In this note I develop two approaches to rule out the Overconsumption paths in the Ramsey model with irreversible capital. The first focuses on the multiplier of the irreversible constraint and is applied to the situation where preferences are CES and the production function is Cobb-Douglas.  The second relies on a revealed preference argument and is used to rule out overconsumption paths when the preferences are strictly concave and the initial level of perspective capital is below its steady state level

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ECONOMIC RESEARCH REPORTSC.V. Starr Center for Applied EconomicsDepartment of EconomicsFaculty of Arts and ScienceNew York University269 Mercer Street, 3rd FloorNew York, New York 10003-6687Two Ways to Rule Outthe Overconsumption Paths in theRamsey Model with IrreversibleInvestmentbyDiego CominRR#:  2000-17                                            October 2000Two Ways to Rule Out the OverconsumptionPaths in the Ramsey Model with IrreversibleInvestment.Diego Com¶in¤First Version: January, 1998This Version: October, 2000.AbstractIn this note I develop two approaches to rule out the overconsumptionpaths in the Ramsey model with irreversible capital. The ¯rst focuses onthe multiplier of the irreversible constraint and is applied to the situationwhere preferences are CES and the production function is Cobb-Douglas.The second, relies on a revealed preference argument and is used to ruleout overconsumption paths when the preferences are strictly concave andthe initial level of per e®ective capital is below its steady state level.JEL: O4, C6.Keywords: Ramsey growth model, Irreversible capital, Overconsump-tion paths.1. IntroductionIn the in¯nite horizon Ramsey model with reversible investment all overconsump-tion paths of the Ramsey model lead to the consumption of all the capital, atsome ¯nite time. This cannot be optimal since the marginal product of capital is¤I thank Robert Barro for suggesting me this exercise and him and John Leahy for usefuldiscussions. I also gratefully acknowledge the ¯nancial support of the Bank of Spain. Commentsare very welcome at dcomin@kuznets.fas.harvard.edu. The usual disclaimer applies.in¯nite at k = 0 and the neoclassical agent would have been better o® by investingthe last unit of capital instead of consuming it. In short, overconsumption pathscan be ruled out because they violate the Euler equation.1 This argument doesnot extend to the situation where capital is irreversible (i.e. c · f (k)). Then, thelocus c = f(k) is reached at some ¯nite time and from then on capital follows thedi®erential equation_k = ¡(± + n + x)kwhere ± is the depreciation rate, k is capital per e®ective unit of labor (i.e.KLA), and x and n are the exogenous rates of technological and population growth.For future reference, c is going to denote consumption per e®ective unit of labor(i.e. CLA). Note that with this law of motion the k = 0 locus is reached onlyasymptotically and the usual argument cannot be made.In this note I present two arguments that, under di®erent circumstances, ruleout the overconsumption paths in the Ramsey model with irreversible capital.The ¯rst approach is applied to the case where the utility function is CES andthe production technology is Cobb Douglas and attacks the problem by showingthat the multiplier of the irreversible constraint eventually becomes negative atany overconsumption path. Remember that the multiplier of any constraint canbe interpreted as the marginal value of relaxing the constraint. Hence the neg-ative multiplier in the irreversible constraint implies that the marginal value ofconsumption is smaller than the marginal value of saving at some point along thec = f(k) locus, i.e. the agent would be better o® by saving more. As a result,overconsumption cannot be