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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
Macroeconomics Set 5
4
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
Macroeconomics Set 5
5
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
Macroeconomics Set 5
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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
Macroeconomics Set 5
7
k
MPK
STRONGDECREASING RETURNSTO CAPITAL
WEAKDECREASINGRETURNS
STRONG AND WEAK DECREASING RETURNS TO CAPITAL
Macroeconomics Set 5
8
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
,
,
11
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
9
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
Macroeconomics Set 5
10
How much of international income differences explained by
“propensity of countries to accumulate”?
Depends on strength of decreasing returns to capital
Macroeconomics Set 5
11
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
Macroeconomics Set 5
12
( )K t
( , )F K Ls
K
*K
t
t
KK
CONVERGENCE UNDER STRONG DECREASING RETURNS TO CAPITAL
Macroeconomics Set 5
13
( )K t
( , )F K Ls
K
*K
t
t
KK
CONVERGENCE AND WEAK DECREASING RETURNS TO CAPITAL
Macroeconomics Set 5
14
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
Macroeconomics Set 5
15
Speed of convergence
• STRONG decreasing returns to capitalFAST convergence to BGP
• WEAK decreasing returns to capitalSLOW convergence to BGP
EMPIRICALLY, using cross-country data
Macroeconomics Set 5
16
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
Macroeconomics Set 5
17
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
Macroeconomics Set 5
18
BUT BUT BUT NO DECREASING RETURNS TO CAPITAL(!)
Y AK
MPK A constant
where A is a CONSTANT
which implies
Macroeconomics Set 5
19
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
Macroeconomics Set 5
20
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
Macroeconomics Set 5
21
( )K t
Y AK KsAK
PERPETUAL CAPITAL ACCUMULATION WITHOUT TECHNOLOGICAL CHANGE
Macroeconomics Set 5
22
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
Macroeconomics Set 5
23
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
Macroeconomics Set 5
24
Moreover,
- Implies that the growth rate of capital does NOT fall as economies accumulate capital
YY
sA
Macroeconomics Set 5
25
( )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 !!
Macroeconomics Set 5
26
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
Macroeconomics Set 5
27
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!Macroeconomics Set 5
28
tK
Ys
K
t
t
KK
Endogenous growth model where GROWTH RATE OF CAPITAL FALLS IN TIME
Macroeconomics Set 5
29
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
Macroeconomics Set 5
30
The problem with the AK model?
• Capital share too large
• Back to the Solow model?
-- externalities
-- human capital
Macroeconomics Set 5
31
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
13
Macroeconomics Set 5
32
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
Macroeconomics Set 5
33
Rivalry and excludability
EXCLUDABLE? YES NO
YES --Banana for personal consumption --Truck for production
-- Crowded highway in Germany -- Sun light
RIVAL?
NO -- NON-crowded highway in Italy (for pay) -- PAY TV
--Car design --New form of organization for production
Macroeconomics Set 5
34
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
Macroeconomics Set 5
35
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
Macroeconomics Set 5
36
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)
Macroeconomics Set 5
37
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
Macroeconomics Set 5
38
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
Macroeconomics Set 5
39
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
Macroeconomics Set 5
40
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)
Macroeconomics Set 5
41
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
Macroeconomics Set 5
42
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
Macroeconomics Set 5
43
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)
Macroeconomics Set 5
44
“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:
Macroeconomics Set 5
50
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
Macroeconomics Set 5
51
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
Macroeconomics Set 5
52
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%
Macroeconomics Set 5
63
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?
Macroeconomics Set 5