Structural change in a multi-sector model of the climate and the economy Gustav Engstr¨ om The Beijer Institute of Environmental Economics Stockholm, December 2012 G. Engstr¨ om (Beijer) Stockholm, December 2012 1 / 26
Motivation (Sterner and Persson (2008)) G. Engstr¨ om (Beijer) Stockholm, December 2012 2 / 26
Moving forward The results in Sterner and Persson (2008) are dramatic and show how relative prices can play a role for optimal mitigation policies! Has received considerable attention from the research community (178 cites according to Google scholar). How can their results be applied in practice when constructing IAMs? What is the environmental good? How do we measure it and how does it relate to other commodities in the economy? What about the role of adaptation? What to do? Why not develop a multi-sector macroeconomic growth model where each sector is impacted differently by climate change and where resources in the economy can be transfered to counteract the heterogeneous impacts? G. Engstr¨ om (Beijer) Stockholm, December 2012 3 / 26
Outline This paper develops a multi-sector growth model of the climate-economy interaction featuring i) endogenous saving ii) emissions from fossil fuel use in production and iii) inter-sectoral factor allocation decisions and iv) allow each sector to be impacted differently by climate change. The purpose of this exercise is to explore (as in Sterner and Persson) the role of relative prices for optimal mitigation policies, but within the context of a multi-sector growth model which allows for calibration to currently available data. By allowing for reallocation of input factors across sectors over time this approach also introduces adaptation into the climate-economy interaction, which reveals a direct relationship between adaptation and optimal fossil fuel taxes. G. Engstr¨ om (Beijer) Stockholm, December 2012 4 / 26
Structural Change Structural change refers to the reallocation of production factors (typically labor) across different sectors of the economy over time. During the process of development, strong structural change takes place with movements of labor and other resources from agriculture to manufacturing and then to services ( Kuznets Facts ). On the aggregate level growth rate of per-capita output, real interest rate, capital-output ratio and the labor income share has remained fairly constant ( Kaldor facts ). Challenge to theory has been to reconcile the non-balanced characteristics of growth at the sector level with balanced picture of growth at the aggregate level. G. Engstr¨ om (Beijer) Stockholm, December 2012 5 / 26
Multi-sector growth models reconciling Kuznets and Kaldor facts of growth Kongsamut, Rebelo and Xie (2001) Structural change is driven by non-homothetic preference which leads to differences in income elasticity of demand across goods. Engel’s law: as a household’s income increases, fraction that it spends on food (agricultural products) declines. Ngai and Pissarides (2007) Structural change is driven by differences in technological growth rates. Acemoglu and Guerrieri (2008) Differences in capital intensities as a driver of structural change through capital deepening. G. Engstr¨ om (Beijer) Stockholm, December 2012 6 / 26
Model description Description of the economy The social planning problem Static efficiency Dynamic efficiency The competitive equilibrium Numerical calibration and simulations Conclude G. Engstr¨ om (Beijer) Stockholm, December 2012 7 / 26
Description of an n -sector economy Representative households preferences are given by ∞ � β t U ( C t ) (1) t =0 The economy produces a unique final good which can be thought of as an aggregate/composite good consisting of the n intermediaries � n � ǫ/ ( ǫ − 1) n w i Y ( ǫ − 1) /ǫ � � Y t = ; w i = 1 (2) i , t i =1 i =1 Production in sector i Y it = Ω i ( S t ) A i , t K α 1 i , t L α 2 i , t E α 3 i , t ; (3) ∀ i Factor inputs can be allocated free of charge across all sectors n n n � � � K t = K i , t ; L t = L i , t ; E t = (4) E i , t i =1 i =1 i =1 G. Engstr¨ om (Beijer) Stockholm, December 2012 8 / 26
Description of an n -sector economy The economy’s aggregate budget constraint is given by K t +1 + C t = Y t + (1 − δ ) K t (5) Fossil fuel use E t of a finite stock R t is governed by ∞ � R t +1 − R t = − E t (6) R 0 ≥ E t ⇒ t =0 Human induced carbon dioxide accumulates in the atmosphere according to S t +1 = (1 − ϕ ) S t + ξ E t (7) G. Engstr¨ om (Beijer) Stockholm, December 2012 9 / 26
