Intermediate Macroeconomics - Lecture 4 - Growth models...

Intermediate MacroeconomicsLecture 4 - Growth models beyond Solow

Zsofia L. Barany

Sciences Po

2014 February

Recap of Solow model

I key: the description of the dynamic evolution of saving -investment - capital accumulation

I prediction: each economy converges to its steady state whichfeatures

I constant capital and output per workerI constant consumption per worker

I output and consumption only grow on the path to the steadystate, during the transition

I changes in the saving rate, the growth rate of population, andthe level of technology change the steady state

I → maybe sustained productivity growth leads to a sustainedgrowth in output per worker?

The Solow model with technological progress

I the basic Solow model with constant technology cannotexplain long-run growth in output per head

I but one of the key reasons why our living standard is muchhigher than a century ago is technological progress

I we next extend the basic Solow model to allow fortechnological progress

I technological progress: given the capital and labor inputs, animprovement in technology leads to an increase in output

I in general technological progress can be labor-augmenting,capital-augmenting or neutral

I we will focus on the type of technological progress that islabor-augmenting: it increases the efficiency, the productivityof workers

Labor augmenting technological progress

Until now the production function was given by:

Y = zF (K ,N)

here z is total factor productivity (TFP), increases in z increasethe productivity of both labor and capital → neutral technologicalprogress

A labor-augmenting technology yields the following productionfunction:

Y = F (K ,AN)

where A is the technology → assume that it grows at constant rateg : At+1 = At(1 + g)

An example: the Cobb-Douglas production function

we can write it as:Y = zKαN1−α

this is equivalent toY = Kα(AN)1−α

if z = A1−α.

In the case of a Cobb-Douglas production function, neutral andlabor-augmenting technological progress are equivalent.

For other neoclassical production functions this is not the case.

A trick to simplifying our model

In the standard Solow model we converted every variable to its perworker version:

Y = zF (K ,N) ⇒ y = zf (k)

Here, in the model with labor-augmenting technological progress,

we convert everything into efficiency worker units:

y ≡ Y

AN=

F (K ,AN)

AN= F

(K

AN, 1

)= f (k)

k ≡ K

AN

we can do this (just as before) due to the constant returns to scaleproperty of F .

Finding the equilibrium of the model

1. Write down the capital accumulation equation

Kt+1 = (1− d)Kt + It

2. Use the clearing of the capital market: It = St = sYt

Kt+1 = (1− d)Kt + sF (Kt ,AtNt)

3. Convert everything into efficiency units of labor

4. Analyze what happens to k over time

The capital accumulation equation

need to convert the following into efficiency labor units:

Kt+1 = (1− d)Kt + sF (Kt ,AtNt)

divide by AtNt and manipulate:

Kt+1

AtNt= (1− d)

Kt

AtNt+ s

F (Kt ,AtNt)

AtNt

At+1Nt+1

At+1Nt+1

Kt+1

AtNt= (1− d)kt+sf (kt)

(1 + g)At(1 + n)Nt

AtNt

Kt+1

At+1Nt+1= (1− d)kt + sf (kt)

(1 + g)(1 + n)︸︷︷︸≈(1+g+n)

kt+1 = (1− d)kt + sf (kt)

We have the following then

(1 + g + n)kt+1 = (1− d)kt + sf (kt)

Compare to the same equation from the Solow model withouttechnological progress:

(1 + n)kt+1 = (1− d)kt + szf (kt)

going back to the model with technological progress, subtract(1 + g + n)kt from both sides:

(1 + g + n)(kt+1 − kt) = sf (kt) + (1− d − (1 + g + n))kt

simplifying

(1 + g + n)(kt+1 − kt) = sf (kt)− (d + g + n)kt

(1 + g + n)(kt+1 − kt) = sf (kt)︸︷︷︸actual investment

− (d + g + n)kt︸︷︷︸break-even investment

k

i

sf (k)

(δ + n + g)k

k∗

if sf (k) > (d + n + g)k ⇒ k increasesif sf (k) < (d + n + g)k ⇒ k decreases

As before, the assumptions about the shape of the productionfunction guarantee that the economy converges to a unique, stablesteady state.

In the steady state: kt+1 = kt = k∗.

