Macroeconomics and Economic PolicyGregory de Walque

Chapter 12: exercises

12.1 .

(a) The statement is not true. First, there has been a decline in tech-nological progress during the years 70 to 85 but since then techno-logical progress has again risen. Furthermore, even though thereis a clear link between R&D spending and technological progress,there can be a delay before the R&D spending indeed translateinto technological progress. Actually technological progress canalso result from a new way to organize activities within and be-tween firms, with firms specializing on their core business andoutsourcing all the other activities.

(b) True. Remember from the capital accumulation equation adaptedfor taking population growth and technological progress into ac-count, that gA (growth rate of technological progress) acts asa "negative force" for the accumulation of capital per effi cientworker. If one of the three "negative forces" increases, the savingrate has to be increased in order to keep the same capital stockper effi cient labour unit:

(1+gA)(1+gN)·(

Kt+1

At+1Nt+1− Kt

AtNt

)= s·F

(Kt

AtNt

)−(δ + gA + gN)

Kt

AtNt

If you want to see it "graphically", an increase in gA means thatthe line of slope δ + gA + gN becomes more sloppy. If s remainsunchanged it will cross the s·F

(KAN

)curve at the left of the initial

intersection, determining a lower steady state KAN. In order to keep

KAN

unchanged, it is thus required to shift up the s ·F(KAN

)curve,

which can be done by increasing the saving rate s.

(c) False. At the steady state, which is actually now a steady growthpath, output per worker grows at the same rate as technologicalprogress, i.e. gA which is different from the population growthrate gN . Output per effi cient worker grows at rate zero and outputgrows at the rate gN + gA.

(d) True. See answer in (c) above.

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(e) True. Indeed, if YAN

= x and that A becomes A′ = A(1 + 0.05)while Y remains unchanged, the only possibility is that N shiftsto N ′ = (1− 0.05) such that

Y

AN= x =

Y

A′N ′

' Y

A(1 + 0.05) ·N(1− 0.05)

=Y

AN

1

(1− 0.005) 'Y

AN

(f) True, since it will not be possible for the private firm to get afinancial return out of this theorem. But it is not the only reason.Actually basic research is at the source of many applied innova-tions, but the transformation from one step to the other can takea very long time, and it is always diffi cult to know in advancewhich outcome of fundamental research will be helpful in the end,and which will not. Time and uncertainty about the possibility totransform fundamental research into useful and marketable tech-nological innovations make that private firms do not enter thispath.

(g) True. As in chapter 11, the long run growth rate of output isindependent of the saving rate. Actually, the long run growth rateof output is gY = gA + gN showing that it only depend of thegrowth rate of population and of the growth rate of technology.

(h) The question is actually badly stated. If indeed the potential re-turns of investing in R&D are equal to those of investing in K,then it should be the case that both investment face the samelevel of risk. Actually, investing in R&D is much more risky thaninvesting inK since when you invest inK, you already know whatthe new machine will help you to produce and sell. When investingin R&D, you are never sure of the outcome of the research. It canbe nothing or it can be the jackpot.

12.2 .

(a) For the advanced economies (or old industrialized countries) themain source of technological progress is innovation, that comesfrom the R&D process. Since they already implemented the "old"technologies, the only way to continue to make output per workergrow is to continue to create "new" technologies. Less advanced

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countries can benefit from technological progress created in themore advanced economies, by importing capital goods endowedwith the technologies that have been developed elsewhere. Thishas been for example the development strategy of Japan at thebeginning of the XXth century. However, since then Japan hasjoined the club of economic leaders and relies on R&D to developnew technologies that will help maintain its growth rate of outputper worker.

(b) Developing countries may have an interest in having a loose patentprotection. Indeed, the protection of patents favour more the de-veloped countries who produce these patents. By not endorsingthe patent protection legislation, developing countries somehowauthorizes their economic actors to use technological innovationsdeveloped by others without having to pay for it. The danger thenis that they can only sell the goods produced thanks to this pro-tected innovation in their own market. If they try to export thesegoods, they will be suited by the owner of the patents.

12.3 .

