Income Distribution and Macroeconomics Oded Galor and Joseph Zeira Galor-Zeira Inequality and Growth 1 / 57 The Galor-Zeira Model The Galor-Zeira Model Galor-Zeira Inequality and Growth 2 / 57 The Galor-Zeira Model The Galor-Zeira Model Overlapping-Generations economy Galor-Zeira Inequality and Growth 2 / 57 The Galor-Zeira Model The Galor-Zeira Model Overlapping-Generations economy t = 0, 1, 2, 3, ... Galor-Zeira Inequality and Growth 2 / 57 The Galor-Zeira Model The Galor-Zeira Model Overlapping-Generations economy t = 0, 1, 2, 3, ... One good Galor-Zeira Inequality and Growth 2 / 57 The Galor-Zeira Model The Galor-Zeira Model Overlapping-Generations economy t = 0, 1, 2, 3, ... One good 3 factors: Galor-Zeira Inequality and Growth 2 / 57 The Galor-Zeira Model The Galor-Zeira Model Overlapping-Generations economy t = 0, 1, 2, 3, ... One good 3 factors: K Galor-Zeira Physical capital Inequality and Growth 2 / 57 The Galor-Zeira Model The Galor-Zeira Model Overlapping-Generations economy t = 0, 1, 2, 3, ... One good 3 factors: K Physical capital Ls Skilled Labor Galor-Zeira Inequality and Growth 2 / 57 The Galor-Zeira Model The Galor-Zeira Model Overlapping-Generations economy t = 0, 1, 2, 3, ... One good 3 factors: K Physical capital Ls Skilled Labor Lu Unskilled Labor Galor-Zeira Inequality and Growth 2 / 57 The Galor-Zeira Model Production Galor-Zeira Inequality and Growth 3 / 57 The Galor-Zeira Model Production Total output produced Yt = Yts + Ytu Galor-Zeira Inequality and Growth 3 / 57 The Galor-Zeira Model Production Total output produced Yt = Yts + Ytu Production in the skilled-intensive sector: Yts = F (Kt , Lst ) Galor-Zeira Lst f (kt ); Inequality and Growth kt Kt /Lst 3 / 57 The Galor-Zeira Model Production Total output produced Yt = Yts + Ytu Production in the skilled-intensive sector: Yts = F (Kt , Lst ) Lst f (kt ); kt Kt /Lst Production in the unskilled-intensive sector: Ytu = aLut Galor-Zeira Inequality and Growth 3 / 57 The Galor-Zeira Model Inverse Demand for Factors Capital: rt = f 0 (kt ) Galor-Zeira Inequality and Growth r (kt ) 4 / 57 The Galor-Zeira Model Inverse Demand for Factors Capital: rt = f 0 (kt ) r (kt ) Skilled labor: wts = f (kt ) Galor-Zeira f 0 (kt )kt Inequality and Growth w s (kt ) 4 / 57 The Galor-Zeira Model Inverse Demand for Factors Capital: rt = f 0 (kt ) r (kt ) Skilled labor: wts = f (kt ) f 0 (kt )kt w s (kt ) Unskilled labor: wtu = a Galor-Zeira Inequality and Growth wu 4 / 57 The Galor-Zeira Model Factor Prices Small open economy Galor-Zeira Inequality and Growth 5 / 57 The Galor-Zeira Model Factor Prices Small open economy World interest = r Galor-Zeira Inequality and Growth 5 / 57 The Galor-Zeira Model Factor Prices Small open economy World interest = r =) Galor-Zeira rt = r kt = f 0 1 (r ) wts = w s (k ) Inequality and Growth k ws 5 / 57 The Galor-Zeira Model Factor Prices Small open economy World interest = r =) rt = r kt = f 0 1 (r ) wts = w s (k ) k ws =) (rt , wts , wtu ) = (r , w s , w u ) Galor-Zeira Inequality and Growth 8t 5 / 57 The Galor-Zeira Model Individuals Individuals Galor-Zeira Inequality and Growth 6 / 57 The Galor-Zeira Model Individuals Individuals Continuum of measure 1 Galor-Zeira Inequality and Growth 6 / 57 The Galor-Zeira Model Individuals Individuals