Development Economics Unified Growth Theory: From Stagnation to Growth Andreas Schäfer University of Leipzig Institute of Theoretical Economics WS 10/11 Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 1 / 36
Contents 2.1 Introduction - Unified Growth Theory 2.2 Historical Evidence 2.3 The Model - Production of Final Output 2.4 The Model - Preferences and Budget Constraints 2.5 The Model - Production of Human Capital 2.6 Optimization 2.7 Some Results 2.8 Technological Progress 2.9 Effective Resources 2.10 Technology and Education 2.11 Dynamics Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 2 / 36
2.1 Introduction - Unified Growth Theory Lit.: Oded Galor and David N. Weil, Population Technology, and Growth: From Malthusian Stagnation to the Demographic Transition and beyond.american Economic Review, Sep. 2000, Vol. 90(4), pp. 806-828. Oded Galor, From Stagnation to Growth: Unified Growth Theory. Handbook of Economic Growth, 2005, North-Holland. capture the process of development over the entire course of human history. That is, capturing the epoch of Malthusian stagnation that characterized most of human history, the contemporary era of modern growth and the forces that triggered the transition between the two regimes. What is the justification for the selective use of observations which characterize only the contemporary growth process? Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 3 / 36
2.2 Historical Evidence The Malthusian model has two key components: 1 Fixed factor of production (land) implying decreasing returns to scale for all other factors. 2 Positive effect of the standard of living on the growth rate of population. Malthus: when population size is small, the standard of living is high, and population will grow as a natural result of passion between sexes. When population size is high, the standard of living is low. Population will be reduced by preventive checks or positive checks (disease, malnutrition, etc.). Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 4 / 36
2.2 Historical Evidence Implications of the Malthusian model: In the absence of technological progress or in the availability of land the size of population will be self-equilibrating, increases in available resources will be offset by the size of population, countries with superior technologies will have denser populations, but a similar standard of living (China). These predictions are consistent with the evolution of of technology, population, and output per capita for most of human history. Maddison (1982): growth rate of per-capita GDP between 500 and 1500 was zero. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 5 / 36
2.2 Historical Evidence The Post-Malthusian Regime: The Malthusian mechanism linking higher income to higher population growth is still at work, but the diluting effect on resources per capita was counteracted by technological progress. In western Europe population growth was 40 percent as large as total output growth between 1820-1870, and only 20 percent as large between 1929-1990. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 6 / 36
2.2 Historical Evidence The Modern Growth Regime: is characterized by steady growth in both income per capita and the level of technology. there is a negative relationship between the level of output and the growth rate of population. In England live births per 1,000 women aged 15-44 fell from 153.6 in 1871-1880 to 109.0 in 1901-1910 (Wrigley, 1969). The reversal of the Malthusian relationship corresponded to an increase in the resources invested in each child. The average number of years of schooling in England and Wales rose from 2.3 for a cohort born between 1801-1805 to 9.1 for the cohort born between 1897-1906. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 7 / 36
2.2 Historical Evidence Implications: Key event that separates the Malthusian and Post-Malthusian Regimes is the acceleration in the pace of technological progress. Event that separates the Post-Malthusian and the Modern Growth Regime is the demographic transition that followed the industrial revolution. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 8 / 36
2.3 The Model - Production of Final Output Production is subject to constant returns to scale and endogenous technological progress, such that where α (0, 1) Y t = H α t (A t X) 1 α, (1) X: fixed quantity of land - without property rights over land(!) the return to land is zero H t = h t L t : efficiency units of labor A t : technological level in period t A t X : effective resources employed in production Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 9 / 36
2.3 The Model - Production of Final Output Output per worker produced at time t reads y t = h α t x1 α t, (2) where h t = Ht L t : level of efficiency units of labor per worker, x t = AtX L t : level of effective resources per worker at time t. Without property rights over land, the wage rate per efficiency unit of labor equals its average product. Hence, w t =(x t /h t ) 1 α. (3) Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 10 / 36
