From Malthus to Modern Growth

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					Introduction     Economy                Equilibrium        From Malthus to Modern Growth




               From Malthus to Modern Growth


                               Omar Licandro


                           European University Institute




                                  April 2009




                                                                                    1 / 19
Introduction   Economy    Equilibrium   From Malthus to Modern Growth



                         Aims




                                                                 2 / 19
Introduction          Economy          Equilibrium      From Malthus to Modern Growth



                                    Aims

           • Evidence: From Malthusian stagnation to modern growth




                                                                                 2 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                                       Aims

           • Evidence: From Malthusian stagnation to modern growth
               • In Western Europe and until the 19th century, income and
                 population where no growing, then




                                                                                      2 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                                       Aims

           • Evidence: From Malthusian stagnation to modern growth
               • In Western Europe and until the 19th century, income and
                 population where no growing, then
               • Industrial Revolution: Income growth rates increase
                 substantially




                                                                                      2 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                                       Aims

           • Evidence: From Malthusian stagnation to modern growth
               • In Western Europe and until the 19th century, income and
                 population where no growing, then
               • Industrial Revolution: Income growth rates increase
                 substantially
               • Demographic Transition: Population growth initially
                 increases, then reduces




                                                                                      2 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                                       Aims

           • Evidence: From Malthusian stagnation to modern growth
               • In Western Europe and until the 19th century, income and
                 population where no growing, then
               • Industrial Revolution: Income growth rates increase
                 substantially
               • Demographic Transition: Population growth initially
                 increases, then reduces
           • Related issue: Exceptionality of the industrial revolution




                                                                                      2 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                                       Aims

           • Evidence: From Malthusian stagnation to modern growth
               • In Western Europe and until the 19th century, income and
                 population where no growing, then
               • Industrial Revolution: Income growth rates increase
                 substantially
               • Demographic Transition: Population growth initially
                 increases, then reduces
           • Related issue: Exceptionality of the industrial revolution
           • Theory: Galor and Weil (2000) unify (be more precise)
               • Malthus theory




                                                                                      2 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                                       Aims

           • Evidence: From Malthusian stagnation to modern growth
               • In Western Europe and until the 19th century, income and
                 population where no growing, then
               • Industrial Revolution: Income growth rates increase
                 substantially
               • Demographic Transition: Population growth initially
                 increases, then reduces
           • Related issue: Exceptionality of the industrial revolution
           • Theory: Galor and Weil (2000) unify (be more precise)
               • Malthus theory
               • Modern growth theory: Solow




                                                                                      2 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                                       Aims

           • Evidence: From Malthusian stagnation to modern growth
               • In Western Europe and until the 19th century, income and
                 population where no growing, then
               • Industrial Revolution: Income growth rates increase
                 substantially
               • Demographic Transition: Population growth initially
                 increases, then reduces
           • Related issue: Exceptionality of the industrial revolution
           • Theory: Galor and Weil (2000) unify (be more precise)
               • Malthus theory
               • Modern growth theory: Solow
               • Becker’s population theory




                                                                                      2 / 19
Introduction   Economy   Equilibrium   From Malthus to Modern Growth




                                                                3 / 19
Introduction   Economy       Equilibrium   From Malthus to Modern Growth



                         Population




                                                                    4 / 19
Introduction          Economy          Equilibrium       From Malthus to Modern Growth



                                 Population


           • Discrete time model of successive generations




                                                                                  4 / 19
Introduction           Economy            Equilibrium          From Malthus to Modern Growth



                                   Population


           • Discrete time model of successive generations

               • Individuals live two periods, childhood and adulthood
               • but, have no independent activity during childhood




                                                                                        4 / 19
Introduction             Economy          Equilibrium          From Malthus to Modern Growth



                                   Population


           • Discrete time model of successive generations

               • Individuals live two periods, childhood and adulthood
               • but, have no independent activity during childhood

           • Notation:




                                                                                        4 / 19
Introduction            Economy            Equilibrium          From Malthus to Modern Growth



                                    Population


           • Discrete time model of successive generations

                • Individuals live two periods, childhood and adulthood
                • but, have no independent activity during childhood

           • Notation:
               • Cohort t is adult at time t




                                                                                         4 / 19
Introduction            Economy             Equilibrium          From Malthus to Modern Growth



                                     Population


           • Discrete time model of successive generations

                • Individuals live two periods, childhood and adulthood
                • but, have no independent activity during childhood

           • Notation:
               • Cohort t is adult at time t
                • The size of adult population is Nt , with Nt+1 = nt Nt




                                                                                          4 / 19
Introduction            Economy              Equilibrium          From Malthus to Modern Growth



                                      Population


           • Discrete time model of successive generations

                • Individuals live two periods, childhood and adulthood
                • but, have no independent activity during childhood