Continuum of measure 1 Each Individual has 1 parent and 1 child Galor-Zeira Inequality and Growth 6 / 57 The Galor-Zeira Model Individuals Individuals Continuum of measure 1 Each Individual has 1 parent and 1 child Identical in: Preferences Innate abilities Galor-Zeira Inequality and Growth 6 / 57 The Galor-Zeira Model Individuals Individuals Continuum of measure 1 Each Individual has 1 parent and 1 child Identical in: Preferences Innate abilities Di¤er in: Parental income ) Inv’t in HC Galor-Zeira Inequality and Growth 6 / 57 The Galor-Zeira Model Individuals Member of Generation t: Period of Life Galor-Zeira Inequality and Growth 7 / 57 The Galor-Zeira Model Individuals Member of Generation t: Period of Life First period of life (Period t ): [invest in HC] or [work as unskilled] Galor-Zeira Inequality and Growth 7 / 57 The Galor-Zeira Model Individuals Member of Generation t: Period of Life First period of life (Period t ): [invest in HC] or [work as unskilled] Second period of life (Period t + 1): [work as unskilled / no inv’t in HC] or [work as skilled / inv’t in HC in 1st period] Galor-Zeira Inequality and Growth 7 / 57 The Galor-Zeira Model Individuals Member of Generation t: Endowment and Preferences Galor-Zeira Inequality and Growth 8 / 57 The Galor-Zeira Model Individuals Member of Generation t: Endowment and Preferences Time endowment: 1 units of time in each period Galor-Zeira Inequality and Growth 8 / 57 The Galor-Zeira Model Individuals Member of Generation t: Endowment and Preferences Time endowment: 1 units of time in each period Capital endowment: bt Galor-Zeira capital inherited in 1st period Inequality and Growth 8 / 57 The Galor-Zeira Model Individuals Member of Generation t: Endowment and Preferences Time endowment: 1 units of time in each period Capital endowment: capital inherited in 1st period bt Preferences: u t = α ln ct +1 + (1 Galor-Zeira α) ln bt +1 Inequality and Growth α 2 (0, 1) 8 / 57 The Galor-Zeira Model Individuals Member of Generation t: Budget Constraint Second period budget constraint: c t + 1 + bt + 1 Galor-Zeira ω t +1 Inequality and Growth 9 / 57 The Galor-Zeira Model Individuals Member of Generation t: Budget Constraint Second period budget constraint: c t + 1 + bt + 1 ct + 1 Galor-Zeira ω t +1 consumption Inequality and Growth 9 / 57 The Galor-Zeira Model Individuals Member of Generation t: Budget Constraint Second period budget constraint: c t + 1 + bt + 1 ct + 1 consumption bt + 1 transfers to o¤spring Galor-Zeira ω t +1 Inequality and Growth 9 / 57 The Galor-Zeira Model Individuals Member of Generation t: Budget Constraint Second period budget constraint: c t + 1 + bt + 1 ω t +1 ct + 1 consumption bt + 1 transfers to o¤spring ω t +1 wealth in period t + 1 Galor-Zeira Inequality and Growth 9 / 57 The Galor-Zeira Model Individuals Member of Generation t: Optimization fct +1 , bt +1 g = arg max[α ln ct +1 + (1 s.t. Galor-Zeira ct + 1 + b t + 1 Inequality and Growth α) ln bt +1 ] ω t +1 10 / 57 The Galor-Zeira Model Individuals Member of Generation t: Optimization bt + 1 = ( 1 α ) ω t +1 ct +1 = αω t +1 Galor-Zeira Inequality and Growth 11 / 57 The Galor-Zeira Model Individuals Member of Generation t: Optimization bt + 1 = ( 1 α ) ω t +1 ct +1 = αω t +1 Indirect Utility: =) vt