2.4 The Model - Preferences and Budget Constraints Each generation (born in t 1) consists of L t identical individuals and joins the labor force. Each individual has a single parent. Agents live for two periods: childhood and parenthood. Parents are endowed with one unit of time which they allocate between labor and child rearing. Moreover, parents choose quantity and quality of children, optimally. Preferences of parents are defined over consumption, c t,abovea subsistence level c >0, as well as over the quality and quantity of their offspring, such that u t =(c t ) 1 γ (w t+1 n t h t+1 ) γ γ (0, 1). (4) w t+1 n t h t+1 : intergenerational altruism. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 11 / 36
2.4 The Model - Preferences and Budget Constraints Population economics: 1 Ben-Porath (1976), Eswaran (1998) - interaction between fertility n and mortality d d n economic reasoning: Old-age security motive (Srinivasan (1988), Sah (1991)... 2 Becker (1960), Barro und Becker (1989) - optimal fertility choice - q-q trade off 1 Interaction between technological progress, human capital, and n (Galor und Weil, 2000). 2 Opportunity costs of childrearing time (Galor und Weil, 1996) interaction with inequality (de la Croix und Doepke, 2003, Schfer, 2005). 3 Reversed direction of intergenerational transfers - Caldwell - Hypothese (Blackburn und Cipriani, 2005) 4 Habits and population dynamics (Schäfer and Valente, 2009) Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 12 / 36
2.4 The Model - Preferences and Budget Constraints Fertility choices follow the standard model introduced by Becker (1960): Household chooses the number of children and their quality dividing resources between child-raising and labor market activities. Here, the only input for child rearing is time, such that the time cost for raising one child to adulthood with educational level e t+1 amount to τ q + τ e e t+1, (5) where τ q : pure child rearing costs as a fraction of the household s unit of time endowment. τ e : fraction of individual s unit time endowment necessary for each unit of education per child. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 13 / 36
2.4 The Model - Preferences and Budget Constraints In the second period of life, the budget constraint reads w t h t n t (τ q + τ e e t+1 )+c t w t h t. (6) Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 14 / 36
2.5 The Model - Production of Human Capital Technological progress raises the value of education in producing human capital. The level of human capital is determined by their education and the technological environment. Technological progress g t+1 =(A t+1 A t )/A t reduces the adaptability of existing human capital for the technological environment erosion effect. Education lessens the adverse effects of technological progress technology complements skills in the production of human capital Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 15 / 36
2.5 The Model - Production of Human Capital formally, with h t+1 = h(e t+1,g t+1 ), (7) h e (e t+1,g t+1 ) > 0, h ee (e t+1,g t+1 ) < 0, h g (e t+1,g t+1 ) < 0, h gg (e t+1,g t+1 ) > 0, h eg (e t+1,g t+1 ) > 0, (e t+1,g t+1 ) 0 Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 16 / 36
{n t,e t+1 } = argmax { (w t h t [1 n t (τ q + τ e e t+1 )]) 1 γ (8) 2.6 Optimization The optimization problem of a household born in t 1 reads as subject to (w t+1 n t h(e t+1,g t+1 )) γ } w t h t [1 n t (τ q + τ e e t+1 )] c, (9) (n t,e t+1 ) 0. (10) Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 17 / 36
2.6 Optimization If potential income z = w t h t is sufficiently high in order to ensure c t > c, we yield c t = (1 γ)w t h t, (11) n t [τ q + τ e e t+1 ] = γ (12) Since γ represents the share of time devoted to child raising, and the time budget is normalized to one, we know that 1 γ is the fraction of time devoted to labor market participation. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 18 / 36
2.6 Optimization If z t z, the subsistence constraint becomes binding. Hence, the fraction of time allocated to labor market participation is larger than 1 γ and the fraction of time devoted to child rearing below γ. Therefore, { n t [τ q + τ e e t+1 ]= γ if z t z 1 [ c/(w t h t )] if z t z (13) Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 19 / 36
2.6 Optimization TimeSpent RaisingChildren 1 IncomeExpansionPath c ct Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 20 / 36
2.6 Optimization Using (8), the optimization with respect to e t+1 implies the implicit functional relationship between e t+1 and g t+1 G(e t+1,g t+1 ) (τ q + τ e e t+1 )h e (e t+1,g t+1 ) τ e h(e t+1,g t+1 ) { =0 if et+1 > 0 0 if e t+1 =0. Moreover,itisassumedthatG(0, 0) < 0. Since, h eg > 0 and h g < 0 lim gt+1 G(0,g t+1 ) > 0. As in addition G(0, 0) < 0, there exists a ĝ, such that G(0, ĝ) =0and e t+1 > 0 (e t+1 =0for g t+1 < ĝ). Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 21 / 36