(a) Spending in R&D is the only way to reach technological progresswhich is actually the only possibility to ensure the growth of in-come per capita. In the absence of investment in research and de-velopment (through private and public funds) the economy wouldgo back to a period of great stagnation. The appropriability ofresearch is an important and essential incentive for firms to makeresearch. In the absence of the possibility for the private sectorto be rewarded for its research efforts, they would not enter thisway. One of the best known way for research to be rewarded isthrough the system of patents, in which the owner of a patenthas got a monopoly on the patented innovation for a given periodof time. Such patents can of course be sold to other firms butin this case the seller lose the right to use the innovation. Thefertility of research is one of the most important things observedsince the industrial revolution. The largest and the most connectedthe community of researchers, the more innovations cross-fertilizeacross domains that can appear a priori very different. For exam-ple, innovations in physics and chemicals are key for the computerscience.

(b) Such an international treaty would of course favour appropriabil-ity, as the outcome of research would then be protected all over

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the world through a unified system of patents. As it would givemore incentive for private firms to develop research programs, andthis all over the world (i.e. even in developing and emergingeconomies), increase potentially the number of researchers andthe possibilities of cross-fertilization of ideas, leading towards newdevelopment and new processes.

(c) A tax credit for each $ spent in R&D will have no effect for theappropriability of research. It is not because research is partiallysubsidized that the outcome of research will be better protected.This is pretty different. On the other hand, the fiscal incentiveto invest in R&D can help make grow the size and numbers ofresearch programs, which can only be good for fertility.

(d) A decrease in subsidies financing conferences between universitiesand corporations: will have no effect on the possibilities of appro-priation of the benefits of research by the firms, but will affect thepossibilities of fertility as it reduces one channel of communicationbetween researchers in academia and in the private sector.

(e) The elimination of patents on breakthrough drugs in order to sellthem at low cost: this clearly destroy the possibilities for firms toobtain benefits out of their research. Their incentive for researchwill definitely be reduced. This reduction in research activities bythe firms will also affect fertility: less volume of research reducesthe possibilities of cross-fertilization.

12.4 .

(a) a permanent reduction in the growth rate of technological progressfrom gA to g′A < gA will

- lead to a progressive increase in the level of output per effi cientlabor unit through a higher capital accumulation per effi cientlabour unit. Therefore, in the first years after the decrease ingA it is possible that output increases more than its initialgrowth rate gA + gN . However, this is only very temporary.

- when the output per effi cient worker has stabilized to its newequilibrium level, the long run growth rate of the economy isg′A+gN < gA+gN . Because of this, the output level in comingdecades will be lower than it would have been if technologicalprogress had continued to grow at the same rate gA.

(b) a permanent reduction in the saving rate from s to s′ < s will

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- lead to a progressive decrease in the level of output per ef-ficient labour unit through a lower capital accumulation pereffi cient labour unit. In the first years after the shift from sto s′, the GDP will grow at a pace lower than the long runequilibrium growth rate (equal to gA + gN).

- but after some decades, the economy will converge to a newsteady state characterized by a permanently lower GDP pereffi cient labour unit. When the economy has reached this newsteady state, the GDP grows at the same pace as initially, i.e.gY = gA + gN .

12.5 .

(a) The nominal GDP in each year is respectively equal to

GDP nom1 = (P h1 ×Qh1) + (P b1 ×Qb1)= (10× 100) + (10× 200)= 3000

GDP nom2 = (P h2 ×Qh2) + (P b2 ×Qb2)= (12× 100) + (12× 230)= 3960

(b) Using the prices of year 1, the real GDP of years 1 and 2 arerespectively

GDP real1 = (P h1 ×Qh1) + (P b1 ×Qb1)= (10× 100) + (10× 200)= 3000

GDP real2 = (P h1 ×Qh2) + (P b1 ×Qb2)= (10× 100) + (10× 230)= 3300

and the growth rate of real GDP is

GDP real2 −GDP real1

GDP real1

=3300− 3000

3000= 10%

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(c) The GDP deflator can be computed as

GDP nom1

GDP real1

× 100 = 100

andGDP nom2

GDP real2

× 100 = 3960

3300× 100 = 120

From these values we can compute the inflation rate as

π =P2 − P1P1

=120− 100100

= 20%

(d) Using year 1 prices, the real GDP per worker can be computed as

GDP real1

W1

=3000

(W h1 +W b

1 )=3000

100= 30

GDP real2

W2

=3300

(W h2 +W b

2 )=3300

110= 30

As the real GDP per worker is unchanged between year 1 and2, it is the case that labour productivity is unchanged as well(since labour productivity is nothing else than real GDP dividedby labour units). Therefore labour productivity growth is zero.The increase in real GDP is coming from the increase in the num-ber of workers.

