advanced macroeconomics - the solow model

Advanced MacroeconomicsThe Solow Model

Micha l Brzoza-Brzezina / Marcin Kolasa

Warsaw School of Economics

Micha l Brzoza-Brzezina / Marcin Kolasa (WSE) Ad. Macro - Solow model 1 / 21

Introduction

Authors: Robert Solow (1956) and Trevor Swan (1956)

Very simple framework for analyzing the mechanics of economicgrowth

Engines of economic growth:

Technological change (exogenous)Capital accumulation (endogenous)

NOT a framework for explaining deep sources of economic growth

Departure point for more elaborate analyses of growth

Our discussion:

General equilibrium frameworkDiscrete time


Basic setup

Closed economy

No government

One homogeneous final good

Price of the final good normalized to 1 in each period (all variablesexpressed in real terms)

Two types of agents in the economy:

FirmsHouseholds

Firms and households identical: one can focus on a representativefirm and a representative household, aggregation straightforward


Firms I

Final output produced by competitive firms

Production functionYt = F (Kt ,AtLt) (1)

where:

Kt - capital stockLt - labour input (population)At - technology

Capital and labour inputs rented from households

Technology grows at a constant rate g > 0:

At+1 = (1 + g)At

Remarks:

F must be neoclassicalTechnology must be of labour-augmenting type (Harrod neutral)Technology is available for free


Firms II

Neoclassical production function FConstant returns to scale (wrt. capital and labour)Positive and decreasing marginal products of capital and labourInada conditions(Essentiality)

E.g. for given Lt and At :

Kt

Yt


Firms III

Maximization problem of firms:

maxLt ,Kt

{F (Kt ,AtLt)−WtLt − RK ,tKt}

Firms maximize their profits, taking factor prices (wage rate Wt andcapital rental rate RK ,t) as given (competitive factor markets)

First order conditions:

Wt =∂F

∂Lt(2)

RK ,t =∂F

∂Kt(3)

Remarks:

Firms’ problem is staticFirms make zero profits


Households

Own production factors (capital and labour), so earn income onrenting them to firms

Labour supplied inelastically, grows at a constant rate n > 0:

Lt+1 = (1 + n)Lt

Capital is accumulated from investment It and subject to depreciation(at a constant rate δ > 0):

Kt+1 = (1− δ)Kt + It (4)

Total income of households can be split between consumption orsavings (equal to investment):

WtLt + RK ,tKt = Ct + St = Ct + It

Save a constant fraction s of income Yt (do NOT optimize):

It = sYt Ct = (1− s)Yt (5)


General equilibrium

Market clearing conditions:

Output produced by firms must be equal to households’ total spending(on consumption and investment):

Yt = Ct + It

Labour supplied by households must be equal to labour inputdemanded by firmsCapital supplied by households must be equal to capital inputdemanded by firms

Definition of equilibrium

A sequence of {Kt ,Yt ,Ct , It ,Wt ,RK ,t}∞t=0 for a given sequence of{Lt ,At}∞t=0 and an initial capital stock K0, such that: Kt satisfies (4), Yt

is given by (1), Ct and It are given by (5), and Wt and RK ,t are given by(2) and (3), respectively.


Production function in intensive form

Using CRS assumption for F :

yt ≡Yt

AtLt=

1

AtLtF (Kt ,AtLt) = F (

Kt

AtLt, 1) = f (kt)

Variables divided by AtLt are ”per unit of effective labour”

Model is easier to analyze if working with normalized variables

f ′(kt) = ∂F∂Kt

(marginal product of capital)

Factor prices (Wt from Euler’s theorem):

RK ,t = f ′(kt)Wt = At (f (kt)− f ′(kt)kt)

Capital share in output is equal to output elasticity wrt. capital:

α(kt) =RK,tKt

Yt=

∂F∂Kt

Kt

Yt= f ′(kt)kt

f (kt)

Labour share: 1− α(kt)


Production function in intensive form II

Properties of f (kt):

f ′(kt) > 0f ′′(kt) < 0Inada conditions: lim

kt→0f ′(kt) =∞ lim

kt→∞f ′(kt) = 0

kt

f(kt)


Fundamental equation of the Solow model

Capital law of motion using normalized variables

kt+1 =1− δ

(1 + g)(1 + n)kt +

s

(1 + g)(1 + n)f (kt) (6)

Graphically:

kt

kt+145

o

k*

k*


Steady state equilibrium (balanced growth) I

If kt = k∗ then:

∆Kt+1

Kt= Kt+1

Kt− 1 = At+1Lt+1k∗

AtLtk∗− 1 = (1 + n)(1 + g)− 1 ≈ n + g

∆Yt+1

Yt= At+1Lt+1f (k∗)

AtLt f (k∗) − 1 = (1 + n)(1 + g)− 1 ≈ n + g

∆Ct+1

Ct= do it yourself

∆It+1

It= do it yourself

∆Wt+1

Wt= do it yourself

RK ,t = do it yourself


Steady state equilibrium (balanced growth) II

Per capita Kt , Ct and Yt :

e.g.∆(

Yt+1Lt+1

)(

YtLt

) =

(Yt+1Lt+1

)(

YtLt

) − 1 =

(Yt+1Yt

)(

Lt+1Lt

) − 1 = (1+n)(1+g)(1+n) − 1 = g

Factor shares (elasticities):

α(k∗) = f ′(k∗)k∗

f (∗) = const


Long-run implications

Fundamental equation (6) rewritten:

∆kt+1

kt=

1

(1 + g)(1 + n)

[sf (kt)

kt− (g + n + δ + ng)

](7)

Capital per unit of effective labour is constant in steady state, whichimplies:

sf (k∗)

k∗= g + n + δ + ng (8)

We can use this relationship to assess long-run implications of theSolow model


Permanent change in the savings rate

Differentiation of the long-run relationship (8) yields:

∂k∗

∂s=

k∗

s(1− α(k∗))(9)

Output elasticity wrt. the savings rate:

∂f (k∗)

∂s

s

f (k∗)=

s

f (k∗)f ′(k∗)

∂k∗

∂s(10)

Using (9) and (10), after rearrangement, yields:

∂f (k∗)

∂s

s

f (k∗)=

α(k∗)

1− α(k∗)(11)

The long-run effect of a change in s depends positively on α(k∗)


Golden rule of capital accumulation

What is the savings rate maximizing the steady-state consumption?

Steady state consumption is equal to

c∗ = (1− s)f (k∗) (12)

FOC obtained by differentiating RHS of this equation wrt. s:

(1− s)f ′(k∗)∂k∗

∂s− f (k∗) = 0 (13)

Using (9) and rearranging yields:

s = α(k∗)

Steady state consumption is at the highest level if s = α(k∗)


Transitional dynamics

kt

kt+145

o

k*

k*

k0

Steady state equilibrium is a stable long-run equilibrium


Conditional convergence

∆kt+1

kt=

1

(1 + g)(1 + n)

[sf (kt)

kt− (g + n + δ + ng)

]

kt

kt+1

kt

k*


Speed of convergence I

First-order approximation of the fundamental equation (7) aroundln kt = ln k∗ (log-linearization):

ln

(kt+1

kt

)=

1

(1 + g)(1 + n)

[sf (kt)

kt− (g + n + δ + ng)

]=

=s

(1 + g)(1 + n)

f ′(k∗) (k∗)2 − f (k∗)k∗

(k∗)2(ln kt − ln k∗) =

=s

(1 + g)(1 + n)

f (k∗)

k∗

(f ′(k∗)k∗

f (k∗)− 1

)(ln kt − ln k∗) =

=g + n + δ + ng

(1 + g)(1 + n)(α(k∗)− 1) ln

(ktk∗

)(14)

Speed of convergence depends negatively on α(k∗)


Speed of convergence II

We can write equation (14) in compact form as:

ln

(kt+1

kt

)= −ω ln

(ktk∗

)(15)

where: ω = g+n+δ+ng(1+g)(1+n) (1− α(k∗))

Equation (15) applies also for yt :

ln

(yt+1

yt

)= −ω ln

(yty∗

)(16)

Equations (15) (or (16)) imply that each period 100ω per cent of thegap between capital (or output) and its steady-state level disappears:

ln

(yt+1

y∗

)= (1− ω) ln

(yty∗

)(17)


Main implications of the Solow model

Long-run growth (of output per capita) possible only withtechnological progress

But: technology exogenous in the model

We should observe conditional, but not necessarily unconditional,convergence

In line with the data

For standard values of parameters (taken from the data), in particularα(k∗) ≈ 0.33:

Elasticity of output wrt. savings rate rather low

Far too low to explain differences in income per capita in the worldDifferences in technology important

Speed of convergence rather high

Implies too high pace of catching-up

Better fit if capital is defined in broader sense, i.e. Kt also includeshuman capital so that α(k∗) ≈ 0.66


advanced macroeconomics - the solow model

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