The Solow Growth Model

October 2007

Development facts

from the Summers-Heston data set

I Enormous variation of per capita income across countries:The poorest countries have about 5% of per capita GDP ofUS per capita GDP.→ dispersion in income levels

Out of 104 countries 37 in 1990 and 38 in 1960 had perworker incomes of less than 10% of the US level.

Development facts

I Enormous variation in growth rates of per worker income.average: 1% to 3%some > 6%some had negative growth rates.

Development facts

I Growth rates determine economic fate of a country overlonger periods of time.A rule of thumb: if a country grows at average rate of g%per year, it takes 72/g years for it to double its per capitaGDP.

I Countries change their relative position in the(international) income dispersion.

Theories of Economic Growth

(1) Real income differences across countries

(2) The welfare consequences of LR growth

The principal conclusion of Solow model

Accumulation of physical capital cannot account for either thevast growth or the vast income differences across countries.

Why?The model treats other potential sources of difference in realincomes as either exogenous (e.g. tech progress) or absentaltogether (e.g. positive externalities from capital).

Relaxing Solow model’s assumptions

1. Optimization: make saving endogenous→ discuss welfare issue.

2. Make tech progress endogenous→ decisions to invest in knowledge accumulation.

3. The role of K: physical K + human K.Positive externalities from k accumulation.

Assumptions

I Y : output; K: capital; L: labor;A: Knowledge (effectiveness of labor).

I Production function at time t:Y (t) = F (K(t), A(t)L(t)).

(1) Production function does not vary over time. Y changesonly when K, L or A changes.

(2) A · L: effective labor.

Assumptions on Production Function

I F (·) has constant returns to scale in K and AL.F (cK, cAL) = cF (K, AL), ∀c ≥ 0.

I Let c = 1AL

, then F ( KAL

, 1) = 1AL

F (K,AL) = YAL

Define f(k) = F (k, 1)⇒ y = f(k).

Assumptions on Production Function

I f(0) = 0, f ′(k) > 0, f ′′(k) < 0Inada conditions:limk→0f

′(k) = ∞, limk→∞f ′(k) = 0

I f ′(k): marginal product of capital.Note:F (K, AL) = ALf( K

AL) ⇒ ∂F

∂K= AL

ALf ′( K

AL) = f ′(k).

Example: Cobb–Douglas Production function

F (K, AL) = Kα(AL)1−α, 0 < α < 1

⇒ Intensive form:f(k) = kα

The evolution of inputs into production

I The initial stock of K, L, A are given.L and A grow at constant rates:

˙L(t) = nL(t)

˙A(t) = gA(t)

n and g: exogenous parameters

I Output is divided b/w consumption and investment:

˙K(t) = s · Y (t)− δK(t)

s and δ: exogenous parameters

The Dynamics of the Model

The dynamics of k:

˙k(t) = sf(k(t))− (n + g + δ)k(t)

(n + g + δ)k: break-even investment.

(figures)

The Balanced Growth Path

The solow model implies that regardless of its starting point,the economy converges to a balanced growth path –Each variable of the model is growing at a constant rate –the growth rate of output per worker is determined solely bythe rate of technology progress.

The Balanced Growth Path

When k = k∗:k = 0, but k = K

AL⇒ K = ALk

I K/K = n + gAL/AL = n + gCRS of f(·) ⇒ Y /Y = n + g.

I K/L and Y/L (output per worker) grow at rate g.

Math aside

I ddt

et = et

x = a · ert

dxdt

= dxd(rt)

· d(rt)dt

= aert · r

I d ln V (t)dt

= dV/dtV

= VV

I K = ALkln K = ln A + ln L + ln k⇒ d

dtln K = d

dt[ln A + ln L + ln k]

⇒ K/K = A/A + L/L + k/k⇒ K/K = g + n

The Impact of a Change in the Saving Rate

I When k = 0, output per worker Y/L grows at g.

I when s ↑, k ↑ ⇒ Y/L grows rate > g.

(Figure)

I At k∗1, k = 0 ⇒ Y/L grows rate at g.

The Impact of a Change in the Saving Rate

I A change in the saving rate has a level effect but not agrowth effect.

I In the Solow model only changes in the rate of technologyprogress have growth effect; all other changes have onlylevel effect.

The Impact on consumption

I On the balanced growth path:

c∗ = f(k∗)− (n + g + δ)k∗

Where k∗ = k∗(s, n, g, δ)

I At k∗0, when s ↑ → c ↓, but as k ↑ → c ↑ again.

(Figure)

The Impact on consumption

∂c∗

∂s= [f ′(k∗(s, n, g, δ)− (n + g + δ)]

∂k∗(s, n, g, δ)

∂s

=⇒ ∂c∗

∂sR 0 as f ′(k∗) R n + g + δ

f ′(k∗): additional output from 4k

n + g + δ: additional output needed to maintain 4k

f ′(k∗) = n + g + δ ⇒ k∗: golden-rule level of the capitalstock (from maximizing c∗).

