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The Environment and Directed Technical Change

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The Environment and Directed Technical Change VOL. VOL NO. ISSUE SHORT TITLE FOR RUNNING HEAD B-1 The Environment and Directed Technical Change By DARON ACEMOGLU, PHILIPPE AGHION, LEONARDO BURSZTYN, AND DAVID HEMOUS Web Appendix APPENDIX B: OMITTED PROOFS AND FURTHER DETAILS (NOT FOR PUBLICATION) B1. Allocation of scientists in laissez-faire equilibrium when the inputs are complementary (" < 1) Under Assumption 1 and if " < 1, there is a unique equilibrium in laissez-faire where innovation first occurs in the clean sector, then occurs in both sectors, and asymptotically the share of scientists devoted to the clean sector is given by sc / the D d c C d I long-run growth rate of dirty input production in this case is , where / . c d c d This proposition is proved using the following lemma: C When " < 1, long-run equilibrium innovation will bee in both sectors,e so that the equilibrium share of scientists in the clean sector converges to sc /(η /. D d c C d Suppose that at time t innovation occurred in both sectors so that 5ct /5dt 1. Then from (18), we have D ' 1 5ct 1 1 csct 1 1 csct C C C C . 5dt 1 D 1 d sdt 1 1 d sdt C C C C Innovation will therefore occur in both sectors at time t 1 whenever the equilibrium C allocation of scientists .sct 1, sdt 1/ at time t 1 is such that C C C 1 ' 1 1 csct 1 1 csct C (B.1) C C C . 1 d sdt 1 D 1 d sdt C C C This equation defines sct 1. 1 sdt 1/ as a function of sct . 1 sdt /. We next claim that C D C D this equation has an interior solution sct 1 .0, 1/ when sct .0, 1/ (i.e., when sct is itself C 2 2 1=.' 1/ interior). First, note that when ' > 0 (that is, " < 1), the function z.x/ x C x is strictly decreasing for x < 1 and strictly increasing for x > 1. Therefore,D x 1 is the unique positive solution to z.x/ 0. Second, note also that the function D D 1 csct 1 csct X.sct / C C , D 1 sdt D 1 .1 sct / C d C d 1 is a one-to-one mapping from .0, 1/ onto . 1 d , 1 c/. Finally, it can be veri- 1 C C 1=.' 1/ 1 fied that whenever X . 1 , 1 /, we also have X C . 1 , 1 2 C d C c 2 C d C B-2 THE AMERICAN ECONOMIC REVIEW MONTH YEAR 1 1=.' 1/ c/. This, together with (B.1), implies that if sct .0, 1/, then sct 1 X .X.sct / C / .0, 1/, proving the claim at the beginning of this paragraph.2 C D 2 From Appendix A, when ' > 0, the equilibrium allocation of scientists is unique at each t. Thus as t , this allocation must converge to the unique fixed point of the !1 1 1 function Z.s/ X .X .s// ' 1 , which is D C d sc . D c C d This completes the proof of the lemma. Now given the characterization of the equilibrium allocations of scientists in Appendix A, under Assumption 1 the equilibrium involves sdt 0 and sct 1, i.e., innovation oc- D D curs initially in the clean sector only. >From (11), Act /Adt will grow at a rate c, and in .' 1/=' 1=' finite time, it will exceed the threshold 1 c C c/ηd . Lemma B.B1 im- C .' 1/=' 1=' 1=' .' 1/=' 1=' plies that when this ratio is in the interval 1 C /η , /η 1 C /η , C c c d c d C d c d equilibrium innovation occurs in both sectors, i.e., s > 0 and s > 0, and from this dt ct point onwards, innovation will occur in both sectors and the share of scientists devoted to the clean sector converges to d /(ηd c/. This completes the proof of Proposition B.B1. C B2. Speed of disaster in laissez-faire >From the expressions in (19), dirty input production is given by: ' ' ' C ' Adt ' Ydt Act Adt ActC Adt ' . D C D ' C' 1 Adt C Act When the two inputs are gross substitutes (" < 1), we have ' 'su < 0, whereas when they are complements (" > 1), we have ' 'co > 0 . Since allD innovations occur in the dirty sector in the substitutability case, butD not in the complementarity case, if we start su co with the same levels of technologies in both cases, at any time t > 0 we have Adt > Adt su co su co and Act < Act , where Akt and Akt denote the average productivities in sector k at time t respectively in the substitutability and in the complementarity case, starting from the same initial productivities Asu Aco. k0 D k0 Assumption 1 implies that su co Asu ' Aco ' dt < d dt Asu ≤ Aco ct c ct VOL. VOL NO. ISSUE SHORT TITLE FOR RUNNING HEAD B-3 so that su su su su Su ' ' su Adt Act Adt Y > co 1 dt su su 1 ' su D Su ' ' C Aco C Act Adt ct ! 