Macroeconomics and Economic PolicyGregory de Walque

Chapter 10: exercises

10.1 .

(a) True. If a variable increases continuously at a 5% rate, i.e. XtXt1Xt1

=0:05 8t, this variable will evolve with a continuously increasingslope, i.e. a prole of an exponential type. However, if we take thelogarithm of this variable, ln(Xt), this logarithm will evolve witha constant slope equal to the growth rate of Xt. Indeed, rememberthat

Xt Xt1Xt1

=XtXt1

1 ' ln

XtXt1

= ln(Xt) ln(Xt1)

and ln(Xt) ln(Xt1) is nothing else than the slope of the variableln(Xt). The slope of the logarithm ofXt is thus constant and equalto 0:05, which is equivalent to say that it is a straight line slope0:05. (see also exercise 10.10 hereunder as an illustration of this).

(b) False. The price of food and basic goods tend to be lower incountries with lower output per person. This is because in suchcountries, the share of the population in the agricultural sector ismuch bigger in the economy, while in countries with high outputper person, this is the other way round (about 5% in the US). Buton the other side, food and basic goods make it for a much largershare of consumption in countries with a low output per worker.

(c) False. If we look at rich countries, the share of person revealingvery/mildly/little happiness is relatively constant whatever theGDP/capita. For example, even though an average American cit-izen is much richer today than it was 50 years ago, (s)he doesntseem to be more happy. The perception of happiness is much morerelated to the income of one person relative to this of its directneighbors. In this sense, within a country, richer people tend tobe more happy than poor people.

(d) False. This catching up process is only observed in the countriesthat have entered the capital accumulation process that goes alongwith industrialization. This is only true of the industrialized coun-tries and emerging countries. Some countries are still left in thebig stagnation.

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(e) True. This is called the Malthusian trap, after Reverend Malthus:In the view of Malthus, population growth is directly related tothe health of the population, which itself is positively related toincome per person. If this is true, any event that brings an in-crease in income per person will, in the long run yield an increasein population that will in the end counteract the initial increasein income per capita. This always brings back the economy toits starvation equilibrium where population is so numerous withrespect to the available natural resources that income per personis very low such that population is just able to reproduce itself,but not to grow anymore.

(f) True because of the diminishing returns of the capital productionfactor. In the long run, as we have seen in chapter 11 on the Solowgrowth model, in the absence of technological progress, output perperson is only determined by the saving rate s and the depreci-ation rate . Capital accumulation only plays a role on the levelof output during the period of convergence towards the long runequilibrium. We then saw in chapter 12 that actually the initialSolow model can be enlarged somewhat to take into account pop-ulation growth rate and technological progress. In that case, wedo not speak of a steady state of the economy, but of a steadygrowth path, where GDP per capita grows at the same pace astechnological progress.

(g) True. The aggregate production function usually writes down as

Yt = F (Kt; Nt)

10.2 .

(a) US consumption per capita can be computed as

($2 2; 000) + ($3 3; 000) = $13; 000(b) Mex. consumption per capita can be computed as

(P10 800) + (P30 300) = P17; 000(c) If the exchange rate is such that $1 = P10, then one can compute

the consumption of the average Mexican in $ as

P17; 000 = $1; 700

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(d) If we now use the PPP method using US prices, we would computethe consumption of the average Mexican in $ as

($2 800) + ($3 300) = $2; 500

(e) We thus observe that the standard of living computed for Mexicousing the PPP method is signicantly larger than if we simplyuse the exchange rate. The reason for this is that basic goods(which form a larger share of consumption in poor countries) areless expensive in poor countries than in rich countries and theother way round for goods of a higher status. Using the sameprice as in the US helps better compare the consumption bundleof both countries, since it eradicates the price bias resulting fromthe dierent stage of economic development of the two countries.

