The Solow Model Assumptions Aggregate neoclassical production - PowerPoint PPT Presentation

The Solow Model

Assumptions • Aggregate neoclassical production function: Y t = F ( K t , A t L t ) → labour augmenting technical change , → constant returns to scale: , F ( λ K t , λ A t L t ) = λ F ( K t , A t L t ) = λ Y t . • Example: Cobb–Douglas t ( A t L t ) 1 − α Y t = K α

• Say’s Law and the aggregate capital stock: ú K t = sY t − δ K t . • Say’s Law and employment growth ú L t = n L t • Technical progress: ú A t = g A t

The Intensive Form • Let λ = A t L t , so that 1 µ K t ¶ Y t = F , 1 A t L t A t L t y t = f ( k t ) where k t = A t L t and y t = K t Y t A t L t → Cobb–Douglas case: , y t = k α t • Inada conditions: f (0) = 0 , f 0 ( k ) > 0 , f 00 ( k ) < 0 k → 0 f 0 ( k ) = ∞ , k →∞ f 0 ( k ) = 0 . lim lim

• Growth rate of capital stock: ú ú k t K t = − g − n k t K t Multiplying through by k t yields ú K t ú k t = − ( n + g ) k t A t L t = sY t − δ K t − ( n + g ) k t A t L t = sy t − ( n + g + δ ) k t

Dynamics of the Model • Dynamics of Capital Stock: ú k t = sf ( k t ) − ( n + g + δ ) k t . • Steady–state or balanced growth path (BGP) when ú k t = 0 : sf ( k ∗ ) = ( n + g + δ ) k ∗ . • Stability: If sf ( k t ) > ( n + g + δ ) k t then ú k t > 0 If sf ( k t ) < ( n + g + δ ) k t then ú k t < 0 .

Properties of the BGP • Long–run growth path is independent of initial conditions → given similar values of s , n , δ and g, poor economies catch up , • Capital stock grows at the same rate as income. • Income per worker increasing in s and decreasing in n • Growth of income per worker depends only on g

(n+g+ δ )k Investment sf(k) k k* 1.The Solow Growth Model

(n+g+ δ )k Investment sf(k) ∆ k k k 0 k* 2.Dynamics of the Solow Model

(n+g+ δ )k Investment sf(k) ∆ k ∆ k k k 1 k 0 k* 3.Dynamics of the Solow Model

(n+g+ δ )k Investment sf(k) ∆ k ∆ k k k 1 k 2 k 0 k* 4.Dynamics of the Solow Model

(n+g+ δ )k Investment sf(k) k k 1 k 2 k 0 k* 5.Dynamics of the Solow Model

Formal Analysis of Convergence (Cobb–Douglas Case) • Dynamics of capital: ú k t = sk α − 1 − ( n + g + δ ) t k t → let x t = ln k : , dx t dt = se ( α − 1) x t − ( n + g + δ ) • Recall Þ rst–order TSE around the steady–state, x ∗ = ln k ∗ : h ( x t ) ' h ( x ∗ ) + h 0 ( x ∗ )( x t − x ∗ ) → in this case , h ( x ∗ ) = 0 h 0 ( x ∗ ) ' ( α − 1) se ( α − 1) x t and

→ and so , dx t dt ' − λ ( x t − x ∗ ) where λ = (1 − α ) se ( α − 1) x ∗ • Solution to this differential equation: x t = x ∗ + e − λ t ( x 0 − x ∗ ) → and so , ln k t = ln k ∗ + e − λ t (ln k 0 − ln k ∗ ) where λ = (1 − α ) sk ∗ ( α − 1) = (1 − α ) ( n + g + δ ) • Note that ln y t = α ln k t , and so ln y t = ln y ∗ + e − λ t (ln y 0 − ln y ∗ )

Evaluation of the Basic Solow Model 1. Unconditional Convergence Baumol (1986) — strict interpretation De Long (1988) — “selection bias”. Penn World Tables — no convergence

Per Capita Income Rich Poor Time 10.Unconditional Convergence

2. Conditional analysis In Cobb–Douglas case µ ¶ α Y i s i 1 − α = A i y i = A i n i + g + δ L i Taking logs: ln Y i α = ln A i + 1 − α [ln s i − ln( n i + g + δ )] . L i Mankiw, Romer and Weil (1992) estimate: ln Y i = a + b ln s i + c ln( n i + 0 . 05) + ε i L i Results: • R 2 = 0 . 59 • ˆ b > 0 and ˆ c < 0 and signi Þ cant. • BUT implied α very large ( > 0.6) and restriction that ˆ c is rejected b = − ˆ

The Augmented Solow Model • Aggregate production function given by t H β Y t = K α t ( A t L t ) 1 − α − β • Evolution of physical and human capital ú K t = s K Y t ú H t = s H Y t , • Intensive form: t h β y t = k α t . ú t h β k t = s K k α t − ( n + g ) k t ú t h β h t = s H k α t − ( n + g ) h t