Galor-Zeira = α ln αω t +1 + (1 α) ln ω t +1 Inequality and Growth 11 / 57 The Galor-Zeira Model Individuals Member of Generation t: Optimization bt + 1 = ( 1 α ) ω t +1 ct +1 = αω t +1 Indirect Utility: =) vt = α ln αω t +1 + (1 = [α ln α + (1 Galor-Zeira α) ln ω t +1 α) ln(1 Inequality and Growth α)] + ln ω t +1 11 / 57 The Galor-Zeira Model Individuals Member of Generation t: Optimization bt + 1 = ( 1 α ) ω t +1 ct +1 = αω t +1 Indirect Utility: =) vt = α ln αω t +1 + (1 = [α ln α + (1 α) ln ω t +1 α) ln(1 α)] + ln ω t +1 =) v t is monotonic increasing in 2nd period wealth, ω t +1 Galor-Zeira Inequality and Growth 11 / 57 The Galor-Zeira Model Individuals Member of Generation t: Optimization bt + 1 = ( 1 α ) ω t +1 ct +1 = αω t +1 Indirect Utility: =) vt = α ln αω t +1 + (1 = [α ln α + (1 α) ln ω t +1 α) ln(1 α)] + ln ω t +1 =) v t is monotonic increasing in 2nd period wealth, ω t +1 =) maximization of ω t +1 , is the basis of occupational choices Galor-Zeira Inequality and Growth 11 / 57 The Galor-Zeira Model Occupational Choice Fundamental Assumptions Imperfect Capital Markets: r <i r interest rate for lender i interest rate for borrowers (for inv’t in HC) Galor-Zeira Inequality and Growth (A1) 12 / 57 The Galor-Zeira Model Occupational Choice Fundamental Assumptions Imperfect Capital Markets: r <i (A1) r interest rate for lender i interest rate for borrowers (for inv’t in HC) Fixed cost of education (Indivisibility of inv’t in HC) Weighted average of the payments to teachers, administrators, and maintenance workers in the school system (i.e., weighted average of the wages skilled and unskilled workers): C H = θw s + (1 Galor-Zeira θ )w u h>0 Inequality and Growth θ 2 [0, 1] (A2) 12 / 57 The Galor-Zeira Model Occupational Choice Income: Unskilled Individuals Galor-Zeira Inequality and Growth 13 / 57 The Galor-Zeira Model Occupational Choice Income: Unskilled Individuals ω ut+1 = (w u + bt )(1 + r ) + w u Galor-Zeira Inequality and Growth 13 / 57 The Galor-Zeira Model Occupational Choice Income: Unskilled Individuals ω ut+1 = (w u + bt )(1 + r ) + w u = w u ( 2 + r ) + ( 1 + r ) bt Galor-Zeira Inequality and Growth 13 / 57 The Galor-Zeira Model Occupational Choice Income: Skilled Individuals ω st +1 Galor-Zeira = ( ws (h bt )(1 + i ) if bt h w s + ( bt h )(1 + r ) if bt h Inequality and Growth 14 / 57 The Galor-Zeira Model Occupational Choice Income: Skilled Individuals ω st +1 =) ω st +1 Galor-Zeira = = ( ( ws (h bt )(1 + i ) if bt h w s + ( bt h )(1 + r ) if bt h ws ( 1 + i ) h + ( 1 + i ) bt if bt h ws (1 + r )h + (1 + r )bt if bt h Inequality and Growth 14 / 57 The Galor-Zeira Model Occupational Choice Assumptions Investment in human capital is not bene…cial for individuals who must …nance the entire cost of education via borrowing ws Galor-Zeira (1 + i )h < 0 Inequality and Growth (A3) 15 / 57 The Galor-Zeira Model Occupational Choice Assumptions Investment in human capital is not bene…cial for individuals who must …nance the entire cost of education via borrowing ws (1 + i )h < 0 (A3) Investment in human capital is bene…cial for individuals who can …nance the entire cost of education without borrowing ws Galor-Zeira (1 + r )h > w u (2 + r ) Inequality and Growth (A4) 15 / 57 The Galor-Zeira Model Occupational Choice