           • Notation:
               • Cohort t is adult at time t
                • The size of adult population is Nt , with Nt+1 = nt Nt
                • nt is the fertility rate (number of kids by adult)




                                                                                           4 / 19
Introduction   Economy        Equilibrium   From Malthus to Modern Growth



                         Preferences




                                                                     5 / 19
Introduction           Economy                Equilibrium             From Malthus to Modern Growth



                                       Preferences

           • Preferences are, δ > 0,

                                 ut = ln(ct ) + δ ln(nt wt+1 ht+1 )




                                                                                               5 / 19
Introduction            Economy                Equilibrium             From Malthus to Modern Growth



                                        Preferences

           • Preferences are, δ > 0,

                                  ut = ln(ct ) + δ ln(nt wt+1 ht+1 )


           • The adult member of cohort t cares about:
               • its own consumption ct , assumed to be larger than a
                                    ˜
                  subsistence level c > 0
               • the number of children nt and child earnings wt+1 ht+1
                  [wages times human capital]




                                                                                                5 / 19
Introduction            Economy                Equilibrium             From Malthus to Modern Growth



                                        Preferences

           • Preferences are, δ > 0,

                                  ut = ln(ct ) + δ ln(nt wt+1 ht+1 )


           • The adult member of cohort t cares about:
               • its own consumption ct , assumed to be larger than a
                                    ˜
                  subsistence level c > 0
               • the number of children nt and child earnings wt+1 ht+1
                  [wages times human capital]

           • Altruism: Different from Barro’s dynasties



                                                                                                5 / 19
Introduction   Economy        Equilibrium   From Malthus to Modern Growth



                    Production technology




                                                                     6 / 19
Introduction       Economy            Equilibrium       From Malthus to Modern Growth



                        Production technology
       Production technology
                        ct + nt et = (1 − nt φ) At ht
                                    working time




                                                                                 6 / 19
Introduction            Economy            Equilibrium          From Malthus to Modern Growth



                             Production technology
       Production technology
                             ct + nt et = (1 − nt φ) At ht
                                          working time

           • Production
               • ht is human capital per worker
               • At is the state of technology
               • Individuals offer inelastically one unit of the labor endowment
               • Raising a kid takes a fraction φ ∈ (0, 1) of parents time




                                                                                         6 / 19
Introduction            Economy            Equilibrium          From Malthus to Modern Growth



                             Production technology
       Production technology
                             ct + nt et = (1 − nt φ) At ht
                                          working time

           • Production
               • ht is human capital per worker
               • At is the state of technology
               • Individuals offer inelastically one unit of the labor endowment
               • Raising a kid takes a fraction φ ∈ (0, 1) of parents time
           • Production is allocated to
               • Consumption
               • Educating kids: et is education by kid




                                                                                         6 / 19
Introduction            Economy             Equilibrium           From Malthus to Modern Growth



                             Production technology
       Production technology
                             ct + nt et = (1 − nt φ) At ht
                                           working time

           • Production
               • ht is human capital per worker
               • At is the state of technology
               • Individuals offer inelastically one unit of the labor endowment
               • Raising a kid takes a fraction φ ∈ (0, 1) of parents time
           • Production is allocated to
               • Consumption
               • Educating kids: et is education by kid


       Per capita growth
                                  At − At−1   ht − ht−1
                           gt ≡             +
                                     At−1        ht−1
                                     TFP          human capital                            6 / 19
Introduction   Economy           Equilibrium   From Malthus to Modern Growth



                         Technical progress




                                                                        7 / 19
Introduction           Economy           Equilibrium     From Malthus to Modern Growth



                                 Technical progress
           • Definition: Technical progress is

                                             At − At−1
                                     gAt ≡
                                                At−1




                                                                                  7 / 19
Introduction            Economy           Equilibrium           From Malthus to Modern Growth



                                  Technical progress
           • Definition: Technical progress is

                                              At − At−1
                                      gAt ≡
                                                 At−1


           • Assumption:

                                                          βet−1
                                               ¯
                            gAt = gA (et−1 ) = g + ρ
                                                        1 + βet−1

               ¯
               g > 0 but small, β > 0 and ρ > 0




                                                                                         7 / 19
Introduction            Economy           Equilibrium           From Malthus to Modern Growth



                                  Technical progress
           • Definition: Technical progress is

                                              At − At−1
                                      gAt ≡
                                                 At−1


           • Assumption:

                                                          βet−1
                                               ¯
                            gAt = gA (et−1 ) = g + ρ
                                                        1 + βet−1

               ¯
               g > 0 but small, β > 0 and ρ > 0

                                                                         ¯
           • Exogenous learning makes technical progress positive, gAt ≥ g




                                                                                         7 / 19
Introduction            Economy           Equilibrium           From Malthus to Modern Growth