optimal. The second technique relies on a revealedpreference argument. In particular, if the constraint on the irreversibility of in-vestment is not binding at any instant along the optimal path for the unrestrictedproblem, and this optimal path is unique, it must also solve the problem oncethe constraint is introduced. As a corollary of this proposition, when the irre-versibility constraint on investment is not binding along the solution to Ramseyproblem with reversible investment, overconsumption paths are not optimal inthe restricted problem. In order to compare the applicability of the two meth-ods I show that when the initial level of e±ciency units of capital is below thesteady state level the irreversible constraint on investment does not bind alongthe optimal path in the traditional Ramsey problem.1See Blanchard and Fisher (1989) or Barro and Sala-i-Mart¶in (1995) among others.2The exercise conducted in this paper is related to two other papers. In hisseminal analysis of the Ramsey model, Cass [1965] imposes the constraint on irre-versible investment but he does not rule out overconsumption paths. Arrow andKurz [1970] are interested in understanding when the economies with irreversiblecapital that converge to the steady state from above consume all the output. Theirmain conclusion is that, in general, it is very hard to characterize the number ofintervals in which the nonnegativity constraint on investment is binding. I amconcerned about the optimality of overconsumption paths. My analysis focusesmore on economies that converge from below, and for the CES utility functionand Cobb Douglas production function case it turns out that, even if the economyconverges from above, the constraint on nonnegative investment is never binding.2. The Ramsey problem with irreversible investmentThe social planner enforces the optimal intertemporal allocation of resources giventhe initial capital (per e®ective unit) (k0), the law of motion for capital accumula-tion (equation 2), the no-Ponzi game restriction on borrowing (inequality 3), andthe irreversibility constraints on investment (inequality 4).2(1) max V =R 10 e¡(½¡n¡x(1¡µ))thc1¡µt ¡e¡xt1¡µidtfct; ktg1t=0s.t.(2) _kt = f (kt)¡ ct ¡ (± + n + x)kt(3) limT!1kT e¡R T0 (f0(kv)¡±¡n¡x)dv ¸ 0(4) f (kt) ¸ ct , 8tgiven k0 > 0where f (kt) is a neoclassical production function.33. The multiplier method.If condition (4) is not imposed, the ¯rst order conditions of this problem are givenby:42Note that for simplicity I write the program with a CES utility function. As will be seenlatter, this is only relevant for the ¯rst approach.3See Barro and Sala-i-Mart¶in (1995), page 16, for a precise description of the properties thatmust satisfy a neoclassical production function.4See Barro and Sala-i-Mart¶in (1995), Chapter 2.3(5) Hc = 0(6) Hk = ¡ _¹t(7) limT!1 ¹TkT = 0(8) ¹t ¸ 0where the Hamiltonian (H) isH = e¡(½¡n¡x(1¡µ)) t·c1¡µt ¡ e¡xt1¡ µ¸+¹t [f (kt)¡ ct ¡ (± + n + x) kt]Hz is the partial derivative of the Hamiltonian with respect to z, ¹t is the Lagrangemultiplier associated with the law of motion of capital (equation 2), equation (7)is the transversality condition, and, for future reference, the mutiplier associatedwith the irreversibility constraint on investment (equation 4), will be denoted by¸t:5When the constraint on the irreversibility of investment (equation (4)) is bind-ing the ¯rst order necessary conditions of the problem are going to be altered inthree ways. The marginal utility of consumption is going to exceed (by ¸t) themarginal utility of investment. The other side of this e®ect is that by increasingthe level of capital per e®ective worker, the representative agent reduces the extentto which her consumption decisions are constrained, by equation (4), in the future.Moreover, the Kuhn-Tucker theorem states that the multiplier