Serge Coulombe, ECO 6120 1 – The Solow Model: theoretical analysis Textbook: Economic Growth second edition by Robert J. Barro and Xavier Sala-i-Martin, MIT Press, 2004. 1 – The Solow Model: theoretical analysis Macro and economic growth Endogenous growth `Growth empirics Economic development Basic model of economic growth Model of capital accumulation `One good model

Key exogenous variables: s, x, n, δ The Solow (1956) Model Key exogenous variables: s, x, n, δ Saving (investment) is a fixed proportion of output sY Population (labour force) growing at rate n: L(t)=L(0)ent Exogenous technological progress: A(t)=A(0)ext Focus on the dynamics of capital accumulation:

A concave production function

Inada conditions

The production function y=f(k) f(k) The production function k Decreasing slope

Cobb-Douglas

The evolution of the Capital/Labour ratio

The dynamics of capital accumulation Since:

(n+x+δ)k sf(k) k k1 k* k2 If k=k1, sf(k1) > (n+x+δ)k1, Δk > 0 If k=k2, sf(k2) < (n+x+δ)k2, Δk < 0

Phase diagram in the Solow model k* k

The steady-state k*: sf(k*)=(n+x+δ)k* Investment per unit of effective worker (n+x+δ)k sf(k) k k* The steady-state k*: sf(k*)=(n+x+δ)k*

Effect of an increase in the savings rate, s(1)>s(0) Investment per unit of effective worker (n+x+δ)k s(1)f(k) s(0)f(k) k k(0) k(1) Effect of an increase in the savings rate, s(1)>s(0)

s t t k t t(0)

ln(Y/L) t c t t(0)

Since On steady state: Δk* = 0 The golden rule Since On steady state: Δk* = 0 C* is maximized at dc*/dk*=0, then:

Golden rule in the Solow Model f(k*) f’(k*) (n+x+δ)k* c*max k* k*gold

Effect of a decrease in population growth, n(1) < n(0) Investment per unit of effective worker (n(0)+x+δ)k (n(1)+x+δ)k sf(k) k k(0) k(1) Effect of a decrease in population growth, n(1) < n(0)