Population Dynamics and Utilitarian Criteria in the Lucas-Uzawa Model * Simone Marsiglio † Davide La Torre ‡ Forthcoming in Economic Modelling Abstract This paper introduces population growth in the Uzawa-Lucas model, analyzing the implications of the choice of the welfare criterion on the model’s outcome. Traditional growth theory assumes population growth to be exponential, but this is not a realistic assumption (see Brida and Accinelli, 2007). We model exogenous population change by a generic function of population size. We show that a unique non-trivial equilibrium exists and the economy converges towards it along a saddle path, independently of population dynamics. What is affected by the type of population dynamics is the dimension of the stable manifold, which can be one or two, and when the equilibrium is reached, which can happen in finite time or asymptotically. Moreover, we show that the choice of the utilitarian criterion will be irrelevant on the equilibrium of the model, if the steady state growth rate of population is null, as in the case of logistic population growth. Then, we show that a closed-form solution for the transitional dynamics of the economy (both in the case population dynamics is deterministic and stochastic) can be found for a certain parameter restriction. Keywords: Population Change, Utilitarian Criteria, Uzawa-Lucas model, Transitional Dynamics, Stochas- tic Shocks, Closed-Form Solution JEL Classification: O40, O41, J13 1 Introduction In standard economic growth theory, population is assumed to grow at an exogenous and exponential rate. This assumption has been firstly introduced by the Solow-Swan model (1956) and it has been applied also to following models with optimizing behavior, as the single-sector Ramsey-Cass-Koopmans (1965) model and the two-sector Lucas-Uzawa (1988) model. Such an assumption however is not without consequences for the analysis of growing economies. In fact, exponential population growth models imply unconstrained growth of population size. However, most populations are constrained by limitations on resources, at least in the short run, and none is unconstrained forever. For this reason, firstly Malthus (1798) discusses about the inevitable dire consequences of exponential growth of the human population of the earth. Recently, Brida and Accinelli clearly state: “The simple exponential growth model can provide an adequate approximation to such growth only for the initial period because, growing exponentially, as t →∞, labor force will approach infinity, which is clearly unrealistic. As labor force becomes large enough, crowding, food shortage and environmental effects come into play, so that population growth is naturally bounded. This limit for the population size is usually called the carrying capacity of the environment”. * We thank the participants of the workshop held at Monash University (Melbourne Graduate Student Conference in Eco- nomics 2010, July 2010) for helpful comments and suggestions † Department of Public Policy and Public Choice, University of Eastern Piedmont Amedeo Avogadro, via Cavour 84, I-15121, Alessandria, Italy; simone.marsiglio@sp.unipmn.it ‡ University of Milan, Department of Economics, Business and Statistics, via Conservatorio, 7, I-20122 Milan, Italy; da- vide.latorre@unimi.it 1