of the irreversibil-ity constraint on investment must be non-negative and will be strictly positiveonly when the constraint is binding. Lemma 3.1 formalizes these intuitions whichare proved in the appendix.Lemma 3.1. The F.O.N.C. of the Ramsey model with irreversible investmentand CES utility function are given by(5') Hc = ¸t(6') Hk = ¡ _¹t ¡ ¸tf 0(kt)(7') limT!1 ¹TkT = 0;5The solution to this problem is given by the system_ctct= 1µ [f0(kt) ¡ ± ¡ ½ ¡ µx]_kt = f (kt) ¡ ct ¡ (± + n + x)ktwith boundary conditionskt = k0 > 0; at t = 0:limT !1 kT e¡R T0 (f0(kv)¡±¡n¡x)dv = 0See Barro and sala-i-Mart¶in (1995), page 71.4(8') ¹t ¸ 0(9.a') f(kt) ¸ ct; (9.b') ¸t ¸ 0; (9.c') ¸t [f(kt) ¡ ct] = 0:Since the conditions stated in Lemma 1 are necessary for optimality, in orderto rule out the overconsumption paths it is su±cient to show that condition (9.b')is eventually violated. By following this strategy, we are able to proof that themarginal utility of consumption at the overconsumption paths is eventually lowerthan the marginal utility of saving and therefore the agent would be better o®by increasing her saving rate along these paths. This result is stated and provedin Proposition 3.2 for the particular case of a Cobb-Douglas production functionand CES utility function. The argument used in the proposition does not depend,a priori, on these speci¯cations; however, it involves several ¯rst order di®erentialequations and the assumed functional forms are very convenient.Proposition 3.2. (Ruling out overconsumption paths) Suppose that the prefer-ences are CES, and the production function is Cobb-Douglas. Then, the over-consumption paths in the Ramsey model with irreversible investment are notoptimal.Proof. Figure 3.1 illustrates two possible overconsumption paths drawn in the(k; c) space. Overconsumption paths in an in¯nite horizon world are characterizedby reaching in a ¯nite time the ct = f(kt) locus. Let's denote by t¤ this instant.Inspecting the phase diagram we can see that the ct = f(kt) locus is above the_k = 0 locus and therefore an economy on an overconsumption path will eventuallyreach the Northern quadrants. Once the economy is in this region the dynamicsimplied by the F.O.N.C. tend to reduce the capital stock per e®ective unit oflabor. From t¤ to in¯nity, the representative agent will consume ct = f(kt): Wecan use the law of motion for capital together with this restriction to pin downthe path of capital starting at t¤: In particular, ct = f (kt) for t ¸ t¤ together withequation (2) imply that_kk= ¡(± + x+ n); 8t ¸ t¤and therefore(10) kt = kt¤e¡(±+n+x)t; for all t ¸ t¤:A Cobb-Douglas production function takes the form f(kt) = Ak®t : Hence fort ¸ t¤; ct = Ak®t¤e¡(±+n+x)® t:5         c=f(k)     .      c=0  c                   .           k=0                                         k                Figure 3.1: Phase DiagramNow we can rewrite conditions (5') to (9') using the de¯nition of the Hamil-tonian and the expressions of ct and kt : In particular, for t ¸ t¤(11) (Ak®t¤ )¡µe¡(½¡n(1+®µ)¡x(1¡µ+®µ)¡®µ±)t = ¹t + ¸t(12) (¹t + ¸t)®Ak®¡1t¤ ¡ ¹t(± + n + x) = ¡ _¹t(13) limT!1 ¹Tkt¤e¡(±+n+x)T = 0;(8') ¹t ¸ 0(14.a) ct = Ak®t¤e¡(±+n+x)® t; (14.b) ¸t ¸ 0:Only three steps remain to conclude the proof. First I derive the evolution of ¹tand then I use the TVC to restrict the parameters. Finally, I reach a contradictionby showing that under this condition inequality (14.b) cannot hold.Plugging equation (11) into (12) we obtain the di®erential equation((Ak®t¤)¡µe¡(½¡n(1+®µ)¡x(1¡µ+®µ)¡®µ±)t)®Ak®¡1t¤ ¡ ¹t(± + n + x) = ¡ _¹t; 8t ¸ t¤The solution to this di®erential equation takes the form¹t = ¡®A1¡µk®(1¡µ)¡1t¤ e(±+n+x)t·e»t» + b¸; 8t ¸ t¤ (15)6where b is an integration constant which must be pin down by some boundarycondition, and » ´ ¡½ + n+ x(1¡ µ) ¡ ®(1¡ µ)(± + n+ x):Using this expression for ¹; and the expression for the level of physical capital;the transversality condition (13) can be written aslimT!1 ¡®A1¡µk®(1¡µ)t¤·e»T»+ b¸= 0 (16)Since both b and » are ¯nite constants, the only possibility for the TVC tohold is that » < 0; and b = 0:Finally, using these restrictions and equations (11) and (15) I can obtain anexpression for the multiplier of the irreversibility constraint.