Optimal policy in OG models

Ghiglino, Christian; Tvede, Mich Working Paper Optimal policy in OG models Working paper, No. 8-98 Provided in Cooperation with: Department of Economics, Copenhagen Business School (CBS) Suggested Citation: Ghiglino, Christian; Tvede, Mich (1998) : Optimal policy in OG models, Working paper, No. 8-98, Copenhagen Business School (CBS), Department of Economics, Frederiksberg, https://hdl.handle.net/10398/7548 This Version is available at: https://hdl.handle.net/10419/208401 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/ Institut for Nationaløkonomi Handelshøjskolen i København Working paper 8-98 OPTIMAL POLICY in OG MODELS Christian Ghiglino Mich Tvede Department of Economics - Copenhagen Business School Nansensgade 19, 5. DK - 1366 København K. Optimal Policy in OG Models¤ Christian Ghiglino Copenhagen Business School Mich Tvede University of Copenhagen Abstract In the present paper general stationary overlapping generations economies with many commodities in every period and many different consumers in every generation are considered. A government maximizes an utilitarian social welfare function, that is the sum of weighted averages of utilities for generations, through ¯scal policy, i.e. monetary transfers and taxes. Both situations with and without time discounting are considered. It is shown that if the discount factor is su±ciently close to one then the optimal policy stabilizes the economy, i.e. the equilibrium path has the turnpike property. Moreover the ¯scal policy is shown to be time-consistent. Keywords: Overlapping Generations Economies, Economic Policy, Turnpike Property, Discounting. JEL-classi¯cation: D51, D91, E32. Correspondence: Christian Ghiglino, Department of Economics, Copenhagen Business School, Nansensgade 19, 5. DK-1366 Kobenhavn K ¤ The authors are grateful to an Associate Editor and an anonymous referee for helpful and constructive comments. We also thank David Cass, Karl Shell, Bruce Smith and Henry Wan for helpful comments. 1 1Introduction In the heart of any study of normative aspects of economic policy is the discussion of the choice of a welfare criterion. In the early work of Diamond (see [8]) the criterion of Pareto optimality is applied, but this criterion is not very potent from the perspective of economic policy because the set of Pareto optima typically is quite large and contains allocations that treats consumers very di®erently with respect to welfare. However as noted by Samuelson in [16] the normative aspects of economic policy should rather be analyzed in terms of an utilitarian social welfare function. With utilitarian social welfare functions a unique equilibrium path is typically selected allowing discussions of the dynamic properties as well as implementation of this equilibrium path. In [16] Samuelson addressed questions with relation to stationarity of equilibrium paths in simple stationary, one-commodity, one-consumer overlapping-generations economies. He showed that the equilibrium path associated with maximization of an utilitarian social welfare function that is the discounted sum of utilities typically di®ers from the Pareto optimal equilibrium reached under laissez faire so implementation of the equilibrium path demands an active government. More recently Calvo and Obstfeld (see [7]) considered a continuous time version of Diamond's stationary overlapping generations economies with production where there are two commodities every period (consumption good/capital and labour) and one consumer in every generation and this consumer has time separable preferences. Calvo and Obstfeld considered the same social welfare function as Samuelson did and their results are