The Facts of Economic Growth and the Introdution tothe Solow Model

Lorenza Rossi

Goethe University 2011-2012

Course Outline

FIRST PART - GROWTH THEORIES

Exogenous GrowthThe Solow ModelThe Ramsey model and the Golden Rule

Introduction to Endogenous Growth modelsThe AK model - Romer (1990)Two sector model of Endogenous growth

SECOND PART - BUSINESS CYCLE

Introduction to NK modelThe BMW model as a static approximation of a forward-looking NKmodelThe BMW model in a closed economy: in�ation targeting versus TaylorrulesThe BMW model in an open economy: comparisons with the MundelFleming modelChristiano, Eichenbaum and Evans model (2005)

TODAY

Brief Review Growth Stylized Facts

Introduction to the Solow Model

DerivationsThe model performance and stylized facts

STYLIZED FACT 1

There is an enormous variation in the per capita income acrosseconomies. The poorest countries have per capita income that areless than 5 percent of per capita incomes in the riches countries.

The Rich and the Poor

The Rich and the Poor

World Population by GDP per worker

PER CAPITA GDP 1960

PER CAPITA GDP 2000

The world distribution of income in 1970

The world distribution of income in 2000

STYLIZED FACT 2

Rates of Economic Growth vary substantially across countries

STYLIZED FACT 3

Growth rates are not generally constant over time. For the world aswhole, growth rates were close to zero over most of the history buthave increased sharply in the twentieth century. For individualcountries, growth rates also change over time.

STYLIZED FACT 4

A country relative position in the world distribution of per capitaincomes is not immutable. Countries can move from being poor tobeing rich, or viceversa.

EXAMPLE: Venezuela vs Italy.

Fifteen Growth Miracles

Fifteen Growth Disasters

OTHER STYLIZED FACTS. FACT 5

In the US over the last century,

1 the real rate of return to capital, r , shows no trend upward ordownward;

2 the shares of income devoted to capital, rK/Y , and the share ofincome devoted to labor, wL/Y , show no trend;

3 the average growth rate of output per person has been positive andrelatively constant over time, i.e. the US exhibits steady, sustainedper capita income growth.

Real Per Capita GDP in the US

OTHER STYLIZED FACTS: FACT 6

Growth in output and growth in the volume of international tradeare closely related

Growth in trade and GDP 1960-1990

OTHER STYLIZED FACTS: FACT 7

Both skilled and unskilled workers tend to migrate from poor to richcountries or regions

ROBERT LUCAS: this movements of labor tell us something about realwages. The returns of both skilled and unskilled

Kaldor Stylized Facts and the Solow Model

The Solow Model

Solow seminal paper (1956). "A Contribution to the Theory ofEconomic Growth"

Check for the ability of the model to explain the stylized facts

Neoclassical model

Countries produce and consume one single good (units of GDP);There is no international trade (since there is only one good)Technology is exogenousPerfect competition in all markets

The Solow Model

The basic model is characterized by two equations1 a production function;2 a capital accummulation equation

The Solow Model

The Neoclassical Aggregate Production Function

Y (t) = F (A (t) ,K (t) , L (t))

where K (t) is physical capital, L (t) is labor and A (t) is a exogenoustechnology shift (TFP)

Technology is free; it is publicly available as a non-excludable,non-rival good.

The Solow Model: Key Assumption

Assumptions

F exhibits constant return to scale in K and L =) F is linearhomogeneous (homegeneous of degree 1)

The Solow Model: Key Assumption

Capital accumulation

K = I � d �K .

I = sY where s represents a constant savings rate.

d is capital depreciation rate

The Solow Model

The Inada Conditions

limK!0

FK (�) = ∞ and limK!∞

FK (�) = 0 for all L > 0 and all A

limL!0

FL (�) = ∞ and limL!∞

FL (�) = 0 for all K > 0 and all A

Important in ensuring the existence of interior equilibria.

The Solow Model

Firms�pro�ts maximization

FOCs

From the assumption of homogeneity of degree 1

and thus �rms pro�ts are zero!

The Solow Model

The production function can be speci�ed as follows:

Yt = K αt (AtLt )

1�α with 0 < α < 1

it is Cobb-Douglas production function with constant return to scale,where A is a technology variable labour augmenting. AL are thee¢ cient units of labor. The rate of growth of technological progress isexogenous and de�ned as

dAdT

1A=ddtln (A) =

AA= g

The Solow Model

The Solow Model

The demand for capital implies

Rt = αK α�1t (AtLt )

1�α

and thus the capital share is

RKY= α

K α�1 (AL)1�α KY

= α constant

The labor demand implies

wt = (1� α)K αt (AtLt )

�α At

and thus the labor share is

wLY= (1� α)

K α (AL)�α ALY

= 1� α constant

The labor share and the capital share are constant in the longrun, in accordance with Kaldor stylized facts.

