Ψ(Rt)]}, so 1 + πt = (1 + µt)/[(1 + gt)(1 + ψt)]. It is mathemati-cally true that if µt, gt and ψt are small decimals, πt ≈ µt − gt − ψt.

1

where gt = Yt+1/Yt − 1 we might think of the gt term as capturing the effect of outputgrowth on the growth for the demand for real balances, φt. So we would expect a positiverelationship. We end up with the following estimate

µt − πt = 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

πt = −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 opposedto indexed bonds) is hurt if inßation turns out to be less than expected. In this case, it meansthe borrower will expect his payments to be more costly in real terms than he previouslyexpected. This means that if there is a decline in inßation expectations between when theborrower 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 seeprepayment 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 wouldnot 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 morevolatile.

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8.2 If someone gives up a unit of consumption at date t, they gain Pt dollars, which can beused to purchase bonds maturing at t+ 1 that will pay Pt(1 +R) dollars. These additionaldollars, in turn, will allow the person to purchase Pt(1 +R)/Pt+1 = (1+R)/(1 + π) = 1+ rextra 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 compute1 +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 borrowtoo much to finance higher consumption, and output would be too low because labor supplywould be lower.

8.8 The answer here depends on how you define revenue. If you define it in dollar termsthen the answer is yes. The government can always dump as much money into the economyas it wants, and its revenue is Mt −Mt−1, so the more money it pours in the more revenueit gets. But if you ask the question in terms of the purchasing power of the government’srevenue then the answer is no.Notice that the purchasing power of the government’s money is

Mt −Mt−1Pt

.

Take the example where Y and r and the money growth rate, µ, are all constant. In thatcase, 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−1Pt

=Mt

Pt− Mt−1Pt−1

Pt−1Pt

= Φ(r + µ, Y ) 1− 1

1 + π= Φ(r + µ, Y )

µ

1 + µ.

An increase in πwill make the Φ(r + µ, Y ) term smaller, while making the µ/(1 + µ) termbigger (you should verify the latter statement by checking that µ/(1 + µ) goes up if µ goesup). One effect makes revenue smaller while the other makes it bigger. For small values ofµ, 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 seeΦ(r + µ, Y )→ 0, whereas µ/(1 + µ)→ 1, implying that revenue might actually get close to0 for very large money growth rates.

1

8.9 (a) If you could hold goods between periods (i.e. they were not perishable) and theirreal value depreciated at the rate δ between periods, then the “nominal interest rate” toholding them would be given by

1 +RG =Pt+1(1− δ)

Pt= (1 + π)(1− δ).

So RG ≈ π − δ. The interest rate on bonds would still be R = r + π, so it seems unlikelythat anyone would actually store the goods as long as R > RG, or r > −δ. The demand formoney should still depend on R, and should remain unaffected as long as r > −δ.(b) Since R = r + πe, if πe 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 tostoring goods as a way of saving. At that point, the demand for money would depend on thespread 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 πe 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 inR 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 (WaterlowCo.) to deliver them fake notes the swindlers were able to use the fake notes to purchase realgoods from the Portuguese economy. In a sense they were able to take some of what wouldotherwise have been the Portuguese government’s “tax” revenue and use it to purchase goodsthe Portuguese government could otherwise have acquired. Although printed money is not aclaim to anything concrete, it has a value determined by people’s willingness to hold it. Asa result, the swindlers were able to acquire goods by doing something only the Portuguesegovernment was allowed to do: i.e. printing money.

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9.4 No, because we know that individual households can borrow in order to Þnance in-vestment. Of course, since aggregate borrowing must be zero, at the aggregate level higherinvestment requires higher saving. Saving always equals investment in equilibrium. If savingwere 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 declinein the MPK reduces investment demand and therefore shifts the aggregate demand curveto 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 consum-ables equals the price of capital because there is no real distinction between the investmentgood and the consumption good. If someone wants a consumption widget or an investmentwidget they just go to the single widget market and purchase the widget for whatever pur-pose they want to use it for. If the prices were different then nobody would buy the moreexpensive 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 sameprice because otherwise producers would only produce the more expensive one.(b) If for some reason investment demand was so low that aggregate Id was actually

negative, then the price for investment goods would fall below the price of consumptiongoods, 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, thegovernment 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+1MPKt+Pt+1(1−a)(1− δ) because Barro assumesyou have to give back the same percentage tax credit to the government if you resell thecapital. Hence, the nominal return to capital is

Pt+1MPKt + Pt+1(1− a)(1− δ)(1− a)Pt

which must equal 1 +Rt in equilibrium. Hence we get the result

MPKt = (rt + δ)(1− a) (1)

1

whereas if the tax credit was not in place we would have

MPKt = rt + δ.