compatible with earlier ¯ndings and in particular they show that equilibrium paths associated with maximization of their social welfare function has the turnpike property. In the present paper general stationary overlapping generations economies with long-lived consumers and production are considered. Moreover the government is supposed to maximize an utilitarian social welfare function, that is a sum of generations' social welfare functions where these are weighted sums of the utility functions of the consumers in the generations, through ¯scal policy, i.e. monetary transfers. However the government has to respect behavior of consumers as well as ¯rms and market clearing, i.e. the government can only in°uence agents through monetary transfers. Questions concerning dynamic properties such as dynamic consistency, the turnpike property and continuity of ¯scal policy with regard to the state of the 2 economy, and implementation of the equilibrium path are studied. In the present analysis both situations with and without time discounting by the government are considered. Indeed, an important feature of the overlapping generation model is the existence of equilibria in which future consumption continues to plays a role, i.e. the Samuelson case  . Clearly, the only way not to exclude these equilibria from the set of solutions to the government's problem is to consider the undiscounted case. Also no assumptions will be made with regard to time-separability of preferences because time-separability eliminates important sources of intertemporal links. For example, for pure exchange economies time-separability implies that the equilibrium path associated with maximization of the social welfare function is stationary which implies that all consumers consume the same because there are no intertemporal links. Furthermore, no restriction on the lifespan of consumers will be made, so that °uctuations with a period shorter than the lifespan are not excluded a priori . Concerning the turnpike property of equilibrium paths associated with maximization of utilitarian social welfare functions when the government considers feasibility constraints rather than maximization by agents and equilibrium conditions the answer is rather simple: If the discount factor is suf- ¯ciently close to one then the equilibrium has the turnpike property, i.e. it converges to a stationary allocation as time tends to in¯nity. This property of the equilibrium path contrasts the fact that equilibria can be quite complex for overlapping generations economies under laissez faire (see [4, 11, 12]). For the implementation part the answer is equally simple: Fiscal policy in terms of monetary transfers implements equilibria that are associated with maximization of utilitarian social welfare functions. Moreover dynamic consistency and continuity of the equilibrium path with respect to the state of the economy are usually obtained for free. It seems to be a common view that economic policy should be directed toward stabilization (see [13, 15]) even though °uctuations may be Pareto optimal as in [12]. The present paper supports this view, but the support is based on intergenerational justice through the utilitarian social welfare function rather than some sort of abstract \social cost" of °uctuations. The paper is structured as follows: In section 2 a stationary overlapping-  We thank the Associate Editor for pointing out this important fact. This remark is due to Karl Shell. 