Macroeconomics and Economic PolicyGregory de Walque

Chapter 11: exercises

11.1 .

(a) True in the simplied framework of the Solow model where

- the economy is assumed to be closed (no foreign saving)- the government budget is supposed to be in equilibrium (G =T )

such that the only possible source of investment is the privatesaving. Therefore It = St = sYt.

(b) False. A higher investment (=saving) rate will temporarily in-crease the growth rate of output. This eect is however limitedin time and the economy will converge towards a new long runequilibrium with a higher output per worker but a zero growthrate.

(c) True. If capital never depreciated, then an increase in capital stockwill always have a higher return in terms of GDP than the de-preciation of capital (equal to zero by assumption), even in thepresence of decreasing returns in capital. Such an economy wouldnever reach its long run equilibrium and would be growing forever.

(d) False. The higher the saving rate, the higher the GDP per worker.However, this is dierent from consumption per worker. In par-ticular with s = 1, we have that consumption is nil since Ct =(1 s) Yt. There exists a particular saving rate sGR, namedthe Golden Rule saving rate, such that consumption per worker ismaximized.

(e) False for two reasons.

i. Shifting from a pay-as-you-go (PAYG) pension system to afully funded (FF) one cannot increase consumption in theshort run since the pensioners will initially not receive theirpension benet anymore (as workers will now save throughbanks instead of paying a pension to the elderly in expecta-tion of the same treatment by the next generation when theywill be old). Therefore aggregate consumption will be initiallydecreased.

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ii. In the long run, it is not obvious that shifting from a PAYGpension system to a FF pension system would increase con-sumption. This can only be the case if s < sGR. If s > sGR,the economy is already saving too much and shifting towardsa FF system will make it even worse.

(f) This is actually the viewpoint defended since a long time by Mar-tin Feldstein (Harvard University) and some followers. However,the computation of the ideal (I mean the Golden Rule) savingrate is not trivial. It depends on several parameters (, the exactfunctional form of the production function, and as seen in chap-ter 12, of gN and gA). Feldsteins computation for the US indeedadvocates for an increase of the saving rate which can be engi-neered through a fully funded pension system. However, since thepension system is now mainly pay-as-you-go, switching from onesystem to the other cannot be done at once since it will makesurfer strongly the actual generation, which will have to pay forthe pension of their parents (pay-as-you-go pension system) andat the same time save in nancial assets for their own pension.This process must therefore be spread over several generations.The transition does not need to be done through tax breaks forsavings. This is one possibility (which is used at very little scalein Belgium) but not the only one. It could also be done through agradual (very slow) reduction of the pension benets paid underthe PAYG regime and the accumulation of the excess into a trustfund. Note however that PAYG and FF pension system have boththeir own dangers:

- in a PAYG pension system, if the population stop to grow,or even start to decrease, the burden of the pension of theelderly on the shoulders of the population in age to work getsheavier and this can become unsustainable

- in a FF system, the saving are placed in some fund through thenancial system. In the case of a major nancial disruptionas we observed in 2008 and since then, the nancial assetsserving to save may disappear partially or totally. There isthus an intrinsic risk associated with this form of pensionsystem, which is due to the nature of the nancial assets.Diversication of the portfolio surely helps but is of little helpin the presence of a systemic crisis.

(g) True. Education is a particular form of investment, an investmentin human capital. Education does not only help the educated per-

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son to get a better productivity and then a higher wage, but thisperson furthermore helps to improve the global production level,with positive externalities for the society as a whole. It is the verynature of positive externalities to be insu ciently nanced by theprivate sector. Therefore a subsidy is fully justied, as long as it islower than the derived advantage given by the positive externalityproducing good (here education).

11.2 It is clear from this chapter that in the long run there is no paradoxof saving. Indeed, we have seen that a higher saving rate will alwaysyield to a higher GDP per worker in the long run. Indeed, a highersaving rate means more capital per worker which is a production factorallowing to produce more.

11.3 We agree that the long run growth rate of the economy is independent ofthe saving rate. Indeed, the Solow model establishes that in the long runthe economy converges towards an equilibrium growth path such thatgY = gN + gA. This growth path is totally independent of the savingrate. However, we must disagree with the statement that one shouldntworry about the saving rate. Even though the long run growth rate ofthe economy is independent of s, the level of the GDP/per worker inthe long run is directly related to the saving rate, and this is even moretrue for the consumption per worker, since the choice of the saving ratecan help maximize the aggregate consumption per worker.