Clearly the higher the tax credit, 0 ≤ a < 1, the lower MPKt will be, and hence the higherdesired capital and investment demand will be.

9.11 (a) Separating ownership of the capital from the use of the capital might be usefulsimply because it might generate returns from specialization. Not all people who are skilledin managing a portfolio of owned capital are also skilled in capital use. So there might begains from specialization if the owners of capital (shareholders) allowed others to be theusers (managers) of the capital. The downside to such an arrangement is that the ownersstill 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 qualitieswhich 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 marketeconomy we have described. Robinson would now have a way to save from one period tothe 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 QuestionsCraig BurnsideEconomics 302


10.2 On reason to reject an offer of a wage above wu is that job search is costlier if youare are working. So there may be some value in staying unemployment in order to wait fora 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 eachother. 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 numberof workers moving into employment (findings) matches the number of workers moving out ofemployment (separations). The natural rate can change if any of the permanent factors thataffect the finding and separation rates change. The unemployment rate itself can change dueto a change in the natural rate, but also because of temporary changes in job finding andseparation rates.

10.5 (a) By raising the reservation wage, wu, an increase in unemployment benefits reducesthe 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’tbe 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 durationof 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 greaterthan wu comes in.

10.6 Use the formulas

Lt+1 = (1− σ)Lt + φUt (1)

Ut+1 = σLt + (1− φ)Ut. (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 millionunemployed 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-runLt 92.0 92.7 93.2 93.6 95.2Ut 8.0 7.3 6.8 6.4 4.8

1



11.1 No. The change in the capital stock is ∆K = sf(K,L) − δK. As can be seen inFigure 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 theincome level (or the capital-labor ratio and output-labor ratio) converge to long-run targetvalues. Absolute convergence refers to the situation where all countries are converging to-wards the same long-run targets. Relative, or conditional, convergence refers to the situationwhere different countries may have different long-run target values.

11.6 The association between the real interest rate and the capital stock depends onwhether the convergence process is still happening or not. For example, suppose that the cap-ital stock is growing both because of convergence effects (because the actual capital stockis below the long-run target path) and because of technological progress and populationgrowth (the long-run target path is rising). In this case, the convergence effects imply thatthe MPK is higher than the MPK 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 thelong-run path, the MPK falls and gets closer to its long-run path, and the real interest ratefalls.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 ratewill remain constant. So the theory is not at odds with the data if you think of 1840-1900as the US period of convergence, and 1900-2000 as a period in which the US had alreadyconverged very close to its long-run target path.

11.9 (a) sf(K,L) = δK becomes

sAKαL1−α = δK

which, if you solve it for K implies that

K∗ = (sA/δ)1/(1−α) L.

Of course this meansY ∗ = f(K∗, L) = A1/(1−α) (s/δ)α/(1−α) L.

1

(b) Steady state investment and consumption are

I∗ = δK∗ = δ−α

1−α (sA)1/(1−α) L

C∗ = Y ∗ − sY ∗ = (1− s)A1/(1−α) (s/δ)α/(1−α) L(c)

∆K

K= sAKα−1L1−α − δ

To know whether this growth rate rises or falls with the capital stock we need to find thederivative of the growth rate with respect to K:

∂(∆K/K)

∂K= (α− 1)sAKα−2L1−α < 0

because 0 < α < 1. So the growth rate of the capital stock falls as K rises.(d) If L is constant then

∆Y

Y= α

∆K

K= α(sAKα−1L1−α − δ).

