Lesson 3 : Economic Growth
Luigi Iovino
Fall 2013
Introduction
Introduction
GDP per capital in China grew at the average rate of 9% per
year during the last 20 years. It corresponds now to about 10% of
the US level. At this rate, it would catch up with the current US
level in 25 years
In Algeria, which is currently around the same level of GDP per
capita as China, GDP per capital grew at the average rate of 1.5%
per year during the last 20 years. At this rate, it would catch up
with the current US level in 140 years only
Lucas : “Once you start thinking about [economic growth issues],
it’s hard to think about anything else”
First application of a dynamic general equilibrium model
Outline
1. The Solow model
2. The Ramsey model
Solow : Basic hypotheses
Neoclassical growth model(s) :
I
1 representative household, supplying labor L and capital K
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1 representative …rm, supplying good Y
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Perfect competition : …rm and household are price-takers
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Neoclassical production function (see next slide)
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Exogenous growth rate of productivity and labor
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Continuous time (convenient)
Solow model : Households save a fraction s of their income. No
optimal consumption/saving behavior (contrary to the Ramsey
model)
Solow : Neoclassical production Function
Yt = F (Kt , At Lt )
Y is GDP, K is capital, and L is labor. A is labor augmenting
technology, shifter of production. Free, non rival, non excludable.
AL represents “e¢ cient labor”
F is increasing and concave in both K and L. Diminishing
marginal returns
F has constant returns to scale, i.e. F (λK , λAL) = λF (K , AL)
Classic case is Cobb-Douglas : F (K , AL) = K α (AL)1
α
Solow : Neoclassical production Function
Assume technology and labor grow at constant rates :
Ȧt = gAt ,
L̇t = nLt
Denote normalized variables with lower case : x = X /AL
Constant returns to scale :
yt =
Yt
=F
At L t
Kt
,1
At Lt
= F (kt , 1) = f (kt )
where f (k ) = F (k, 1). f 0 > 0 and f 00 < 0
It is su¢ cient to study the dynamics of k
Solow : Representative …rm pro…ts
The representative …rm maximizes its pro…ts
F (Kt , At Lt )
wt Lt
rt Kt
Notice price of …nal good is normalized to one, and wages and
interest rates are takes as given by the …rm (perfect competition)
We obtain
wt = At FL (Kt , At Lt ),
rt = FK (Kt , At Lt )
Inputs are paid their marginal product
Solow : Euler theorem
Standard result under perfect competition and constant returns
to scale : there are no pro…ts
π t = F (Kt , At Lt )
= F (Kt , At Lt )
=0
wt Lt
rt Kt
At FL (Kt , At Lt )Lt
Exercise : Check with Cobb-Douglas
FK (Kt , At Lt )Kt
Solow : Law of motion of capital
Law of motion of capital :
K̇t = It
δKt
with K̇ = dK /dt
This is the continuous-time version of Kt +1
Kt = It
δKt
To understand the link, take a smaller time unit dt :
Kt +dt
Kt = It dt
δKt dt )
Kt +dt Kt
= It
dt
δKt
Solow : Law of motion of capital
We are in a closed economy, which means that whatever is
invested domestically has to come from domestic savings :
St = It
Therefore :
K̇t = St
δKt
Savings are crucial because they a¤ect the evolution of the capital
stock
Solow : Savings
Households consume and save their income :
wt Lt + rt Kt = Ct + St
Using the Euler theorem, we get : F (Kt , At Lt ) = Ct + St
The KEY assumption in the Solow/Swan model is that of
constant, given savings rate :
St = sF (Kt , At Lt )
The resulting law of motion of capital is a di¤erential equation in
K :
K̇t = sF (Kt , At Lt ) δKt
Solow : Steady state
Divide the LOM of capital by At Lt :
K̇t
= sF
At L t
Kt
,1
At Lt
Notice that Ẋt /Xt = d log(Xt )/dt
Product rule :
X t˙Y t
Xt Yt
=
X˙t
Xt
+
Y˙t
Yt
δ
Kt
At L t
Solow : Steady state
We get the law of motion of capital per e¢ cient unit of labor :
k̇t = sf (kt )
(δ + g + n)kt
Steady state : capital per e¢ cient unit of labor ceases to grow,
which happens for k̇t = 0 :
sf (k ) = (δ + g + n )k
(at the s.s., investment is exactly enough to replenish capital per
unit of AL)
k̇t = 0 corresponds to “balanced growth” : K grows at rate g + n
Solow : Steady state
Cobb-Douglas case :
k =
Comparative statics
s
δ+g +n
1/(1 α)
Solow : Steady state
Solow : Golden rule
Agents care about consumption
Steady-state consumption is c = f (k )
k de…ned by sf (k ) = (δ + g + n )k
(δ + g + n)k with
What is the level of s that maximizes consumption ? The one
that sets ∂c /∂k = 0, i.e. f 0 (k ) = δ + g + n (tangency
property)
Solow : Golden rule
Solow : Empirical implications
This simple framework has wide-ranging testable implications. Now
review them, and their link with theory
1. Growth Accounting
2. Convergence
3. Level Accounting
Solow : Growth accounting
Consider the production function in its general form :
Yt = F (Kt , At Lt )
Take a total di¤erentiation and divide through by Y :
Ẏ
F K K̇
AF L L̇ LFL A Ȧ
= K
+ L
+
Y
Y K
Y L
Y A
Solow : Growth accounting
Now denote growth rates by gY = Ẏ /Y , gK = K̇ /K and
gL = L̇/L
Let x = LFYL A ȦA be the contribution of technology to growth.
