Lecture 1: Basic Models of Growth

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 tend to diminish Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 2 / 12

Some Kaldor s Fact 1 Per Capita output grows over time, and its growth rate does not tend to diminish 2 Physical Capital per worker grows over time Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 2 / 12

Some Kaldor s Fact 1 Per Capita output grows over time, and its growth rate does not tend to diminish 2 Physical Capital per worker grows over time 3 The growth rate of output per worker di ers substantially across countries Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 2 / 12

Di erent paths of Growth 80000 70000 60000 50000 40000 30000 20000 10000 0 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 United States Spain Argentina Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 3 / 12

Di erent paths of Growth 80000 70000 60000 50000 40000 30000 20000 10000 0 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 United States Spain Argentina Kaldor facts do not apply to stagnating countries Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 3 / 12

Di erent paths of Growth 80000 70000 60000 50000 40000 30000 20000 10000 0 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 United States Spain Argentina Kaldor facts do not apply to stagnating countries Macroeconomic growth consider growing countries Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 3 / 12

Di erent paths of Growth 80000 70000 60000 50000 40000 30000 20000 10000 0 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 United States Spain Argentina Kaldor facts do not apply to stagnating countries Macroeconomic growth consider growing countries Development and growth proceed ed separately Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 3 / 12

Di erent paths of Growth 80000 70000 60000 50000 40000 30000 20000 10000 0 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002 United States Spain Argentina Kaldor facts do not apply to stagnating countries Macroeconomic growth consider growing countries Development and growth proceed ed separately Generating a unique model for growth and development is still far away Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 3 / 12

Ramsey-Samuelson model Household behavior Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Ramsey-Samuelson model Household behavior size: L(t) = e nt Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Ramsey-Samuelson model Household behavior size: L(t) = e nt Utility U = R 0 u[c(t)]l(t)e ρt dt Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Ramsey-Samuelson model Household behavior size: L(t) = e nt Utility U = R 0 u[c(t)]l(t)e ρt dt per cap. wealth acc.: ȧ = w + ra c na Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Ramsey-Samuelson model Household behavior size: L(t) = e nt Utility U = R 0 u[c(t)]l(t)e ρt dt per cap. wealth acc.: ȧ = w + ra c na no ponzi game: lim t! fa(t) exp[ [r(v) n]dvg 0 R t 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Ramsey-Samuelson model Household behavior size: L(t) = e nt Utility U = R 0 u[c(t)]l(t)e ρt dt per cap. wealth acc.: ȧ = w + ra c na no ponzi game: lim t! fa(t) exp[ [r(v) n]dvg 0 Firms Behavior R t 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Ramsey-Samuelson model Household behavior size: L(t) = e nt Utility U = R 0 u[c(t)]l(t)e ρt dt per cap. wealth acc.: ȧ = w + ra c na no ponzi game: lim t! fa(t) exp[ [r(v) n]dvg 0 Firms Behavior Y = F (K, ˆL) with ˆL = L(t)T (t) R t 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Ramsey-Samuelson model Household behavior size: L(t) = e nt Utility U = R 0 u[c(t)]l(t)e ρt dt per cap. wealth acc.: ȧ = w + ra c na no ponzi game: lim t! fa(t) exp[ [r(v) n]dvg 0 Firms Behavior Y = F (K, ˆL) with ˆL = L(t)T (t) increasing and concave in K and ˆL, R t 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Ramsey-Samuelson model Household behavior size: L(t) = e nt Utility U = R 0 u[c(t)]l(t)e ρt dt per cap. wealth acc.: ȧ = w + ra c na no ponzi game: lim t! fa(t) exp[ [r(v) n]dvg 0 Firms Behavior Y = F (K, ˆL) with ˆL = L(t)T (t) increasing and concave in K and ˆL, Constant Return to scale: R t 0 F (λk, λˆl) = λf (K, ˆL) Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 4 / 12

Static Equilibria Household Optimal choice: limfa(t) exp[ max U(c) c w + ra c na 0 Z t 0 [r(v) n]dvg 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 5 / 12

Static Equilibria Household Optimal choice: limfa(t) exp[ max U(c) c w + ra c na 0 Z t 0 [r(v) n]dvg 0 Euler Condition r = ρ + [ u 00 (c)c u 0 ] ĉ/ĉ (c) Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 5 / 12

Static Equilibria Household Optimal choice: limfa(t) exp[ max U(c) c w + ra c na 0 Z t 0 [r(v) n]dvg 0 Euler Condition r = ρ + [ with u(c) = c 1 θ 1 1 θ (CIES): u 00 (c)c u 0 ] ĉ/ĉ (c) ĉ/ĉ = (1/θ)(r ρ) Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 5 / 12

Static Equilibrium (cont d) Firms optimal Choice max F (K, ˆL) (r + δ)k wl = w max f ( ˆk) (r + δ) ˆk T (t) Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 6 / 12

Static Equilibrium (cont d) Firms optimal Choice max F (K, ˆL) (r + δ)k wl = w max f ( ˆk) (r + δ) ˆk T (t) FOCs f 0 ( ˆk) = r + δ [f ( ˆk) ˆkf 0 ( ˆk)]e xt = w Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 6 / 12