11.4 Shifting from a PAYG pension system towards a FF pension system willincrease the saving rate s since young workers will now save throughbank accounts (and banks will then invest this money) instead of giv-ing it directly to the elder people who use it for their consumption.However, this increase in s does not aect the long run growth rate ofoutput per worker since the latter is independent of s in the long run.However, it will decrease the long run output per worker, as this one isalways growing in s (and is maximized for s = 1). However the answeris much more complicated in what concerns the long run level of con-sumption per worker: it actually depend of the position of the initialsaving rate with respect to the Golden Rule saving rate. If sinit < sGR,then shifting from a PAYG pension system to a FF pension system willincrease s above sinit and bring it closer to sGR, such that consumptionper worker will increase. If sinit > sGR, then shifting from a PAYGpension system to a FF pension system will increase s above sinit andbring it even further from sGR, such that consumption per worker willdecrease.

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11.5 In the long run,

- output per person will be positively aected by the right to ex-clude saving from the tax base. Indeed, this will act as an incentiveto save more since, beside the usual benet from saving it further-more drive a "scal dividend" by allowing to pay less taxes.

- if, for a constant population more women enter the labor marketand thus the labor force, this will have an impact on the levelof output which will increase since the labor force is augmented.However, output per worker will not be modied in the long runsince this higher output will be produced by a higher number ofworkers. But the question relates to output per person which willbe denitely increased since the population is now made of a largernumber of person participating to the production of market goodsand services while less persons stay away from the productionsector (note that women staying home produce lots of goods andservices but these are not computed into the GDP).

11.6Y = 0:5

pKpN

(a) Then output per worker is equal to

Y

N=

0:5pKpN

N

= 0:5

rK

N

and in steady state, we can easily compute the capital stock perworker

Kt+1N KtN

= sYtN Kt

N= 0 8t

, sYtN=

KtN

, s 0:5rKtN=

KtN

, 0:5s=

rKtN

, KtN=0:5

s

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and therefore the steady state output per worker is

Y

N= 0:5

rK

N

= 0:5

r0:5

s

2= 0:25 s

(b) The steady state output per worker has been derived in (a) above.Consumption per worker can be computed by using the followingidentities

Y = C + I

I = sY ) C = (1 s)YTherefore

C

N= (1 s)Y

N

= 0:25 s(1 s)

(c) (d) and (e). Using the answers in (a) and (b) above, we can easilyenter the formulas into an xls spreadsheet and obtain the fol-lowing graphs for the series Y

N(s) and C

N(s) for = 0:05 and

s = 0; 0:01; 0:02; :::1.

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As predicted by the theory of the Solow model, we observe thatthe long run (steady state) level of output per worker is maximizedfor s = 1, while the long run (steady state) level of consumptionper worker is minimized for s = f0; 1g. Steady state C

Nis maxi-

mized for s = 0:5. The reason why consumption per worker is notmaximized when income per worker is maximized is that what issaved and invested cannot be consumed. There is thus a trade-obetween saving (which allows to produce more) and consuming.We could have found the saving rate that maximizes consump-tion per worker (i.e. the golden rule saving rate sGR) through thefollowing computation:

@C=N

@s=

@0:25 s(1s)

@s

= 0:25(1 s)

0:25s

= 0:25(1 2s)

and CNis maximized when @C=N

@sis equal to zero, i.e. if

@C=N

@s= 0:25

(1 2s)

= 0

, s = 0:5

11.7 We assume the following production function

Y = KN1

with =1

3

(a) We can prove that this production function displays constant re-turn to scale. Indeed, if we multiply both K and N by some con-stant x, then output will be multiplied by x as well:

(xK)(xN)1 = xKx1N1

= x+1KN1

= xKN1

= xY

(b) This production function displays decreasing returns to capital.Indeed, the larger capital is, the less an increase in the capital

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stock by one unit will increase total output. In mathematics,

@Y

@K=

@ (KN1)@K

= K1N1

=

N

K

1and you directly observe that if K = 0, then @Y

@K= 1, while if

K !1, then @Y@K= 0.

(c) The same can be said about labour. This production functiondisplays decreasing returns to labour. Indeed, the larger labouris, the less an increase in the number of workers by one unit willincrease total output. In mathematics,

@Y

@N=

@ (KN1)@N

= (1 ) KN

= (1 ) K

N

and you directly observe that if N = 0, then @Y

@N= 1, while if

N !1, then @Y@N= 0.

