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 Theoretical Economics Letters, 2012, 2, 351-354 http://dx.doi.org/10.4236/tel.2012.24064 Published Online October 2012 (http://www.SciRP.org/journal/tel) On the Closed-Form Solution to the Endogenous Growth Model with Habit Formation Ryoji Hiraguchi Faculty of Economics, Ritsumeikan University, Kyoto, Japan Email: [email protected] Received June 25, 2012; revised July 23, 2012; accepted August 20, 2012 ABSTRACT We study the AK growth model with external habit formation. We show that there exists a unique solution path expressed in terms of the Gauss hypergeometric function. Using the closed-form solution, we also show that the opti mal path con- verges to a balanced growth path. Keywords: Endogenous Growth; Closed-Form Solution; Habit Formation 1. Introduction The Gauss hypergeometric functions are typically used in mathematical physics, but are not so common in eco- nomics. As far as we know, Boucekkine and Ruiz- Tamarit [1] are the first to find that the functions are also useful in dynamic macroeconomics. They obtain an ex- plicit solution path to Lucas-Uzawa two-sector endoge- nous growth model by using the hypergeometric function. They express the optimal path as a system of four dif- ferential equations and two transversality conditions and then use the hypergeometric functions for solving the system of equations. Several authors get analytical solution paths to the exogenous growth models. Pérez-Barahona [2] investi- gates the model with non-renewable energy resources and find that the optimal path has a closed form solution path by using the hypergeometric function. Hiraguchi [3] finds that a solution path to the neoclassical growth model with endogenous labor is also represented by the special function. Boucekkine and Ruiz-Tamarit [1] argue (see page 34) that the hypergeometric functions will also be useful in the investigation of the endogenous growth models. They guess that the transition dynamics of the models will be easier to understand if we can use the special functions. However, there is only a few literature that applies the special functions to the endogenous growth models other than the Lucas-Uzawa model. One ex- ample is Guerrini [4] who uses the special functions and obtains a closedform solution path to the AK model with logistic population growth. Broad applicability of the hypergeometric functions to the endogenous growth models is uncertain at this point, and more investiga- tions are needed. In this paper, we study the AK endogenous model with external habit formation. The model has been investi- gated by many authors including Carroll et al. [5] and Gómez [6]. The utility function of the agent depends on both the absolute level of consumption and the ratio be- tween consumption and habit stock. Here the habit for- mation is external and the level of the habit stock is ex- ogenous to each agent. We show that there exists a unique solution path and it is represen ted by the hypergeometric function. Habit formation in consumption is now popular in modern macroeconomics. Authors have explained some empirical facts by incorporating habits into the dynamic macroeconomic models. Abel [7] and Gal [8] show that habit formation can solve the equity premium puzzle in asset pricing models and Carroll et al. [9] provide an explanation of strong correlations between saving and growth. Some authors characterize the properties of the optimal paths in these models. Alvarez-Cuadrado et al. [10], Alonso-Carrera et al. [11] and Gómez [6] in- vestigate the transitio nal dyn amics and the stability of the optimal paths in endogenous growth models with habits, both analytically and numerically. The problem of the previous papers is that they assume the existence and the uniqueness of the optimal path without proof. These properties are not at all obvious here, because there exists no general theorem on the existence of a solution path in an infinite horizons opti- mization problem with externalities. Here we utilize the special functions to show that their assumptions are in fact correct. The note is organized as follows. Section 2 describes C opyright © 2012 SciRes. TEL  R. HIRAGUCHI 352 the model and obtains the first order conditions. Section 3 obtains the closed-from solution path. The conclusions are in Section 4. Proofs of the propositions are in Ap- pendix. 2. Set-Up In this section, we construct the one-sector endogenous growth model with external habit formation and obtains the first order conditions. There is a continuu m of agents with unit measure. There is no population growth. The instantaneous utility func tion of each agent is 11 111 ,= = 11 ttt tt tt cch ch uch . Here is his own consumption, t is the habit stock, t c >0 h is the parameter on the utility curvature and shows the importance of relative con- sumption level 0, 1 tt ch on the utility function. When there =0 and the utility is time-separable, the pa- rameter coincides with the coefficient of the relative risk aversion. The habit stock is exogenous to the con- sumer and is accumulated by the following differential equation: = ttt hch . (1) Here t c is the average level of consumption and >0 is a parameter. The parameter is high, the habit stock responds to the recent consumption quickly. The consumer solves the following problem: 0 , :,d,s.t.