Abstract
We use an overlapping generations model with physical and human capital to ascertain the consequences for optimality of a social planner adopting a welfare criterion that treats all generations alike and is respectful of individual preferences. In particular, we consider a social planner who maximizes a non-discounted sum of individual utilities à la Ramsey, with consumption levels expressed in terms of output per unit of efficient labour. We show that the optimal growth path does not depend on the precise cardinalization of preferences (i.e., the degree of homogeneity of the utility function) and that it converges to the “Golden Rule” defined in this endogenous growth framework. The instruments available to the social planner are subsidies to the investment in education by the younger generation and lump-sum taxes on the middle-aged and the retirees. Decentralizing the optimum trajectory requires that subsidies to investment in education be negative (i.e., taxes), and that pensions to the elderly be positive along the entire optimal growth path. These results hold regardless of the initial conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This issue has been the subject of some attention in a “static” optimal taxation framework. As pointed out by Stiglitz (1987, p. 1017) “there are many alternative ways of representing the same family of indifference maps [...each yielding...] a different optimal income tax. The literature has developed no persuasive way for choosing among these alternative representations”. The relevance of this point seems to be even greater in “dynamic” settings as the current one, where the welfare levels of individuals born at different time periods are involved.
In particular, the requirement that a positive amount of input is needed to obtain a positive amount of output translates into \(e(0)=1\). In words, if there is no investment of output in education in period \(t-1\), the growth rate of productivity in period t will be zero. This, together with the strict concavity of the e(.) function, provides a sufficient condition for the average product of investment in human capital to be greater than the marginal one.
Note that \(c_{t}^{m}L_{t}\) and \(c_{t}^{o}L_{t-1}\) are measured in units of output. Since middle-aged individuals supply one unit of natural labour, \( c_{t}^{m}\) and \(c_{t}^{o}\) are expressed in units of output per unit of natural labour. The interpretation of \({\tilde{c}}_{t}^{m}\) and \({\tilde{c}}_{t}^{o}\) in terms of units of output per unit of efficient labour follows naturally.
It is well known that in overlapping generations models with only physical capital à la Diamond (1965), providing conditions for existence, uniqueness and stability of equilibria is not straightforward. See Galor and Ryder (1989), De La Croix and Michel (2002) and Li and Lin (2012). The same considerations apply in the current endogenous growth setting.
In addition, as mentioned above, in the presence of productivity growth, the precise cardinalization of the utility function will affect both the characterization of the social optimum and the policies that support it if we adopt this criterion (see Del Rey and Lopez-Garcia 2012).
Of course, to be fully coherent with the approach followed by Ramsey, (18) could be written as \((-{\tilde{W}})=\sum _{t=0}^{\infty }\left[ {\tilde{U}}_{*}-U( {\tilde{c}}_{t}^{m},{\tilde{c}}_{t+1}^{o})\right] \), the purpose of the social planner being to minimize the non-discounted sum of the divergences of the amount by which utility falls short of the bliss level. Obviously both are equivalent, as maximizing \({\tilde{W}}\ \) is tantamount to minimizing (\(-{\tilde{W}})\).
Indeed, using the double subscript \(*\) to denote the optimal balanced growth path in Docquier et al. (2007), where the social planner maximizes (with a social discount factor \(\gamma \)) a discounted sum of utilities defined over consumption per unit of natural labour, the optimal balanced growth path is characterized by the Modified Golden Rule, i.e., \(\gamma f^{\prime }({\tilde{k}}_{**})=[e({\tilde{d}}_{**})]^{1-j}(1+n)\), and the marginal product of physical capital will be greater than the economy’s growth rate. The presence of the degree of homogeneity j in this expression makes it apparent that different cardinalizations of the same ordinal preferences will entail different optimal balanced growth paths (and, consequently, different optimal configurations of the tax parameters designed to decentralize them). Notice that the same kind of objection arises in the framework suggested by Caballé (1995) with altruistic individuals.
The degree of homogeneity of the utility function is also irrelevant in the model used in Bishnu (2013), that relates human capital accumulation to consumption externalities. Balanced growth paths therein, however, display no productivity growth so that demography is the only source of long-run growth. As a consequence, the optimal balanced growth path is characterized by the same Modified Golden Rule as in standard overlapping generation models à la Diamond (1965).
A full characterization of the optimal policy along this track also requires that the optimal lump-sum taxes and education tax in period 0 be identified. Thus, given \({\tilde{k}}_{0}\ \)(and thus \({\tilde{s}}_{-1}\), \(w_{0}\) and \(r_{0}\) ), \({\tilde{c}}_{0}\) and \({\tilde{d}}_{-1}\), (28), (29) and (30) become:
$$\begin{aligned} \theta _{*0}= & {} -\frac{e^{\prime }({\tilde{d}}_{-1})\Lambda _{0}}{f^{\prime }( {\tilde{k}}_{0})} \\ {\tilde{z}}_{*0}^{m}= & {} w_{0}-(1+r_{0}){\tilde{d}}_{-1}(1-\theta _{*0})/e( {\tilde{d}}_{-1})-{\tilde{s}}_{*0}-{\tilde{c}}_{*0}^{m} \\ {\tilde{z}}_{*0}^{o}= & {} (1+r_{0}){\tilde{s}}_{-1}-{\tilde{c}}_{0}^{o} \end{aligned}$$Together, they provide three equations to be solved in \(\theta _{*0}\), \( {\tilde{z}}_{*0}^{m}\) and \({\tilde{z}}_{*0}^{o}\).
