The Solow Model Assumptions Aggregate neoclassical production - - PowerPoint PPT Presentation

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–Douglas

Yt = Kα

t (AtLt)1−α

• Say’s Law and the aggregate capital stock:

ú Kt = sYt − δKt.

• Say’s Law and employment growth

ú Lt Lt = n

• Technical progress:

ú At At = g

The Intensive Form

• Let λ =

1 AtLt, so that

Yt AtLt = F µ Kt AtLt , 1 ¶ yt = f(kt)

where kt =

Kt AtLt and yt = Yt AtLt

, → Cobb–Douglas case: yt = kα

t

• Inada conditions:

f(0) = 0, f 0(k) > 0, f 00(k) < 0 lim

k→0 f0(k) = ∞,

lim

k→∞ f0(k) = 0.

• Growth rate of capital stock:

ú kt kt = ú Kt Kt − g − n

Multiplying through by kt yields

ú kt = ú Kt AtLt − (n + g)kt = sYt − δKt AtLt − (n + g)kt = syt − (n + g + δ)kt

Dynamics of the Model

• Dynamics of Capital Stock:

ú kt = sf(kt) − (n + g + δ)kt.

• Steady–state or balanced growth path (BGP) when ú

kt = 0: sf(k∗) = (n + g + δ)k∗.

• Stability:

If sf(kt) > (n + g + δ)kt then ú

kt > 0

If sf(kt) < (n + g + δ)kt then ú

kt < 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:

ú kt kt = skα−1

t

− (n + g + δ) , → let xt = ln k : dxt dt = 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 dxt dt ' −λ(xt − x∗)

where

λ = (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∗)

where

λ = (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 Capita Income Time Rich Poor

10.Unconditional Convergence

• 2. Conditional analysis

In Cobb–Douglas case

Yi Li = Aiyi = Ai µ si ni + g + δ ¶ α

1−α

Taking logs:

ln Yi Li = ln Ai + α 1 − α [ln si − ln(ni + g + δ)] .

Mankiw, Romer and Weil (1992) estimate:

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

Results:

• R2 = 0.59
• ˆ

b > 0 and ˆ c < 0 and signiÞcant.

• BUT implied α very large (> 0.6) and restriction that ˆ

b = −ˆ c is rejected

The Augmented Solow Model

• Aggregate production function given by

Yt = Kα

t Hβ t (AtLt)1−α−β

• Evolution of physical and human capital

ú Kt = sKYt ú Ht = sHYt,

• Intensive form:

yt = kα

t hβ t .

ú kt = sKkα

t hβ t − (n + g)kt

ú ht = sHkα

t hβ t − (n + g)ht

k h (k=0) (h=0) k* h* . .

8.Phase Diagram for Augmented Solow model

• Stable BGP where ú

kt = ú ht = 0: k = Ã s1−β

K sβ H

n + g !

1 1−α−β

and h =

µsα

Ks1−α H

n + g ¶

1 1−α−β

. ⇒ output per effective worker: y = " sα

Ksβ H

(n + g)α+β #

1 1−α−β

Empirical Evaluation In logs we have

ln Y L = ln A + 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

ln Yi Li = 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 state

are 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 ln µ si ni + g + δ ¶ − (1 − e−λt) ln y0

Per Capita Income Time High s, low n Low s, high n

7.

, → Since yt = Yt/AtLt: ln Yt Lt − ln Y0 L0 = gt + (1 − e−λt) α 1 − α ln µ si ni + g + δ ¶ −(1 − e−λt) ln Y0 L0 + (1 − e−λt) ln A0

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

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

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

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

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

Problems with MRW Methodology

• 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) = Kα

i (AtLi)1−α

• Cost minimization:

AtFL FK = wt qt .

In Cobb–Douglas case, this implies that

Kt Lt = µ α 1 − α ¶ wt qt

• r

kt = µ α 1 − α ¶ wt Atqt

(*)

K L Isoquant Isocost Line K* L* w/q

2.Cost Minimization

• Goods market competition ⇒ zero proÞts:

Kα

i (AtLi)1−α = wLi + qKi

It follows that

At µKi Li ¶α = w + qKi Li At µµ α 1 − α ¶ wt qt ¶α = wt + µ α 1 − α ¶ wt

and so

w1−α

t

qα

t = αα(1 − α)1−αA1−α t

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

wt = (1 − α) Atkα

t

= marginal product of labour

• Implied user cost of capital

qt = αkα−1

t

= 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:

rI rUS = µ kI kUS ¶α−1 = µyUS yI ¶1−α

α

If α = 0.3:

rI rUS = µyUS yI ¶2 = µYUS/LUS YI/LI × AI AUS ¶2

If AI = AUS, then

rI rUS = 202 = 400

• The augmented Solow model resolves this problem

y = kαhβ , → high rate of return on capital in poor countries due to diminishing

returns 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 developing

countries:

wH = βkαhβ−1 , → can’t get away from the effects of diminishing returns

y=f(k , h

R)

k y ∆yR ∆yP ∆k=1 ∆k=1 Rich Poor ∆yP < ∆yR y=f(k , h

P)

9.Implication for Rates of Return Conditional on Human Capital