                                  Technical progress
           • Definition: Technical progress is

                                              At − At−1
                                      gAt ≡
                                                 At−1


           • Assumption:

                                                          βet−1
                                               ¯
                            gAt = gA (et−1 ) = g + ρ
                                                        1 + βet−1

               ¯
               g > 0 but small, β > 0 and ρ > 0

                                                                         ¯
           • Exogenous learning makes technical progress positive, gAt ≥ g
           • Technical progress results from an externality (education)


                                                                                         7 / 19
Introduction            Economy           Equilibrium           From Malthus to Modern Growth



                                  Technical progress
           • Definition: Technical progress is

                                              At − At−1
                                      gAt ≡
                                                 At−1


           • Assumption:

                                                          βet−1
                                               ¯
                            gAt = gA (et−1 ) = g + ρ
                                                        1 + βet−1

               ¯
               g > 0 but small, β > 0 and ρ > 0

                                                                         ¯
           • Exogenous learning makes technical progress positive, gAt ≥ g
           • Technical progress results from an externality (education)
                                              ¯
           • Technical progress is bounded by g + ρ
                                                                                         7 / 19
Introduction   Economy      Equilibrium     From Malthus to Modern Growth



                 Human capital technology




                                                                     8 / 19
Introduction         Economy           Equilibrium   From Malthus to Modern Growth



                       Human capital technology



       Education technology:

                               ht+1 = µ(θ + et )η

       µ > 0, θ > 0 and η ∈ (0, 1)




                                                                              8 / 19
Introduction           Economy           Equilibrium      From Malthus to Modern Growth



                         Human capital technology



       Education technology:

                                 ht+1 = µ(θ + et )η

       µ > 0, θ > 0 and η ∈ (0, 1)

           • There is a minimum level of human capital µθ η




                                                                                   8 / 19
Introduction           Economy           Equilibrium      From Malthus to Modern Growth



                         Human capital technology



       Education technology:

                                 ht+1 = µ(θ + et )η

       µ > 0, θ > 0 and η ∈ (0, 1)

           • There is a minimum level of human capital µθ η
           • In the limit, when e → ∞, gh = ηg




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Introduction   Economy       Equilibrium      From Malthus to Modern Growth



                  Structure of the solution




                                                                       9 / 19
Introduction          Economy           Equilibrium        From Malthus to Modern Growth



                         Structure of the solution

       Generation t receives {At , ht } from the previous generation and
       transfers {At+1 , ht+1 } to the following generation




                                                                                    9 / 19
Introduction          Economy           Equilibrium        From Malthus to Modern Growth



                         Structure of the solution

       Generation t receives {At , ht } from the previous generation and
       transfers {At+1 , ht+1 } to the following generation

       [the dynamics is purely backward]




                                                                                    9 / 19
Introduction          Economy           Equilibrium        From Malthus to Modern Growth



                         Structure of the solution

       Generation t receives {At , ht } from the previous generation and
       transfers {At+1 , ht+1 } to the following generation

       [the dynamics is purely backward]

       The economy, depending on per capita income, At ht , may be in
       different regimes




                                                                                    9 / 19
Introduction          Economy           Equilibrium        From Malthus to Modern Growth



                         Structure of the solution

       Generation t receives {At , ht } from the previous generation and
       transfers {At+1 , ht+1 } to the following generation

       [the dynamics is purely backward]

       The economy, depending on per capita income, At ht , may be in
       different regimes

                                 ˜
           • Modern regime: ct > c and et > 0




                                                                                    9 / 19
Introduction          Economy           Equilibrium        From Malthus to Modern Growth



                         Structure of the solution

       Generation t receives {At , ht } from the previous generation and
       transfers {At+1 , ht+1 } to the following generation

       [the dynamics is purely backward]

       The economy, depending on per capita income, At ht , may be in
       different regimes

                                 ˜
           • Modern regime: ct > c and et > 0

                                          ˜
           • Post-Malthusian regime: ct = c and et > 0




                                                                                    9 / 19
Introduction          Economy           Equilibrium        From Malthus to Modern Growth



                         Structure of the solution

       Generation t receives {At , ht } from the previous generation and
       transfers {At+1 , ht+1 } to the following generation

       [the dynamics is purely backward]

       The economy, depending on per capita income, At ht , may be in
       different regimes

                                 ˜
           • Modern regime: ct > c and et > 0

                                          ˜
           • Post-Malthusian regime: ct = c and et > 0

                                     ˜
           • Malthusian regime: ct = c and et = 0


                                                                                    9 / 19
Introduction   Economy            Equilibrium   From Malthus to Modern Growth



                         Individual behavior




                                                                        10 / 19
Introduction             Economy                   Equilibrium   From Malthus to Modern Growth