3 generations economy with an active government is presented; in section 3 the equilibrium associated with maximization of the social welfare function is characterized. Since the undiscounted case is not excluded from the analysis, the meaning of maximal is not straightforward. As in the growth literature, in those circumstances maximality is characterized through the Ramsey-Weizsacker overtaking criterion. Irrespective of the discounting, the characterization is obtained through 3 steps: Transformation of the overlapping generations economy into an optimal growth model; showing that the solution to the planner's problem in the optimal growth model can be implemented as equilibrium in the overlapping generations economy through monetary transfers and; application of a turnpike theorem to the planner's problem for the optimal growth model in order to show that the solution has the turnpike property. The application of the turnpike theorem is not straight forward because the transformation of the overlapping generations economy into an optimal growth model results in production technologies that exhibit decreasing returns to scale as well as constant returns to scale - a case that does not seem to be covered by the various turnpike theorems in the literature. On the one hand, when the future is discounted, the proof by Bewley in [6] can be applied with some modi¯cations. On the other hand, when the government does not discount the future, the proofs of Gale in [10] and McKenzie in [14] are used. Section 5 contains the proofs while some concluding remarks can be found in section 4. 2TheModel In the present paper stationary overlapping generations economies with production and ¯scal policies are considered. Time extends from zero to in¯nity with t2N[f 0g=N  . First the consumers and the ¯rms are introduced, second equilibria and steady states are de¯ned and thirdly the government is introduced. 2.1 Commodities, Consumers and Firms Let Lbe a set of lcommodities in every period. Moreover let L c ½Ldenote asetofl c consumption commodities, L o ½Ldenote a set of l o primary commodities and L p =LnL o denote a set of l p producible commodities. 4 Let Ibe a set of mconsumers in every generation. Consumers live for S periods except for the ¯rst S¡1 generations where the ¯rst generation lives for one period, the second generation lives for two periods,.., the S¡1'th generation lives for S¡1 periods and all subsequent generations live for Speriods. The consumers are described by their consumption sets, endowments, utility functions and shares in ¯rms. Hence the consumers are described by ((X i (t);! i (t);u i (t);µ i;j (t)) ¡ 1 t = ¡ S +1 ;(X i ;! i ;u i ) s 2 N  ); where ¡ 1 : t = ¡ S +1 : i 2 I µ i;j (t)=1 thus only \middle-aged" and \old" consumers are supposed to own shares at the start of the economy. For t2N 0 consumers are supposed to satisfy the following assumptions (A.1) Consumption sets, X i , are the nonnegative orthants of R Sl c ,i.e. X i =R Sl c + : (A.2) Endowments, ! i , are primary commodities, i.e. ! i 2R Sl o + : (A.3) Endowments of consumption commodities are positive, i.e. ! s;k i 2R ++ for some s2f 1;:::;Sgand some k2l c \l o . (A.4) Utility functions, u i , are twice di®erentiable with positive ¯rstorder derivatives and negative de¯nite Hessian matrix. For t2f¡ S+1;:::;¡1gconsumers are de¯ned as follows (A.1') X i (t)=R ( S + t ) l c + (A.2') ! i (t)=(! 