The Solow Model

Production function in terms of output per worker:

y = kαA1�α

where y = Y /L and k = K/LProduction funtion in terms of e¢ ecient unit of labor per worker is

y = kα

where y = Y / (AL) and k = K/ (AL) .

The Solow Model

To see the growth implications of the model, we take the log andthen di¤erentiate y = Akα, we �nd

yy= (1� α)

AA+ α

kk

notice however that kk =��K

L

� LK . which is equal to

��KL

�LK=LKKL� LKL2

thenkk=KK� LL

where the labor force growth rate is exogenous and given by LL = n.

The Solow Model

We can express capital in terms of e¢ cient unit of labor per worker

k =KAL

then�kk =

KK �

LL �

AA

The Solow Model

Remember that K = I � d �K . This means that KK can be writtenas,

KK= s

YK� d

if YK is constant, then KK is also constant.

While from kk =

KK �

LL we can rewrite,

KK=kk+LL= s

Y /LK/L

� d

thenk = sy � (d + n) k = skαA1�α � (d + n) k

is the law of motion of capital per worker, while�k = sy � (g + d + n) k = skα � (g + d + n) k

is the law of motion of capital per unit of e¢ cient labor.

The Solow Model

An economy start with a given stock of capital per worker k0, a givenpopulation growth rate, n, a given technology growth rate, g , and agiven investment rate.

1 How does output per worker, y , (or output per e¢ cient units of laborper worker, y) evolve over time?

2 How does the economy growth?3 How does output per worker compare in the long-run between twoeconomies that have di¤erent investment rate?

4 How does output per worker compare in the long-run between twoeconomies that have di¤erent technology growth rate?

The Solow Model

With AA = 0, then capital per worker is

k = skα � (d + n) k

in the steady state k = 0 and

k� =�

sd + n

� 11�α

thus

y � = (k�)α =

�s

d + n

� α1�α

In the Solow model countries with higher savings/investment rate willtend to be richer, ceteris paribus. Such countries have more capitalper worker and thus more output per worker.Countries with high population growth rate, in contrast will tend tobe poorer according to the Solow model.

The Solow Model

The Solow diagram determines the steady state value of output perworker

The dynamics converges to the steady state value of capital perworkers k�.

The Solow Model

The Solow diagram and household consumption

The Solow Model. The Golden rule

Consumption isct = F (kt )� sF (kt )

At the steady state consumption is

c� = F (k�)� sF (k�) = (k�)α � (d + n) k�

consumption is maximum if

∂c∂k= 0 : F 0 (k)� (d + n) = 0

The Solow Model. The Golden rule

∂c∂k = 0 implies

α (k)α�1 � (d + n) = 0and thus, solving for k

kGR =�

α

d + n

� 11�α

we call kGR the Golden rule capital stock, that is the value of k suchthat ∂c

∂k = 0 and consumption is maximum.

The Solow Model. The Golden rule

The Solow Model. The Golden rule

DYNAMIC EFFICIENCY. If kGR > k�1 =) s1 < sGR and c�1 < cGR .Increasing savings increases also the steady state per capitaconsumption

DYNAMIC INEFFICIENCY. If kGR < k�2 =) s2 > sGR andc�2 < c

GR . The economy is oversaving and is said to be dynamicallyine¢ cient.

The Solow Model

An increase in the investment rate s

the steady state value of capital per worker increases.

The Solow Model

An increase in population growth

The Solow Model

Consider now AA 6= 0 = g . In the steady state

�k = 0 and

k� =�

sd + n+ g

� 11�α

thus

y � = kα =

�s

d + n+ g

� α1�α

while output per worker is

y � (t) =�

sd + n+ g

� α1�α

A (t)

where t is included just to remind that A is an exogenous growingvariable.

The Solow Model

Consider now AA 6= 0 = g . The Solow diagram becomes

The Solow Model

An increase in the savings rate

The Solow Model

An increase in the savings rate and conditional convergence.

The Solow Model

Output per worker along the balanced growth path isdetermined by technology, investment rate and the populationgrowth rateChanges in the investment rate and the population growth ratea¤ect the long-run level of output per worker, but do not a¤ectthe long-run growth rate of output per worker.Policy changes do not have long-run growth e¤ects.Policy changes can have level e¤ects, that is a permanentpolicy change can permanently raise (or lower) the level of percapita output.Conditional convergence.