(e) The table becomesyear 1 year 2P1 Q1 W1 P2 Q2 W2

Haircut 10 100 50 12 100 50Banking 10 200 50 10 0 0Telebanking 13 0 0 12 230 60

and you can easily check that, using year 1 prices, the real GDPfor year 2 is now computed as

GDP real2 = (P h1 ×Qh2) + (P b1 ×Qb2) + (P tb1 ×Qtb2 )= (10× 100) + (10× 0) + (13× 230)= 3990

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The growth rate of real GDP can now be re-computed as

GDP real2 −GDP real1

GDP real1

=3990− 3000

3000= 33%

which is significantly higher than the 10% growth rate computedhigher.

(f) The rate of inflation deserves to be re-computed on this new basis.The GDP deflator can be computed as

GDP nom1

GDP real1

× 100 = 100

andGDP nom2

GDP real2

× 100 = 3960

3990× 100 = 99.25

From these deflator values we can compute the inflation rate as

π =P2 − P1P1

=99.25− 100

100= −0.75%

(g) Labour productivity has now increased compared to what hadbeen computed in (d) above. Indeed, labour productivity is equalto real GDP per worker and we compute that

GDP real1

W1

=3000

(W h1 +W b

1 +W tb1 )=3000

100= 30

GDP real2

W2

=3990

(W h2 +W b

2 +W tb2 )=3990

110= 36.273

and labour productivity has now grown by

36.273− 3030

= 21%

(h) If we do not take into account the introduction of the telebankingservices, we will indeed overestimate inflation as we will considerthat banking services prices increase from 10 to 12, while actuallyit should be considered that telebanking prices shrinks from 13 to12. It also yields a bad assessment of real GDP since telebank-ing services are priced the same way as banking services, whileactually, telebanking services are more expensive than bankingservices. All this is pretty clear when comparing (b) with (e), (c)with (f) and (d) with (g).

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12.6 .

(a) geographic location should, according to the Solow model, have norole on the steady state of output. However, we will all agree thatlabour is easier under some climate conditions than others. In gen-eral temperate climates are assumed better suited for productionthan too hot or too cold ones.

(b) education is important first for the human capital, determining theeffi ciency of workers. Furthermore, a higher level of education willinduce better research and more technological innovation. Thiswill thus also have an impact on the level of technology.

(c) Protection of property rights in general give investors an incentiveto invest. Therefore it will affect positively the capital stock. If thisprotection extents to ideas and the outcome of research, it will alsogive an incentive for research and affect the level of technology aswell.

(d) A priori, according to the Solow model (which is a closed econ-omy model) openness to trade should affect none of the threedimensions A, K, or H. However, if it is possible to exchangegoods with other countries, it is possible to by investment goodsendowed with particular technologies that have been developedin foreign countries. This is a very powerful way to introduce newtechnologies in the country without having to pay the full researchand development costs required for it.

(e) Low tax rates have no effect on any of the three dimensions A, K,or H.

(f) Good public infrastructure gives firms an incentive to invest, sincethe public infrastructure is present and helps to carry the producedgoods from the place where they are produced to the place wherethey are consumed.

(g) A low population growth facilitates the acquisition of capital percapita (the slope of the δ + gN + gA line is less steep and so thatit crosses the s · Y

Ncurve further to the right, which corresponds

to a higher GDP per effi cient worker and therefore a higher GDPper worker). Therefore it will help to reach a higher level of GDPper capita.

12.7 Consider the following production function

Y =√K√AN

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with the parametrization of the economy

s = 16%

δ = 10%

gN = 2%

gA = 4%

(a) .

i. We may compute the steady state capital stock per effi cientlabour unit as follows

0 = s ·√K√AN− (δ + gN + gA)

K

AN

⇔ s ·√K√AN

= (δ + gN + gA)K

AN

⇔ s

δ + gN + gA=

√K√AN

⇔ K

AN=

(s

δ + gN + gA

)2⇔ K

AN=

(0.16

0.10 + 0.02 + 0.04

)2⇔ K

AN= 1

ii. From there the output per effi cient labour unit can be com-puted as follows

Y

AN=

√K√AN

=

√K

AN

=√1

= 1

iii. The long run equilibrium growth rate of YAN

is zero...iv. ... while Y

Nin the long run equilibrium grows at a rate of

gA = 4%...

v. ... and output grows at gA + gN = 6%.