Planning problem ∞ � β t U ( C t ) max { K t +1 , R t +1 , S t +1 E t , C t , { K i , t , L i , t , E i , t } ∀ i } t =0 subject to K t +1 = Y t − C t + (1 − δ ) K t S t +1 = (1 − ϕ ) S t + ξ E t R t +1 = R t − E t n n n � � � K t = K i , t ; L t = L i , t ; E t = E i , t i =1 i =1 i =1 where � n � ǫ/ ( ǫ − 1) w i Y ( ǫ − 1) /ǫ Y i , t = Ω i ( S t ) A i , t K α 1 i , t L α 2 i , t E α 3 � Y t = ; i , t ; ∀ i i , t i =1 G. Engstr¨ om (Beijer) Stockholm, December 2012 10 / 26
Static efficiency From the planing problem the following static efficiency conditions follow = ∂ Y j , t /∂ Y i , t = ∂ Y j , t /∂ Y i , t = ∂ Y j , t /∂ Y i , t p i , t ; ∀ i , j (8) ∂ K j , t ∂ K i , t ∂ L j , t ∂ L i , t ∂ E j , t ∂ E i , t p j , t 1 ǫ denotes the price of good i in the decentralized where p i , t ≡ w i ( Y t / Y it ) market equilibrium. The following proposition follows Proposition 1 Given equal factor income shares across sectors and constant returns to scale production functions the intra temporal resource allocation at time t is determined by � ǫ � Ω i ( S t ) � ǫ − 1 � w i K i , t = L i , t = E i , t A i , t = ≡ Ψ i , j ( S t ) , ∀ i , j Ω j ( S t ) K j , t L j , t E j , t w j A j , t G. Engstr¨ om (Beijer) Stockholm, December 2012 11 / 26
Static efficiency By proposition 1 we can also show that relative factor inputs shares also equal the expenditure share of production in sector i relative to sector j . � ǫ � Ω i ( S t ) � ǫ − 1 � w i p i , t Y i , t A i , t = = Ψ i , j ( S t ) , ∀ i , j (9) Ω j ( S t ) p j , t Y j , t w j A j , t This corresponds exactly to condition (10) of Ngai and Pissarides (2007) which in their model corresponds to the ratio of consumption expenditure in good i relative to sector m . Relative prices between sector i and j will be determined by p i , t = Ω j ( S t ) A j , t , ∀ i , j (10) Ω i ( S t ) p j , t A i , t G. Engstr¨ om (Beijer) Stockholm, December 2012 12 / 26
Static efficiency Define, ˜ A i , t ≡ Ω i ( S t ) A i , t as the climate externality adjusted TFP growth rate. Then by proposition 1, taking the logs of both sides and differencing � � ˜ ˜ � L i , t +1 � � L j , t +1 � A j , t +1 A i , t +1 ln − ln = (1 − ǫ ) ln − ln , ∀ i , j (11) ˜ ˜ L i , t L j , t A j , t A i , t which gives us Proposition 2 Necessary and sufficient conditions for structural change are that ǫ � = 1 and that ln(˜ A j , t +1 / ˜ A j , t ) � = ln(˜ A i , t +1 / ˜ A i , t ) for some i. Let the climate adjusted TFP growth rate be smaller in sector i compared to sector j when ǫ < 1 , or alternatively let i have a larger climate adjusted TFP growth rate when ǫ > 1 . In both case this implies that sector i expands faster over time relative to sector j and that prices of good i increase at a faster pace relative to good j. G. Engstr¨ om (Beijer) Stockholm, December 2012 13 / 26
Static efficiency Finally by proposition 1 get the maximized value of current output ˜ Y t given the capital, fossil fuel and carbon dioxide stock at time t Proposition 3 Based on proposition 1 and the market clearing conditions the composite production function, or final good, can be written as Y t = Γ t ( S t ) K α 1 ˜ t L α 2 t E α 3 (12) t with � n ǫ ǫ − 1 1 � n ǫ − 1 � ˆ � Γ t ( S t ) ≡ w i (Ω i ( S t ) A it Ψ i , j ( S t )) ; Ψ j ≡ Ψ i , j (13) ǫ ˆ Ψ j i =1 i =1 G. Engstr¨ om (Beijer) Stockholm, December 2012 14 / 26
Dynamic efficiency The static solution simplifies the planning problem. ∞ � β t U ( C t ) max { C t , K t +1 , R t +1 , E t , S t +1 } t =0 K t +1 = ˜ s.t Y t − C t + (1 − δ ) K t ; (14) R t +1 − R t = E t ; S t +1 = (1 − ϕ ) S t + ξ E t with R 0 ; K 0 ; S 0 ; R t ≥ 0 First order conditions w.r.t C t , K t +1 ˜ U ′ ( C t ) Y t +1 β U ′ ( C t +1 ) = α 1 + (1 − δ ) (15) K t +1 G. Engstr¨ om (Beijer) Stockholm, December 2012 15 / 26
Dynamic efficiency The necessary optimality conditions w.r.t. S t +1 implies ˜ λ S , t +1 Y t +1 α 3 E t +1 + ξ U ′ ( C t ) U ′ ( C t +1 ) β U ′ ( C t +1 ) = (16) ˜ λ S , t Y t α 3 E t + ξ U ′ ( C t ) which denotes the present value of the marginal damages (Lagrangian multiplier of S t ) F.o.c. for R t +1 and E t results in ∞ ˜ (1 − ϕ ) s − 1 β s U ′ ( C t + s ) ∂ Γ t + s Y t + s � λ S , t = (17) ∂ S t + s Γ t + s s =1 which is an externality adjusted Hotelling type formula where λ S , t G. Engstr¨ om (Beijer) Stockholm, December 2012 16 / 26
Recommend
More recommend