                                       Individual behavior
       The representative adult of cohort t

                         max ln(ct ) + δ ln(nt ) + δη ln(θ + et )
                       {ct ,nt ,et }

       s.t.
                                ct + nt et + nt φAt ht = At ht
                                               ˜
                                          ct ≥ c      et ≥ 0
       given At , ht




                                                                                         10 / 19
Introduction             Economy                   Equilibrium   From Malthus to Modern Growth



                                       Individual behavior
       The representative adult of cohort t

                         max ln(ct ) + δ ln(nt ) + δη ln(θ + et )
                       {ct ,nt ,et }

       s.t.
                                ct + nt et + nt φAt ht = At ht
                                               ˜
                                          ct ≥ c      et ≥ 0
       given At , ht

       [At equilibrium, wages are equal to At ]




                                                                                         10 / 19
Introduction             Economy                   Equilibrium   From Malthus to Modern Growth



                                       Individual behavior
       The representative adult of cohort t

                         max ln(ct ) + δ ln(nt ) + δη ln(θ + et )
                       {ct ,nt ,et }

       s.t.
                                ct + nt et + nt φAt ht = At ht
                                               ˜
                                          ct ≥ c      et ≥ 0
       given At , ht

       [At equilibrium, wages are equal to At ]

       Trade-off between fertility and education



                                                                                         10 / 19
Introduction             Economy                   Equilibrium   From Malthus to Modern Growth



                                       Individual behavior
       The representative adult of cohort t

                         max ln(ct ) + δ ln(nt ) + δη ln(θ + et )
                       {ct ,nt ,et }

       s.t.
                                ct + nt et + nt φAt ht = At ht
                                               ˜
                                          ct ≥ c      et ≥ 0
       given At , ht

       [At equilibrium, wages are equal to At ]

       Trade-off between fertility and education

                                 θ
                   ˜
       Assumption: c <          ηφ     < (1 + δ)˜
                                                c
                                                                                         10 / 19
Introduction   Economy         Equilibrium   From Malthus to Modern Growth



                         Modern growth




                                                                     11 / 19
Introduction         Economy           Equilibrium       From Malthus to Modern Growth



                               Modern growth
       When per capita income is large enough, At ht ≥ (1 + δ)˜, the
                                                              c
       solution is interior




                                                                                 11 / 19
Introduction           Economy             Equilibrium     From Malthus to Modern Growth



                                 Modern growth
       When per capita income is large enough, At ht ≥ (1 + δ)˜, the
                                                              c
       solution is interior

           • A constant fraction of income is allocated to consumption

                                            At ht
                                    ct =           ˜
                                                  ≥c
                                            1+δ




                                                                                   11 / 19
Introduction           Economy              Equilibrium    From Malthus to Modern Growth



                                 Modern growth
       When per capita income is large enough, At ht ≥ (1 + δ)˜, the
                                                              c
       solution is interior

           • A constant fraction of income is allocated to consumption

                                             At ht
                                     ct =           ˜
                                                   ≥c
                                             1+δ

               • A property of log preferences




                                                                                   11 / 19
Introduction           Economy              Equilibrium        From Malthus to Modern Growth



                                 Modern growth
       When per capita income is large enough, At ht ≥ (1 + δ)˜, the
                                                              c
       solution is interior

           • A constant fraction of income is allocated to consumption

                                             At ht
                                     ct =           ˜
                                                   ≥c
                                             1+δ

               • A property of log preferences

                      ˜
               • ct = c at the optimum when At ht = (1 + δ)˜
                                                           c




                                                                                       11 / 19
Introduction           Economy                 Equilibrium               From Malthus to Modern Growth



                                   Modern growth
       When per capita income is large enough, At ht ≥ (1 + δ)˜, the
                                                              c
       solution is interior

           • A constant fraction of income is allocated to consumption

                                                At ht
                                        ct =           ˜
                                                      ≥c
                                                1+δ

               • A property of log preferences

                      ˜
               • ct = c at the optimum when At ht = (1 + δ)˜
                                                           c

           • The complement is allocated to raising and educating kids

                                                              δ
                                 nt et + nt φAt ht =             At ht
                                                             1+δ
                                                                                                 11 / 19
Introduction   Economy        Equilibrium      From Malthus to Modern Growth



                   Becker’s fertility theory




                                                                       12 / 19
Introduction          Economy           Equilibrium         From Malthus to Modern Growth



                          Becker’s fertility theory
       Richer parents prefer less, but more educated kids




                                                                                    12 / 19
Introduction             Economy            Equilibrium        From Malthus to Modern Growth



                             Becker’s fertility theory
       Richer parents prefer less, but more educated kids

           • Quantity vs quality: The ratios of marginal costs and marginal
               benefits equalize
                                   et + φAt ht      1/nt
                                               =
                                       nt        η/(θ + et )




