Advanced Macroeconomics Endogenous Growth Marcin Kolasa Warsaw School of Economics Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 1 / 18
Introduction The Solow and Ramsey models are exogenous growth models: Long-run growth (of output per capita) not possible without technological progress Technological progress governed by an exogenous process (not explained by the model) Two generations of endogenous growth models: 1st: Long-run growth without technological progress 2nd: Endogenous technological change Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 2 / 18
AK model - basic setup Production function: Y t = F (K t ) = AK t where: A > 0 is a constant (i.e. there is no technological progress, g = 0) Production function in intensive form (per capita): y t = f (k t ) = Ak t where: y t = Yt L t, k t = Kt L t Production function is then not neoclassical: F K = f (k) = A Otherwise identical to the Ramsey model Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 3 / 18
Equilibrium dynamics The equilibrium dynamics of the model at any time t can be characterized by 3 equations (compare to the Ramsey model): ( ct+1 ) θ = β(a + 1 δ) (1) k t+1 k t c t = A + 1 δ 1 c t 1 + n 1 + n k t (2) ( ) 1 + n t = 0 A + 1 δ (3) lim t k t+1 Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 4 / 18
Two technical assumptions Positive consumption growth requires: A > 1 β 1 + δ (A.1) Developing the condition for lifetime utility to be bounded: From (1): c t = [β (A + 1 δ)] t θ c 0 = (1 + γ c ) t c 0 (4) where γ c = [β (A + 1 δ)] 1 θ 1 is the growth rate of c so the lifetime utility becomes: U 0 = B 0 t=0 1 θ = L 0c 1 θ [ 0 β(1 + n)(1 + γc ) 1 θ] t 1 θ β t c1 θ t which means that U 0 is bounded if t=0 β(1 + n)(1 + γ c ) 1 θ < 1 (A.2) Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 5 / 18
Initial consumption level Capital accumulation using (4): k t+1 = A + 1 δ k t c 0 1 + n 1 + n (1 + γ c) t (5) Equation (5) is a first-order difference equation, which can be solved by iteration to yield: ( ) k t = a t k 0 c 0 1 ( 1+γc a ) t (6) 1 + n a (1 + γ c ) where a = A+1 δ 1+n Plugging (6) into the transversality condition (3) and using assumption (A.2) yields an explicit formula for c 0 given k 0 : c 0 = (1 + n)(a (1 + γ c ))k 0 (7) Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 6 / 18
Capital dynamics and the savings rate Substituting for c 0 from (7) into (6) gives: k t = (1 + γ c ) t k 0 (8) Conclusions: Equations (4), (8) and (7) give the full solution of the model for any given k 0 Consumption, capital (and hence output) per capita grow at the same rate all the time: c t+1 = k t+1 = y t+1 = [β (A + 1 δ)] 1 θ c t k t y t Hence, the savings rate is constant and equal to: s = (1 + n)β 1 θ (A + 1 δ) 1 θ A 1 + δ (9) Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 7 / 18
Sustained growth via capital accumulation Capital accumulation graphically: k t+1 = (1 + γ c )k t k t+1 45 o k 0 k t Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 8 / 18
The role of policy Long-run growth with taxes (see the Ramsey model for derivations) c t+1 = y [ t+1 = β 1 + τ ] 1 i(1 δ) + (1 τ k )(1 τ f )(A δ) θ c t y t (1 + τ i ) Changes in taxes have a permanent effect on the growth rate of output per capita (in the Ramsey model only the long-run level of output is permanently affected, the effect on the growth rate is only transitory) (10) Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 9 / 18
Adding human capital I Problems with the standard AK model Assuming that only physical capital matters for production seems unrealistic It implies that the share of capital in national income should be 1 (since wages, as the marginal product of labour, are 0) Let us instead assume that there are two inputs to the production function: physical capital K t and human capital H t : Y t = F (K t, H t ) where F exhibits all standard neoclassical properties Constant returns to scale allow us to write alternatively: Y t = K t f (H t /K t ) where f exhibits all properties of a neoclassical production function in itensive form Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 10 / 18
Adding human capital II An important difference to the Solow or Ramsey model: both physical and human capital are subject to accumulation: K t+1 = (1 δ K )K t + I K,t H t+1 = (1 δ H )H t + I H,t Output can be used on a one-for-one basis for consumption C t, investment in physical capital I K,t or investment in human capital I H,t : Y t = C t + I K,t + I H,t Firms maximize profits and factor markets are competitive: F K t = f (H t /K t ) H t /K t f (H t /K t ) = R K,t F H t = f (H t /K t ) = R H,t Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 11 / 18