I capital per effective worker, k∗, is constant

I output per effective worker, y∗ = f (k∗), is constant

I capital per worker is k∗t = At k∗, is growing at rate g

I output per worker is y∗t = At y∗, is growing at rate g

I capital is K ∗t = NtAt k∗, is growing at rate g + n

I output is Y ∗t = NtAt y∗, is growing at rate g + n

The Kaldor facts

Nicholas Kaldor (1958) characterized modern economic growth bythe following facts:

1. The rate of growth of the capital stock per worker is roughlyconstant over long periods of time

2. The rate of growth of output per worker is roughly constantover long periods of time

3. The capital/output ratio is roughly constant over long periodsof time

4. The rate of return to capital is roughly constant over longperiods of time

5. The shares of national income received by labour and capitalare roughly constant over long periods of time

6. The real wage grows over time

Do the predictions of the Solow model with technological progressfit with these stylized facts?

Summary

I the Solow model can explain the experience of developedcountries since the industrial revolution

I it matches the Kaldor facts well

I prediction: sustained growth can only come from sustainedimprovements in technology

Note:

for Cobb-Douglas production fct Y = Kα(AN)1−α, y = kα

rate of return to capital: R = MPK

= αKα−1(AN)1−α = αkα−1

capital share in income: R·KY

= αKα−1(AN)1−α·KKα(AN)1−α

= α

wage rate: W = MPN

= (1− α)Kα(AN)−αA = (1− α)k−αA

labour share in income: W ·NY

= (1−α)Kα(AN)−αA·NKα(AN)1−α

= 1− α

Summary

I the Solow model can explain the experience of developedcountries since the industrial revolution

I it matches the Kaldor facts well

I prediction: sustained growth can only come from sustainedimprovements in technology

Note: for Cobb-Douglas production fct Y = Kα(AN)1−α, y = kα

rate of return to capital: R = MPK = αKα−1(AN)1−α = αkα−1

capital share in income: R·KY = αKα−1(AN)1−α·K

Kα(AN)1−α= α

wage rate: W = MPN = (1− α)Kα(AN)−αA = (1− α)k−αA

labour share in income: W ·NY = (1−α)Kα(AN)−αA·N

Kα(AN)1−α= 1− α

Moving beyond the Solow model

There are at least three reasons why economists have move beyondthe Solow model

1. Lucas asked, ’why doesn’t capital move from rich to poorcountries?’

2. growth accounting shows the important contribution oftechnological progress to growth within a country

3. development accounting shows differences in the levels of’technology’ are needed to account for large cross-countryincome differences

Lucas: why doesn’t capital move from rich to poor?

Lucas (1990 AER) asked, if physical capital accumulation is themain driving force behind economic growth and there isdiminishing marginal productivity, ’why doesn’t capital move fromrich to poor countries?’

The argument is that large differences in Y /N must imply largedifferences in K/N, which – under diminishing MPK – implies thatreturns to capital in the poor countries must be much higher thanin the rich

⇒ capital should move from rich to poor countries

Lucas: why doesn’t capital move from rich to poor? II.

To quantify this, Lucas uses a Cobb-Douglas production function:

Y = zKαN1−α ⇒ y =Y

N=

zKαN1−α

N= z

Kα

Nα= zkα

rearranging

y = zkα ⇔ k =(y

z

) 1α

the marginal product of capital

MPK =∂y

∂k= zαkα−1 = zα

((y

z

) 1α

)α−1 = zα

yα−1α

zα−1α

= αz1α y

α−1α

Lucas: why doesn’t capital move from rich to poor? III.

Lucas considers the income per capita ratio between India and theUS, which was about 15 in 1988

I set α = 0.4 which is the average of US and Indian capitalshares ⇒ 1−α

α = 1.5

I if z is the same and y is 15 times higher in the US then theMPK in India must be 151.5 = 58 times higher

So if the return to capital is 58 times higher in India, then whydoesn’t capital flow from the US to India?

Potential answers:

I maybe we don’t account for a type of capital in the model:human capital → we’ll see a model of human capital andlearning-by-doing

I TFP (z) might be different across countries

Why is TFP different across countries?

1. one interpretation of TFP is the level of technology → we willgo through a model with research and development

2. but poor countries do not need to carry out R&D, they canadopt the technology of the rich countries → why don’t theyadopt?

