Through Scarcity to Prosperity and Beyond: A Theory of the Transition to Sustainable Growth Pietro F. Peretto Duke University March 2013 Peretto (Duke University) Transition to sustainable growth March 2013 1 / 30
The literature: Two seemingly disconnected ideas Resource economics emphasizes the role of exhaustible resources in generating diminishing returns to other inputs that worsen over time as resources run out. Two classic questions: Is level of consumption per capita sustainable? (Solow 1974) Is growth of consumption per capita sustainable? (Stiglitz 1974) Growth economics emphasizes the role of land, a non-exhaustible resource, in generating diminishing returns to labor that allow construction of Matlhusian equilibria (Galor-Weil 1996, 2000, Lucas 2002, Galor 2005, 2011). It then asks: How did interaction of endogenous technological change and fertility choice drive escape from Malthusian trap of low consumption per capita? transition to sustained growth of consumption per capita? Peretto (Duke University) Transition to sustainable growth March 2013 2 / 30
Are these two ideas connected? Resource economics studies the future behavior of economy under increasing scarcity. As we run out of natural inputs, and diminishing returns to man-made inputs reduce output per capita, how can we sustain our current — and growing — standards of living? Growth economics studies the past transition to sustained growth experienced by advanced economies. It emphasizes population-technology interactions but ignores exhaustible resources, perhaps because modeling escape from the Malthusian regime requires that sustained growth be feasible in the …rst place. So, the two ideas are two sides of the same coin and provide the foundation of a general theory of the interactions of population, resources and technology. Peretto (Duke University) Transition to sustainable growth March 2013 3 / 30
This paper Integrates fertility choice and exhaustible resource dynamics in model of endogenous innovation to develop a theory of the transition from resource-based to knowledge-based growth . Initial phase where agents build up the economy by exploiting exhaustible natural resources to support population growth. Intermediate phase where agents turn on Schumpeterian engine of innovation-led growth in response to market expansion. Terminal phase where growth becomes driven by knowledge accumulation and no longer requires growth of physical inputs. Last part is crucial: not only economics growth no longer requires growth of physical inputs, but also knowledge accumulation compensates for the exhaustion of the natural resource. The paper thus proposes a theory of de-coupling. Peretto (Duke University) Transition to sustainable growth March 2013 4 / 30
A Schumpeterian model with exhaustible resources Final producers: Homogeneous good that is consumed, used to produce intermediate goods, or invested in R&D. (One-sector structure.) This good is the numeraire , so P Y � 1. Intermediate producers: Develop new goods and set up operations to serve market (variety innovation or entry) and, when already in operation, invest in R&D internal to …rm (quality innovation). Households: Consume, save and set optimal path of population growth and resource use. For simplicity, they play the role of the extraction sector. (Alternative assumptions are feasible, of course.) Peretto (Duke University) Transition to sustainable growth March 2013 5 / 30
Representative …nal producer (i) Technology: Z N Z N � i Z 1 � α L γ R 1 � γ � 1 � θ di , 1 Y = N ( σ � 1 )( 1 � θ ) X θ Z α Z � N Z j dj . i 0 0 where: 0 < θ , γ < 1 standard parameters that map into factor shares; i Z 1 � α vertical technology index, with α 2 [ 0 , 1 ) measure of private Z α returns to quality and 1 � α measure of social returns to quality; N is horizontal technology index, with σ 2 [ 0 , 1 ) measure of social returns to variety (love-of-variety e¤ect in production). Peretto (Duke University) Transition to sustainable growth March 2013 6 / 30
Representative …nal producer (ii) Demand for product i : � θ � 1 1 � θ X i = N σ � 1 Z α i Z 1 � α L γ R 1 � γ . P i Factor payments: Z N N � PX = P i X i di = θ Y ; 0 wL = γ ( 1 � θ ) Y ; pR = ( 1 � γ ) ( 1 � θ ) Y . Peretto (Duke University) Transition to sustainable growth March 2013 7 / 30