= max ttttt t ck tt PeuchtkAk .c (2) here >0 is the discount factor, Equation (2) is the resource constraint, t is physical capital and is the technology parameter. We assume that there is no capital depreciation. The initial capital stock 0 and the initial habit stock are given. In what follows, we denote the growth rate of a variable as k 0 h >0A k t x ˆ= ttt x xx . The current value Hamiltonian is 1 =11 tttt t H chAkc where t is the multiplier. The first order conditions (FOCs) and the transversality condition (TC) are FOC(k) :=, t t A (3) (1 ) FOC(c) :=, tt t ch (4) TC:() =0. lim t tt tke (5) Here FOC(x) means the FOC on the variable x . In equilibrium, the individual consumption is equal to t c and the habit stock is accu- mulated according to =. ttt h hc (6) The path ,, ttt ckh )-(6) for sois optimal if Eq and only if it satisfies uations (2me 0 t . The next lemma shows that when t A he productivity is too high, the interior optimal path does not exist1. Lemma 1. If 11>A , the optimal path do Appendix. is too low and satisfies es not exists. Proof. See the Moreover, if the productivity > A , Equations (3) and (4) together imply that the ed growth rate of consumption (and also habit stock) is negative. Thus we impose the following re- striction on the parameters to ensure that the optimal path is interior and that the optimal growth rate is positive: balanc 11<<. A A (7) 3. Closed-Form Solution , we first obtain a linear To characterize the optimal path differential equation on the habit-consumption ratio = ttt zhc. Note that ˆˆ ˆ= ttt zhc. Substitution ˆ ˆ=. tt zh A of Equation (4) into Equation (3) yields (8) where =1 >0 . written as On the other hand, Equation (6) is 1 t z. Thus Equation (8) implies ˆ= t h 1 1 t zA ˆ=1 t z he equation by t z, we get. Multiplying both sides of t a linear differential equation = tt zzz ith w =A nd >0 a=z >0A . (9) Since The solution is 0 =. t t zze zz >0 atio z con- , the habit consumption r ve t rges to z as t goes to . Next wuse Euation (9)to e q obtain the equilibrium consumption path. Since ˆ ˆˆ= tt t czh, Equation (8) is re- expressed as ˆ= t z ˆt c A . Thus 00 =A tt cce t zz consump and the tion is t c 100 ,zh (10) = gt tt cez where the parameter g is defined as ==> 1 AA g0. (11) Note that by definition, 00 =chz 0 . Since =< tt zz lim , the co growth rate co we show that the growth rate of the nsumptionn- verges to g. Later 1It is not a simple task to prove Lemma 1 only by using the transversal- ity condition (5). This is because we cannot easily obtain the asymp- totic growth rate of physical capital by using the system of the differ- ential Equations (2)-(6). the aver age consumptio n t c Copyright © 2012 SciRes. TEL  R. HIRAGUCHI 353 equilibrium capital also converges to the same value g. In Equation (10), 0 z is unknown. To fix its value, we ha n (10), we get ve to use the transversality condition (5). Here the re- source constraint (2) can be written as 00 =d t At As ts kek ecs . Using Equatio 1 tAg s 00 0 0 =d. ts hez s Under the parametric restriction (7), At ke zk 1=1 1<AA and then >=Ag A . Thus the term 1 0d te constant as t goes lemma simplifies the transversality condition (5). Lemma 2. The transversality condition (5) holds Ag ss zs to . The next if an converges to a finite d only if 1 000 0 =d Agss kzhezs . (12) There exists a unique that satisfies the condition. (H the equilibrium capital as 0 z enere 0 k and 0 h are giv.) Proo See the ppendix. f. A Using Lemma 2, we can re-write 1 00 =d Ags At ts t kezhe zs . We now express the equilibrium capital without using the integral. As Hiraguchi (2012) shows, the hyper- geometric function 21 F,,;=abcz =0 ! n nnn n abcz n with a satisfies ()=()/ () n aan 3 a y 12 12 3 111 12 2 21 3 1221 d =F,,1;, ay a x a ax ax ebbe y ebaab ae aaab (13) where , , , and are 1>0a t2. A sim2>0a ilar equ3>0a, ion is a1>0b o prove2 b ouconstan atlsd by Bcek- kine and Ruiz-Tamarit [1] (see Proposition 1 in page 40). Thus we can express the integral part of t k as /1 d Ags ezs /1 21 0 =F 1,,1;. s t Ag t t ze Ag zz Ag Ag e z Recall that 0 =t t zzezz . Finally we get the following proposition. Proposition 1. The optimal path exists, is unique and is expressed as ,, ttt chk 1 00 =, gt t t ceze zzzh 0 (14) 00 =={( )}() gt t ttt hzcezezzz h 0 , (15) 1 gt t zh Ag k 0 0 0 t zz Ag ez (16) The parameters are 21 =F1, ,1;.ezA g =1 >0 , =>0 , =>A 0 and zA =1gA . The value of termined by Equation (12). It is known that for any a, b and c, Thus the growth rate of the capital also c Therefore the optimal path to a balanced growth path 0 z is de- 21 F,,;0=1abc. onverges to g. always converges h rate g. ,, ttt chk with the growt If =0 (no habit =formation), then n Equa and the first te the hypergeetric function itio is rm ofomn (16) 1=0 . Thus the olly: ptimal consumption path and capital path grows exponentia 00 =, g t t cz he (17) 1 0. 