1 ¡ t i ;:::;! S i ) (A.3') There exists x i (t)2R ¡ tl c + such that u i (t)isde¯nedby u i (t)(x)=u i (x i (t);x): 5 Hence the ¯rst S¡1 generations are truncations of subsequent generations. All assumptions are more or less standard within the di®erentiable framework. With some abuse of notation endowments and consumption plans are from time to time taken to be vectors in R Sl or R l rather than R Sl c or R l c . Let Jbe a set of n¯rms. Firms are in¯nitely lived and they are described by their production sets and stocks of output in period 0. Firms are supposed to satisfy the following assumptions (A.5) Production sets, Y j , allow transformations of some commodities in one period into producible commodities in the subsequent period, i.e. Y j ½R l  j ¡ £R l j + where L 1 j ½Lis a set of l 1 j commodities and L 2 j ½L p is a set of l 2 j commodities. (A.6) Zero production is possible, i.e. 02Y j : (A.7) Primary commodities are necessary for production, i.e. suppose that (y 1 ;y 2 )2Y j then y 1 ;k =0forallk2L o )y 2 =0: (A.8) There exist twice di®erentiable functions, f j :R l  j ¡ £R l j + , with positive ¯rst-order derivatives and positive de¯nite Hessian matrix on the orthogonal complement to Df j ,suchthat y2Y j ,f j (y)2R ¡ : As for the consumers all assumptions are more or less standard within the di®erentiable framework. With some abuse of notation production plans are from time to time taken to be vectors in R 2 l or R l rather than R l  j £R l j , R l  j or R l j . Total resources and the aggregate production set, 2 j 2 J Y j , are supposed to satisfy the following assumptions 6 (A.9) There is a positive amount of every primary commodity, i.e. != : i 2 I S : s =1 ! s i 2R l o ++ (A.10) There exist production plans, (y 1 j ;y 2 j ) j 2 J , such that all commodities are available, i.e. : i 2 I S : s =1 ! s i + : j 2 J (y 1 j +y 2 j )2R l ++ : Let p t 2R l + be commodity prices, q t 2R n + asset prices and p m;t money price in period t. The government controls the ¯scal policy, i.e. it makes lump-sum transfers and taxes to the consumers as well as to the ¯rms. Only ¯scal policies that are compatible with the price of money being positive are considered. Let (((¿ i;t ) i 2 I ) 1 t = ¡ S +1 ;((¾ j;t ) j 2 J ) 1 t =0 ) be a ¯scal policy, where ¿ i;t =(¿ s i;t ) S s =1 is the transfer to consumer iin generation tand ¾ j;t the transfer to ¯rm jin period t: Consumers maximize their utilities subject to their budget constraints. For t2N 0 the consumers solve the following problems max u i (x 1 ;:::;x S ) s.t. S : s =1 (p s + t ¡ 1 ¢(x s ¡! s i )¡p m;s + t ¡ 1 ¿ s i;s + t ¡ 1 ) + S ¡ 1 : s =1 µ s ¢(¼ s + t +q s + t ¡q s + t ¡ 1 )=0: where µ s 2R n is the portfolio and ¼ s + t 2R n is the net pro¯t of the ¯rms. For t2f¡ S+1;:::;¡1gthe consumers solve the following problems max u i (x i (s);x 1 ¡ t ;:::;x S ) s.t. S : s =1 ¡ t (p s + t ¡ 1 ¢(x s ¡! s i )¡p m;s + t ¡ 1 ¿ s i;s + t ¡ 1 ) + S ¡ 1 : s =1 ¡ t µ s ¢(¼ s + t +q s + t ¡q s + t ¡ 1 )=µ i (t)¢(¼ 0 +q 0 ): 7 It should be emphasized that the limit of optimal policies when the discount rate goes to zero coincides with the solution found for zero discounting (see [9] for a proof). This fact provides a way to obtain the optimal policy when there is no discounting which is sometimes easier than looking directly at the overtaking criterion. The focus of the paper is on the characterization of the global optimum of the planner. Of course whether or not it is implementable depends on the available information and instruments, so that the optimum may be in this sense infeasible. Indeed, if the government has incomplete information about the types of individual consumers, then ¯scal policies of the form considered in the present paper become infeasible. However, with a suitable notion of distance at hand, a similar analysis allows to select the best policy given the constraints. The present model can also be interpreted as an overlapping generation economy with bequests. In this case the welfare weights are endogenously determined by the market and the turnpike property concerns the competitive equilibria reached under laissez faire. The condition for convergence is then that consumers are su±ciently concerned by future generations. 