Steady state Dynamics (k = a ) ˆk = f ( ˆk) ĉ (x + n + δ) ˆk ĉ/ĉ = (1/θ)(f 0 (k) δ ρ θx) limfa(t) exp[ Z t 0 [r(v) n]dvg = 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 7 / 12

Steady state Dynamics (k = a ) ˆk = f ( ˆk) ĉ (x + n + δ) ˆk Equilibrium ĉ/ĉ = (1/θ)(f 0 (k) δ ρ θx) limfa(t) exp[ Z t 0 [r(v) n]dvg = 0 ĉ = 0! f 0 (k ) = δ + ρ + θx ˆk = 0! ĉ = f ( ˆk ) (x + n + δ) ˆk with ŷ ŷ = α ˆk ˆk = 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 7 / 12

Steady state analysis Growth in steady state ĉ ĉ ˆk ˆk ŷ ŷ = C (t) t e (n+x )t K (t) e (n+x )t / C (t) = ċ x = 0 e (n+x )t = / K (t) t e = k x = 0 (n+x )t ˆk = ẏ x = α ˆk = 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 8 / 12

Steady state analysis Growth in steady state ĉ ĉ ˆk ˆk ŷ ŷ = C (t) t e (n+x )t K (t) e (n+x )t / C (t) = ċ x = 0 e (n+x )t = / K (t) t e = k x = 0 (n+x )t ˆk = ẏ x = α ˆk = 0 y (per capita income) in steady state grows with T (t) = e xt GROWTH IS EXOGENOUS Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 8 / 12

AK model Consumer behavior exactly as in Ramsey Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 9 / 12

AK model Consumer behavior exactly as in Ramsey Firms behavior and static equilibrium Y = AK y = f (k) = Ak Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 9 / 12

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... Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 9 / 12

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 = 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 9 / 12

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 = 0 r = A δ Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 9 / 12

AK model Steady state Dynamics (k = a ) k = (A δ n) c/k ċ/c = (1/θ)(A δ ρ) limfk(t)e (A δ ρ)t g = 0 Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 10 / 12

AK model Steady state Dynamics (k = a ) k = (A δ n) c/k ċ/c = (1/θ)(A δ ρ) limfk(t)e (A δ ρ)t g = 0 Equilibrium ċ/c = cons k/k = ċ/c ẏ/y = k/k = cons Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 10 / 12

Model with Human capital Firms producey = F (H, K ), Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 11 / 12

Model with Human capital Firms producey = F (H, K ), let Y = Kf (H/K ) Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 11 / 12

Model with Human capital Firms producey = F (H, K ), let Y = Kf (H/K ) Market determines R H, R K Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 11 / 12

Model with Human capital Firms producey = F (H, K ), let Y = Kf (H/K ) Market determines R H, R K Depreciation rates δ H, δ K Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 11 / 12

Model with Human capital Firms producey = F (H, K ), let Y = Kf (H/K ) Market determines R H, R K Depreciation rates δ H, δ K In equilibrium unique value for H/K. f (H/K ) f 0 (H/K )(1 + H/K ) = δ K δ H Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 11 / 12

Model with Human capital Firms producey = F (H, K ), let Y = Kf (H/K ) Market determines R H, R K Depreciation rates δ H, δ K In equilibrium unique value for H/K. f (H/K ) f 0 (H/K )(1 + H/K ) = δ K δ H De ne A = f (H/K ) and we obtain a AK model Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 11 / 12

Model with learning by doing and spillover Y i = F (K i, KL i ) = L i F (k i, K ) Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 12 / 12

Model with learning by doing and spillover Y i = F (K i, KL i ) = L i F (k i, K ) K is the aggregate (physical or human) capital, since k i = k then K = kl Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 12 / 12

Model with learning by doing and spillover Y i = F (K i, KL i ) = L i F (k i, K ) K is the aggregate (physical or human) capital, since k i = k then K = kl F (k, K )/k = f (K /k) = f (L) and F 1 (k, K ) = f (K /k) f 0 (K /k) K k 2 i k i = f (L) f 0 (L)L private marginal product of capital is non decreasing in k.. Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 12 / 12

Model with learning by doing and spillover Y i = F (K i, KL i ) = L i F (k i, K ) K is the aggregate (physical or human) capital, since k i = k then K = kl F (k, K )/k = f (K /k) = f (L) and F 1 (k, K ) = f (K /k) f 0 (K /k) K k 2 i k i = f (L) f 0 (L)L private marginal product of capital is non decreasing in k.. ċ/c = (1/θ)[f (L) Lf 0 (L) δ ρ] constant Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 12 / 12

Model with learning by doing and spillover Y i = F (K i, KL i ) = L i F (k i, K ) K is the aggregate (physical or human) capital, since k i = k then K = kl F (k, K )/k = f (K /k) = f (L) and F 1 (k, K ) = f (K /k) f 0 (K /k) K k 2 i k i = f (L) f 0 (L)L private marginal product of capital is non decreasing in k.. ċ/c = (1/θ)[f (L) Lf 0 (L) δ ρ] constant Generates long run growth Eugenio Proto () Lecture 1: Basic Models of Growth February 18, 2009 12 / 12