Income from Being Unskilled Worker ωtu+1 = wu (2 + r ) + (1 + r )bt wu ( 2 + r ) bt Galor-Zeira Inequality and Growth 16 / 57 The Galor-Zeira Model Occupational Choice Income from Being Unskilled Worker ωtu+1 = wu (2 + r ) + (1 + r )bt ωtu+1 wu ( 2 + r ) (1 + r ) bt Galor-Zeira Inequality and Growth 17 / 57 The Galor-Zeira Model Occupational Choice Income from Being Skilled Worker: Borrowers ws − ( 1 + i)h < 0 (A3) ωtu+1 wu ( 2 + r ) (1 + r ) bt w s − (1 + i )h Galor-Zeira Inequality and Growth 18 / 57 The Galor-Zeira Model Occupational Choice Income from Being Skilled Worker: Borrowers ωts+1 = ws − (1 + i )h + (1 + i )bt if bt ≤ h ωtu+1 wu ( 2 + r ) (1 + r ) bt w s − (1 + i )h Galor-Zeira Inequality and Growth 19 / 57 The Galor-Zeira Model Occupational Choice Income from Being Skilled Worker: Borrowers ωts+1 = ws − (1 + i )h + (1 + i )bt if bt ≤ h ωtu+1 ws wu ( 2 + r ) (1 + r ) (1 + i ) w − (1 + i )h s Galor-Zeira h Inequality and Growth bt 20 / 57 The Galor-Zeira Model Occupational Choice Income from Being Skilled Worker: Borrowers ws − ( 1 + r)h > wu( 2 + r) (A4) ωtu+1 ws wu (2 + r ) + (1 + r )h wu ( 2 + r ) (1 + r ) (1 + i ) w − (1 + i )h s Galor-Zeira h Inequality and Growth bt 21 / 57 The Galor-Zeira Model Occupational Choice Income from Being Skilled Worker: Lenders ωts+1 = ws − (1 + r )h + (1 + r )bt if bt ≥ h (1 + r ) ws wu ( 2 + r ) ωts+1 ωtu+1 (1 + r ) (1 + i ) w − (1 + i )h s Galor-Zeira h Inequality and Growth bt 22 / 57 The Galor-Zeira Model Occupational Choice Bequest and Occupational Choice ωts+1 = ω s (bt ) ωtu+1 = ω u (bt ) wu ( 2 + r ) w s − (1 + i )h Galor-Zeira f h Inequality and Growth bt 23 / 57 The Galor-Zeira Model Occupational Choice Bequest and Occupational Choice ωts+1 = ω s (bt ) ωtu+1 = ω u (bt ) f Galor-Zeira h Inequality and Growth bt 24 / 57 The Galor-Zeira Model Occupational Choice Bequest and Occupational Choice bt ( Galor-Zeira <f ! xtu+1 > xts+1 (individual t becomes unskilled) >f ! xtu+1 < xts+1 (individual t becomes skilled) Inequality and Growth 25 / 57 The Galor-Zeira Model Occupational Choice Bequest and Occupational Choice bt ( <f ! xtu+1 > xts+1 (individual t becomes unskilled) >f ! xtu+1 < xts+1 (individual t becomes skilled) where f = Galor-Zeira w u (2 + r ) [w s i r (1 + i )h ] Inequality and Growth >0 25 / 57 The Galor-Zeira Model Bequest Dynamics Bequest Dynamics bt + 1 = ( 1 Galor-Zeira α)xt +1 Inequality and Growth 26 / 57 The Galor-Zeira Model Bequest Dynamics Bequest Dynamics bt + 1 = ( 1 bt + 1 8 (1 > > < (1 = > > : (1 Galor-Zeira α)xt +1 α)[w u (2 + r ) + (1 + r )bt ] bt 2 [0, f ] α)[w s ( 1 + i ) h + ( 1 + i ) bt ] bt 2 [ f , h ] α)[w s (1 + r )h + (1 + r )bt ] bt 2 [h, ∞] Inequality and Growth 26 / 57 The Galor-Zeira Model Bequest Dynamics Bequest Dynamics: Su¢ ciet Conditions for Multiplicity of Steady-Sate (1 α)(1 + r ) < 1 (1 α)(1 + i ) > 1 (A5) (1 Galor-Zeira α )w s > h Inequality and Growth (A6) 27 / 57 The Galor-Zeira Model Bequest Dynamics Bequest Dynamics bt +1 φ (bt ) (1 − α ) ws (1 − α )(1 + r ) (1 − α )(1 + i ) (1 − α ) wu (2 + r ) (1 − α )(1 + r ) f Galor-Zeira h Inequality and Growth bt 28 / 57 The Galor-Zeira Model Bequest Dynamics Bequest Dynamics: Multiple Steady-State Equilibrium bt +1 φ (bt ) bu Galor-Zeira g Inequality and Growth bs bt 29 / 57 