5Proofs 5.1 Proof of Theorem 1 Consider ¯rst the planner's problem where the objective of the government is to maximize its social welfare function subject to feasibility conditions rather than equilibrium conditions. Then the planner's problem, denoted (PP), is to ¯nd a sequence that cannot be overtaken by any other sequence subject to the feasability conditions, i.e. ((± t : i 2 I ¸ i u i (x i (t);x i;t )) ¡ 1 t = ¡ S +1 ;(± t : i 2 I ¸ i u i (x i;t )) t 2 N  ) maximal s.t. 8 > > > > > > > > > < > > > > > > > > > : S X s =1 X i 2 I (x s i;t ¡! s i )= X j 2 J 2 X s =1 y s j;t f j (y 1 j;t ;y 2 j;t +1 )0 for all j K 0 is ¯xed: (PP) 14 Clearly the maximal value of the planner's problem is at least as high as the maximal value of the government's problem because the planner has more direct instruments at hand than the government The proof then relies on the transformation of the overlapping generations economy into an optimal growth model. This involves three steps: Transformation of the overlapping generations economy with Speriods of life into an economy with 2 periods of life (this is the transformation that was introduced in [2]); transformation of the overlapping generations economy with 2 periods of life into an optimal growth model and; introduction of arti¯cial ¯rms 3 . For the transformation of the economy with Speriods of life into an economy with 2 periods of life let the periods (S¡1)t; :::; (S¡1)(t+1)¡1be identi¯ed with period tfor all t2N 0 . Then there are (S¡1)lcommodities in every period, (S¡1)mconsumers in every generation and (S¡1)n¯rms. For i2f 1;:::;m glet ! 1 i =  * * ( ! 1 i . . . ! S ¡ 1 i  + + ) ; x 1 i;t =  * * ( x 1 i; ( S ¡ 1) t . . . x S ¡ 1 i; ( S ¡ 1)( t +1) ¡ 1  + + ) ; ! 2 i =  * * * * ( ! S i 0 . . . 0  + + + + ) ; x 2 i;t +1 =  * * * * ( x S i; ( S ¡ 1)( t +1) 0 . . . 0  + + + + ) ; and u i =¸ i u i . The variables related to all other consumers are de¯ned in a ! David Cass has informed the authors that uni¯ed models of overlapping generations economies and models of capital accumulation were used by L. Benveniste in the early eighties in an unpublished paper. 15 similar way. Then for i2f (S¡2)m+1;:::;(S¡1)mglet ! 1 i =  * * * * ( 0 . . . 0 ! 1 i ¡ ( S ¡ 2) m  + + + + ) ; x 1 i;t =  * * * * ( 0 . . . 0 x 1 i ¡ ( S ¡ 2) m; ( S ¡ 1)( t +1) ¡ 1  + + + + ) ; ! 2 i =  * * ( ! 2 i ¡ ( S ¡ 2) m . . . ! S i ¡ ( S ¡ 2) m  + + ) ; x 2 i;t +1 =  * * ( x 2 i ¡ ( S ¡ 2) m; ( S ¡ 1)( t +1) . . . x S i ¡ ( S ¡ 2) m; ( S ¡ 1)( t +2) ¡ 1  + + ) ; and u i =± S ¡ 2 ¸ i ¡ ( S ¡ 2) m u i ¡ ( S ¡ 2) m . For the ¯rst S¡1 generations ! 1 i =x 1 i;t =0 and ! 2 i and x 2 i;t is de¯ned as for all other generations and utility functions are de¯ned by v i =± s ¡ 1 ¸ i ¡ ( s ¡ 1) m u i ¡ ( s ¡ 1) m (x i ¡ ( s ¡ 1) m (S¡s);¢) for i2f (s¡1)m+1;:::;sm gand s2f 1;:::;S¡1g. For j2f 1;:::;n glet y 1 j;t =  * * * * ( y 1 j; ( S ¡ 1) t 0 . . . 0  + + + + ) ; y 2 j;t =  * * * * * * * ( 0 y 2 j; ( S ¡ 1) t +1 0 . . . 0  + + + + + + + ) ; with (y 1 j;t ; y 2 j;t )2Y j ,f j (y 1 j;t ; y 2 j;t )0 so production takes place within periods. The variables related to all other ¯rms up to i=(S¡2)nare de¯ned in a similar while for j2f (S¡2)n+ 1;:::;(S¡1)nglet y 1 j;t =  * * * * ( 0 . . . 0 y 1 j ¡ ( S ¡ 1) n +1 ; ( S ¡ 1)( t +1) ¡ 1  + + + + ) ; y 2 j;t =  * * * * ( y 2 j ¡ ( S ¡ 1) n +1 ; ( S ¡ 1) t 0 . . . 0  + + + + ) ; 16 with (y 1 j;t ; y 2 j;t +1 )2Y j ,f j (y 1 j;t ; y 2 j;t +1 )0 so that for these ¯rms production takes place between periods. Finally K t = ( S ¡ 1) n : j =( S ¡ 2) n +1 y 2 j;t is the stock of output in period t. For the transformation of the economy with 2 periods of life into an optimal growth model let the consumption of consumers in generation tbe identi¯ed with consumption of consumers in period t, but consumption in the ¯rst period of life are considered to be di®erent commodities. So there are (S¡1) 2 lm +(S¡1)lcommodities in every period, (S¡1)mconsumers and (S¡1)n¯rms. For i=1let ! i =  * * * * ( 0 . . . 