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MacroeconomicsBGSE/UPF

LECTURE SLIDES SET 5

Professor Antonio Ciccone

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III. Economic Growth with Human Capital and

Externalities

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Outline

1. THE IMPORTANCE OF THE ROLE PLAYED BY CAPITAL IN PRODUCTION

2. A SIMPLE MODEL OF ENDOGENOUS GROWTH

3. EXTERNALITIES AND GROWTH

4. HUMAN CAPITAL AND GROWTH

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1. THE IMPORTANCE OF THE ROLE PLAYED BY CAPITAL IN PRODUCTION

Let us return to the Solow model

• Savings a constant fraction s of income• Depreciation rate of capital is

• Population growth n• Rate of echnological progress a

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PRODUCTION FUNCTION with DECREASING RETURNS TO CAPITAL

DECREASING RETURNS TO CAPITAL

CLOSE TO ZERO: STRONG DECREASING RETURNS

CLOSE TO UNITY: WEAK DECREASING RETURNS

1( , ) ( )F K L K AL

0 1

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COBB-DOUGLAS PRODUCTION FUNCTION

1( )Y K AL

1 1 1( )MPK K AL k

(1 )MPK K MPK k

K MPK k MPK

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k

MPK

STRONGDECREASING RETURNSTO CAPITAL

WEAKDECREASINGRETURNS

STRONG AND WEAK DECREASING RETURNS TO CAPITAL

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Effect of savings rate on BGP income/capital under STRONG and WEAK decreasing

returns to capital

11,BGP t

BGPt t

K sk

A L n a

,

,

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BGP t

BGP t

K ss K

• STRONG DECREASING RETURNS TO CAPITALSmall BGP effects of savings rate

• WEAK DECREASING RETURNS TO CAPITAL Large BGP effects of savings rate

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1

BGPBGPs

y kn a

,

, 1BGP t

BGP t

Y ss Y

• STRONG DECREASING RETURNS TO CAPITALSmall BGP effects of savings rate

• WEAK DECREASING RETURNS TO CAPITAL Large BGP effects of savings rate

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How much of international income differences explained by

“propensity of countries to accumulate”?

Depends on strength of decreasing returns to capital

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Convergence to the BGP under WEAK and STRONG decreasing returns to capital

EQUILIBRIUM CAPITAL ACCUMULATION EQUATION

K sF(K , L) K

KK

sF(K , L)

K

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( )K t

( , )F K Ls

K

*K

t

t

KK

CONVERGENCE UNDER STRONG DECREASING RETURNS TO CAPITAL

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( )K t

( , )F K Ls

K

*K

t

t

KK

CONVERGENCE AND WEAK DECREASING RETURNS TO CAPITAL

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INCOME CONVERGENCE EQUATION (CLOSE to balanced growth path)

(1 )( )(ln * ln )tt

t

ya n a y y

y

(1 )( ) ln *

(1 )( ) ln

t

t

t

ya n a y

y

n a y

growth between and

(determinants of BGP income)

convergence parameter ln(initial income)

t t T

f

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Speed of convergence

• STRONG decreasing returns to capitalFAST convergence to BGP

• WEAK decreasing returns to capitalSLOW convergence to BGP

EMPIRICALLY, using cross-country data

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REMEMBER THAT IN THE SOLOW MODEL

Elasticity of output with respect to capital

= Capital income share

= 1/3 (empirically)

=STRONG DECREASING RETURNS: Fast convergence to BGP

Small BGP level effects of savings rate

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2. A SIMPLE MODEL OF ENDOGENOUS GROWTH

Return to the Solow model

• Savings a constant fraction s of income• Depreciation rate of capital is

• No population growth• No technological change

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BUT BUT BUT NO DECREASING RETURNS TO CAPITAL(!)

Y AK

MPK A constant

where A is a CONSTANT

which implies

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THIS PRODUCTION FUNCTION ALSO IMPLIES THAT

Elasticity of output with respect to capital

= Capital income share

• which is evidently in CONTRADICTION with empirical observation

• but let’s see where it leads us

1

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EQUILIBRIUM CAPITAL ACCUMULATION EQUATION

K sAK K

K sA K

-- if sA>, CAPITAL per WORKER and therefore OUTPUT per WORKER grow forever, even if there is NO TECHNOLOGICAL PROGRESS

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( )K t

Y AK KsAK

PERPETUAL CAPITAL ACCUMULATION WITHOUT TECHNOLOGICAL CHANGE

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Is there a BALANCED GROWTH PATH?(path where all variables grow at constant rate)

tK sAK K

KsA

K

To growth rate of capital

From equilibrium accumulation equation

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To growth rate of output

Hence in this ENDOGENOUS GROWTH MODEL

1) long run growth in absence of technological progress

2) a higher savings rate means FASTER GROWTH IN the SHORT, MEDIUM, and LONG run

Y KsA

Y K

Y=AK

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Moreover,

- Implies that the growth rate of capital does NOT fall as economies accumulate capital

YY

sA

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( )K t

Y AKs s sA

K K

t t

t t

K YK Y

GROWTH RATE OF CAPITAL (AND OUTPUT) STAYSCONSTANT IN TIME

same macro fundamentals (s,A,), same growth rate, no matter what initial conditions !!