And, of course, with overlapping generations models with exogenous growth à la Diamond (1965). It is well known that in these models, the nature of the optimal social security that supports the Golden Rule depends on whether the laissez-faire capital-labour ratio is higher or lower than the optimum one. In other words, it will be a pay-as-you-go system [resp. a “reverse” pay-as-you-go system or a “more-than-fully-funded” one] when along the laissez-faire balanced growth path the marginal product of capital is less [resp. greater] than the economy’s growth rate (Samuelson 1975b).
References
Barro RJ (1974) Are government bonds net wealth? J Polit Econ 82(6):1095–1117
Bishnu M (2013) Linking consumption externalities with optimal accumulation of human and physical capital and intergenerational transfers. J Econ Theory 148:720–742
Boldrin M, Montes A (2005) The intergenerational state, education and pensions. Rev Econ Stud 72:651–664
Burbidge JB (1983) Government debt in an overlapping-generations model with bequests and gifts. Am Econ Rev 73(1):222–27
Caballé J (1995) Endogenous growth, human capital and bequests in a life-cycle model. Oxf Econ Pap 47:156–181
Carmichael J (1982) On Barro’s theorem of debt neutrality: the irrelevance of net wealth. Am Econ Rev 72(1):202–13
Cass D (1965) Optimum growth in an aggregative model of capital accumulation. Rev Econ Stud 37:233–240
Cass D (1966) Optimum growth in an aggregative model of capital accumulation: a turnpike theorem. Econometrica 66:833–850
Chamley C (1986) Optimal taxation of capital income in general equilibrium with infinite lives. Econometrica 54(3):607–622
De La Croix D, Michel P (2002) A theory of economic growth: dynamics and policy in overlapping generations. Cambridge University Press, Cambridge
Del Rey E, Lopez-Garcia MA (2012) On welfare criteria and optimality in an endogenous growth model. J Public Econ Theory 14:927–943
Del Rey E, Lopez-Garcia MA (2013) Optimal education and pensions in an endogenous growth model. J Econ Theory 148(4):1737–1750
Diamond PA (1965) National debt in an neoclassical growth model. Am Econ Rev 55:1126–1150
Docquier F, Paddison O, Pestieau P (2007) Optimal accumulation in an endogenous growth setting with human capital. J Econ Theory 134:361–378
Galor O, Ryder H (1989) Existence, uniqueness, and stability of equilibrium in an overlapping-generations model with productive capital. J Econ Theory 49(2):360–375
Judd KL (1985) Redistributive taxation in a simple perfect foresight model. J Public Econ 28(1):59–83
Kawamoto K (2007) Preferences for educational status, human capital accumulation, and growth. J Econ 91(1):41–67
Kemnitz A, Wigger BU (2000) Growth and social security: the role of human capital. Eur J Polit Econ 16:673–683
Koopmans TC (1965) On the concept of optimal economic growth. In: The econometric approach to development planning. North-Holland Publ. Co. and Rand McNally, 1966 a reissue of Pontificiae Academiae Scientarum Scripta Varia, vol 28, 1965, pp 225–300
Koopmans TC (1967) Objectives, constraints and outcomes in optimal growth models. Econometrica 35(1):1–15
Li J, Lin S (2012) Existence and uniqueness of steady-state equilibrium in a generalized overlapping generations model. Macroecon Dyn 16(Supplement 3):299–311
Malinvaud E (1965) Les croissances optimales. Cah Sémin d’Econ 8:71–100
Michel P (1990) Some clarifications of the transversality condition. Econometrica 58(3):705–723
Phelps ES (1961) The golden rule of accumulation: a fable for growthmen. Am Econ Rev 51:638–43
Ramsey FP (1928) A mathematical theory of saving. Econ J 38(152):543–559
Samuelson PA (1958) An exact consumption loan model of interest with or without the social contrivance of money. J Polit Econ 66:456–482
Samuelson PA (1965) A catenary turnpike theorem involving consumption and the golden rule. Am Econ Review 55(3):486–496
Samuelson PA (1968) The two-part golden rule deduced as the asymptotic turnpike of catenary motions. West Econ J 6:85–89
Samuelson P (1975a) The optimum growth rate for population. Int Econ Rev 16(3):531–538
Samuelson P (1975b) Optimum social security in a life-cycle growth model. Int Econ Rev 16(3):539–544
Stiglitz JE (1987) Chapter 15: Pareto efficient and optimal taxation and the new new welfare economics. In: Auerbach AJ, Feldstein M (eds) Handbook of public economics, vol 2. Elsevier, Amsterdam, pp 991–1042
Tran-Nam B, Truong CN, Van Tu PN (1995) Human capital and economic growth in an overlapping generations model. J Econ 61(2):147–173
Acknowledgements
The authors are indebted to two anonymous referees for their helpful comments. This work has also benefited from discussion with Michael Kaganovich. Needless to say, the usual caveat applies. We acknowledge financial support from the Institute of Fiscal Studies (Ministry of Finances, Spain), the Spanish Ministry of Economy and Competitiveness through Research Grants ECO2016-76255-P and ECO2015-67999-R, the Autonomous Government of Catalonia through Research Grants 2014SGR-1360 and 2014SGR-327, the MOMA Network under Project ECO2014-57673-REDT and XREPP (Research Reference Network for Economics and Public Policies).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Del Rey, E., Lopez-Garcia, MA. Optimal public policy à la Ramsey in an endogenous growth model. J Econ 128, 99–118 (2019). https://doi.org/10.1007/s00712-018-0643-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00712-018-0643-z