Endogenous Technological Change October 2007 () Endogenous - PowerPoint PPT Presentation

Endogenous Technological Change October 2007 () Endogenous Techological Change October 2007 1 / 19

Basic idea Economic growth ! increased market size ! higher pro…ts ! justi…es the …xed costs of knowledge creation ! knowledge growth ! higher productivity of labour/capital ! economic growth Two classes of model: (1) Growth sustained by increased specialization of labour across an increasing variety of activities (A. Smith, 1776, Romer, 1990) (2) Growth through creative destruction (Schumpeter, 1935, Aghion and Howitt, 1992) () Endogenous Techological Change October 2007 2 / 19

A baseline GE model with a variety of products Utility maximizing households , ! …xed population L with CES preferences , ! assets are claims to pro…ts of …rms (no capital) Competitive …nal goods producers i combine labour L i and intermediates to produce output Y i Monopolistically competitive intermediate producers j 2 f 1 , N g protected by patents R&D …rms allocate resources to invention, then sell ideas to producers () Endogenous Techological Change October 2007 3 / 19

Producers of Final Ouput Production function: N Y i = AL 1 � α X α ∑ 0 < α < 1 i ij j = 1 where X ij = quantity of intermediate j used by …nal output producer i All …nal output is assumed identical with price = 1 (numeraire) Pro…t: N ∑ Y i � wL i � P j X ij j = 1 FOCs: ∂ Y i ( 1 � α ) Y i = = w ∂ L i L i ∂ Y i α AL 1 � α X α � 1 = = P j 8 j i ij ∂ X ij () Endogenous Techological Change October 2007 4 / 19

Demand function for X ij : � A α � 1 1 � α X ij = L i P j Aggregate demand for labour L i = 1 � α Y i = ( 1 � α ) Y ( t ) L = ∑ ∑ w ( t ) w i i () Endogenous Techological Change October 2007 5 / 19

Intermediate Good Producers Simpli…cation: perpetual monopoly rights conferred by patent Present value of pro…ts: Z ∞ V ( t ) = D ( t , s ) π j ( s ) ds t where π j ( s ) is ‡ow of pro…ts at time s and D ( t , s ) = e � R s t r ( ω ) d ω () Endogenous Techological Change October 2007 6 / 19

Note that Z ∞ dD ( t , s ) dV dt = π j ( s ) ds � D ( t , t ) π j ( t ) dt t Z ∞ ˙ V ( t ) = r ( t ) D ( t , s ) π j ( s ) ds � π j ( t ) t ˙ V ( t ) = r ( t ) V ( t ) � π j ( t ) This can be expressed as ˙ r ( t ) = π j ( t ) V ( t ) V ( t ) + V ( t ) , ! rate of return on claims to pro…ts = dividend yield plus capital gains () Endogenous Techological Change October 2007 7 / 19

Cost of producing one unit of intermediate = one unit of output , ! pro…t is then π j = [ P j � 1 ] X j where � A α � � A α � 1 1 1 � α 1 � α X j = ∑ ∑ X ij = L i = L P j P j i i Pro…t-maximization ) � A α � 1 1 � α max π j = [ P j � 1 ] L P j P j � � 1 1 1 � � 1 1 � α L 1 � α 1 � α = P � P ( A α ) j j () Endogenous Techological Change October 2007 8 / 19

FOC is �� � � � � d π j 1 1 1 1 � � 1 � α � 1 1 1 � α L = 0 1 � α = 1 � P + P ( A α ) j j 1 � α 1 � α dP j , ! this simpli…es to � � � � 1 1 P � 1 1 � + = 0 j 1 � α 1 � α , ! multiply through by 1 � α and solve for P j P j = 1 α > 1 , ! monopoly price is a constant mark-up over marginal cost Quantity of each intermediate j used in …nal goods industry i : � A α 2 � 1 1 � α L i X ij = () Endogenous Techological Change October 2007 9 / 19

Final output in industry i � A α 2 � α 1 � α L α AL 1 � α Y i = N i i 1 2 α 1 � α α 1 � α NL i = A Aggregate output: 1 2 α 1 � α α 1 � α LN ( t ) Y ( t ) = A , ! output must grow at the same rate as N Pro…ts are � 1 � α � � A α 2 � 1 1 � α L π j = π = α Present discounted value of pro…ts � 1 � α � � Z ∞ A α 2 � 1 1 � α L V ( t ) = D ( s ) ds α t () Endogenous Techological Change October 2007 10 / 19