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MAIN RESULTS:

• perpetual accumulation-driven growth: capital accumulation alone can be the “engine of economic growth”

• savings rate has long-run growth effects: an increase in the savings rate increases the growth rate of capital and output forever

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Endogenous growth and convergence

The AK model has two interesting features:

(A) a poor economy will NOT achieve the income per capita of a rich economy even if has the same macro fundamentals

(B) holding deep parameters or macro fundamentals constant as economies become richer, growth does not slow down

are these two linked? NO!

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tK

Ys

K

t

t

KK

Endogenous growth model where GROWTH RATE OF CAPITAL FALLS IN TIME

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Endogenous growth and convergence

(A) a poor economy will NOT achieve the income per capita of a rich economy even if has the same macro fundamentals

(B) holding deep parameters or macro fundamentals constant as economies become richer, growth MAY STILL slow down

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The problem with the AK model?

• Capital share too large

• Back to the Solow model?

-- externalities

-- human capital

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3. EXTERNALITIES AND ENDOGENOUS GROWTH

In the Solow model we have• perfect competition• no externalities

As a result

which we said was around

YMPK r

K

( )" " CAPITAL INCOME SHARE

Y K r KK Y Y

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Why ?

Because the RESULTS of INVESTMENT are assumed to be

– EXCLUDABLE (only the INVESTOR benefits directly)

But sometimes investments by one particular firm yields results that are

– NON-EXCLUDABLE

– NON-RIVAL

CAPITAL INCOME SHAREY KK Y

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Rivalry and excludability

EXCLUDABLE? YES NO

YES --Banana for personal consumption --Truck for production

-- Crowded highway in Germany -- Clean air in city

RIVAL?

NO -- NON-crowded highway in Italy (for pay) -- PAY TV

--Car design --New form of organization for production

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What if investment has a non-rival, non excludable element?

PRIVATE

Yr

K

(SOCIAL)ECONOMY WIDE

Yr

K

1CAPITAL INCOME SHARE=

3Y KK Y

Externalities: real world has SLOWER convergence than Solow model, but not as slow as in endogenous growth model

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Non-excludability, non-rivalry in the Solow model?

• Technological progress!

• But fell from heaven; or to put it differently COMES WITH THE PASSAGE OF TIME, not with investment

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The Solow model with externalities

• Capital income share reflects the internal return to capital

• Elasticity of aggregate output wrt to capital reflects the social return to capital (private plus external return)

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Solow model with externalities

1( , ) ( )f f f fF K L K EL

where f is an index for firms: f=1,…,N

E Ak

where A grows at rate a; and there are positive externalities to aggregate capital accumulation if and only if > 0

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Solve:

• Optimal behavior of each firm (rental of capital and labor)

• Aggregate production as a function of aggregate inputs (capital and labor)

• Solow and non-Solow dynamics

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4. HUMAN CAPITAL AND ENDOGENOUS GROWTH

In the Solow model we have

• perfect competition

• no externalities

• only ONE TYPE OF CAPITAL: PHYSICAL CAPITAL

As a result

CAPITAL INCOME SHAREY KK Y

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But what about HUMAN CAPITAL?

What is human capital?

• knowledge in people that makes them more productive

In many ways similar to physical capital

• first INVEST (go to school; get some training) • then GET A RETURN (higher wage)

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1CAPITAL INCOME SHARE=

3Y BroadCapital

BroadCapital Y

Human capital (like capital externalities):

• real world has SLOWER convergence than Solow model, but not as slow as in endogenous growth model

• capital and savings explains more of international differences in income than in the Solow model

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Level and growth effects of HC

• Level effect of HC: more HC raises output (“neoclassical view of HC”)

• Growth effect: human capital may determine the rate of technological progress:

may affect growth rate in BGP

or have transitional growth effects only

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Growth effects of HC (A)

• Lucas, JME, 1988: human capital can produce output or “technology”:

,

, " ",

,

( )c t

c t LEARNINGc t c

c t

Aa h HC

A

increasing HC allocated to learning may therefore increase the BGP growth rate(the downside is that output is reduced in the short and medium run)

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“Growth” effects of HC (B)

Nelson and Phelps, AER, 1966

, ,,

,

( ) frontier t c tc t c

frontier t

A Aa h HC

A

, ,c t frontier ta a

,,

,

( ) 1 c tfrontier t c

frontier t

Aa h HC

A

1

( )frontierc

frontier c

aA

A h HC

BGP:

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Empirical work on link between human capital and growth

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The human capital “level” effect

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FROM ELASTICITIES to AGGREGATE RATES OF RETURNTO SCHOOLING

Much of the aggregate work estimates:

1% increase in average years of schoolingincome per capita growth(?)

Formally:increase y

yelasticity=

increase in SS

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Something that is easier to interpret intuitively would be:

1 YEAR increase in average years of schoolingincome per capita growth(?)

*

increase yy

Aggr. Return= Sincrease in S

S

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Elasticity Aggr. Return

0.1 1.25%

0.2 2.5%

0.3 3.75%

0.4 5%

0.5 6.25%

0.7 8.75%

1 12%

1.2 15%

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HUMAN CAPITAL QUALITY

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Human capital externalitiesMoretti, AER 2004

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Estimatingexternalities:

PLANT

INDUSTRY

CITY

-- does output IN THE PLANT(controlling for inputs in plant and industry)INCREASE with THE SHARE OF COLLEGE WORKERSoutside of INDUSTRY but inside CITY?

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Estimating equation:

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Data:

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Benchmark results:

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Physical capital externalities?

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