0 ! 1 i +! 2 i  + + + + ) ; x i;t =  * * * * * * * ( x 1 i;t 0 . . . 0 x 2 i;t +1  + + + + + + + ) ; and u i =u i . The variables related to all other consumers are de¯ned in a similar way. Then for i=(S¡1)mlet ! i =  * * * * ( 0 . . . 0 ! 1 i +! 2 i  + + + + ) ; x i;t =  * * * * * * * ( 0 . . . 0 x 1 i;t x 2 i;t +1  + + + + + + + ) ; and u i =u i and v i =v i . For the ¯rms let y 1 j;t =  * * * * ( 0 . . . 0 y 1 j;t +1  + + + + ) ; y 2 j;t =  * * * * ( 0 . . . 0 y 2 j;t +1  + + + + ) ; 17 for all j2f 1;:::;(S¡1)ng. So for j2f 1;:::;(S¡2)ngproduction takes place within periods and for j2f (S¡2)n+1;:::;(S¡1)ngproduction takes place between periods. Note that production only involves the last (S¡1)l commodities. Finally let K ¡ 1 = ( S ¡ 1) n : j =( S ¡ 2) n +1 y 2 j; ¡ 1 and K t = ( S ¡ 1) n : j =( S ¡ 2) n +1 y 2 j;t + ( S ¡ 1) lm : k =1 z 2 k;t be the stock of output in period t. Hence in order to produce the ¯rst commodities (S¡1)larti¯cial ¯rms are introduced for every consumer so (S¡1) 2 lm arti¯cial ¯rms are introduced. Actually many of the arti¯cial ¯rms are not needed because consumers only consume Sl c commodities rather than 2(S¡1)lcommodities but in order to keep the notation at a reasonable level all (S¡1)larti¯cial ¯rms are introduced for every consumer. They produce the ¯rst commodities by use of the last (S¡1)lcommodities with one-to-one technologies. Their production sets are Z k ½R ( S ¡ 1) lm +( S ¡ 1) l ¡ £R ( S ¡ 1) lm +( S ¡ 1) l + , their inputs are z 1 k;t and their outputs are z 2 k;t for all k2f 1;:::;(S¡1) 2 lmgwith z 1 ;k  k;t =0fork 0 6=k¡(S¡1)l(i¡1) + (S¡1) 2 lm z 2 ;k  k;t =0fork 0 6=k for all k2f (S¡1)l(i¡1) + 1;:::;(S¡1)ligand all i2f 1;:::;(S¡1)mg. Moreover (z 1 k;t ; z 2 k;t +1 )2Z k ,g k (z 1 k;t ; z 2 k;t +1 )0 where g k (z 1 k;t ; z 2 k;t +1 )=z 1 ;k ¡ ( S ¡ 1) l ( i ¡ 1)+( S ¡ 1) lm k;t +z 2 ;k k;t +1 for all k2f (S¡1)l(i¡1) + 1;:::;(S¡1)ligand all i2f 1;:::;(S¡1)mg. For all arti¯cial ¯rms production takes place between periods. It may seem a little strange to distinguish between consumption in the ¯rst period of life for consumers in the same generation but in the proof of the turnpike property it turns out to be very useful as will become clear - hopefully. A di±culty is that some of the production technologies exhibit decreasing returns to scale while other production technologies exhibit constant returns to scale. However in the literature on turnpike theory in optimal growth 18 models all production technologies are assumed to exhibit either decreasing returns to scale as in [6] or constant returns to scale as in [17, 18]. A useful property of the model that is not altered by the inclusion of arti¯cial ¯rms is that for any given initial capital feasible consumptions are bounded, i.e. there exists an upper bound, a2Rsuch that if (x t ; y t ; z t ) t 2 N  is an equilibrium allocation sup t 2 N  jx t j a: Indeed, unbounded consumptions may only be caused by the real ¯rms (with decreasing returns to scale) and not by the arti¯cial ¯rms so that Lemma 7.1 in [6] applies (it is also possible to use Lemma 1 and Lemma 3 in [17]). In the following, the arti¯cial technologies are transformed into decreasing returns to scale technologies. First, modify the arti¯cial ¯rms such that g k (z 1 ; z 2 )=z 1 ;k ¡ ( S ¡ 1) l ( i ¡ 1)+( S ¡ 1) lm ¡h ¯ (¡z 2 ;k ) for all k2f (S¡1)l(i¡1) + 1;:::;(S¡1)ligand all i2f 1;:::;(S¡1)mg where