                                                                                       12 / 19
Introduction             Economy             Equilibrium       From Malthus to Modern Growth



                             Becker’s fertility theory
       Richer parents prefer less, but more educated kids

           • Quantity vs quality: The ratios of marginal costs and marginal
               benefits equalize
                                   et + φAt ht      1/nt
                                               =
                                       nt        η/(θ + et )

           • Education
                                           ηφAt ht − θ
                                    et =               >0
                                             1−η




                                                                                       12 / 19
Introduction             Economy             Equilibrium                From Malthus to Modern Growth



                             Becker’s fertility theory
       Richer parents prefer less, but more educated kids

           • Quantity vs quality: The ratios of marginal costs and marginal
               benefits equalize
                                   et + φAt ht      1/nt
                                               =
                                       nt        η/(θ + et )

           • Education
                                           ηφAt ht − θ
                                    et =               >0
                                             1−η

           • Number of kids
                                                                   −1
                                          δ                 θ
                          nt = (1 − η)             φ−                   >0
                                         1+δ               At ht

                                                                                                12 / 19
Introduction             Economy             Equilibrium                From Malthus to Modern Growth



                             Becker’s fertility theory
       Richer parents prefer less, but more educated kids

           • Quantity vs quality: The ratios of marginal costs and marginal
               benefits equalize
                                   et + φAt ht      1/nt
                                               =
                                       nt        η/(θ + et )

           • Education
                                           ηφAt ht − θ
                                    et =               >0
                                             1−η

           • Number of kids
                                                                   −1
                                          δ                 θ
                          nt = (1 − η)             φ−                   >0
                                         1+δ               At ht

       [Assumption: having kids requires parents time, but education not]                       12 / 19
Introduction   Economy       Equilibrium    From Malthus to Modern Growth



                   Post-Malthusian regime




                                                                    13 / 19
Introduction        Economy              Equilibrium      From Malthus to Modern Growth



                        Post-Malthusian regime
                                 θ
       When per capita income   ηφ   ≤ At ht < (1 + δ)˜
                                                      c




                                                                                  13 / 19
Introduction           Economy            Equilibrium         From Malthus to Modern Growth



                           Post-Malthusian regime
                                  θ
       When per capita income    ηφ   ≤ At ht < (1 + δ)˜
                                                       c

                                                          ˜
           • Consumption is at the subsistence level ct = c




                                                                                      13 / 19
Introduction           Economy             Equilibrium        From Malthus to Modern Growth



                           Post-Malthusian regime
                                   θ
       When per capita income     ηφ   ≤ At ht < (1 + δ)˜
                                                        c

                                                          ˜
           • Consumption is at the subsistence level ct = c
           • Additional income is allocated to having more kids
                                                         ˜
                                                 At ht − c
                                 nt = (1 − η)
                                                 φAt ht − θ




                                                                                      13 / 19
Introduction           Economy             Equilibrium        From Malthus to Modern Growth



                           Post-Malthusian regime
                                   θ
       When per capita income     ηφ   ≤ At ht < (1 + δ)˜
                                                        c

                                                          ˜
           • Consumption is at the subsistence level ct = c
           • Additional income is allocated to having more kids
                                                         ˜
                                                 At ht − c
                                 nt = (1 − η)
                                                 φAt ht − θ

           • and provide them with more education
                                         ηφAt ht − θ
                                 et =                >0
                                           1−η




                                                                                      13 / 19
Introduction           Economy             Equilibrium        From Malthus to Modern Growth



                           Post-Malthusian regime
                                   θ
       When per capita income     ηφ   ≤ At ht < (1 + δ)˜
                                                        c

                                                          ˜
           • Consumption is at the subsistence level ct = c
           • Additional income is allocated to having more kids
                                                         ˜
                                                 At ht − c
                                 nt = (1 − η)
                                                 φAt ht − θ

           • and provide them with more education
                                         ηφAt ht − θ
                                 et =                >0
                                           1−η

               • et = 0 at the optimum when At ht =       θ
                                                         ηφ




                                                                                      13 / 19
Introduction           Economy             Equilibrium        From Malthus to Modern Growth



                           Post-Malthusian regime
                                   θ
       When per capita income     ηφ   ≤ At ht < (1 + δ)˜
                                                        c

                                                          ˜
           • Consumption is at the subsistence level ct = c
           • Additional income is allocated to having more kids
                                                         ˜
                                                 At ht − c
                                 nt = (1 − η)
                                                 φAt ht − θ

           • and provide them with more education
                                         ηφAt ht − θ
                                 et =                >0
                                           1−η

               • et = 0 at the optimum when At ht =       θ
                                                         ηφ