3. a more general interpretation of TFP is the level of effectivetechnology, i.e. countries might have access to the sametechnology but the technology operates with differentefficiencies in different countries→ why?

I maybe institutionsI there might be a gap between the marginal product of capital

and private incentives to save/invest due to taxation,corruption, risk of expropriation, . . .

Institutions

Institutions is a broad concept that captures a set of rules whichgovern the process of decision making

I focus on the set of institutions that encourage ’productiveactivities’ such as investment in physical and human capital ortechnology adoption, over ’diversion or rent-seeking’

I perhaps the gap between the marginal product of capital andprivate incentives to save/invest is bigger in poorer countries

Protection from expropriation and GDP per capita

Baumol (1990 JPE): policies affect the rules of the game

I Ancient Rome: inventor turned to the emperor for a reward,instead of turning to an investor for capital with which to puthis invention into production

I Medieval China: aim was to climb the ladder through imperialexaminations such as in Confucian philosophy and calligraphy

I Medieval Europe: wealth and power were pursued primarilythrough military activities

Olson (1996 JEP): the role of institutions can be seen from thegrowth experiences of divided countries

I in 1990, GDP per capita in West Germany was 2.5 timeshigher than in East Germany

I in 1997, GDP per capita in Hong Kong was 10 times higherthan in China

I North and South Korea

Growth accounting

growth accounting: measures how much of the growth inaggregate output is accounted for by

I factor accumulation → measurableI capitalI laborI human capital

I increases in total factor productivity → not directlymeasurable

→ decompose growth into growth of inputs vs improvements intechnology

Why is this interesting?

growth accounting allows us to understand what is the source ofgrowth

I if the fast growth is due to a high rate of technologicalprogress (high g), then it implies higher steady state growth,i.e. it will last in the long run

I if the fast growth is due to high capital accumulation, thengrowth in the short run exceeds the steady state growth, g ,but will eventually fall back to it due to the diminishingmarginal products of capital

→ decompose growth into growth of inputs vs improvements intechnology

Growth accounting in practice

take the Cobb-Douglas example:

Y = zKαN1−α

take the natural logarithm:

ln Y = ln z + α ln K + (1− α) ln N

the above equation for two consecutive periods:

ln Yt+1 = ln zt+1 + α ln Kt+1 + (1− α) ln Nt+1

ln Yt = ln zt + α ln Kt + (1− α) ln Nt

and their difference

ln Yt+1−ln Yt = ln zt+1−ln zt+α(ln Kt+1−ln Kt)+(1−α)(ln Nt+1−ln Nt)

Growth accounting in practice

ln Yt+1 − ln Yt︸︷︷︸% change in Y

= ln zt+1 − ln zt︸︷︷︸% change in z

+

α (ln Kt+1 − ln Kt)︸︷︷︸% change in K

+(1− α) (ln Nt+1 − ln Nt)︸︷︷︸% change in N

rearrange to get

ln zt+1 − ln zt︸︷︷︸unobservable

= ln Yt+1 − ln Yt︸︷︷︸observable

− (α(ln Kt+1 − ln Kt) + (1− α)(ln Nt+1 − ln Nt))︸︷︷︸weighted average of % change in K and N, observable

choose α to match the capital share of incomeall RHS variables are observable → can construct the % change inz as a residual, known as the Solow residual

Growth rate of measured GDP, capital, labor and theSolow residual

The Solow residual in the US

Growth accounting for the Asian tigers

⇒ high growth rate due to high rate of capital accumulation, andlabor growth

I high rates of growth in capital were caused by high rates ofinvestment

I high rates of growth in labor were driven by populationgrowth and increases in labor force participation

⇒ what does this imply for output growth in the long-run?

Development accounting

One of our objectives is to use the growth model as a tool tounderstand the large cross-country differences in income andproductivity

I technology is exogenous in the Solow model

I assume it is the same across countriesI ⇒ the model predicts that a country is poorer if it has

I a lower saving rateI a higher population growth rateI a higher depreciation rate

I in the data, population growth and depreciation rates are notvery different across countries, but saving rates areapproximately 5 times lower in poor countries

The quantitative effect of lower saving rate

How much lower income per capita does a 5 times lower savingrate imply?