(Recall with competition r = FK and w = AFL )
We obtain well known estimates of Total Factor Productivity
(“Solow residual”) :
x = gY
α K gK
αL gL
where we de…ne factor shares αK = rK /Y and αL = wL/Y
Generalization of the well-known Cobb Douglas case. Crucial
assumption is perfect competition
With observed (time-varying) factor shares and observed
(time-varying) growth rates in Y , K and L, can get x
Solow : Growth accounting
Solow : Growth accounting
Pitfalls :
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Measurement of αK and αL
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Measurement of gK and gL
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Technological regress ?
Solow : Convergence
How fast does a country converge to k ?
Take a Taylor expansion of k̇ around k = k :
k̇ ' k̇ jk =k +
∂k̇
j k =k (k
∂k
k )
After rearranging :
k̇t '
(1
|
αK )[δ + n + g ](kt
{z
}
λ
k )
Solow : Convergence
This implies :
ẏt '
λ(yt
y )
Hence, in level, output satis…es : yt = y
e
λt (y
y0 )
What is the value of λ ?
δ + n + g ' 0 (δ = 3 4%, n = 1
αK ' 1/3, so λ ' 4%
2% and g = 1
2%) and
y and k take 18 years to cross half of their distance to y and
k :very fast ! At odds with reality
Solow : Unconditional convergence ?
Approximate growth regression in discrete time (Baumol (1986))
gi ,t,t
1
= b0 + b1 log(yi ,t
1 ) + ei ,t
ei ,t is a stochastic term capturing all omitted in‡uences
Baumol (1986) : OECD countries 1870-1979, get b1 negative
and signi…cant
DeLong (1988) : But not for the whole world. No unconditional
convergence
May be too demanding. If countries do di¤er, they should
converge to their own steady state
Convergence ?
Not always !
Solow : Conditional convergence ?
If countries di¤er, a more appropriate regression may be :
gi ,t,t
1
= b0,i + b1 log(yi ,t
1 ) + ei ,t
Barro and Sala-i-Martin (2004) estimate models where b0,i is
assumed to be a function of country-speci…c variables :
gi ,t,t
1
= Xi0,t β + b1 log(yi ,t
1 ) + ei ,t
These regressions tend to show negative estimates of b1 , but with
much lower magnitudes than what is implied by Cobb-Douglas
calibration
Solow : Level accounting
Consider a simpli…ed production function :
Yj = Kjα (Aj Hj )1
α
Get cross country information on Kj (permanent inventory
method) and Hj (years of schooling multiplied by labor input).
Choose α = 1/3. Construct "predicted" income at a point in time
using :
Ỹj = Kj1/3 (AUS Hj )2/3
1/3
AUS is computed so that ỸUS = KUS
(AUS HUS )2/3
Then one can compare each Ỹj with the actual series. Gap
between the two represents the contribution of technology
Solow : Level accounting
Solow : Level accounting
Di¤erences in physical and human capital still matter a lot
However, shows signi…cant technology (productivity) di¤erences
Fit of the model seems to deteriorate over time
Based on many assumptions : Cobb-Douglas, identical factor
shares, perfect competition, range of assumptions to compute K
and H...
Solow : Conclusions
Evidence in favor of convergence overwhelming - even with
econometric caveats. Supports diminishing marginal returns, i.e. a
steady state
If so, policy recommendations clear :
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Saving, Investment, Human Capital have only level e¤ects,
not growth e¤ects. Consistent with level accounting
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Long Run growth derives from TFP growth. About which
model is silent
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Does NOT mean one "should" target golden rule saving level.