       Population grows when per-capita income is high
                                                                                      13 / 19
Introduction   Economy           Equilibrium   From Malthus to Modern Growth



                         Malthusian regime




                                                                       14 / 19
Introduction         Economy           Equilibrium        From Malthus to Modern Growth



                               Malthusian regime


                                                      θ
                                    ˜
       When per capita income is in c ≥ At ht <      ηφ




                                                                                  14 / 19
Introduction           Economy           Equilibrium          From Malthus to Modern Growth



                                 Malthusian regime


                                                        θ
                                    ˜
       When per capita income is in c ≥ At ht <        ηφ

                                                          ˜
           • Consumption is at the subsistence level ct = c




                                                                                      14 / 19
Introduction           Economy           Equilibrium          From Malthus to Modern Growth



                                 Malthusian regime


                                                        θ
                                    ˜
       When per capita income is in c ≥ At ht <        ηφ

                                                          ˜
           • Consumption is at the subsistence level ct = c
           • There is no education et = 0, implying ht = µθ η > 0




                                                                                      14 / 19
Introduction           Economy           Equilibrium          From Malthus to Modern Growth



                                 Malthusian regime


                                                        θ
                                    ˜
       When per capita income is in c ≥ At ht <        ηφ

                                                          ˜
           • Consumption is at the subsistence level ct = c
           • There is no education et = 0, implying ht = µθ η > 0
           • Any additional income is allocated to having more kids

                                                    ˜
                                            At ht − c
                                     nt =
                                             φAt ht




                                                                                      14 / 19
Introduction            Economy           Equilibrium         From Malthus to Modern Growth



                                  Malthusian regime


                                                         θ
                                    ˜
       When per capita income is in c ≥ At ht <         ηφ

                                                          ˜
           • Consumption is at the subsistence level ct = c
           • There is no education et = 0, implying ht = µθ η > 0
           • Any additional income is allocated to having more kids

                                                     ˜
                                             At ht − c
                                      nt =
                                              φAt ht
               Population grows when per-capita income is high




                                                                                      14 / 19
Introduction   Economy      Equilibrium   From Malthus to Modern Growth



                         Dynamics




                                                                  15 / 19
Introduction         Economy          Equilibrium   From Malthus to Modern Growth



                                Dynamics
       The model is purely backward




                                                                            15 / 19
Introduction           Economy          Equilibrium   From Malthus to Modern Growth



                                  Dynamics
       The model is purely backward


           • In the Malthusian regime




                                                                              15 / 19
Introduction            Economy           Equilibrium         From Malthus to Modern Growth



                                    Dynamics
       The model is purely backward


           • In the Malthusian regime
                       ¯
                • gt = g > 0, implying that At is permanently growing
                • The economy moves slowly to the following regime




                                                                                      15 / 19
Introduction            Economy           Equilibrium         From Malthus to Modern Growth



                                    Dynamics
       The model is purely backward


           • In the Malthusian regime
                       ¯
                • gt = g > 0, implying that At is permanently growing
                • The economy moves slowly to the following regime

           • In the Post-Malthusian regime




                                                                                      15 / 19
Introduction            Economy           Equilibrium         From Malthus to Modern Growth



                                    Dynamics
       The model is purely backward


           • In the Malthusian regime
                       ¯
                • gt = g > 0, implying that At is permanently growing
                • The economy moves slowly to the following regime

           • In the Post-Malthusian regime
                                                 ¯
                • Positive education raises gt > g and makes human capital
                  grow




                                                                                      15 / 19
Introduction            Economy           Equilibrium         From Malthus to Modern Growth



                                    Dynamics
       The model is purely backward


           • In the Malthusian regime
                       ¯
                • gt = g > 0, implying that At is permanently growing
                • The economy moves slowly to the following regime

           • In the Post-Malthusian regime
                                                 ¯
                • Positive education raises gt > g and makes human capital
                  grow
                • The economy moves to the following regime faster




                                                                                      15 / 19
Introduction            Economy           Equilibrium         From Malthus to Modern Growth



                                    Dynamics
       The model is purely backward


           • In the Malthusian regime
                       ¯
                • gt = g > 0, implying that At is permanently growing
                • The economy moves slowly to the following regime

           • In the Post-Malthusian regime
                                                 ¯
                • Positive education raises gt > g and makes human capital
                  grow
                • The economy moves to the following regime faster

           • Modern growth regime




                                                                                      15 / 19
Introduction            Economy           Equilibrium         From Malthus to Modern Growth



                                    Dynamics
       The model is purely backward


           • In the Malthusian regime
                       ¯
                • gt = g > 0, implying that At is permanently growing
                • The economy moves slowly to the following regime

           • In the Post-Malthusian regime
                                                 ¯
                • Positive education raises gt > g and makes human capital
                  grow
                • The economy moves to the following regime faster