In the steady state: sf (k∗) = (n + d + g)k∗

with the Cobb-Douglas production function f (k∗) =

z

(k∗)α,

k∗ =

(s

z

d + n + g

) 11−α⇒ y∗ =

z

(s

z

d + n + g

) α1−α

for α = 1/3, the ratio of y∗ in rich to poor countries is equal to

y rich

ypoor=

z rich

(srich

z rich

d+n+g

) α1−α

zpoor

(spoor

zpoor

d+n+g

) α1−α

=

(srich

spoor

) α1−α

︸︷︷︸=51/2≈2

(z rich

zpoor

) 11−α

→ much smaller than observed cross-country income differences⇒ a difference in z is needed (what about human capital?)

The quantitative effect of lower saving rate

How much lower income per capita does a 5 times lower savingrate imply?

In the steady state: sf (k∗) = (n + d + g)k∗

with the Cobb-Douglas production function f (k∗) = z(k∗)α,

k∗ =

(sz

d + n + g

) 11−α⇒ y∗ = z

(sz

d + n + g

) α1−α

for α = 1/3, the ratio of y∗ in rich to poor countries is equal to

y rich

ypoor=

z rich(srichz rich

d+n+g

) α1−α

zpoor(spoor zpoor

d+n+g

) α1−α

=

(srich

spoor

) α1−α

︸︷︷︸=51/2≈2

(z rich

zpoor

) 11−α

→ much smaller than observed cross-country income differences

⇒ a difference in z is needed (what about human capital?)

Productivity across countries

Source: Weil p193 Table 7.2 (y = Akαh1−α)

Summary

I development accounting shows differences in the level oftechnology are needed to account for large cross-countyincome differences

I so why is technological progress different across countries?

Three reasons for technological progress:

1. learning-by-doing: acquire knowledge as a byproduct whileworking

2. human capital: accumulated skills and talent of workers thatputs ideas into production

3. research and development that produces new ideas andknowledge

→ endogenous growth models: long-run growth is endogenouslydetermined in the model

Beyond the Solow Model

I before we move on to endogenous growth models, let’s stepback and ask:why is long run growth exogenous in the Solow model?

I the growth mechanism that is endogenous to the Solow modelis capital accumulation

I → only present during the transition to the steady state

I → due to the diminishing marginal returns to capital: theincentive to accumulate capital declines as the capital stockincreases

I different endogenous growth models try to tackle this fromdifferent angles

From the Solow model to the AK model

Using the Cobb-Douglas production function, Y = AKαN1−α:

MPK = αAKα−1N1−α = αAkα−1

To make it easier for comparison with the ‘AK’-model, we now use‘A’ to denote TFP.

I if α < 1 and A is constant ⇒ MPK falls as k rises, so theincentive to accumulate capital will eventually disappear⇒ no growth in the steady state of the basic Solow model

I if α < 1 and A grows at a constant rate g ⇒ MPK might notbe diminishing as k increases→ if the fall in kα−1 is exactly offset by the increase in A ⇒MPK is constant⇒ constant growth in the steady state of the Solow modelwith labor-augmenting technological progress

The AK model

Production function Y = AKαN1−α ⇒ MPK = αAkα−1

I if α = 1 ⇒ Y = AK and MPK = A⇒ independent of k⇒ MPK not diminishing even if A constant⇒ there is growth in the steady state even if A is constant

To see this look at the dynamic equation for capital (with constantA, i.e. g = 0):

(1+n)(k ′−k) = sAf (k)−(n+d)k = sAk−(n+d)k = (sA− (n + d)) k

I if sA < d + n ⇒ k is falling and converges to zero

I if sA > d + n ⇒ k is growing at a constant rate even if A isconstant

The AK model

k

i

(d + n)k

sAk

sAk

if sA < d + n ⇒ k is falling and converges to zeroif sA > d + n ⇒ k is growing at a constant rate even if A isconstant

The AK model

k

i

(d + n)k

sAk

sAk

if sA < d + n ⇒ k is falling and converges to zero

if sA > d + n ⇒ k is growing at a constant rate even if A isconstant

The AK model

I Graphically, the savings curve becomes a straight line so thereis no steady state for k, i.e. there is long run growth even if Ais constant.

I This is referred to as the ‘AK model’ because the productionfunction is Y = AK .

I Different endogenous growth models interpret the reasonsfor the constant MPK (for the AK assumption) differently.

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