¸t = e»t0BB@(Ak®t¤ )¡µ| {z }> 0+®A1¡µk®(1¡µ)¡1t¤»e(±+n+x)t| {z }< 01CCA ; 8t ¸ t¤ (17)Note that since (±+n+x) > 0 the second term in the parenthesis will eventuallydominate the ¯rst and therefore 9T^ s.t. ¸t < 0; 8t > T^ :¤The intuition behind this result comes from equation (11). When ¸t is nega-tive, the marginal utility of investment (¹t) exceeds the marginal utility of con-sumption ((Ak®t¤)¡µe¡(½¡n(1+®µ)¡x(1¡µ+®µ)¡®µ±)t) and therefore the agent would bebetter of by reducing her consumption (i.e. not following an overconsumptionpath).4. The revealed preference argumentA second approach consists in contrasting the solution to the traditional Ramseymodel with the Ramsey model with irreversible investment. If the constraint doesnot bind along the optimal path, and this is unique,6 the unrestricted solutioncoincides with the solution to the problem once constraint (4) is incorporated.Next I state precisely and prove this result.Theorem 4.1. Whenever the initial level of capital per e®ective unit of labordoes not exceed the steady state level (i.e. k0 · ks:s) and the instantaneous utility6A su±cient condition for uniqueness of the optimal consumption path is that the instanta-neous utility function is strictly concave.7function is strictly concave, the solution to the Ramsey model with irreversibleinvestment is exactly the same as the solution to the traditional Ramsey model.Proof. Let c(t) be the optimal path of consumption in the problem de¯nedby (1) to (3) with a given k0; and let cR(t) be the optimal path for the con-sumption level per unit of e®ective worker of the representative agent when theirreversibility constraint on investment (4) is added to the problem. Since the setof possible paths for consumption for the unrestricted problem (C) is a supersetof the set of possible paths for the restricted problem (CR); 7the utility attainedby cR(t) cannot exceed the utility of an agent that consumes according to c(t),i.e. V (c(t)) ¸ V (cR(t)):Next I use the law of motion for capital , equation (2), to show that theirreversibility constraint on investment (4) does not bind along the optimal pathfor the unrestricted problem. We know from the analysis of the reversible Ramseymodel that if k0 · kss; then _kt ¸ 0;8t: But, from equation (2), _kt ¸ 0 implies thatf(kt) ¸ ct;and therefore there is no need to consume the existing capital alongthe optimal path of the unconstrained problem.Hence c(t) 2 CR;and since V (c(t)) ¸ V (cR(t)); and the optimal path is unique(by assumption) then it must be the case that c(t) = cR(t):¤But since overconsumption is suboptimal in the Ramsey problem with re-versible capital so is when capital is irreversible.Corollary 4.2. The overconsumption paths in the in¯nite horizon Ramsey modelwith irresversible investment are not optimal when k0 < kss.5. ConclusionsIn this note, I have presented two approaches to rule out overconsumption pathsin the Ramsey model with irreversible investment. The ¯rst is based in showingthat along any overconsumption path the multiplier of the irreversible constraintis eventually negative and therefore the marginal utility of saving exceeds themarginal utility of consumption. The second, relies on a revealed preference argu-ment. In particular, if the solution to the unconstrained Ramsey problem is uniqueand along the optimal path there is no need to consume the existing capital, theoptimal path of the constrained and