10.3 Consider the following production function

Y =pKpN

(a) Then ,if K = 49 and N = 81, then we can compute that

Y =p

49p

81

= 7 9= 63

(b) and (c) If the quantities of both capital and labor are doubled,output will also double with such a type of production function.The production function Y =

pKpN is said to display constant

return to scale:

if Y =pKpN =

pK N

thenpxKpxN =

px2 K N = x

pK N = xY

and forx = 2, xY = 2Y

d. Output per worker can be computed as follows

Y =pKpN

) YN

=

pKpN

N=

pKpN

=

rK

N

which shows how the production function can be used to establisha relationship between output per worker and capital per worker.

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e. and if K=N = 4, then using the relationship established in d.above, we get

Y

N=

rK

N=p

4 = 2

If we now double K=N to 8, we get

Y

N=

rK

N=p

8 = 2:83

which is clearly lower than 4. This clearly shows that output perworker does not exhibit constant return to scale, but instead, di-minishing return to scale. Output per worker is thus a concavefunction of capital per worker.

g. The answer in (f) diers from this in (c) for the very reason thatoutput and output per worker do not measure the same thing.To clearly show this, let us rst consider that both K and N aremultiplied by x:

pxKpxN = x

pKpN = xYr

xK

xN=Y

N

This clearly shows that if both K and N are multiplied by x,output is multiplied by the same factor while of course output perworker remains constant. Let us now look at what happens if it isK=N that is multiplied by x :r

xK

N=px

rK

N=pxY

N

which clearly shows that output per worker is then not multipliedby x but only by

px.

h. .

Y

N=

rK

N

4

0 10 20 30 40 500

1

2

3

4

5

6

7

8

K/N

Y/N

This indeed shows that an increase in K=N leads to smaller in-crease in Y=N if K=N is large than if K=N is small.

10.4 We continue to consider the production function

Y =pKpN

but we now assume that N = 1:

(a) Under this assumption, the production function simplies to

Yt =pKt = K

0:5t

Applying the approximation

z = xa ) gz ' a gxto this production function, we obtain that

if Y = K0:5

then gy ' 0:5 gk(b) Given what we obtained in a., in order to obtain gy = 2%, we

need to have gk such that

2% ' 0:5 gk, gk ' 4%

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(c) Let us compute the capital/output ratio:

K

Y=

K

K0:5= K0:5 = Y

It directly follows that

gK=Y = gy = 2%

(d) No it is not possible for output to continuously grow at a pace of2% a year in this economy. The reason is that there is no techno-logical progress. If we use the Solow model developed in chapter11, we can compute the long run equilibrium of this economy

Yt = K0:5t

Kt+1 = (1 )Kt + sK0:5t, Kt+1 Kt = sK0:5t Kt

The long run equilibrium is reached when Kt+1 = Kt 8t, that iswhen

sK0:5 = K

, K0:5 = s

, K =s

2, Y = s

and since s and are constant, Y is constant in the long run,meaning that its long run equilibrium growth rate is gy = 0.

10.5 This is simply due to the catching process coming from the conse-quences of the second World War. In France, Germany and Japanthe second World War has had as consequence that the capital stock(plants, cities, railways, roads, all the dierent networks - water, tele-phone, electricity ...-, airports, ports...) was heavily destroyed. Even ifwe assume that these countries were at the same economic developmentlevel as the US before the 2nd WW, for sure they strongly regressedafter it. The strong economic growth process observed during the 50sand 60s in these three countries comes from the capital accumulationprocess and de decreasing returns of capital: when the capital stock isrelatively small, an increase of this capital stock yields a much strongerincrease in output than when the capital stock is large.

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10.6 Eects of a technological innovation in the long run in the Malthusmodel:

y = Y/N

L/N=lN/N

We conclude from this long run analysis that a technological innovationis not able to ensure a higher output per worker in the long run. Itsonly eect is to reduce the surface of agricultural land per inhabitant.

In the short run, this is dierent. The technological innovation (fromA to A0) will indeed make labour more e cient, allowing an increasein output per worker:

YtNt

= AF

LNt

) Yt+1

Nt+1= A0F

LNt+1

> A0F

LNt

if A0 > A .