Obviously the growth rate of Y falls as K rises, because the growth rate of K falls as Krises. 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 thatthe convergence process will be fast in the early stages of development but it will slow downmore 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 convergenceprocess 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 thismodel 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 wasonly physical capital non-diminishing returns wouldn’t make much sense, because the moremachines you add to the production process with the same amount of labor, you shouldeventually 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 increasetogether, it is not as obvious that diminishing returns will kick in.

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12.3 The real interest rate rises with a temporary increase in G because the aggregatedemand curve shifts to the right more than the aggregate supply curve. In order to makedemand 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 rightby β∆Gbecause output is directly increased by the increase in G. But the demand shift isgreater than the supply shift because α+ β < 1.

12.4 Crowding out is the description of what happens to consumption and investmentdemand as the result of an increase in government purchases. Part of this is the directsubstitution effect that happens because households don’t want to change their effectiveconsumption, CE = C + αG, so if G rises they reduce C. The other part of this is theintertemporal substitution effect that when the real interest rate rises, CE and I both fall.There is no direct substitution effect on investment in the model we developed in class, butwe could easily write down a model where there was one, if we thought of G as includedgovernment investment purchases, not just consumption purchases.

12.8 In this question the permanent increase in government purchases will happen in thefuture. So people will perceive much the same wealth effects of the future increase in govern-ment purchases that they would if the increase happened right away. (They would discountthem 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+MPL∆L, where ∆L represents the increase in labor supply due to the wealtheffect. 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 thatMUC×MPL = −MUL, that the shift of the consumption demand curve has to be negativeotherwise 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 leftonly due to the wealth effect.The supply curve shifts to the right because of the increase in labor supply because of the

wealth effect, byMPL∆L. There is no direct effect on supply because G does not rise rightaway. 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 consumptiondemand will be affected by offsetting forces (a negative wealth effect, but a positive intertem-poral 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 thatdecreases 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 MPL. We know the MPL schedule is unaffected by aG-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 MPL falls and w/P falls.(b) For the permanent increase in Gwe found that r is unchanged, but wealth declines.

Therefore, L rises, so MPL falls and w/P falls.

2

r

Y

dC

dY

dI

FIGURE 1Short-run Effects of a Prospective

Permanent Increase in G

G

LMPL∆sY

LMPLG ∆+∆−+ )1( αβ



13.2 If we didn’t hold tax revenue constant then an increase in the tax rate would mixsubstitution effects with negative wealth effects.

13.3 A rise in the tax rate discourages consumption and investment demand and shiftsaggregate demand more to the left than it shifts the aggregate supply curve. Therefore, theafter-tax real interest rate declines. Although the after-tax real interest rate declines, thenet effect is a decrease in investment demand which reduces the capital stock in the long-runbecause the tax change is permanent. Notice that the condition

(1− τ)(MPK − δ) = r̃

implies that it is still true thatMPK − δ = r

since r̃ = r(1−τ). The long-run decline in K raisesMPK, and therefore must be associatedwith a long-run rise in r.

13.4 Since a rise in τ reduces (1− τ)MPL 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 belower. This raises the long-run MPK as in the previous question. But it should lower thelong-runMPLbecause having less capital to work with should lower the marginal product oflabor (this effect is absent in the short-run because it takes time for the reduction in capitalto 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 exemptionthat leaves total taxes paid equal to 0, causes people to reduce consumption demand andlabor supply (as we saw in the chapter). One of these changes lowers utility while the otherraises utility. However, the tax change can’t we welfare improving. If it were, the householdcould 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 PtGt+Vt =Tt +Mt −Mt−1, with Tt/Pt = τ(Ct −Et). The household budget constraint becomes

yt + (1 +Rt−1)bt−1Pt

+mt−1Pt

+vtPt− τ(ct − et) = ct + it + bt

Pt+mt

Pt. (1)

which we can rewrite as

yt + (1 +Rt−1)bt−1Pt

+mt−1Pt

+vtPt+ τet = (1 + τ)ct + it +

btPt+mt

Pt. (2)

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 interestand 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 onwidget purchases, so the household ends up being able to purchase 1 + rt extra widgets attime t. So we get our old condition

MUCt = (1 + rt)MUCt+1 (3)

Suppose the household works harder by raising lt by 1 unit of time: this leads to a utilityloss = −MULt. From (2) that the household’s extra unit of work translates into MPLtextra units of output. These extra widgets worth of output allow the household to buyMPLt/(1 + τ) extra widgets for consumption.. If the household consumes these it gets autility gain =MUCt ×MPLt/(1 + τ). So MPLt ×MUCt/(1 + τ) = −MULt.Investment decisions aren’t distorted so we get the old condition

MPKt − δ = rt.

