The Solow Model

Assumptions

� Aggregate neoclassical production function:

Yt = F (Kt,AtLt)

,→ labour augmenting technical change

,→ constant returns to scale:

F (λKt,λAtLt) = λF (Kt,AtLt) = λYt.

� Example: Cobb�DouglasYt = K

αt (AtLt)

1−α

� Say�s Law and the aggregate capital stock:úKt = sYt − δKt.

� Say�s Law and employment growthúLtLt= n

� Technical progress:úAtAt= g

The Intensive Form

� Let λ = 1AtLt, so that

YtAtLt

= F

µKtAtLt

, 1

¶yt = f (kt)

where kt = KtAtLt

and yt =YtAtLt

,→ Cobb�Douglas case:yt = k

αt

� Inada conditions:

f (0) = 0, f 0(k) > 0, f 00(k) < 0limk→0

f 0(k) = ∞, limk→∞

f 0(k) = 0.

� Growth rate of capital stock:úktkt=úKtKt− g − n

Multiplying through by kt yields

úkt =úKt

AtLt− (n+ g)kt

=sYt − δKtAtLt

− (n + g)kt

= syt − (n + g + δ)kt

Dynamics of the Model

� Dynamics of Capital Stock:úkt = sf (kt)− (n + g + δ)kt.

� Steady�state or balanced growth path (BGP) when úkt = 0:

sf(k∗) = (n + g + δ)k∗.

� Stability:

If sf(kt) > (n + g + δ)kt then úkt > 0If sf(kt) < (n + g + δ)kt then úkt < 0.

Properties of the BGP

� Long�run growth path is independent of initial conditions,→ given similar values of s, n, δ and g, poor economies catch up

� Capital stock grows at the same rate as income.

� Income per worker increasing in s and decreasing in n

� Growth of income per worker depends only on g

k

Investment

sf(k)

(n+g+δ)k

k*

1.The Solow Growth Model

k

Investment

sf(k)

(n+g+δ)k

k*

∆k

k0

2.Dynamics of the Solow Model

k

Investment

sf(k)

(n+g+δ)k

k*

∆k

k0

∆k

k1

3.Dynamics of the Solow Model

k

Investment

sf(k)

(n+g+δ)k

k*

∆k

k0

∆k

k1 k2

4.Dynamics of the Solow Model

k

Investment

sf(k)

(n+g+δ)k

k*k0 k1 k2

5.Dynamics of the Solow Model

Formal Analysis of Convergence(Cobb�Douglas Case)

� Dynamics of capital:úktkt= skα−1t − (n + g + δ)

,→ let xt = ln k :

dxtdt= se(α−1)xt − (n + g + δ)

� Recall Þrst�order TSE around the steady�state, x∗ = ln k∗:

h(xt) ' h(x∗) + h0(x∗)(xt − x∗),→ in this case

h(x∗) = 0 and h0(x∗) ' (α− 1) se(α−1)xt

,→ and so

dxtdt' −λ(xt − x∗)

whereλ = (1− α) se(α−1)x∗

� Solution to this differential equation:

xt = x∗ + e−λt(x0 − x∗)

,→ and so

ln kt = ln k∗ + e−λt(ln k0 − ln k∗)

whereλ = (1− α) sk∗(α−1) = (1− α) (n + g + δ)

� Note that lnyt = α ln kt, and so

ln yt = ln y∗ + e−λt(ln y0 − ln y∗)

Evaluation of the Basic Solow Model

1. Unconditional Convergence

Baumol (1986) � strict interpretation

De Long (1988) � �selection bias�.

Penn World Tables � no convergence

Per CapitaIncome

Time

Rich

Poor

10.Unconditional Convergence

2. Conditional analysisIn Cobb�Douglas case

YiLi= Aiyi = Ai

µsi

ni + g + δ

¶ α1−α

Taking logs:

lnYiLi= lnAi +

α

1− α [ln si − ln(ni + g + δ)] .

Mankiw, Romer and Weil (1992) estimate:

lnYiLi= a + b ln si + c ln(ni + 0.05) + εi

Results:

� R2 = 0.59� �b > 0 and �c < 0 and signiÞcant.� BUT implied α very large (> 0.6) and restriction that �b = −�c is rejected

The Augmented Solow Model

� Aggregate production function given by

Yt = Kαt H

βt (AtLt)

1−α−β

� Evolution of physical and human capital

úKt = sKYtúHt = sHYt,

� Intensive form:

yt = kαt h

βt .

úkt = sKkαt h

βt − (n + g)kt

úht = sHkαt h

βt − (n + g)ht

k

h

(k=0)

(h=0)

k*

h*

.