h ¯ (z)=(1 ¡¯)z+¯ln(1 + z) with ¯2[0;1]. Second, modify the utility functions such that u i (x)=u i ((h ¡ 1 ¯ (x k )) ( S ¡ 1) lm k =1 ;(x k ) ( S ¡ 1) lm +( S ¡ 1) l k =( S ¡ 1) lm +1 ): Hence as ¯increases output of the arti¯cial ¯rms decreases and the utilities of output from arti¯cial ¯rms increases. Clearly, consumptions of the last (S¡1) commodities, productions, inputs of arti¯cial production and the maximal value of the planner's stationary problem are independent of ¯. Then there exists ¯ 0 such that if ¯2]0;¯ 0 [ then the modi¯ed utility functions have negative de¯nite Hessian matrix for all xwith jxja,whereais the upper bound on consumption found in the previous paragraph. For ±<1;the planner's problem for the overlapping generations economy with Speriods of life, (PP), translates into the following objective for the 19 optimal growth model max ° ¡ 1( S ¡ 1) m : i =1 v i (x i; ¡ 1 )+ : t 2 N  ° t ( S ¡ 1) m : i =1 u i (x i;t ) s.t. 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : ( S ¡ 1) m X i =1 (x i;t ¡! i )= ( S ¡ 1) n X j =1 (y 1 j;t +y 2 j;t )+ ( S ¡ 1) lm X k =1 (z 1 k;t +z 2 k;t ) f j (y 1 j;t ; y 2 j;t )0 for all j2f 1;:::;(S¡2)ng f j (y 1 j;t ; y 2 j;t +1 )0 for all j2f (S¡2)n+1;;:::;(S¡1)ng g k (z 1 k;t ; z 2 k;t +1 )0 for all k K ¡ 1 is ¯xed; where °=± S ¡ 1 . The case ±= 1 can be stated easily with the help of the overtaking criterion. In order to transform the optimal growth model with (S¡1)mconsumers into an optimal growth model with one consumer, let the maximal aggregate utility of the consumers be the utility of the aggregate consumer, i.e. let != ( S ¡ 1) m X i =1 ! i ; u(x)=max ( S ¡ 1) m X i =1 ¸ i u i (x i );s.t. ( S ¡ 1) m X i =1 x i =x; and let vbe de¯ned as u. Then the optimal growth model with (S¡1)m consumers is transformed into an optimal growth model with one consumer. The sequel of the proof of Theorem 1 depends on whether the government discounts the future or not. In the case ±<1;according to [5, 6] the optimal growth model with 1 consumer has an equilibrium, (p t ) 1 t = ¡ 1 , and the equilibrium allocation, (x t ; y t ; z t ) 1 t = ¡ 1 , is Pareto optimal. Clearly the equilibrium allocation solves the planner's problem for the optimal growth model with one consumer. 20 When there is no discounting, [5, 6] cannot be applied. In this case it is useful to restate the model in its reduced form and use the fact that strict concavity of the reduced utility function directly follows from the strict concavity of production as well as of the original utility function. The other usual assumptions of the turnpike literature with no discounting follow directly from our assumptions and from the fact that feasible consumptions are bounded (see above). Then Theorem 6.1 in [14] or Theorem 9 in [10] ensure the existence of the optimal path, at least for t¸0:It is then straightforward to complete this path to obtain an optimal path for the problem with t¸¡ 1. Implementation in the overlapping generations economy of the equilibrium for the optimal growth model has to be considered. Let ((x t ) 1 t = ¡ S +1 ; (y t ) t 2 N  ) be the associated allocation for the overlapping generations economy - this allocation is found by going through the steps used in transforming the overlapping generations economy into an optimal growth economy in reverse order. The associated commodity prices, (p t ) t 2 N  , are obtained in the usual way by considering the gradients of the individual utilities at the given allocation (a proof that mimics the proof of the second welfare theorem). Then the allocation solves the planner's problem, (PP), because otherwise the equilibrium allocation, (x t ; y t ; z t ) 1 t = ¡ 1 , would not be a