           • Modern growth regime
                                                           ¯
              • Converges monotonically to a BGP with gA = g + ρ and
                       η
                gh = 1−η (¯ + ρ)
                          g

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Introduction   Economy         Equilibrium   From Malthus to Modern Growth



                         Key assumption




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Introduction           Economy           Equilibrium   From Malthus to Modern Growth



                                 Key assumption



           • An adult generation leaves for 25 years




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Introduction              Economy         Equilibrium      From Malthus to Modern Growth



                                    Key assumption



           • An adult generation leaves for 25 years
           • The initial time t = 0 is year 400, the end of the Roman
               Imperium




                                                                                   16 / 19
Introduction              Economy         Equilibrium      From Malthus to Modern Growth



                                    Key assumption



           • An adult generation leaves for 25 years
           • The initial time t = 0 is year 400, the end of the Roman
               Imperium
           • The economy is initially at the Malthusian regime




                                                                                   16 / 19
Introduction              Economy         Equilibrium      From Malthus to Modern Growth



                                    Key assumption



           • An adult generation leaves for 25 years
           • The initial time t = 0 is year 400, the end of the Roman
               Imperium
           • The economy is initially at the Malthusian regime
           • Education becomes positive in year 1650, t = 50




                                                                                   16 / 19
Introduction              Economy         Equilibrium      From Malthus to Modern Growth



                                    Key assumption



           • An adult generation leaves for 25 years
           • The initial time t = 0 is year 400, the end of the Roman
               Imperium
           • The economy is initially at the Malthusian regime
           • Education becomes positive in year 1650, t = 50
           • The modern regime is reached in year 1850, t = 58




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Introduction   Economy      Equilibrium      From Malthus to Modern Growth



               The mechanics of transition




                                                                     17 / 19
Introduction          Economy          Equilibrium          From Malthus to Modern Growth



                       The mechanics of transition


                                 ¯
           • Learning makes gt = g > 0 from the beginning




                                                                                    17 / 19
Introduction           Economy            Equilibrium        From Malthus to Modern Growth



                        The mechanics of transition


                                 ¯
           • Learning makes gt = g > 0 from the beginning
           • Initially, additional income is spent on more kids [Malthus]




                                                                                     17 / 19
Introduction           Economy            Equilibrium        From Malthus to Modern Growth



                        The mechanics of transition


                                 ¯
           • Learning makes gt = g > 0 from the beginning
           • Initially, additional income is spent on more kids [Malthus]
           • Around 1650, individuals can afford positive education




                                                                                     17 / 19
Introduction           Economy            Equilibrium        From Malthus to Modern Growth



                        The mechanics of transition


                                 ¯
           • Learning makes gt = g > 0 from the beginning
           • Initially, additional income is spent on more kids [Malthus]
           • Around 1650, individuals can afford positive education
           • Human capital and technical progress slightly accelerate




                                                                                     17 / 19
Introduction           Economy            Equilibrium        From Malthus to Modern Growth



                        The mechanics of transition


                                 ¯
           • Learning makes gt = g > 0 from the beginning
           • Initially, additional income is spent on more kids [Malthus]
           • Around 1650, individuals can afford positive education
           • Human capital and technical progress slightly accelerate
           • But, people are still constrained by subsistence [Malthus]




                                                                                     17 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                         The mechanics of transition


                                 ¯
           • Learning makes gt = g > 0 from the beginning
           • Initially, additional income is spent on more kids [Malthus]
           • Around 1650, individuals can afford positive education
           • Human capital and technical progress slightly accelerate
           • But, people are still constrained by subsistence [Malthus]
           • Around 1850, consumption becomes interior and the
               Beckerian mechanism starts working




                                                                                     17 / 19
Introduction            Economy           Equilibrium        From Malthus to Modern Growth



                         The mechanics of transition


                                 ¯
           • Learning makes gt = g > 0 from the beginning
           • Initially, additional income is spent on more kids [Malthus]
           • Around 1650, individuals can afford positive education
           • Human capital and technical progress slightly accelerate
           • But, people are still constrained by subsistence [Malthus]
           • Around 1850, consumption becomes interior and the
               Beckerian mechanism starts working
           • Stationary population: Parameters are chosen st. nt → 1
               when t → ∞




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Introduction   Economy        Equilibrium   From Malthus to Modern Growth



                         Calibration




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Introduction            Economy        Equilibrium   From Malthus to Modern Growth



                                  Calibration
             ˜
           • c = 1.25




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Introduction              Economy            Equilibrium          From Malthus to Modern Growth



                                      Calibration
             ˜
           • c = 1.25
           • The time allocated to raising kids is φ =     1
                                                           4   and the total
                                                  1
               weight of kids in utility is δ =   2




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Introduction               Economy           Equilibrium          From Malthus to Modern Growth