R&D Sector The cost of producing one new idea = η units of output Free Entry condition: V ( t ) � η If V ( t ) < η ) no resources allocated to R&D. We focus on case where V ( t ) = η . From our asset pricing equation ˙ π V ( t ) r ( t ) = V ( t ) + V ( t ) , ! it follows that � 1 � α � � A α 2 � r ( t ) = r = 1 1 1 � α L η α () Endogenous Techological Change October 2007 11 / 19

Households As usual, their optimal decision is characterized by ˙ c ( t ) ˙ C ( t ) C ( t ) = r � ρ c ( t ) = = γ θ where C ( t ) = c ( t ) L . Dynamic budget constraint: z ( t ) = w ( t ) + rz ( t ) � c ( t ) ˙ where z ( t ) = assets . () Endogenous Techological Change October 2007 12 / 19

In aggregate ˙ Z = wL + rZ � C Total assets = market value of …rms Z ( t ) = V ( t ) N ( t ) = η N ( t ) It follows that Z = η ˙ ˙ N and so η ˙ N = wL + r η N � C , ! divide through by N and note that w ( t ) L = ( 1 � α ) Y ( t ) ˙ N N = ( 1 � α ) Y ( t ) + r η � C ( t ) η N ( t ) N ( t ) () Endogenous Techological Change October 2007 13 / 19

Along a BGP, ˙ N / N is constant. Since Y / N and r are constant: ˙ ˙ ˙ Y N C Y = N = C = g It follows that output growth is given by � 1 � 1 � α � � � A α 2 � γ � = 1 1 1 � α L � ρ θ η α Note that the transversality condition requires that r > γ � () Endogenous Techological Change October 2007 14 / 19

Implications A , θ and ρ a¤ect growth in the same way as in the AK model A decrease in the unit cost of R&D, η , raises the rate of return r and thereby raises the growth rate Scale e¤ect : larger the labour endowment ) higher growth rate , ! demand: L " ) larger “market” for ideas ) more incentive to invent , ! supply: L " ) opportunity cost of invention is lower Cross-country, post WWII data ) no scale e¤ect (Jones, 1999) Long term, regional data ) evidence of scale e¤ect (Kremer, 1993) More recent extensions ) endogenous technical change without scale e¤ects () Endogenous Techological Change October 2007 15 / 19

Comparison to the Social Optimum Because of imperfect competition, the decentralized equilibrium is not Pareto optimal. Consider a a hypothetical social planner that chooses aggregates to maximize household utility. Output is either consumed, used to produce intermediates or used in R&D: Y = AL 1 � α N 1 � α X α = C + X + η ˙ N and so � � N = 1 AL 1 � α N 1 � α X α � cL � X ˙ η The Hamiltonian for the planner’s problem is J = e � ρ t c 1 � θ � � 1 � θ + λ 1 AL 1 � α N 1 � α X α � cL � X η , ! here there are 2 control variables, c and X and 1 state variable, N . () Endogenous Techological Change October 2007 16 / 19

The Hamiltonian conditions are e � ρ t c � θ � λ dJ = η L = 0 (1) dc � � = 0 dJ λ 1 α AL 1 � α N 1 � α X α � 1 � 1 = (2) dX η dJ λ 1 η ( 1 � α ) AL 1 � α N � α X α = � ˙ = λ (3) dN , ! as usual, from (1) we have ˙ � ρ � θ ˙ c λ c = λ , ! condition (2) implies X 1 1 � α L N = ( α A ) () Endogenous Techological Change October 2007 17 / 19

It follows that (3) can be written ˙ 1 λ α 1 � α = � η ( 1 � α ) A ( α A ) λ ! substituting out ˙ , λ / λ we get ρ + θ ˙ c c = 1 1 α 1 � α α η ( 1 � α ) A 1 � α It follows that the optimal growth rate of consumption is � 1 � 1 � α � � γ P = 1 1 1 � α L � ρ ( A α ) θ η α Recall that � 1 � 1 � α � � � A α 2 � γ � = 1 1 1 � α L � ρ θ η α , ! since α < 1, γ P > γ � () Endogenous Techological Change October 2007 18 / 19

Implications The decentralized growth rate is too low relative to the social optimum , ! output of monopolists is too low ) reduces return to innovation � γ P � γ � � " Note that α # ) monopoly mark-up 1 / α " and ) monopoly pro…ts are necessary for growth, but greater monopoly power is not optimal Could achieve social optimum by subsidizing purchases of intermediates () Endogenous Techological Change October 2007 19 / 19