. (h=0) k . (k=0) k* h h* 8.Phase Diagram for Augmented Solow model

• Stable BGP where ú k t = ú h t = 0 : Ã ! µ s α ¶ 1 1 s 1 − β K s β 1 − α − β K s 1 − α 1 − α − β H and h = H k = . n + g n + g ⇒ output per effective worker: " # 1 K s β 1 − α − β s α H y = ( n + g ) α + β

Empirical Evaluation In logs we have ln Y α β α + β 1 − α − β ln( n + g ) . L = ln A + 1 − α − β ln s K + 1 − α − β ln s H + Mankiw, Romer and Weil (1992) estimate ln Y i = a + b ln s Ki + c ln s Hi + d ln( n i + 0 . 05) + ε i . L i Results: • R 2 = 0 . 79 • b > 0 , c > 0 and d < 0 and signi Þ cant • Implied values factor shares are α = 0 . 31 and β = 0 . 28 . • Restriction that b + c = − d , cannot be rejected at the 5% level.

Conditional Convergence • Previous estimates assume that deviations from a country’s steady state are random. MRW (1992) also test convergence properties. • Recall the convergence equation: ln y t = ln y ∗ + e − λ t (ln y 0 − ln y ∗ ) ln y t − ln y 0 = (1 − e − λ t ) ln y ∗ − (1 − e − λ t ) ln y 0 → substituting for y ∗ : , µ ¶ α s i ln y t − ln y 0 = (1 − e − λ t ) − (1 − e − λ t ) ln y 0 1 − α ln n i + g + δ

High s, low n Per Capita Income Low s, high n Time 7.

→ Since y t = Y t /A t L t : , µ ¶ ln Y t − ln Y 0 α s i = gt + (1 − e − λ t ) 1 − α ln L t L 0 n i + g + δ − (1 − e − λ t ) ln Y 0 + (1 − e − λ t ) ln A 0 L 0 • MRW estimate growth equation (with t = 25 ): ln Y i − ln Y i, 0 = a + b ln s i + c ln ( n i + 0 . 05) + d ln Y i, 0 + ε i L i L i, 0 L i, 0 • Same basic idea carries over to the augmented Solow model: ln Y i − ln Y i, 0 = a + b K ln s Ki + b H ln s Hi + c ln ( n i + 0 . 05) + d ln Y i, 0 + ε i L i L i, 0 L i, 0 • MRW argue that results are consistent with the augmented model.

Problems with MRW Methodology • Endogeneity bias. • Omitted variable bias — Howitt (2000). • Proxy for s H is arbitrary – Klenow and Rodriguez–Clare (1997) → other proxies suggest a large role for residual TFP , • TFP growth rates are signi Þ cantly correlated with savings rates — Bernanke and Gurkaynak (2001) → see below ,

Competitive Markets in the Solow Model • Production of Þ rm i : i ( A t L i ) 1 − α X i = F ( K i , A t L i ) = K α • Cost minimization: A t F L = w t . F K q t In Cobb–Douglas case, this implies that µ ¶ w t K t α = L t 1 − α q t or µ ¶ w t α (*) k t = 1 − α A t q t

K Isocost Line K* Isoquant w/q L* L 2.Cost Minimization

• Goods market competition ⇒ zero pro Þ ts: i ( A t L i ) 1 − α = wL i + qK i K α It follows that µ K i ¶ α = w + qK i A t L i L i µµ ¶ w t ¶ α µ ¶ α α A t = w t + w t 1 − α q t 1 − α and so w 1 − α q α t = α α (1 − α ) 1 − α A 1 − α t t

• Using (*) to sub out q t we get the implied real wage w t = (1 − α ) A t k α t = marginal product of labour • Implied user cost of capital q t = α k α − 1 t = marginal product of capital = r t + δ • Additional predictions: → real interest rate shows no secular trend in long run, , → real wage grows at rate g . ,

Cross–country rates of return and the Solow Model Lucas (1990) — why doesn’t capital ß ow from rich to poor countries?. Example: µ k I ¶ α − 1 µ y US ¶ 1 − α r I α = = r US k US y I If α = 0 . 3 : µ y US ¶ 2 µ Y US /L US ¶ 2 r I × A I = = r US y I Y I /L I A US If A I = A US , then r I = 20 2 = 400 r US

• The augmented Solow model resolves this problem y = k α h β → high rate of return on capital in poor countries due to diminishing , returns is offset by low level of human capital: r = α k α − 1 h β BUT it introduces another problem (see Assignment #1) → implies the marginal product of human capital is higher in developing , countries: w H = β k α h β − 1 → can’t get away from the effects of diminishing returns ,

y Rich ∆ y R y=f(k , h R ) y=f(k , h P ) Poor ∆ y P k ∆ k=1 ∆ k=1 ∆ y P < ∆ y R 9.Implication for Rates of Return Conditional on Human Capital