solution to the planner's problem in the optimal growth model with one consumer. Profits can be taxed away by a suitable negative transfer to the ¯rms 4 . In this case, the non-arbitrage condition, (NA);implies that the prices of assets are constant. Therefore, assets behave exactly like (bubble) money. Since their prices can be chosen arbitrarily, let q j;t =0;j 2J; and all t2N 0 . The budget constraint is then S : s =1 ¡ t p s + t ¡ 1 ¢(x s i;s + t ¡ 1 ¡! s i )= S : s =1 ¡ t ¿ s i;s + t ¡ 1 for all i2Iand all t2f¡ S+1;:::;¡1gand S : s =1 p s + t ¡ 1 ¢(x s i;s + t ¡ 1 ¡! s i )= S : s =1 ¿ s i;s + t ¡ 1 " When ± = 1, zero pro¯ts is a necessary condition for implementation of the optimal path, but not for existence of the solution to the government's problem. On the other hand, these taxes are super°uous when there is discounting. 21 for all i2Iand all t2N  . Then (p t ;q t ;¿ t ;¾ t ) t 2 N  is an equilibrium with ((x t ) 1 t = ¡ S +1 ;(y t ) t 2 N  ) as the associated equilibrium allocation and this allocation solves the problem of the government because it is Pareto optimal for the associated general equilibrium growth model. Q.E.D. Remark In theorem 1 the sum of the transfers to every consumer is determined while the pro¯le of transfers over periods is undetermined. This fact is related to the irrelevance of government budget de¯cits when lump-sum taxes and transfers are available. 5.2 Proof of Theorem 2 The transformation of the stationary overlapping generation economy into an optimal growth model results in an optimal growth model that is stationary - except for the ¯rst period where utility functions di®er. If the ¯rst period is disregarded then the planner's stationary problem is obtained, and in the case ±<1itreads max : t 2 N  ° t ( S ¡ 1) m : i =1 u i (x i;t ) s.t. 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : ( S ¡ 1) m X i =1 (x i;t ¡! i )= ( S ¡ 1) n X j =1 (y 1 j;t +y 2 j;t )+ ( S ¡ 1) lm X k =1 (z 1 k;t +z 2 k;t ) f j (y 1 j;t ; y 2 j;t )0 for all j2f 1;:::;(S¡2)ng f j (y 1 j;t ; y 2 j;t +1 )0 for all j2f (S¡2)n+1;;:::;(S¡1)ng g k (z 1 k;t ; z 2 k;t +1 )0 for all k K 0 is ¯xed. When ±=1;maximality is stated with the help of the overtaking criterion. This problem is studied in the literature on turnpikes where it is proven that astationary solution exists. 22 Lemma 1 Consider the planner's stationary problem then there exist ± ¤ 2 [0;1[,K ¤ 2R ( S ¡ 1) lm +( S ¡ 1) l + and (x ¤ ; y ¤ ; z ¤ )such that if ±2]± ¤ ;1] and ( S ¡ 1) n : j =( S ¡ 2) n +1 y 2 j; 0 + ( S ¡ 1) lm : k =1 z 2 k; 0 =K ¤ and (x t ; y t ; z t ) t 2 N  solves the planner's stationary problem then (x t ; y t ; z t )=(x ¤ ; y ¤ ; z ¤ ) for all t2N 0 . Proof For ±<1 it follows directly from [6]. When ±=1;the result follows from Theorem 6.1 in [14]. Q.E.D. Under (A.11) and (A.12) a second lemma is also obtained. Lemma 2 Consider the planner's stationary problem and let (x ¤ ; y ¤ ; z ¤ )be the associated stationary allocation. Then for every compact set, C½R ( S ¡ 1) lm +( S ¡ 1) l ++ ; there exists ± ¤ C 2[0;1[ such that if (x t ; y t ; z t ) t 2 N  solves the planner's stationary problem then lim t !1 j(x t ; y t ; z t )¡(x ¤ ; y ¤ ; z ¤ )j=0 for all K2C. Proof When ±<1;the proof is an application of theorem (4.5) in [6]. Indeed, since Cis compact and K2Cthere exists an upper bound, a2R; such that if (x t ; y t ; z t ) t 2 N  is an equilibrium allocation associated with K2C then sup t 2 N  jx t j a: Therefore there exists ¯ C such that if ¯2]0;¯ C [ then the modi¯ed utility functions have negative de¯nite Hessian matrix for all xwith jxja. Hence for ¯2]0;¯ C [ theorem (4.5) in [6] can be applied to the modi¯ed optimal growth model in order to obtain lemma 2. 23