Lecture 1: Basic Models of Growth - University of WarwickSome Kaldor™s Fact 1 Per Capita output...

Lecture 1: Basic Models of Growth

Eugenio Proto

February 18, 2009

Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 1 / 12

Some Kaldor�s Fact

1 Per Capita output grows over time, and its growth rate does not tendto diminish

2 Physical Capital per worker grows over time3 The growth rate of output per worker di¤ers substantially acrosscountries




2 Physical Capital per worker grows over time

3 The growth rate of output per worker di¤ers substantially acrosscountries




2 Physical Capital per worker grows over time3 The growth rate of output per worker di¤ers substantially acrosscountries


Di¤erent paths of Growth

0

10000

20000

30000

40000

50000

60000

70000

80000

1950

1954

1958

1962

1966

1970

1974

1978

1982

1986

1990

1994

1998

2002

UnitedStatesSpain

Argentina

Kaldor facts do not apply to stagnating countriesMacroeconomic growth consider growing countriesDevelopment and growth proceed ed separatelyGenerating a unique model for growth and development is still faraway



0

10000

20000

30000

40000

50000

60000

70000

80000

1950

1954

1958

1962

1966

1970

1974

1978

1982

1986

1990

1994

1998

2002

UnitedStatesSpain

Argentina

Kaldor facts do not apply to stagnating countries

Macroeconomic growth consider growing countriesDevelopment and growth proceed ed separatelyGenerating a unique model for growth and development is still faraway



0

10000

20000

30000

40000

50000

60000

70000

80000

1950

1954

1958

1962

1966

1970

1974

1978

1982

1986

1990

1994

1998

2002

UnitedStatesSpain

Argentina

Kaldor facts do not apply to stagnating countriesMacroeconomic growth consider growing countries

Development and growth proceed ed separatelyGenerating a unique model for growth and development is still faraway



0

10000

20000

30000

40000

50000

60000

70000

80000

1950

1954

1958

1962

1966

1970

1974

1978

1982

1986

1990

1994

1998

2002

UnitedStatesSpain

Argentina

Kaldor facts do not apply to stagnating countriesMacroeconomic growth consider growing countriesDevelopment and growth proceed ed separately

Generating a unique model for growth and development is still faraway



0

10000

20000

30000

40000

50000

60000

70000

80000

1950

1954

1958

1962

1966

1970

1974

1978

1982

1986

1990

1994

1998

2002

UnitedStatesSpain

Argentina

Kaldor facts do not apply to stagnating countriesMacroeconomic growth consider growing countriesDevelopment and growth proceed ed separatelyGenerating a unique model for growth and development is still faraway


Ramsey-Samuelson model

Household behavior

size: L(t) = ent

Utility U =R ∞0 u[c(t)]L(t)e

�ρtdtper cap. wealth acc.: a = w + ra� c � nano ponzi game: limt!∞fa(t) exp[�

R t0 [r(v)� n]dvg � 0

Firms�Behavior

Y = F (K , L) with L = L(t)T (t)increasing and concave in K and L,Constant Return to scale:

F (λK ,λL) = λF (K , L)



Household behavior

size: L(t) = ent


�ρtdt

per cap. wealth acc.: a = w + ra� c � nano ponzi game: limt!∞fa(t) exp[�

R t0 [r(v)� n]dvg � 0

Firms�Behavior


F (λK ,λL) = λF (K , L)



Household behavior

size: L(t) = ent


�ρtdtper cap. wealth acc.: a = w + ra� c � na

no ponzi game: limt!∞fa(t) exp[�R t0 [r(v)� n]dvg � 0

Firms�Behavior


F (λK ,λL) = λF (K , L)



Household behavior

size: L(t) = ent



R t0 [r(v)� n]dvg � 0

Firms�Behavior


F (λK ,λL) = λF (K , L)



Household behavior

size: L(t) = ent



R t0 [r(v)� n]dvg � 0

Firms�Behavior

Y = F (K , L) with L = L(t)T (t)

increasing and concave in K and L,Constant Return to scale:

F (λK ,λL) = λF (K , L)



Household behavior

size: L(t) = ent



R t0 [r(v)� n]dvg � 0

Firms�Behavior

Y = F (K , L) with L = L(t)T (t)increasing and concave in K and L,

Constant Return to scale:

