Neoclassical Models of Endogenous Growth October 2007 () Endogenous Growth October 2007 1 / 20
Motivation What are the determinants of long run growth? Growth in the "e¤ectiveness of labour" should depend on economic incentives , ! decision makers who make A grow must be rewarded , ! BUT since F ( K , AL ) exhibits CRS when A is exogenous, it must exhibit IRS when A is a separate factor , ! not all factors can be paid their marginal products , ! inconsistent with perfect competition and, hence, the neoclassical framework. () Endogenous Growth October 2007 2 / 20
Alternative Paradigms of Endogenous Growth Neoclassical or AK paradigm , ! e¤ectively assumes that raw labour, L , is not a factor of production , ! emphasizes knowledge that is embodied in the work force , ! growth promoting factor (human capital) is a private , rival good with no dynamic externalities Endogenous technological change paradigm , ! incorporates IRS by allowing for imperfect competition in a GE framework , ! emphasizes knowledge that is disembodied , ! growth promoting factor (ideas) is a non–rival , public good with dynamic externalities . () Endogenous Growth October 2007 3 / 20
Basic AK Model Ramsey model with capital share α = 1 and no technical change: y ( t ) = Ak ( t ) Household’s optimal consumption path: c ( t ) c ( t ) = r ( t ) � ρ ˙ θ Perfect competition ) r ( t ) = A � δ Consumption growth is then c ( t ) c ( t ) = A � δ � ρ ˙ θ () Endogenous Growth October 2007 4 / 20
Aggregate resource constraint: c ( t ) + ˙ k ( t ) + δ k ( t ) = Ak ( t ) . , ! dividing by k ( t ) , we get ˙ c ( t ) k ( t ) k ( t ) + k ( t ) = A � δ . ˙ k ( t ) Along a BGP k ( t ) is constant ) c / k must be constant ˙ ) ˙ c ( t ) k ( t ) c ( t ) = k ( t ) Since y ( t ) = Ak ( t ) it follows that ˙ y ( t ) ˙ k ( t ) k ( t ) = A � δ � ρ y ( t ) = = g θ () Endogenous Growth October 2007 5 / 20
. c k=0 Saddlepath k Figure: Phase Diagram for the AK Model () Endogenous Growth October 2007 6 / 20
What about the transversality condition? D ( T ) k ( T ) = e � rT e gT k ( 0 ) goes to zero as T becomes large if and only if > r g A � δ � ρ A > θ () Endogenous Growth October 2007 7 / 20
Implications Simplest possible endogenous growth model ) long–run growth rate depends on level of MP of capital (net of depreciation) relative to discount rate ) growth increases with willingness of households to substitute consumption across time BUT most estimates …nd diminishing returns to physical capital and wages/salaries ' 2/3 of output , ! this simple model does not conform well with basic observations Also implies no conditional convergence () Endogenous Growth October 2007 8 / 20
A One-Sector Model with Physical and Human Capital Could expand de…nition of "capital" as in augmented Solow model Resource constraint: Y = AK α H 1 � α = C + I K + I H where ˙ ˙ K = I K � δ K and H = I H � δ H Implications are very similar to basic AK model (see Barro ch. 5) () Endogenous Growth October 2007 9 / 20
A Two-Sector Model with Physical and Human Capital Uzawa–Lucas Model Based on “The Mechanics of Economic Development” (Lucas, 1988) , ! emphasizes the central role of human capital accumulation in driving long-run growth Simpli…ed version: no population growth and no externalities Focus on balanced (steady state) growth path Sectors producing human and physical capital di¤er () Endogenous Growth October 2007 10 / 20
Assumptions Aggregate output is produced according to Y ( t ) = AK ( t ) α H ( t ) 1 � α , where H ( t ) = u ( t ) h ( t ) L ( t ) . and u ( t ) = fraction of labour time allocated to working h ( t ) = human capital per worker In per capita terms: y ( t ) = Ak ( t ) α [ u ( t ) h ( t )] 1 � α (1) where y ( t ) = Y ( t ) / L ( t ) , etc. () Endogenous Growth October 2007 11 / 20