0 = g t zh t ke zA g (18) It is well-kown that the basic AK growth model does not have transitional dynamics and n the optimal growth rate is always constant. 4. Conclusion In this paper, we obtain a closed-form so the AK growth model with habit formation. As Boucek- ki s are applicable to many kinds of economic models. As a future study, igate the different kinds of the endo- lution path to ne and Ruiz-Tamarit [1] claim, the hypergeometric functions are actually very useful in the investigation of the endogenous growth models. We guess that the Gauss hypergeometric function the dynamic macro we hope to invest genous growth models, especially the growth models with R & D. REFERENCES [1] R. Boucekkine and R. Ruiz-Tamarit, “Special Functions for the Study of Economic Dy namics: The Case of the Lu- cas—Uzawa Model,” Journal of Mathematical Econom- ics, Vol. 44, No. 1, 2008, pp. 33-54. doi:10.1016/j.jmateco.2007.05.001 [2] A. Pérez- 2If we let , we can easily show that the integral is equal to 1 =ay ze 1 11 ax bb 21 3 132 20d a e aaa abz z . As many authors have already shown, the integral can be obtained by using the hy- d c b az z p ergeometric function. Barahona, “Nonrenewable Energy Resources as Copyright © 2012 SciRes. TEL  R. HIRAGUCHI Copyright © 2012 SciRes. TEL 354 Input for Phytion: A New Ap- ing Up with the Joneses,” American Economic Review, Vol. 80, No. 2, 1990, pp. 38-42. [8] J. Gal, “Keeping up with the Jone sical Capital Accumula proach,” Macroeconomic Dynamics, Vol. 15, No. 1, 2011, pp. 1-30. doi:10.1017/S1365100509090415 [3] R. Hiraguchi, “A Note on the Analytical Solution to the Neoclassical Growth Model with Leisure,” Macroeco- nomic Dynamics, 2012, pp. 1-7. ses: Consumption Ex- ternalities, Portfolio Choice, and Asset Prices,” Journal of Money, Credit, and Banking, Vol. 26, No. 1, 1994, pp. 1- 8. doi:10.2307/2078030 doi:10.1017/S1365100512000442 [4] L. Guerrini, “Transitional Dynamics in the Ramsey Mod- el with AK Technology and Logistic Population Change,” Economics Letters, Vol. 109, No. 1, 2010, pp. 17-19. [9] C. Carroll, J. Overland and D. Weil, “Saving and Growth with Habit Formation,” American Economic Review, Vol. 90, No. 3, 2000, pp. 341-355. doi:10.1257/aer.90.3.341 [10] F. Alvarez- Cuad ra do, G. Mo nte ir o and S. T urn ov sky , “Ha bit doi:10.1016/j.econlet.2010.07.002 [5] C. Carroll, J. Overland and D. Weil, “Comparison Utility in a Growth Model,” Journal of Economic Growth, Vol. 2 No. 4, 1997, pp. 339-367. Formation, Catching up with the Joneses, and Economic Growth,” Journal of Economic Growth, Vol. 9, No. 1, 2004, pp. 47-80. , doi:10.1023/A:1009740920294 doi:10.1023/B:JOEG.0000023016.26449.eb [11] J. Alonso-Carrera, J. Caballé and X. Raurich, “Growth, Habit Formation, and Catching up with the Joneses,” Euro- pean Economic Re 1691. [6] M. A. Gómez, “Convergence Speed in the AK Endoge- nous Growth Model with Habit Formation,” Economics Letters, Vol. 100, No. 1, 2008, pp. 16-21. doi:10.1016/j.econlet.2007.10.022 view, Vol. 49, No. 6, 2005, pp. 1665- doi:10.1016/j.euroecorev.2004.03.005 [7] A. Abel, “Asset Prices under Habit Formation and Catch- Appendix . Proof of Lemma 1 Let Consider a path s th is feasible since the as 1and then the term t tt ke ality condition can be expressed as. the transversholds if and At tt ke only if Thus =1 1>0A . e pa ritten uch that =/2 >0 tt ck . Th resource constraint can be w 1 00 0 00. t =d= lim limAg s At ts tt ekkzh ezs Under Equation (7) > A g and then =2>0 t kk the path, we can easily show tA . that Along 1d Ags ezs is finite. Thus the above equation 0s equivalent to EqWe next show exists a unique Equation (12). If we de- fin is that thereuation (1 0 satisfying 2). z e a function t G as ˆˆ ˆ==2 lim tt tt ckA= lim lim t t h the instantaneous utility as . If we denote = t U 11(1 ) ttt ech , o. The growth rate of converges to a positive constant be- cause 1 =1 tt t Gzze zze , Equation (12) iwritten as then the intertemporal utility is equal t 0d t Ut t U s 00 0 0 =d Agt t kheGz t . ˆ=11 2=2>0 limtt UA . Thus Since 10 t e , the function 1 =1 tt t Gzezze 0d= t Ut . Then the optimal path does not exist. oof of Lem 2 quivalent to Equa- f the multiplier 1 tt eez z g fuof. Mor and is a strictly decreasin nction eover, 0= t G =0 zt G. Thus there exts a uniqueng is 0 z satisfyi 00 0 0 =t G 2. Prmakh zdt . We first show that Equation (12) is e is A tion (5). The growth rate ot |