The Galor-Zeira Model Bequest Dynamics Bequest Dynamics: Stability of High Bequest Equilibrium bt +1 φ (bt ) bu Galor-Zeira g Inequality and Growth bs bt 30 / 57 The Galor-Zeira Model Bequest Dynamics Bequest Dynamics: Stability of Steady- State Equilibria bt +1 φ (bt ) bu Galor-Zeira g Inequality and Growth bs bt 31 / 57 The Galor-Zeira Model Inequality and Skill Composition The Distribution of the Inheritance in Period t ξ (b0 ) bt Galor-Zeira Inequality and Growth 32 / 57 The Galor-Zeira Model Inequality and Skill Composition Income Distribution and the Long Run Decomposition of the Labor Force ξ t ( bt ) Distribution of inheritance at time t =) Lt = Galor-Zeira Z ∞ 0 ξ (bt )dbt Inequality and Growth 1 33 / 57 The Galor-Zeira Model Inequality and Skill Composition The Distribution of the Inheritance in Period t L0 ≡ 1 ξ (b0 ) bt Galor-Zeira Inequality and Growth 34 / 57 The Galor-Zeira Model Inequality and Skill Composition Income Distribution of the Long Run Decomposition of the Labor Force limt !∞ ltu = limt !∞ lts = Galor-Zeira Rg 0 R∞ g ξ t (bt )dbt l̄ u ξ t (bt )dbt l̄ s Inequality and Growth 35 / 57 The Galor-Zeira Model Inequality and Skill Composition Income Distribution of the Long Run Decomposition of the Labor Force limt !∞ ltu = limt !∞ lts = where Rg 0 R∞ g ξ t (bt )dbt l̄ u ξ t (bt )dbt l̄ s ∂l̄ s /∂g < 0 Galor-Zeira Inequality and Growth 35 / 57 The Galor-Zeira Model Inequality and Skill Composition Income Distribution of the Long Run Decomposition of the Labor Force limt !∞ ltu = limt !∞ lts = where Rg 0 R∞ g ξ t (bt )dbt l̄ u ξ t (bt )dbt l̄ s ∂l̄ s /∂g < 0 and g= Galor-Zeira (1 α)[(1 + i )h (1 α)(1 + i ) Inequality and Growth ws] >0 1 35 / 57 The Galor-Zeira Model Inequality and Skill Composition Income Distribution of Skill Composition bt +1 φ (bt ) bu l lu s g Galor-Zeira bt bs Inequality and Growth ξ (b0 ) bb t t 36 / 57 The Galor-Zeira Model Inequality and Skill Composition Income Distribution of Skill Composition l l u s g Galor-Zeira Inequality and Growth ξ (b0 ) bt 37 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Income Per Capita in the Long Run Income of a skilled individual in the second period of life (wage and capital income) I2s = w s + (b̄ s Galor-Zeira Inequality and Growth h )r 38 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Income Per Capita in the Long Run Income of a skilled individual in the second period of life (wage and capital income) I2s = w s + (b̄ s h )r Income of an unskilled individual in the second period of life (wage and capital income) I2u = w u + (b̄ u + w u )r Galor-Zeira Inequality and Growth 38 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Income Per Capita in the Long Run Income of a skilled individual in the second period of life (wage and capital income) I2s = w s + (b̄ s h )r Income of an unskilled individual in the second period of life (wage and capital income) I2u = w u + (b̄ u + w u )r Income of an unskilled individual in the …rst period of life (only wage income) I1u = w u Galor-Zeira Inequality and Growth 38 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Income Per Capita in the Long Run Aggregate income in the steady-state Ȳ = I2s l̄ s + I2u l̄ u + I1u l̄ u Galor-Zeira Inequality and Growth 39 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Income