No optimal consumption/investment decision
Need for a theory of TFP
Need for a framework with optimal, forward looking behaviour :
Ramsey model
Ramsey model
Optimal behaviour (consumption/saving)
Key for policy analysis
Demographics, taxation policy, institutions surely all a¤ect
individual behaviour. And Consumption/Saving decision
We will not solve a dynamic optimization program in continuous
time. Take optimal discrete time decisions to the limit
To simplify, we assume no growth rate in labour : Lt = 1
Ramsey : Euler equation
A representative household maximizes :
∞
Vt =
∑
s =0
1
1+ρ
s
U ( Ct + s )
subject to :
Kt +1 = (1 + rt
δ)Kt + wt Lt
Ct
Solving the problem with a Lagrangean gives us the Euler
equation :
1 + rt +1 δ 0
U 0 ( Ct ) =
U ( Ct + 1 )
1+ρ
Ramsey : Euler equation
If we take shorter time intervals dt, the Euler equation becomes :
U 0 ( Ct ) =
1 + (rt +dt δ) dt 0
U (Ct +dt )
1 + ρ dt
Constant elasticity of substitution : U (C ) = C 1
The Euler equation becomes :
Ct +dt
1 + (rt +dt δ) dt
=
Ct
1 + ρ dt
γ
1/γ / (1
1/γ)
Ramsey : Euler equation
In logs :
log(Ct +dt )
log(Ct ) = γ[log[1 + (rt +dt
δ) dt ]
log(1 + ρ dt )]
As dt goes to zero, this can be approximated :
log(Ct +dt )
log(Ct ) = γ(rt +dt
ρ) dt
δ
and we can write :
Ċt
d log(Ct )
= γ(rt
=
dt
Ct
δ
ρ)
Ramsey : Euler equation
Need to solve for rt . With competitive factor markets,
h
i
Kt
∂
A
F
,
1
t
At
∂F (Kt , At )
rt =
=
= f 0 (kt )
∂Kt
∂Kt
Here we have used pro…t maximization by the …rm and Lt = 1
Thus,
Ċt
= γ[f 0 (kt )
Ct
δ
ρ]
Ramsey : Euler equation
As for k, de…ne c = C /A. The Euler equation becomes :
ċt
= γ[f 0 (kt )
ct
δ
ρ
g /γ]
Key is that preference parameters (γ, ρ) now govern
consumption-saving decisions
(1)
Ramsey : LOM of capital
In addition, the continuous-time LOM of capital is :
K̇t = rt Kt + wt Lt
Ct
δKt
Using the Euler theorem and normalizing :
K̇t
= f (kt )
At
δkt
ct
Using the product rule :
k̇t = f (kt )
(δ + g )kt
ct
(2)
Ramsey : Steady state
Equations (1) and (2) de…ne the economy’s optimal dynamics
Notice equation (2) is very similar to case with exogenous
savings. Di¤erence is that ct is now endogenous, and governed by
equation (1), i.e. optimal intertemporal choice of consumption vs.
saving
Where does this economy converge to ? A steady state de…ned
by ċt = 0 and k̇t = 0
This de…nes a balanced growth path where C and K grow at
rate g
Ramsey : Steady-state capital-labor ratio
We must have a steady state capital-labor ratio k such that
f 0 (k ) = δ + ρ + g /γ
This pins down steady state capital-labor ratio
Note again importance of preference parameters : steady state
capital-labor ratio increases in γ and decreases in ρ
Lower discount rate means more patience and thus greater
savings
Higher elasticity of intertemporal substitution means a stronger
reaction of savings to the rate of return to capital
Ramsey : Steady-state consumption
c must verify
c = f (k )
( δ + g )k
Both k and c increase with γ and fall with ρ
Phase diagram
Ramsey : Steady-state consumption
Note this does NOT maximize steady state consumption, which
would verify
f 0 (k̃ ) = δ + g
k̃ is the modi…ed Golden rule capital-labor ratio
k < k̃, but still, k is an OPTIMAL capital-labour ratio
Earlier consumption is preferred to later consumption (so less
capial in s.s.)
Ramsey : Policy analysis
What is the e¤ect of taxation ?
Introduce linear tax policy on capital, at rate τ. Proceeds are
rebated to consumers. Capital accumulation remains :
k̇t = f (kt )
(δ + g )kt
ct
But now rate of return is given by (1 τ )(f 0 (kt )
rate of (normalized) consumption is given by :
ċt
= γ[(1
ct
τ )(f 0 (kt )
δ)
ρ
g /γ]
δ) So growth
Ramsey : Policy analysis
Steady state capital per capita :
f 0 (k ) = δ +
ρ + g /γ
1 τ
Since f 0 (.) is decreasing, higher taxes decrease k
Higher taxes depress capital accumulation and reduce income per
capita
Conclusion
Major contribution : opens the black box of capital accumulation
Can draw OPTIMAL implications Does this generate new
insights about the source of cross-country income di¤erences ?
Largely, no. But model clari…es the nature of the economic
decisions, so that we are in a better position to ask questions
about fundamental causes of economic growth
Now, we can think also about productivity growth in an optimal
framework. E.g. human capital accumulation, innovation...