1 su 1 Aco Act C dt C su su Act Act co > > > Y 'co c0 1 dt. Aco co ' 'c0 C ct Act 1 Aco 1 co C dt Adt C su co Repeating the same argument for t 1, t 2,..., we have that Ydt > Ydt for all t. This establishes that, under AssumptionC 1, thereC will be a greater amount of dirty input production for each t when " > 1 than when " < 1, implying that an environmental disaster will occur sooner when the two sectors are gross substitutes. B3. Proof of Proposition 4 Using the fact that the term • .St / premultiplies all A’s, equation (19) is now be re- placed by: 1 ' 1 1 ' ' C' ' ' ' ' Ydt • .St / 1 A A A C Adt , and Yt • .St / 1 A A Act Adt . D ct C dt ct D ct C dt In particular, as in Section II, under laissez-faire, all innovation is directed towards the dirty sector, Adt grows to infinity. Then, an environmental disaster can only be avoided if 1/.1 / Ydt and thus • .St / Adt remain bounded. Since Adt is growing exponentially, this 1/.1 / is only possible if St converges to 0. Now, suppose that • .St / Adt converges to a finite value as time t goes to infinity. Then there exists > 0 such that for any T there 1/.1 / exists v > T such that • .Sv/ Ad > /ξ. But for v > T sufficiently high, we also have 1/.1 / 1/.1 / Yd • .S / Ad < / .3/ since, asymptotically, Ydt • .St / Adt , and v v v ' .1 / Sv < /3 as St converges to 0. But then (12) gives Sv 1 0, which corresponds C C D to an environmental disaster. Consequently, to avoid a disaster under laissez-faire, it 1/.1 / must be the case that • .St / Adt converges to 0. But this implies that Yt converges to 0 as well, and so does Ct . B4. Proof of Proposition 6 First we need to derive the optimal production of inputs given technologies and the tax implemented. Using (A.9) and (A.10), the shadow values of clean and dirty inputs satisfy 1 " 1 " (B.2) p .pdt .1 t // 1. ct C C D b b B-4 THE AMERICAN ECONOMIC REVIEW MONTH YEAR This, together with (A.15), yields 1 1 Act Adt (B.3) pdt 1 and pct 1 . D ' 1 " ' 1 " D ' 1 " ' 1 " A .1 t / A A .1 t / A ct C C dt ct C C dt b b Using (7), (A.12), (A.16) and (B.3), we obtain 1 " ' .1 t / Act AdtC (B.4) Yct C , and D ' 1 " ' ' ' " ' A .1 t / A A .1 t / A dt C C ct ct C C dt 1 ' ActC Adt (B.5) Ydt . D ' 1 " ' ' ' " ' A .1 t / A A .1 t / A dt C C ct ct C C dt Equation (B.5) implies that the production of dirty input is decreasing in t . Moreover, clearly as t , we have Ydt 0. ! 1 ! We next characterize the behavior of this tax rate and the research subsidy, qt . Recall that to avoid an environmental disaster, the optimal policy must always ensure that Ydt remained bounded, in particular, Ydt .1 / S/ξ. ≤ C Assume " > 1. The proof consists of six parts: (1) We show that, for a discount rate sufficiently low, the optimal allocation cannot feature a bounded Yct , thus Yct must be unbounded as t goes to infinity. (2) We show that this implies that Act must tend towards infinity. (3) We show that if the optimal allocation involves Yct unbounded (i.e lim sup Yct ), then it must be the case that at the optimum Yct as t goes to D 1 ! 1 infinity. (4) We prove that the economy switches towards clean research, that is, sct 1. (5) We prove that the switch in research to clean technologies occurs in finite time,! that is, there exists T such that sct 1 for all t T . (6) We then derive the implied behavior Q D ≥ Q of t and qt . Part 1: To obtain a contradiction, suppose that the optimal allocation features Yct re- maining bounded as t goes to infinity. If Ydt was unbounded, then there would be an environmental disaster, but then the allocation could not be optimal in view of the as- sumption that limS 0 u.C, S/ (equation (2)).
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