F (λK ,λL) = λF (K , L)



Household behavior

size: L(t) = ent



R t0 [r(v)� n]dvg � 0

Firms�Behavior


F (λK ,λL) = λF (K , L)


Static Equilibria

Household Optimal choice:

maxcU(c)

w + ra� c � na � 0

limfa(t) exp[�Z t

0[r(v)� n]dvg � 0

Euler Condition

r = ρ+ [�u00(c)cu0(c)

] � ˙c/c

with u(c) = c1�θ�11�θ (CIES):

˙c/c = (1/θ)(r � ρ)


Static Equilibrium (cont�d)

Firms�optimal Choice

maxF (K , L)� (r + δ)K � wL =

max f (k)� (r + δ)k � wT (t)

FOCsf 0(k) = r + δ

[f (k)� k f 0(k)]ext = w


Steady state

Dynamics (k = a )

˙k = f (k)� c � (x + n+ δ)k˙c/c = (1/θ)(f 0(k)� δ� ρ� θx)

limfa(t) exp[�Z t

0[r(v)� n]dvg = 0

Equilibrium

˙c = 0! f 0(k�) = δ+ ρ+ θx˙k = 0! c� = f (k�)� (x + n+ δ)k�

with˙yy= α

˙kk= 0


Steady state analysis

Growth in steady state

˙cc

=∂�C (t)e (n+x )t

�∂t

/C (t)e(n+x )t

= c � x = 0

˙kk

=∂�K (t)e (n+x )t

�∂t

/K (t)e(n+x )t

= k � x = 0

˙yy

= y � x = α˙kk= 0

y (per capita income) in steady state grows with T (t) = ext

GROWTH IS EXOGENOUS


AK model

Consumer behavior exactly as in Ramsey

Firms behavior and static equilibrium

Y = AK

y = f (k) = Ak

capital = human capital, knowledge, public good...no raw labour , w = 0r = A� δ


AK model



Y = AK

y = f (k) = Ak

capital = human capital, knowledge, public good...

no raw labour , w = 0r = A� δ


AK model



Y = AK

y = f (k) = Ak

capital = human capital, knowledge, public good...no raw labour , w = 0

r = A� δ


AK model



Y = AK

y = f (k) = Ak

capital = human capital, knowledge, public good...no raw labour , w = 0r = A� δ


AK modelSteady state

Dynamics (k = a )

k = (A� δ� n)� c/kc/c = (1/θ)(A� δ� ρ)

limfk(t)e�(A�δ�ρ)tg = 0

Equilibrium

c/c = cons

k/k = c/cy/y = k/k = cons


Model with Human capital

Firms produceY = F (H,K ),

let Y = Kf (H/K )

Market determines RH ,RKDepreciation rates δH , δK

In equilibrium

f (H/K )� f 0(H/K )(1+H/K ) = δK � δH

unique value for H/K .De�ne A = f (H/K ) and we obtain a AK model




let Y = Kf (H/K )

Market determines RH ,RK

Depreciation rates δH , δK

In equilibrium

f (H/K )� f 0(H/K )(1+H/K ) = δK � δH





let Y = Kf (H/K )


In equilibrium

f (H/K )� f 0(H/K )(1+H/K ) = δK � δH





let Y = Kf (H/K )


In equilibrium

f (H/K )� f 0(H/K )(1+H/K ) = δK � δH

unique value for H/K .

De�ne A = f (H/K ) and we obtain a AK model




let Y = Kf (H/K )


In equilibrium

f (H/K )� f 0(H/K )(1+H/K ) = δK � δH



Model with learning by doing and spillover

Yi = F (Ki ,KLi ) = LiF (ki ,K )

K is the aggregate (physical or human) capital, since ki = k thenK = kL

F (k,K )/k = f (K/k) = f (L) and

F1(k,K ) = f (K/k)� f 0(K/k)Kk2iki = f (L)� f 0(L)L

private marginal product of capital is non decreasing in k..

c/c = (1/θ)[f (L)� Lf 0(L)� δ� ρ] constant

Generates long run growth


Lecture 1: Basic Models of Growth - University of WarwickSome Kaldor™s Fact 1 Per Capita output...

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Transcript of Lecture 1: Basic Models of Growth - University of WarwickSome Kaldor™s Fact 1 Per Capita output...