Per Capita in the Long Run Aggregate income in the steady-state Ȳ = I2s l̄ s + I2u l̄ u + I1u l̄ u Aggregate income (note: l̄ s + l̄ u = 1) Y = [w s rh + r b̄ s ]l̄ s + [w u (2 + r ) + r b̄ u ](1 = w u (2 + r ) + r b̄u + [(w s Galor-Zeira rh ) Inequality and Growth l̄ s ) w u (2 + r ) + (b̄ s b̄ u )]l̄ s 39 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Income Per Capita in the Long Run Aggregate income in the steady-state Ȳ = I2s l̄ s + I2u l̄ u + I1u l̄ u Aggregate income (note: l̄ s + l̄ u = 1) Y = [w s rh + r b̄ s ]l̄ s + [w u (2 + r ) + r b̄ u ](1 = w u (2 + r ) + r b̄u + [(w s rh ) l̄ s ) w u (2 + r ) + (b̄ s b̄ u )]l̄ s Income per capita ȳ = Ȳ /2 Galor-Zeira Inequality and Growth 39 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Skill Composition and Income Per Capita in the Long Run An increase in the fraction of skilled workers increases income per capita in the steady-state ∂ȳ = [(w s ∂l̄ s Galor-Zeira rh ) w u (2 + r ) + (b̄ s Inequality and Growth b̄ u )]/2 > 0 40 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Skill Composition and Income Per Capita in the Long Run An increase in the fraction of skilled workers increases income per capita in the steady-state ∂ȳ = [(w s ∂l̄ s since rh ) ws w u (2 + r ) + (b̄ s b̄ u )]/2 > 0 (1 + r )h > w u (2 + r ) b̄ s > b̄ u Galor-Zeira Inequality and Growth 40 / 57 The Galor-Zeira Model Skill Composition and Income Per Capita Skill Composition and Income Per Capita in the Long Run An increase in the fraction of skilled workers increases income per capita in the steady-state ∂ȳ = [(w s ∂l̄ s since rh ) ws w u (2 + r ) + (b̄ s b̄ u )]/2 > 0 (1 + r )h > w u (2 + r ) b̄ s > b̄ u An increase in g reduces income per capita in the steady-state ∂ȳ ∂ȳ ∂l̄ s = s <0 ∂g ∂l̄ ∂g Galor-Zeira Inequality and Growth 40 / 57 Inequality and Economic Development Inequality and Development: Rich Economies bt +1 φ (bt ) bu bs l u g Galor-Zeira bt bt bt Inequality and Growth 41 / 57 Inequality and Economic Development Rich Economies: Inequality is Harmful for Development Inequality reduces human capital formation bt +1 φ (bt ) bu bs l u g bt Galor-Zeira bt Inequality and Growth bt 42 / 57 Inequality and Economic Development Rich Economies: Inequality is Harmful for Development bt +1 φ (bt ) bu bs l u g bt Galor-Zeira bt Inequality and Growth bt 43 / 57 Inequality and Economic Development Rich Economies: Inequality is Harmful for Development l u g Galor-Zeira bt Inequality and Growth bt 44 / 57 Inequality and Economic Development Inequality and Development: Poor Economies bt +1 φ (bt ) bu bs l s bt g Galor-Zeira bt Inequality and Growth bt 45 / 57 Inequality and Economic Development Poor Economies: Inequality may Bene…t Development Inequality stimulates human capital formation bt +1 φ (bt ) bu bs l s bt g Galor-Zeira bt Inequality and Growth bt 46 / 57 Inequality and Economic Development Poor Economies: Inequality may Bene…t Development l bt Galor-Zeira s g Inequality and Growth bt 47 / 57 Inequality and Economic Development Robustness Robustness The qualitative results are robust to: Education cost that is indexed to wages Galor-Zeira Inequality and Growth 48 / 57 Inequality and Economic Development Robustness Robustness The qualitative results are robust to: Education cost that is indexed to wages Labor augmenting technical change Galor-Zeira Inequality and Growth 48 / 57 Inequality and Economic Development Robustness Robustness The qualitative results are robust to: Education cost that is indexed to wages Labor augmenting technical change Shocks the outcome of investment in human capital, as long as wages are endogenous Galor-Zeira Inequality and Growth 48 / 57 Inequality and Economic Development Robustness Robustness The qualitative results are robust to: Education cost that is indexed to wages Labor augmenting technical change Shocks the outcome of investment in human capital, as long as wages are endogenous Concave production function of human capital (Moav (EL, 2002), GalorMoav (RES, 2004)) Galor-Zeira Inequality and Growth 48 / 57 Inequality and Economic Development Robustness Robustness: Technological Progress and Endogenous Education Cost Labor Augmenting Technological Progress: increases the productivity of workers in both the skilled-intensive and the unskilled intensive sector. Production in the skilled-intensive sector Yts = F (Kt , At Lst ) Galor-Zeira At Lst f (kt ); Inequality and Growth kt Kt /At Lst 49 / 57 Inequality and Economic Development Robustness Robustness: Technological Progress and Endogenous Education Cost Labor Augmenting Technological Progress: increases the productivity of workers in both the skilled-intensive and the unskilled intensive sector. Production in the skilled-intensive sector Yts = F (Kt , At Lst ) At Lst f (kt ); kt Kt /At Lst Production in the unskilled-intensive sector Ytu = At aLut Galor-Zeira Inequality and Growth 49 / 57 Inequality and Economic Development Robustness Robustness: Technological Progress and Endogenous Education Cost Labor Augmenting Technological Progress: increases the productivity of workers in both the skilled-intensive and the unskilled intensive sector. Production in the skilled-intensive sector Yts = F (Kt , At Lst ) At Lst f (kt ); kt Kt /At Lst Production in the unskilled-intensive sector Ytu = At aLut Technological progress At + 1 = ( 1 + λ ) At Galor-Zeira Inequality and Growth λ > 0. 49 / 57 Inequality and Economic Development Robustness Robustness: Technological Progress and Endogenous Education Cost Factor Prices wts = At [f (k ) wtu = At a f 0 (k )k ] At w s At w u rt = r Galor-Zeira Inequality and Growth 50 / 57 Inequality and Economic Development Robustness Cost of Education Weighted average of the payments to teachers, administrators, and maintenance workers in the school system Galor-Zeira Inequality and Growth 51 / 57 Inequality and Economic Development Robustness Cost of Education Weighted average of the payments to teachers, administrators, and maintenance workers in the school system ) Weighted average of the wages skilled and unskilled workers Galor-Zeira Inequality and Growth 51 / 57 Inequality and Economic Development Robustness Cost of Education Weighted average of the payments to teachers, administrators, and maintenance workers in the school system ) Weighted average of the wages skilled and unskilled workers CtH = θAt w s + (1 Galor-Zeira θ ) At w u Inequality and Growth At h 51 / 57 Inequality and Economic Development Robustness Income: Unskilled Individuals xtu+1 = (At w u + bt )(1 + r ) + At +1 w u Galor-Zeira Inequality and Growth 52 / 57 Inequality and Economic Development Robustness Income: Unskilled Individuals xtu+1 = (At w u + bt )(1 + r ) + At +1 w u = At w u ( 2 + r + λ ) + ( 1 + r ) b t Galor-Zeira Inequality and Growth 52 / 57 Inequality and Economic Development Robustness Income: Skilled Individuals xts+1 Galor-Zeira = ( At + 1 w s ( At h At +1 w s + (bt bt )(1 + i ) if bt At h At h )(1 + r ) if bt At h Inequality and Growth 53 / 57 Inequality and Economic Development Robustness Income: Skilled Individuals xts+1 = ( At + 1 w s ( At h At +1 w s + (bt bt )(1 + i ) if bt At h At h )(1 + r ) if bt At h =) xts+1 = ( Galor-Zeira At [ w s ( 1 + λ ) ( 1 + i ) h ] + ( 1 + i ) bt if bt At h At [ w s ( 1 + λ ) (1 + r )h] + (1 + r )bt if bt At h Inequality and Growth 53 / 57 Inequality and Economic Development Robustness Threshold level of Bequest for Becoming Skilled Worker in Period t f = Galor-Zeira At f w u ( 2 + r ) [w s (1 + i )h ] (i r ) Inequality and Growth λ (w s w u )g 54 / 57 Inequality and Economic Development Robustness Threshold level of Bequest for Becoming Skilled Worker in Period t f = At f w u ( 2 + r ) ft At f w u ( 2 + r ) = At Galor-Zeira [w s [w s (1 + i )h ] (i r ) (1 + i )h ] (i r ) Inequality and Growth λ (w s λ (w s w u )g w u )g fˆ > 0 54 / 57 Inequality and Economic Development Robustness Threshold level of Bequest for Becoming Skilled Worker in Period t f = At f w u ( 2 + r ) ft At f w u ( 2 + r ) = At [w s [w s (1 + i )h ] (i r ) (1 + i )h ] (i r ) λ (w s λ (w s w u )g w u )g fˆ > 0 for w u (2 + r ) > [w s (1 + i )h ] + λ (w s w u ) Galor-Zeira Inequality and Growth 54 / 57 Inequality and Economic Development Robustness Bequest Dynamics bt + 1 8 (1 > > > < (1 = > > > : (1 Galor-Zeira α)fAt w u (2 + r + λ) + (1 + r )b t g bt 2 [0, f ] s (1 + i )h ] + (1 + i )b t g s (1 + r )h] + (1 + r )bt g bt 2 [At h, ∞] α)fAt [w (1 + λ) α)fAt [w (1 + λ) Inequality and Growth b t 2 [ f , At h ] 55 / 57 Inequality and Economic Development Robustness Bequest Dynamics Let b̂t +1 b̂t +1 = b t + 1 At + 1 8 > > > > < > > > > : Galor-Zeira 1 α 1 +λ fw u (2 + r + λ) + (1 + r )b̂t g b̂t 2 [0, (fˆ )] 1 α 1 +λ f[w s (1 + λ) (1 + i )h] + (1 + i )b̂t g b̂t 2 [fˆ , h ] 1 α 1 +λ f[w s (1 + λ) (1 + r )h] + (1 + r )b̂t g b̂t 2 [h, ∞] Inequality and Growth 56 / 57 Inequality and Economic Development Robustness Bequest Dynamics Let b̂t +1 b̂t +1 = b t + 1 At + 1 8 > > > > < > > > > : 1 α 1 +λ fw u (2 + r + λ) + (1 + r )b̂t g b̂t 2 [0, (fˆ )] 1 α 1 +λ f[w s (1 + λ) (1 + i )h] + (1 + i )b̂t g b̂t 2 [fˆ , h ] 1 α 1 +λ f[w s (1 + λ) (1 + r )h] + (1 + r )b̂t g b̂t 2 [h, ∞] ) The dynamical system is una¤ected qualitatively by labor-augmenting technological progress Galor-Zeira Inequality and Growth 56 / 57 Inequality and Economic Development Robustness Su¢ cient Conditions for Multiple Steady-States (1 α)(1 + r ) < (1 + λ) (1 α)(1 + i ) > (1 + λ) w s (1 + λ ) (1 + i )h < 0 ) The system is characterized by multiple steady-state, where the unstable equilibrium ĝ = Galor-Zeira (1 α)[(1 + i )h w s (1 + λ)] >0 [(1 α)(i + i ) (1 + λ)] Inequality and Growth 57 / 57

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# Income Distribution and Macroeconomics Oded Galor and Joseph Zeira Galor-Zeira Inequality and Growth