The net implication of the changes in three of our key equations is that we should now write

Cd(r−,wealth+,permanent changes in MPL/(1 + τ) schedule

+)

Ls(r+,wealth− ,MPL/(1 + τ) schedule

+)

Id(r−,MPK schedule+

).

andY s(r

+,wealth− ,MPL/(1 + τ) schedule

+, technology level

+).

(d) Now a permanent increase in the tax rate τ should shift leftward consumption demandand output supply by roughly the same amount in a diagram increase with the real interestrate on the vertical axis. Investment demand should be unchanged. Thus Y will fall, rwillstay 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 lowerin equilibrium.

2



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 amountof the increase in taxes. This means people are lending less to the government. The amountthat their lending to the government is reduced by is the same as the amount their disposableincome 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 isthe consequence. This will cause the nominal interest rate to rise but the real interest ratewill 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 interestrate, consumption demand today will be unaffected, but labor supply will decrease now andincrease later through a simple intertemporal substitution effect. This is intuitively clearsince 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 forconsumption is

MUCt = (1 + r̃t)MUCt+1. (1)

This condition is not affected by the future tax cut, so as long as there is no wealth effectthe consumption demand schedule won’t shift. But notice that the intratemporal conditionsfor periods t and t+ 1 are

(1− τ t)MPLt ×MUCt = −MULt.

(1− τ t+1)MPLt+1 ×MUCt+1 = −MULt+1.These conditions together with (1) imply:

−MULt(1− τ t)MPLt

= (1 + r̃t)−MULt+1

(1− τ t+1)MPLt+1.

Since τ t > τ t+1, for a givenMPL schedule and a given real interest rate, the fact that 1− τ tis smaller than 1 − τ t+1 implies that −MULt must be smaller than −MULt+1, so Ltmustfall 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 r̃t, a decline inYt, declines in Ct and It, a decline in Lt (implied by the decline in Yt) and a rise in the realwage rate.

1

14.7 There is no aggregate wealth effect from the tax cut, because the lifetime income ofan infinitely lived household is unaffected. The tax cut now implies a tax increase in thefuture.(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. Ifthey 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 frompeople who are bad credit risks that creditors would have at collecting debt payments fromthem.

14.8 (a) This implies that lump-sum taxes will rise in the future. None of the variablesmentioned 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 realbalances 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, sothey would have the same implications, except for the fact that the timing of taxes wouldbe different. Under the Reagan plan, the fact that taxes were higher in the early part of the3-year phase in plan and lower later, implies that the tax cut would induce intertemporalsubstitution effects similar to what you saw in question 14.5. Relative to the end of thephase-in period, the earlier period would be characterized by lower employment, output andinvestment and a higher real interest rate. An immediate tax-cut to the long-run tax rateswould have induced no intertemporal effects of that kind.

2



17.2 A bank holds checkable deposits as a liability, and things like loans and bonds asearning assets. The spread between the rate of interest on the earning assets and the depositsreflects the bank’s cost of doing business: (i) reserve requirements, (ii) administrative costsof 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 isimminent because banks only keep a fraction of the deposits on reserve. So if a failure takesplace there is a depositor will find himself unable to withdraw his funds when he wants towhen he wants to (or, as in the days before deposit insurance, he may find himself losinghis money). So there is an incentive to withdraw the funds right away. If one person feelsthis way, all people may feel this way, and so everyone may run on the bank. Given that thebank only keeps a fraction of deposits on reserve it will fail. Insurance eliminates the risk oflosing 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, reducesthe demand for the real monetary base. Therefore the price level will rise to reduce the realsupply of money.