Adding human capital III Optimality implies that the rates of return the households earn on both types of capital must be the same: or equivalently: R K,t δ K = R H,t δ H f (H t /K t ) (H t /K t + 1)f (H t /K t ) = δ K δ H which determines a unique, constant value of H/K at all times Conclusions: We get a model with essentially the same dynamics as the standard AK model (think about f (H/K) = A, where A is a constant) Important difference: it generates stable factor distribution of income, with a significant fraction accruing to labour Important observation: there are decreasing returns to accumulation of each factor (physical and human capital) but constant returns to accumulation of broadly defined capital - hence endogenous growth Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 12 / 18
1st generation endogenous growth models - summary Long-run growth possible even without technological progress The engine of growth: capital accumulation not facing decreasing returns (unlike in the Solow or Ramsey models) Consumption and output growth rates are constant; in particular, they do not depend on the initial capital level; hence: There is no transitional dynamics in the model It does not imply conditional nor unconditional convergence Savings rate is constant, but not exogenous as in the Solow model As in the Ramsey model, the competitive equilibrium (without distortionary taxation) is Pareto optimal Changes in preferences or policy parameters can have a permanent effect on the economy s growth rate Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 13 / 18
2nd generation endogenous growth models - preliminaries Basic idea: technological change is costly (requires R&D) and results from innovations driven by profit motives Nonrivalry of ideas underlying technology (Romer, 1986) Problems with the classical assumptions: Standard replication argument implies that the production function exhibits constant returns to scale in capital and labour, but increasig returns to scale in capital, labour and technology Euler s theorem and perfect competition imply that the factor payments exhaust the output, so that each firm s profit equals zero at every time: no funds to finance innovation It follows that: Imperfect competition on the goods market necessary Excludability (at least partial) of ideas needed, otherwise making improvements to technology would not be profitable Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 14 / 18
R&D-based growth model - a sketch A fully-fledged R&D-based growth model requires a detailed modelling of production and innovation sectors, here we only sketch the main concepts Final output production function: Y t = F (K t, A t L Y,t ) (11) where L Y,t is labour input used in the goods production sector Knowledge production function (deterministic for simplicity): A t+1 A t = BL A,t A φ t (12) where L A,t is labour input used in the R&D sector Assume for simplicity that the division of labour between the two sectors is constant: L A,t = a L L t L Y,t = (1 a L )L t where 0 < a L < 1 is an exogenous parameter Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 15 / 18
R&D-based growth model - dynamics I Technological change dynamics: γ A,t+1 = A t+1 A t A t = BL A,t A φ 1 t (13) Romer (1990): φ = 1 Equation (13) becomes: γ A,t+1 = BL A,t The rate of technological progress depends positively on population (larger communities should grow faster: scale effects ) Kremer (1993): this picture fits the human history (since 1 million B.C.); most researchers find this result unattractive With growing population, technological progress should accelerate - at odds with the data Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 16 / 18
R&D-based growth model - dynamics II Jones (1995): φ < 1 Equation (13) implies: γ A,t+1 = Ba LL t A φ 1 t γ A,t Ba L L t 1 A φ 1 = (1 + n)(1 + γ A,t ) φ 1 t 1 The unique positive value of γ A,t such that γ A,t is constant: γ A = (1 + n) 1 1 φ 1 Long-run growth possible only if population is growing No scale effects : the rate of growth does not depend on the size of population (nor its proportion engaged in R&D) There are scale effects in levels: the higher L t (for given a L ), the higher the level of technology (from (13)): A t+1 = A t + Ba L L t A φ t and so the higher output per capita Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 17 / 18
2nd generation endogenous growth models - summary Technological progress endogenous and depends on the amount of resources engaged in R&D Their micro-foundations (not worked out in this course) imply that the decentralized equilibrium does not coincide with social planner s choices and so is not Pareto optimal The amount of resources spent on research usually too small: researchers do not internalize positive effects of their inventions on productivity of future research Long-run properties critically depend on the knowledge production function: knife-edge assumptions needed for sustained growth with constant population Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 18 / 18