This immediate increase in Y=N will also aect the demography: aseconomic agents are richer than before, they are in a better health andhave more children, as shown by the G function

Nt+1Nt

= G

YtNt

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This increase in population then decreases the available land per worker,reducing progressively output per worker until the economy is back toits starvation equilibrium, where Y=N is just su cient to ensure thestability of the population. This is illustrated on the following graphwith time on the horizontal axis:

N

Y/N

N

T

T

10.7 Eects of a medical innovation improving the health of the populationin the long run in the Malthus model:

- a medical innovation has the eect to shift the demographic func-tion G(Y/N) to the left to G(Y/N) since the same income perperson allows now economic agents to have more children sincetheir health condition is improved.

- this yields an increase in population, such that the amount of landper worker is reduced, and therefore income per worker also.

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Therefore, in the short run, a medical innovation will only helppopulation to grow more rapidly, and from this higher increase inpopulation, the economy reaches an new long run starvation equi-librium with a higher population and a lower income per worker.The dynamic graph can be represented this way:

In the short run, the medical innovation yields an increase in the demo-graphic function (from G(Y=N) to G0(Y=N)) such that at the period

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for which this innovation is introduced:

G0YtNt

> G

YtNt

= 1

) Nt+1Nt

> 1

which means that population increases.

As population increases while there is no technological improvement,land per worker decreases and income per worker follows. This increasein population continues until a new starvation equilibrium is reached,where population stabilizes at a higher level while land per worker andincome per worker are permanently decreased.

10.8 The Malthus reasoning is based on two elements:

(a) the only production factor other than labor force is in xed quan-tities

(b) labor force grows accordingly with income per person

Therefore, any increase in income per person will yield population in-crease and less land per worker, with the eect to reduce again in-come per person. This is clearly what is still observed in stagnatingeconomies.

However, in emerging economies, another mechanism is set into place,breaking the vicious circle between limited natural resources and de-mography. Actually, there exists a production factor that can be re-produced and extended through production and accumulation. Let uscall it capital. A higher income per person can then be translated intoa higher capital accumulation, yielding a higher output (and thus in-come) per person. Thanks to the possibility of capital accumulation,the Malthus vicious circle can be broken. Beside the possibility to ac-cumulate capital, it is also essential to note that in the Solow model,population growth is not related anymore to income per person. Thislink assumed by Malthus can be observed in very poor economies, stillclose to the starvation equilibrium. However, once one gets away fromthis starvation equilibrium, the dynamics of population growth seemsindeed to be modied, as parents tend to invest more in "children qual-ity" (through education, etc.) than in "children quantity".

In old industrial economies, the initial Solow growth model states thatin the long run, they reach a high income per person equilibrium, in-compatible with the observed continuous increase in GDP per worker

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in these economies. In order to reconcile this model with the observedfacts, one needs to take into account a continuous increase in techno-logical progress. In the long run, the observed growth rate of GDP perworker is actually equal to the growth rate of technological progress.

10.9 This question is actually answered in the last tow of answer to ques-tion 10.8.

10.10 This question is just a numerical application of what has been statedin the answer to exercise 10.1.a hereabove.

(a) The variable X is such that

X1 = 1

X2 = X1 (1 + 0:03) = 1:03X3 = X2 (1 + 0:03) = 1:061X4 = X3 (1 + 0:03) = 1:093

:::

X200 = X191 (1 + 0:03) = 358:60

and this series is clearly displaying an exponential pattern.

(b) We can then graph this series:

(c) If we then take the logarithm of this series, we observe that theseries behaves as a straight line, the slope of it being 3%. This is

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totally normal since

ln(Xt) = ln [Xt1 (1 + 0:03)]= ln(Xt1) + ln(1 + 0:03)

' ln(Xt1) + 0:03which displays a series growing period after period by 0:03.

(d) see (a) and (c) above, as well as 10.1(a).

10.11 Consider the Malthus economy, with Nt+1Nt

= G(yt) = y0:5t =

pyt while

yt = A F

LNt

= A

LNt

0:8and initially, A is equal to 1.