17.7 (a) Raising the reserve requirement raises the demand for the real demand for basemoney.(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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18.2 (a) Both our market clearing model, and the expectational Phillips curve both suggestthat 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. Sothen the relationship between actual inflation and unemployment would coincide with therelationship between unexpected inflation and unemployment predicted by the expectationalPhillips 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 moneydemand are the result of private decisions. In the market clearing model a rise in supply hasa similar effect to a decline in demand: the price level rises. But if supply and demand roseat the same time it is unclear what the effects would be on the price level. Of course, weexpect neither to have an effect on output.

18.6 Endogenous money refers to the fact that some changes in the money supply aredriven by changes in real activity (not exogenous government behavior). If the governmentresponds to a real event that raises money demand by raising money supply then the moneysupply will appear to have a causal impact on the economy, even though it is the other wayaround.

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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 termsof goods (widgets). Buyers and sellers of good z both care about Pt(z)/Pt. If Pt(z) and Ptboth change by the sane proportion, this leaves Pt(z)/Pt unchanged. Neither the buyer norseller 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 onaverage you will be right, and you will use all information available to you in making yourforecast.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 reces-sions. Then in a recession people would be able to anticipate a rise in the money supply, soit wouldn’t take them by surprise. As a result the price level would rise in proportion to themoney supply increase.

19.5 Persistent deviations of output from trend (due to monetary shocks) could be ex-plained by the fact that investment might change as the result of a monetary shock. Changesin 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)MPKt + Pt+1(1− δ)

Pt

which must equal 1 +Rt = (1 + πt)(1 + rt). Hence we have

(1 + πt)Pt+1(z)

Pt+1MPKt + 1− δ = (1 + πt)(1 + rt)

orPt+1(z)

Pt+1MPKt − δ = rt.

(b) The gross nominal return on a unit of investment in the second example is

Pt+1(z)MPKt + Pt+1(z)(1− δ)

Pt(z)


Pt+1(z)

Pt(z)(MPKt + 1− δ) = (1 + πt)(1 + rt)

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or

MPKt + 1− δ =(1 + πt)(1 + rt)

1 + πt(z).

The gross nominal return on a unit of investment in the third example is

Pt+1(z)MPKt + Pt+1(1− δ)

Pt(z)


Pt+1(z)

Pt+1MPKt + (1− δ) =

(1 + πt)(1 + rt)

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)/P et , to the observed local

price will fall; i.e. P et 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 chapter13.

19.13 (a) If the Fed has this policy then people understand that it has this policy. Thereforethey expect faster money growth when they see a recession happening. Therefore the changesin 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 supplyeven more, etc. etc. In this case, there is no sensible rational expectations equilibrium.

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20.1 (a) The big difference between the theories is that the Keynesian theory puts lessemphasis on lifetime income (wealth) and more emphasis on current income as a determi-nant of consumption demand. In the market-clearing model current income doesn’t matterbecause a household can always borrow or lend to smooth its consumption.(b) The big difference between the models in this case is that the MPK schedule is an

important determinant of investment in the market-clearing model, but in the Keynesianmodel the producer’s forecast of what his additional sales will be in the future is moreimportant.

20.4 In the Keynesian model an increase in the money supply increases the quantity ofreal balances because prices are sticky. So, for a given level of real activity, the real demandfor money must rise to make supply equal demand in the money market. This happens ifthe 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 willtend to fall because Y falls. So it’s bit clear which way S = I tends to fall, even though thequestion 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 onsaving is unclear because the decline in Y causes I to fall, but the decline in r causes it torise. Labor input falls because Y falls.(c) In the market-clearing model we did not really have the analog to changes in au-

tonomous consumption so it is hard to answer this question.

20.8 (a) The multiplicative effect on output comes from the fact that the increase ininvestment leads to an increase in people’s current income, which leads to an increase inforecast future sales, which leads to more investment, etc. Also, the increase in income leadsto increased consumption, which increases current income, which increases consumption andinvestment, 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 onthe slopes of IS and LM. If LM is very steep then the multiplier will be small. If IS is verysteep 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.

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(iii) If money demand is sensitive to the interest rate, then the LM curve is flatter (agiven change in Y requires a smaller change in the real interest rate), so the multiplier isbigger.

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 realinterest 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 isfelt.(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 priceswill be too high in the Keynesian model. As a result, inflation will actually slow down (belowthe 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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