The Impact on output in the LR

∂y∗

∂s= f ′(k∗(s, n, g, δ)

∂k∗(s, n, g, δ)

∂s

Homework:

(1) Find the elasticity of output w.r.t. saving rate.(hint: k∗ is defined by k = 0)

(2) Suppose saving increases from 20 percent of output to 22percent. By about how much does LR output rise relativeto the path it would have followed?

The Solow Model and the Central Questions of

Growth Theory

I Conclusion:Variations in the accumulation of physical K do notaccount for a significant part of either worldwide economicgrowth or cross–country income differences.

I Two Problems with difference in K −→ large difference inincomes.

(1) The required differences in K are far too large.(2) The required differences in the rate of return on K are far

too large.

Math aside

y = kα ⇒α = dy/y

dk/k= d ln y

d ln kelasticity of output w.r.t. capital

y differs by a factor of x (This means dy = x.)⇒ ln y differs by a factor of ln x (This means d ln y = ln x.)

⇒ ln k differs by a factor of ln xα

⇒ k differs by a factor of eln xα = x1/α

Two Problems with difference in K −→ large difference inincomes.

(1) The required differences in K are far too large.YL today

: YL hundred years ago

= 10 : 1requires:KL today

: KL hundred years ago

= 101/αk : 1

but data:KL today

: KL hundred years ago

= 10 : 1

(K/Y is roughly constant over time.)

(2) The required differences in the rate of return on K are fartoo large.f ′(k) = αkα−1 = αy(α−1)/α ⇒ d ln f ′(k)

d ln y= α−1

α

y differs by a factor of x⇒ f ′(k) differs by a factor of x

α−1α

YL rich

: YL poor

= 10 : 1requires:f ′(k)rich : f ′(k)poor = 10

α−1α : 1

But there is only moderate variation over time and acrosscountries.

Effectiveness of labor

I Problems: Solow Model treats A as exog. and does notidentify what it is.

I A model must identifyI what A is;I how it evolves;I how it affects growth;I why it differs cross countries.

Effectiveness of labor

I Abstract knowledge.

I Education and skills of labor force, the strength of propertyrights, the quality of infrastructure, cultural attitudestoward entrepreneurship and work ,...etc.

I The role of K is physical K + human K.Positive externalities from k accumulation.

Empirical Applications

LR growth in Y/L depends only on technology progress butSR growth can result from technology or capital accumulation.→ Determining the sources of SR growth is an empirical issue.

Growth accounting – To break down the growth rate of agg.output into contributions from the growth of input.

Y

Y= αK ·

K

K+ αL ·

L

L+ R

⇒ Y

Y− L

L= αK [

K

K− L

L] + R

R: Solow residual – a measure of the contribution oftechnology progress.

Empirical Applications

I Growth accounting

(1) Qualities of inputs(2) Young(1994)(3) Productivity slowdown(4) An accumulated stock of R&D as an additional input.

I Convergence

I Saving and investment

Kaldor’s Growth facts

– for U.S. and most other industrialized countries

(1) Output per worker y = Y/L and k = K/L grow at arelatively constant and positive rate.

(2) y and k grow at similar rates, so K/Y constant.

(3) Real return to capital r (real interest rate r − δ) constant.

(4) α = rKY

, 1− α = wLY

, capital and labor shares roughlyconstant. ⇒ motivate Solow growth model.

Kaldor’s Growth facts

I Solow model in the balanced growth pathKK

= n + g, ALAL

= n + g

⇒ YY

= n + g⇒ K/L & Y/L grow at rate g.

I The true test of the Solow model is to what extent it canexplain differences in income level and growth rates acrosscountries (development facts).

Competitive identical firms’ problem:

maxK(t),L(t)

F (K(t), A(t)L(t))− r(t)K(t)− w(t)L(t)

F.O.C.s r(t) = FK(·, ·)w(t) = A(t)FL(·, ·)

FK & FL is homogeneous of degree 0

r(t) = Fk(K(t)

A(t)L(t), 1)

w(t) = A(t)FL(K(t)

A(t)L(t), 1)

r(t) = FK(K(t)

A(t)L(t), 1) w(t) = A(t)FL(

K(t)

A(t)L(t), 1)

I In a balanced growth path K(t)A(t)L(t)

= k(t) = k∗ constant⇒ r& r − δ are constant.

I The wage rate increase at the rate of g.

I Capital share α = r(t)K(t)Y (t)

constant

because r(t) constant, Y (t) and K(t) grow at the samerate (g + n)

I Solow model reproduce all four stylized facts reported byKaldor!