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(b) If gA doubles from 4% to 8% the answers computed in (a) become

i.

K

AN=

(s

δ + gN + gA

)2⇔ K

AN=

(0.16

0.10 + 0.02 + 0.08

)2⇔ K

AN= 0.64

ii.

Y

AN=

√K√AN

=

√K

AN

=√0.64 = 0.08

iii. The long run equilibrium growth rate of YAN

is unchanged tozero...

iv. ... while YNin the long run equilibrium grows at a rate of

gA = 8%...

v. ... and output grows at gA + gN = 10%.

(c) If gA = 4% and gN = 6% then the answers computed in (a) become

i.

K

AN=

(s

δ + gN + gA

)2⇔ K

AN=

(0.16

0.10 + 0.06 + 0.04

)2⇔ K

AN= 0.64

which is identical to what we computed in (b).ii.

Y

AN=

√K√AN

=

√K

AN=√0.64 = 0.08

which is also identical.iii. The long run equilibrium growth rate of Y

ANis unchanged to

zero...

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iv. ... while YNin the long run equilibrium grows at a rate of

gA = 4%...

v. ... and output grows at gA + gN = 10%.

Even though the answers look very similar to those computedin (b), people are more happy in the economy (b) than in (c).The reason is that their individual income (i.e. Y

N) grows more

rapidly. What is important for people welfare is income per workerrather than global income (i.e. GDP). It is therefore better thatthe economy grows because of technological progress rather thanbecause of population growth.

12.8 Given the production function

Yt = K1/3t (AtNt)

2/3

we can deduce that

ln(Yt) =1

3ln(Kt) +

2

3ln(At) +

2

3ln(Nt)

and equivalently

ln(Yt−1) =1

3ln(Kt−1) +

2

3ln(At−1) +

2

3ln(Nt−1)

and removing the second equation from the first one, we obtain anequation in growth rate

ln(Yt)− ln(Yt−1) =1

3ln(Kt) +

2

3ln(At) +

2

3ln(Nt)−

1

3ln(Kt−1)−

2

3ln(At−1)−

2

3ln(Nt−1)

=1

3[ln(Kt)− ln(Kt−1)] +

2

3[ln(At)− ln(At−1)] +

2

3[ln(Nt)− ln(Nt−1)]

We remember also that

gX =Xt −Xt−1

Xt−1' ln

(Xt

Xt−1

)= ln(Xt)− ln(Xt−1)

which says that the first difference of a series in logarithm is a fairlygood approximation of the growth rate of this series (as long as thegrowth rate is not too large). Therefore, we obtain

gY =1

3gK +

2

3gA +

2

3gN

⇔ (gY − gN) =1

3(gK − gN) +

2

3gA

which is well the relationship stated in the exercise.

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(a) The quantity gY − gN represents the growth rate of output perworker while gK − gN represents the growth rate of capital perworker. Therefore the expression

(gY − gN) =1

3(gK − gN) +

2

3gA

says that the growth rate of output per worker comes for one thirdfrom the accumulation of capital per worker (i.e. the growth rateof capital per worker), and for two third from the growth rate oftechnological progress. Note also that

(gY − gN) =1

3(gK − gN) +

2

3gA

=1

3(gK − gN − gA) + gA

According to the Solow model, in the long run equilibrium thegrowth rate of capital per effi cient worker is zero (i.e. gK − gN −gA = 0), such that the growth rate of income per capita (i.e. gY −gN) is simply equal to the growth rate of technological progress(i.e. gA).

(b) Or equivalently, the growth rate of capital per worker can be com-puted as

(gK − gN) = 3 · (gY − gN)− 2 · gA(c) Using the figures in Table 12-2, we may compute a crude measure

of the growth rate of capital per worker (i.e. gK − gN) as being- for France:

(gK − gN) = 3 · (gY − gN)− 2 · gA= 3 · 3.2− 2 · 3.1= 3.4%

- for Japan:

(gK − gN) = 3 · 4.2− 2 · 3.8= 5%

- for UK:

(gK − gN) = 3 · 2.4− 2 · 2.6= 2.0%

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- for US:

(gK − gN) = 3 · 1.8− 2 · 2.0= 1.4%

- average:

(gK − gN) = 3 · 2.9− 2 · 2.9= 2.9%

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