2.6 Optimization It follows that e t+1 = e(g t+1 ),where { =0 if gt+1 ĝ, e(g t+1 ) > 0 if g t+1 > ĝ, (14) with e (g t+1 ) > 0 and ĝ>0. children s quality depends only on the rate of technological progress and not on the level of potential income z t Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 22 / 36
2.6 Optimization It follows that n t is n t = { γ τ q +τ e e(g t+1 ) nb (g t+1 ) 1 [ c/(w th t)] τ q +τ e e(g t+1 ) na (g t+1 ) if z t z if z t z (15) Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 23 / 36
2.7 Some Results 1 Technological progress results in a decline of in the number of children and an increase in their quality, i.e., n t / g t+1 0 and e t+1 / g t+1 0. 2 If z t < z, an increase in parental potential income raises n t, but has no effect on children s quality, i.e, n t / z t > 0 and e t+1 / z t =0. 3 If z t > z, an increase in parental potential does not change n t or children s quality, i.e, n t / z t = e t+1 / z t =0. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 24 / 36
2.8 Technological Progress Technological progress g t+1 depends on education of the working cohort, e t, and the size of the population, L t (scale effect a la Jones, 1995). Hence, where g(0,l t ) > 0, g t+1 A t+1 A t A t = g(e t,l t ), (16) g i (e t,l t ) > 0 and g ii (e t,l t ) < 0,i = e t,l t. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 25 / 36
2.9 Effective Resources Effective resources are defined as x t = AtX L t, hence x evolves over time according to where x 0 is historically given. x t+1 = A t+1/a t L t+1 /L t x t = 1+g t+1 n t x t, (17) Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 26 / 36
2.10 Technology and Education The evolution of technology and education follows g t+1 = g(e t ; L) e t+1 = e(g t+1 ). This system is depicted by three different configurations. The economy shifts endogenously from one configuration to another as population increases and g(e t ; L) shifts upwards. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 27 / 36
2.10 Technology and Education Panel 1: For a range of small population sizes, the system is globally stable. The steady state is (ē, ḡ) =(0,g l ). (18) As g(0,l) > 0, the rate of technological change increases with the size of population, whereas the level of education remains unchanged. Panel 2: For a range of moderate population sizes there are three steady states. Two of them are locally stable (ē, ḡ) = (0,g l ) (19) (ē, ḡ) = (e h,g h ). (20) the interior steady state (ē, ḡ) =(e u,g u )is unstable. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 28 / 36
2.10 Technology and Education Panel 3: For a range of large population sizes there is globally stable steady state, that is (ē, ḡ) =(e h,g h ). (21) Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 29 / 36
2.11 Dynamics - Panel 1 g t e t e( g ) 1 t1 l gt 1 g( et; L) l g () L xt x t x 1 t et l x () L Andreas Schäfer (University of Leipzig) Unified el ˆ( ) Growth Theory WS 10/11 30 / 36 et
2.11 Dynamics - Panel 1 Population is low and the implied rate of technological change is small. Parents have no incentive to provide education. Malthusian steady state: effective resources and output per capita are constant. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 31 / 36
2.11 Dynamics - Panel 2 g t h g e t e( g ) 1 t1 m gt1 g( et; L ) u g l g xt u e e t e x h e t 1 t 1 t t 1 t x e e et u e ê Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 32 / 36 h e et
2.11 Dynamics - Panel 2 Multiple history dependent equilibria. Given the initial conditions, the economy remains in the vicinity of the Malthusian steady state. As the rate of technological progress continues to rise, the Malthusian steady state vanishes. There is only one stable steady state Panel 3 Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 33 / 36
2.11 Dynamics - Panel 3 g h e t e( g ) 1 t1 h gt1 g( et; L ) xt x t x 1 t e t h e e 1 t et ê h Andreas Schäfer (University of Leipzig) Unified Growth Theory e WS 10/11 34 / 36 et
2.11 Dynamics - Panel 3 The increase in the rate of technological progress has two opposing effects on the evolution of the population: 1 There are more resources available for child rearing. 2 Reallocation of additional resources towards child quality which lowers fertility. due to a low demand for human capital, the first effect dominated and households increased their family size as well as the quality per child in the Post-Malthusian regime. The interaction between human capital formation and technological progress induced further investments in child quality. Fertility declines, the offsetting effect of population growth on the growth rate of income per capita is eliminated. resources per capita rise, as technological progress outstrips population growth. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 35 / 36
2.11 Dynamics - Panel 3 If population growth is zero, the levels of education and technological progress and the growth rates of resources per capita are constant. The model makes no prediction of the long-run growth rate of the population and of the economy. Andreas Schäfer (University of Leipzig) Unified Growth Theory WS 10/11 36 / 36