(d) Dividing the production function by N we can transform it intoa relationship between capital per worker and output per worker,thanks to the property observed above in (a) of constant returnto scale:

Y = KN1

, YN=KN1

N

, YN= KN

, YN=

K

N

(e) In steady state, there is no more capital accumulation per worker

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such thatKt+1N KtN

= sY

N K

N= 0

, sYN=

K

N

, sK

N

=

K

N

, s=

K

N

1, K

N=s

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(f) From what we computed in (d) and (e) above, we can derive thesteady state output per worker as

Y

N=

K

N

=

s

1

(g) If = 1=3, = 0:08 and s = 0:32, we can then easily computethat the steady state output per worker is

Y

N=

s

1

=

0:32

0:08

1

= 41=31=3

= 41=2

= 2

(h) If s drops now from s = 0:32 to s = 0:16, the steady state outputper worker would drop from 2 to

Y

N=

s

1

=

0:16

0:08

1

= 21=31=3

= 21=2

= 1:41

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which means that halving the saving rate reduces output perworker but by less than a factor 2.

11.8 We continue with the production function

Y = KN1

with =1

3

but now state that = s = 0:1:

(a) From 11.7(e) above, the steady state capital per worker can becomputed as

K

N=

s

11

= 1

(b) And it follows that the steady state output per worker will beequal to unity accordingly since

Y

N=

K

N

=s

1

= 1

(c) If the economy is at its steady state such that KN= Y

N= 1 and that

suddenly the depreciation rate increases from = 0:1 to 0 = 0:2,the economy will move away from its initial long run equilibriumand converge slowly towards a new one. This new steady state willbe reached when

K

N=

s0 11

= 0:512=3

= 0:53=2

= 0:35

andY

N=

s

1

= 0:50:5

= 0:71

Without surprise, as increases to 0, the steady state capitalstock per worker decreases as well as the steady state output perworker.

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(d) Lets say that in period t = 0 the economy is at its initial steadystate K

N= Y

N= 1. Then jumps from 0:1 to 0 = 0:2. In order

to asses the dynamic path of the economy, we have to use theequation of capital accumulation

Kt+1N KtN= s

YtN 0Kt

NWe then compute

K1N K0N

= sY0N 0K0

N

, K1N= s

Y0N+ (1 0)K0

N

, K1N= 0:1 1 + 0:8 1

, K1N= 0:9

andY1N

=

K1N

1=3= 0:91=3 = 0:97

The next period we haveK2N K1N

= sY1N 0K1

N

, K2N= s

Y1N+ (1 0)K1

N

, K2N= 0:1 0:97 + 0:8 0:9

, K2N= 0:82

andY2N

=

K2N

1=3= 0:821=3 = 0:94

and in period t = 3, we getK3N K2N

= sY2N 0K2

N

, K3N= s

Y2N+ (1 0)K2

N

, K3N= 0:1 0:94 + 0:8 0:82

, K3N= 0:75

andY3N

=

K3N

1=3= 0:751=3 = 0:91

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11.9 Let us now use the production function

Y =pKpN

which, you will verify, is actually identical to the production function

Y = KN1

with =1

2

(a) With such a production function, one can easily nd the steadystate capital per worker as well as the steady state output perworker. The procedure is the same as the one applied in 11.7(e)above. The rst step requires to nd the level of capital per workersuch that capital accumulation is nil. From the capital accumula-tion equation we get

Kt+1 = (1 )Kt + s Yt, Kt+1

N= (1 )Kt

N+ s Yt

N

, Kt+1N KtN= s Yt

N Kt

N

and

Kt+1N

=KtN=K

N

, s YN=

K

N

, s YN=

K

N

, s pKpN

N=

K

N

, s pKpN=

K

N

, s=

rK

N

, KN=s

2and therefore the steady state output per worker is

Y

N=

rK

N=s

11

(b) If s = 18% and = 8%, then we easily compute that in steadystate

Y

N=s

=0:18

0:08= 2:25

whileK

N=s

2= 2:252 = 5:06

(c) If the private saving rate remains equal to 18% but that the public(or government) saving rate jumps from 0 to 6%, the nationalsaving rate becomes 18% + 6% = 24%. This increase in s hasthe eect of increasing gradually the capital stock per worker upto a new long run equilibrium. The new steady state that willbe reached in the end of the convergence process can easily becomputed as

YnewN

=snew

=0:24

0:08= 3

KnewN

=snew

2= 32 = 9

We clearly observe that the capital stock per worker has consid-erably increased thanks to this increased saving rate, yielding toa higher output per worker as well.

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