Intermediate producers Technologies: � i Z 1 � α � X i + φ Z α Cost i = 1 � ; ˙ Z i = I i . Firm’s objective: Z ∞ e � R t 0 r ( s ) ds � � i ( t ) Z 1 � α ( t ) � I i ( t ) X i ( t ) ( P i ( t ) � 1 ) � φ Z α V i ( 0 ) = dt . 0 In symmetric equilibrium : ! � 1 � X θ � 1 max P i , I i V i ) r = α � φ � r Z ; Z � 1 � � φ Z � I ˙ ˙ N ) r = X θ � 1 = β Y Y N V max + Y � N � r N . i β Y N Peretto (Duke University) Transition to sustainable growth March 2013 8 / 30
Representative household Chooses C ( t ) , L ( t ) , R ( t ) and b ( t ) to maximize � � C ( t ) � � Z ∞ M ( t ) M η + 1 ( t ) e � ρ t U 0 = + f ( b ( t )) ρ , η > 0 , log dt , 0 subject to: ˙ A = rA + wL + pR � C � Ψ bM , M � L � 0 , ψ > 0 ; ˙ M = M ( b � δ ) , M 0 > 0 , δ > 0 ; Z ∞ ˙ S 0 � R ( t ) dt , R � 0 , S 0 > 0 , S = � R . 0 What’s new here? M evolves according to fertility choice b ; preference for b increasing and (weakly) concave, i.e, f 0 > 0, f 00 � 0. S is stock of exhaustible resource that evolves according to extraction choice R (for simplicity, extraction cost is zero). Peretto (Duke University) Transition to sustainable growth March 2013 9 / 30
Household behavior (i) First-order conditions for control variables C , b , R : f 0 ( b ) + λ M M = λ A Ψ M ; 1 = λ A C ; λ A p = λ S , where the λ s denote the shadow values of A , M and S ; for state variables A , M , S : ˙ λ A r + = ρ ; λ A � ˙ � η ˙ M + λ A ( w � Ψ b ) M λ M + + = ρ ; λ M λ M M ˙ λ S = ρ . λ S Plus usual transversality conditions. Peretto (Duke University) Transition to sustainable growth March 2013 10 / 30
Household behavior (ii) Let c � C / Y and h � λ M M . Conditions for C and A yield Euler equation ˙ ˙ C C = ρ + ˙ c Y r = ρ + c + Y . To simplify, cost of reproduction in units of the …nal good (taken as given since depends on aggregate variables that household does not control): Ψ = ψ Y M . Peretto (Duke University) Transition to sustainable growth March 2013 11 / 30
Household behavior (iii) Conditions for C , b , A and M yield fertility rule f 0 ( b ) + h = ψ c and asset-pricing equation � wM � h = ρ h � η � 1 ˙ � ψ b . c Y Conditions for C , R , S and the Euler equation yield Hotelling rule ˙ C = λ S ) ˙ p p C p = ρ + C = r . Peretto (Duke University) Transition to sustainable growth March 2013 12 / 30
Equilibrium extraction (i) Flow of resource supplied by household equals …nal sector demand pR = ( 1 � γ ) ( 1 � θ ) Y . Log-di¤erentiating and using Hotelling rule � ˙ � ˙ ˙ ˙ R Y Y � ˙ p Y c R = p = Y � r = � c + ρ . Integrating and de…ning average growth rate of extraction ‡ow between � � R t c ( s ) ˙ time 0 and time t as ε ( t ) � 1 c ( s ) + ρ ds yields t 0 R ( t ) = R 0 e � ε ( t ) t . Peretto (Duke University) Transition to sustainable growth March 2013 13 / 30
Equilibrium extraction (ii) Substituting into Z ∞ S 0 = R ( t ) dt 0 yields � Z ∞ � � 1 e � ε ( t ) t dt R 0 = � S 0 . 0 | {z } constant that depends on fundamentals Therefore, the extraction rule at time t is e � ε ( t ) t R ∞ R ( t ) = 0 e � ε ( t ) t dt � S 0 . Peretto (Duke University) Transition to sustainable growth March 2013 14 / 30
Technology and resources: output per capita dynamics In symmetric equilibrium aggregate output is 2 θ Y = κ N σ ZM γ R 1 � γ , κ � θ 1 � θ where N σ Z is TFP. Output per capita is � R � 1 � γ y � Y M = κ N σ Z . M Let n � ˙ N / N , z � ˙ Z / Z and g � ˙ y / y . At time t , output per capita growth rate is: g = σ n + z � ( 1 � γ ) ( m + ˙ c / c + ρ ) . | {z } | {z } TFP growth growth drag Peretto (Duke University) Transition to sustainable growth March 2013 15 / 30
Transition dynamics: di¤erential equation for …rm size Final producer pays total compensation N � PX = θ Y to intermediate producers who set P = 1 / θ . Hence, NX = θ 2 Y . Let µ � P � 1 and x � X / Z . Reduced-form production function yields x = θ 2 Y / NZ = θ 2 κ M γ R 1 � γ / N 1 � σ . Returns to innovation become: r = α ( µ x � φ ) ; � � 1 µ � φ + z + ˙ x r = x + z . βθ 2 x Note: …rm-level decisions depend on quality adjusted …rm size x , which follows di¤erential equation ˙ x ˙ Y x = Y � n � z = γ m � ( 1 � γ ) ( ˙ c / c + ρ ) � ( 1 � σ ) n . | {z } | {z } market growth market fragmentation Peretto (Duke University) Transition to sustainable growth March 2013 16 / 30
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