(a) In the long run equilibrium of this economy, population is con-stant, meaning that Nt+1

Nt= 1 8t. This means that

Nt+1Nt

= G(yt) = y0:5t =

pyt = 1

, yt = YtNt

= 1

Furthermore, as

yt = A LNt

0:8this implies that

A LNt

0:8= 1

,L

Nt=

1

A

10:8

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As it is also stated that A = 1, we get

L

Nt=

1

A

10:8

= 1

(b) If now A jumps suddenly from A = 1 to A = 2, then we will havethat

Nt+1Nt

=pyt = 1) Nt+1 = Nt

thereforeL

Nt+1=

L

Ntand

yt+1 = 2 LNt+1

0:8= 2

LNt

0:8= 2

Then,

Nt+2Nt+1

=pyt+1 =

p2 = 1: 41) Nt+2 = Nt+1 1:41

thereforeL

Nt+2=

L

Nt+1 pyt+1 =L

Nt+1 1:41and

yt+2 = 2 LNt+2

0:8= 2

LNt+1 1:41

0:8= 2

1

1:41

0:8= 1:52

and we can redo this computation recursively into an xls sheetwith the following formulas:

A B C1 t L

Ntyt =

YtNt

2 t+ 0 1 13 t+ 1 =B2 =2*(B2)^0.84 t+ 2 =B3/(C3)^0,5 =2*(B4)^0,85 t+ 3 =B4/(C4)^0,5 =2*(B5)^0,8

::: ... ...13 t+ 11 =B12/(C12)^0,5 =2*(B13)^0,8We then obtain the series that are graphed hereunder:

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This numerical example illustrates perfectly the mechanism ex-plained in the resolution to exercise 10.6 hereabove.

(c) Let us now consider again that A = 2 forever, but that from t+ 1on, the demographic function is altered such that

NtNt1

= y0:5t1 jumps toNt+1Nt

= y0:3t

Then we will have that

Nt+1Nt

= y0:3t ) Nt+1 = Nt y0:3t = Nttherefore

L

Nt+1=

L

Ntand

yt+1 =

LNt+1

0:8=

LNt

0:8= 1

and we will remain at the same long run starvation equilibriumbecause starting from yt = 1, we have that y0:5t = y

0:3t = 1 since yt

is the neutral for the multiplication operation. In order to obtainthe desired progressive decrease in Nt+1, one should shift to thedemographic function dierently, for example this way:

NtNt1

= y0:5t1 jumps toNt+1Nt

= yt

1:1

0:5

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Then we would obtain

Nt+1Nt

= yt

1:1

0:5) Nt+1 = Nt

yt1:1

0:5= Nt 0:95

thereforeL

Nt+1=

L

Nt yt1:1

0:5 = LNt 0:95 = 1:05and

yt+1 =

LNt+1

0:8=

L

Nt yt1:1

0:5!0:8

= 1:04

And one period later

Nt+2Nt+1

=yt+1

1:1

0:5) Nt+2 = Nt+1

yt+11:1

0:5therefore

L

Nt+2=

L

Nt+1 yt+11:1

0:5and

yt+2 =

LNt+2

0:8=

L

Nt+1 yt+11:1

0:5!0:8

and we can redo this computation recursively into an xls sheetwith the following formulas:

A B C1 t L

Ntyt =

YtNt

2 t+ 0 1 13 t+ 1 =B2/(C2/1.1)^0.5 =(B3)^0.84 t+ 2 =B3/(C3/1.1)^0.5 =(B4)^0,85 t+ 3 =B4/(C4/1.1)^0.5 =(B5)^0,8

::: ... ...13 t+ 11 =B12/(C12/1.1)^0.5 =(B13)^0,8and the numerical computation gives us the following series:We clearly see from this graph that a better birth control in theMalthus model yields a progressive increase in land per worker andin income per worker, until a new long run equilibrium is reachedwhere income per worker is forever higher than initially.

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