Aggregate Resource constraint k ( t ) + δ k ( t ) = Ak ( t ) α [ u ( t ) h ( t )] 1 � α c ( t ) + ˙ (2) Competitive factor markets: � u ( t ) h ( t ) � 1 � α r ( t ) = � δ α (3) k ( t ) � � α k ( t ) w ( t ) = ( 1 � α ) (4) u ( t ) h ( t ) where w ( t ) = wage per unit of human capital () Endogenous Growth October 2007 12 / 20
Representative household preferences: Z ∞ e � ρ t c ( t ) 1 � θ U = 1 � θ dt . 0 Dynamic budget constraint ˙ k ( t ) = r ( t ) k ( t ) + w ( t ) u ( t ) h ( t ) � c ( t ) (5) Human capital accumulation ˙ h ( t ) = B ( 1 � u ( t )) h ( t ) , (6) Boundary conditions: T ! ∞ D ( T ) k ( T ) � 0 and u ( t ) 2 ( 0 , 1 ) lim Note that there are 2 control variables and 2 state variables () Endogenous Growth October 2007 13 / 20
Optimality conditions if both k and h are accumulated Note that there are 2 control variables and 2 state variables Hamiltonian for household’s optimization problem: J = e � ρ t c 1 � θ 1 � θ + λ [ rk + wuh � c ] + µ [ B ( 1 � u ) h ] The Hamiltonian conditions are dJ e � ρ t c � θ � λ = 0 = (7) dc dJ λ r = � ˙ = λ (8) dk dJ = λ wh � µ Bh = 0 (9) du dJ = λ wu + µ B ( 1 � u ) = � ˙ µ (10) dh () Endogenous Growth October 2007 14 / 20
Di¤erentiating (7) w.r.t. time and combining with (8), we get ˙ � ρ � θ ˙ c λ c = λ = � r Di¤erentiating (9) w.r.t. time ˙ w = ˙ λ λ + ˙ w µ µ Substituting out λ w in (10) using (9): µ Bu + µ B ( 1 � u ) = � ˙ µ � ˙ µ = B µ () Endogenous Growth October 2007 15 / 20
It follows that r ( t ) = B + ˙ w ( t ) (11) w ( t ) if both h and k are being accumulated by the household, the rates of return must be equal , ! otherwise only the asset with the highest return will be accumulated () Endogenous Growth October 2007 16 / 20
The Balanced Growth Path , ! situation where all aggregates grow at constant rates (need not be equal) If ˙ h / h is constant ) u ( t ) = u is constant Let ˙ c / c = g Then from the Euler equation r ( t ) = r = θ g � ρ It follows that from (3) that h ( t ) / k ( t ) is constant: ˙ ˙ h k h = k Dividing the (2) by k ( t ) yields � uh ( t ) � 1 � α ˙ c ( t ) k ( t ) k ( t ) + k ( t ) + δ = k ( t ) Since ˙ k / k is constant, c ( t ) / k ( t ) must be constant ) ˙ ˙ c ˙ k h c = k = h = g () Endogenous Growth October 2007 17 / 20
From (1) it follows that ˙ ˙ y ˙ k h y = α k + ( 1 � α ) h = g Since k ( t ) / h ( t ) is constant (4) implies w ( t ) ˙ w ( t ) = 0. But then from (11) we have r = B It follows that the equilibrium growth rate is g = B � ρ . θ () Endogenous Growth October 2007 18 / 20
Implications Similar expression to basic AK model growth rate BUT growth depends on the productivity of human capital sector, B , ! does not depend on marginal product of physical capital Physical capital accumulation is NOT the “engine of growth" here , ! capital stock adjusts so that r = B in the long run Lucas model generates endogenous growth in a competitive model while preserving diminishing returns to physical capital Transitional dynamics ) conditional convergence () Endogenous Growth October 2007 19 / 20
Extension — Human Capital Externalities Suppose the production function (in per capita terms) is y ( t ) = Ak ( t ) α [ uh ( t )] 1 � α h a ( t ) γ , where h a = e¤ect of the average human capital not taken into account by …rms , ! perceived marginal product of human capital: � � α k ( t ) h a ( t ) γ w ( t ) = ( 1 � α ) u ( t ) h ( t ) , ! but, in equilibrium, h a ( t ) = h ( t ) () Endogenous Growth October 2007 20 / 20
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