                                       Calibration
             ˜
           • c = 1.25
           • The time allocated to raising kids is φ =     1
                                                           4   and the total
                                                  1
               weight of kids in utility is δ =   2
           • η=     1
                    4   to n be unity at steady state




                                                                                          18 / 19
Introduction               Economy           Equilibrium          From Malthus to Modern Growth



                                       Calibration
             ˜
           • c = 1.25
           • The time allocated to raising kids is φ =     1
                                                           4   and the total
                                                  1
               weight of kids in utility is δ =   2
           • η=     1
                    4   to n be unity at steady state
           • θ = ηφ(1 + δ − 1/8)˜ ≃ 0.107 to respect Assumption 1
                                c




                                                                                          18 / 19
Introduction               Economy           Equilibrium          From Malthus to Modern Growth



                                       Calibration
             ˜
           • c = 1.25
           • The time allocated to raising kids is φ =     1
                                                           4   and the total
                                                  1
               weight of kids in utility is δ =   2
           • η=     1
                    4   to n be unity at steady state
           • θ = ηφ(1 + δ − 1/8)˜ ≃ 0.107 to respect Assumption 1
                                c
           • Initial conditions: A0 = 1 and h0 chosen to n1 = 1




                                                                                          18 / 19
Introduction               Economy           Equilibrium          From Malthus to Modern Growth



                                       Calibration
             ˜
           • c = 1.25
           • The time allocated to raising kids is φ =     1
                                                           4   and the total
                                                  1
               weight of kids in utility is δ =   2
           • η=     1
                    4   to n be unity at steady state
           • θ = ηφ(1 + δ − 1/8)˜ ≃ 0.107 to respect Assumption 1
                                c
           • Initial conditions: A0 = 1 and h0 chosen to n1 = 1
           • µ ≃ 2.9 makes h0 = µθ η [consistent with e1 = 0]




                                                                                          18 / 19
Introduction               Economy           Equilibrium          From Malthus to Modern Growth



                                       Calibration
             ˜
           • c = 1.25
           • The time allocated to raising kids is φ =     1
                                                           4   and the total
                                                  1
               weight of kids in utility is δ =   2
           • η=     1
                    4   to n be unity at steady state
           • θ = ηφ(1 + δ − 1/8)˜ ≃ 0.107 to respect Assumption 1
                                c
           • Initial conditions: A0 = 1 and h0 chosen to n1 = 1
           • µ ≃ 2.9 makes h0 = µθ η [consistent with e1 = 0]
             ¯
           • g ≃ 0.0615% makes the economy converge to the
               Post-Malthusian regime in year 1650




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Introduction               Economy           Equilibrium          From Malthus to Modern Growth



                                       Calibration
             ˜
           • c = 1.25
           • The time allocated to raising kids is φ =     1
                                                           4   and the total
                                                  1
               weight of kids in utility is δ =   2
           • η=     1
                    4   to n be unity at steady state
           • θ = ηφ(1 + δ − 1/8)˜ ≃ 0.107 to respect Assumption 1
                                c
           • Initial conditions: A0 = 1 and h0 chosen to n1 = 1
           • µ ≃ 2.9 makes h0 = µθ η [consistent with e1 = 0]
             ¯
           • g ≃ 0.0615% makes the economy converge to the
               Post-Malthusian regime in year 1650
           • β = 0.3 such that the economy reaches modern growth in
               year 1850



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Introduction               Economy           Equilibrium          From Malthus to Modern Growth



                                       Calibration
             ˜
           • c = 1.25
           • The time allocated to raising kids is φ =     1
                                                           4   and the total
                                                  1
               weight of kids in utility is δ =   2
           • η=     1
                    4   to n be unity at steady state
           • θ = ηφ(1 + δ − 1/8)˜ ≃ 0.107 to respect Assumption 1
                                c
           • Initial conditions: A0 = 1 and h0 chosen to n1 = 1
           • µ ≃ 2.9 makes h0 = µθ η [consistent with e1 = 0]
             ¯
           • g ≃ 0.0615% makes the economy converge to the
               Post-Malthusian regime in year 1650
           • β = 0.3 such that the economy reaches modern growth in
               year 1850
           • ρ = 0.25 implies a yearly rate of technical progress of around
               1% at steady state
                                                                                          18 / 19
Introduction                    Economy                  Equilibrium                       From Malthus to Modern Growth




               From Malthus to modern growth.nb                                                                   1



                            g                                                n
                                                                          1.3
                         0.3
                                                                         1.25
                        0.25
                                                                          1.2
                         0.2

                        0.15                                             1.15

                         0.1                                              1.1

                        0.05                                             1.05

                                                                  time                                          time
                                1750      1850    1950     2050                  1750   1850    1950     2050




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