unconstrained problem must coincide.7i.e. C ¾ CR :8Both proofs are complementary. I use the ¯rst to rule out overconsumptionpaths when preferences are CES and the production technology is Cobb-Douglas;however, there is nothing speci¯c to these functional forms, apart from beingconvenient to conduct the prove with pencil and paper, that is necessary to usethe multiplier argument. The second method just requires that the initial level ofcapital is below its steady state level. In this sense the multiplier method mightbe more general than the revealed preference argument.References[1] Arrow, K. and M. Kurz (1970), \Optimal Growth with Irreversible Investmentin a Ramsey Model" Econometrica, Volume 38,2 pp. 331-344.[2] Barro, R. and X. Sala-i-Mart¶in Economic Growth, Advanced Series in Eco-nomics. New York; London and Montreal: McGraw-Hill, 1995.[3] Blanchard, O. and S. Fisher Lectures on Macroeconomics, Cambridge, Mass.and London: MIT Press, 1989.[4] Cass, D. (1965), Optimum Growth in an Aggregative Model of Capital Accu-mulation" Review of Economic Studies XXXII (3), No. 91 (July), pp. 233-40.9AppendixIn this appendix I prove Lemma 3.1.Lemma 3.1 The F.O.N.C. of the Ramsey model with irreversible investmentand CES utility function are given by(5') Hc = ¸t(6') Hk = ¡ _¹t ¡ ¸tf 0(kt)(7') limT!1 ¹TkT = 0;(8') ¹t ¸ 0(9.a') f(kt) ¸ ct; (9.b') ¸t ¸ 0; (9.c') ¸t [f(kt) ¡ ct] = 0:Proof. In the proof I follow the perturbation method used by Barro and Sala-i-Mart¶in (1995) (Appendix 1.3).In particular, I start with the ¯nite horizon problem(A.1) max V =R T0 e¡(½¡n¡x(1¡µ))thc1¡µt ¡e¡xt1¡µidtfct; ktgTt=0s.t.(A.2) _kt = f (kt)¡ ct ¡ (± + n + x)kt(A.3) kTe¡R T0 (f0(kv)¡±¡n¡x)dv ¸ 0(A.4) f (kt) ¸ ct , t 2 [0; T ]given k0 > 0Next I de¯ne the Lagrangian (L)L =Z T0e¡(½¡n¡x(1¡µ))t·c1¡µt ¡ e¡xt1¡ µ¸dt+ (A.5)+Z T0h¹t(f (kt)¡ ct ¡ (± + n + x)kt ¡ _kt)idt++vTkT eR T0 (¡f0(k¿)+±+n+x)d¿ +Z T0¸t (f(kt) ¡ ct) dtIntegration by parts of the termR T0 ¹t _ktdt givesZ T0¹t _ktdt = [¹tkt]T0 ¡Z T0_¹tktdt (A.6)10and by plugging it into (A.5) the Lagrangian can be written asL =Z T0(H(ct; kt; t) + _¹tkt) dt+ ¹0k0 ¡ ¹TkT (A.7)+vTkT eR T0 (¡f0(k¿)+±+n+x)d¿ +Z T0¸t (f(kt) ¡ ct) dtwhereH (ct; kt; t) = e¡(½¡n¡x(1¡µ))t·c1¡µt ¡ e¡xt1¡ µ¸+¹t(f(kt) ¡ ct ¡ (± + n + x)kt)Let c¤t and k¤t be the optimal time paths for consumption and capital. If weperturb the optimal path c¤t by an arbitrary perturbation function, p1t; then wecan generate a neighboring path for consumption,ct = c¤t + " p1tWhen ct is perturbed, there must be a corresponding perturbation of kt andkT so as to satisfy the budget constraint:kt = k¤t + " p2tkT = k¤T + " dkTIf the initial paths are optimal then @L@" = 0 when L is evaluated at the optimalpaths. Let's rewrite the Lagrangian as a function of " by substituting ct; kt andkT into (A.7).L =Z T0(H(c¤t + " p1t; k¤t + " p2t; t) + _¹t (k¤t + " p2t))dt +¹0k0+(k¤T + " dkT )³vT eR T0 (¡f 0(k¿)+±+n+x)d¿ ¡ ¹T´+Z T0¸t (f(k¤t + " p2t) ¡ (c¤t + " p1t)) dt@L@" (0) = 0 implies thatZ T0[(Hc ¡ ¸t)p1t + (Hk + _¹t + ¸tf 0(kt)p2t)] dt+dkT (vT eR T0 (¡f0(k¿)+±+n+x)d¿¡¹T) = 0(A.8)11Equation (A.8) can hold for all perturbation paths described by p1t; p2t anddkT if each of the components in the equation vanishes, i.e.Hc = ¸t (A.9)Hk = ¡ _¹t ¡ ¸tf 0(kt) (A.10)vT eR T0 (¡f 0(k¿)+±+n+x)d¿ = ¹T (A.11)Using (A.11), (A.3) and the complementary slackness condition associatedwith (A.3) we can derive that¹TkT = 0If T ! 1 this condition takes the form (7'). Conditions (8') and (9') arenecessary conditions from the Kuhn Tucker Theorem.¤12

Two Ways to Rule Out the Overconsumption Paths in the Ramsey Model with Irreversible Investment

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Two Ways to Rule Out the Overconsumption Paths in the Ramsey Model with Irreversible Investment

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