.

8.Phase Diagram for Augmented Solow model

� Stable BGP where úkt = úht = 0:

k =

Ãs1−βK sβHn + g

! 11−α−β

and h =

µsαKs

1−αH

n+ g

¶ 11−α−β

.

⇒ output per effective worker:

y =

"sαKs

βH

(n + g)α+β

# 11−α−β

Empirical Evaluation

In logs we have

lnY

L= lnA +

α

1− α− β ln sK +β

1− α− β ln sH +α + β

1− α− β ln(n + g).

Mankiw, Romer and Weil (1992) estimate

lnYiLi= a+ b ln sKi + c ln sHi + d ln(ni + 0.05) + εi.

Results:

� R2 = 0.79� b > 0, c > 0 and d < 0 and signiÞcant� Implied values factor shares are α = 0.31 and β = 0.28.� Restriction that b + c = −d, cannot be rejected at the 5% level.

Conditional Convergence

� Previous estimates assume that deviations from a country�s steady stateare random. MRW (1992) also test convergence properties.

� Recall the convergence equation:

ln yt = ln y∗ + e−λt(ln y0 − ln y∗)

ln yt − ln y0 = (1− e−λt) ln y∗ − (1− e−λt) ln y0

,→ substituting for y∗:

ln yt − ln y0 = (1− e−λt) α

1− α lnµ

sini + g + δ

¶− (1− e−λt) ln y0

Per CapitaIncome

Time

High s, low n

Low s, high n

7.

,→ Since yt = Yt/AtLt:

lnYtLt− ln Y0

L0= gt + (1− e−λt) α

1− α lnµ

sini + g + δ

¶−(1− e−λt) ln Y0

L0+ (1− e−λt) lnA0

� MRW estimate growth equation (with t = 25):

lnYiLi− ln Yi,0

Li,0= a+ b ln si + c ln (ni + 0.05) + d ln

Yi,0Li,0

+ εi

� Same basic idea carries over to the augmented Solow model:

lnYiLi− ln Yi,0

Li,0= a + bK ln sKi + bH ln sHi + c ln (ni + 0.05) + d ln

Yi,0Li,0

+ εi

� MRW argue that results are consistent with the augmented model.

Problems with MRWMethodology

� Endogeneity bias.

� Omitted variable bias � Howitt (2000).

� Proxy for sH is arbitrary � Klenow and Rodriguez�Clare (1997),→ other proxies suggest a large role for residual TFP

� TFP growth rates are signiÞcantly correlated with savings rates �Bernanke and Gurkaynak (2001)

,→ see below

Competitive Markets in the Solow Model

� Production of Þrm i:Xi = F (Ki,AtLi) = K

αi (AtLi)

1−α

� Cost minimization:AtFLFK

=wtqt.

In Cobb�Douglas case, this implies that

KtLt=

µα

1− α¶wtqt

or

kt =

µα

1− α¶wtAtqt

(*)

K

L

Isoquant

Isocost Line

K*

L*

w/q

2.Cost Minimization

� Goods market competition⇒ zero proÞts:

Kαi (AtLi)

1−α = wLi + qKi

It follows that

At

µKiLi

¶α= w + q

KiLi

At

µµα

1− α¶wtqt

¶α= wt +

µα

1− α¶wt

and so

w1−αt qαt = αα(1− α)1−αA1−αt

� Using (*) to sub out qt we get the implied real wage

wt = (1− α)Atkαt= marginal product of labour

� Implied user cost of capital

qt = αkα−1t

= marginal product of capital= rt + δ

� Additional predictions:,→ real interest rate shows no secular trend in long run,

,→ real wage grows at rate g.

Cross�country rates of return and the Solow Model

Lucas (1990) � why doesn�t capital ßow from rich to poor countries?.

Example:rIrUS

=

µkIkUS

¶α−1=

µyUSyI

¶1−αα

If α = 0.3:rIrUS

=

µyUSyI

¶2=

µYUS/LUSYI/LI

× AIAUS

¶2If AI = AUS, then

rIrUS

= 202 = 400

� The augmented Solow model resolves this problem

y = kαhβ

,→ high rate of return on capital in poor countries due to diminishingreturns is offset by low level of human capital:

r = αkα−1hβ

BUT it introduces another problem (see Assignment #1)

,→ implies the marginal product of human capital is higher in developingcountries:

wH = βkαhβ−1

,→ can�t get away from the effects of diminishing returns

y=f(k , hR)

k

y

∆yR

∆yP

∆k=1 ∆k=1

Rich

Poor

∆yP < ∆yR

y=f(k , hP)

9.Implication for Rates of Return Conditional on Human Capital