Non-convex aggregate technology and optimal economic growth

This paper examines a model of optimal growth where the aggregation of two separate well behaved and concave production technologies exhibits a basic non-convexity. First, we consider the case of strictly concave utility function: when the discount rate is either low enough or high enough, there will be one steady state toward which the convergence of the optimal path is monotone and asymptotic. When the discount rate is in some intermediate range, we will find sufficient conditions for having either one equilibrium or multiple equilibria steady state. Depending to whether the initial capital per capita is located with respect to a critical value, we show that the optimal paths monotonically converge to one single appropriate equilibrium steady state. Second, we consider the case of linear utility and provide sufficient conditions to have either unique or two steady states when the discount rate is in some intermediate range. In this range, we give conditions under which the above critical value might not exist, and the economy attains one steady state in finite time, then stays at the other steady state afterward. P. Michel passed away when this research was completed. This paper is dedicated to his memory as a friend and colleague. N. M. Hung and C. Le Van thank the referee for vey helpful remarks and criticisms. They are grateful to Takashi Kamihigashi for very fruitful discussions. They also thank J.-F. Leclerc for editing the final version of this paper.     

What to stabilize in the open economy*

We consider the question how “best” to maintain price‐level stability in an open economy, and evaluate three possible policy choices: (a) a constant money growth rate rule; (b) a fixed exchange rate; and (c) a policy of explicit commitment to a price‐level target. In each case we assume that policy is conducted by injecting reserves into or withdrawing reserves from the “banking system.” In evaluating the three regimes, we adopt the criterion that the “best” policy should leave the least scope for indeterminacy and “excessive” economic volatility. In a steady‐state equilibrium, the choice of regime is largely irrelevant; any steady‐state equilibrium under one regime can be duplicated by an appropriate choice of the “control” variable under any other regime. However, we show that the sets of equilibria under the three regimes are dramatically different. When all countries follow the policy of fixing a constant rate of money growth, there are no equilibria displaying endogenously arising volatility and there is no indeterminacy of equilibrium. Under a regime of fixed exchange rates, indeterminacies and endogenously arising fluctuations are impossible if and only if the country with the low “reserve‐to‐deposit” ratio is charged with maintaining the fixed rate. Finally, when one country targets the time path of its price level, under very weak conditions, there will be indeterminacy of equilibrium and endogenously arising volatility driven by expectations.     

Lifetime Earnings, Discount Rate, Ability and the Demand for Post–compulsory Education in Men in England and Wales

Human capital theory suggests educational investments are made based on expected returns over the lifetime. Most other work in this field, particularly using British data, is based on demand models estimated in reduced form, with no earnings measures, or crudely constructed earnings measures, based on one or two earnings observations per individual.
We present a structural model of demand for educational investment which includes estimates of earnings paths for educational options as determinants of educational choice. This provides us with directly interpretable parameter estimates. The discount rate is also determined within our demand model.
Ability controlled earnings profiles are estimated by matching individuals from the General Household Survey to individuals in similar occupations from the National Child Development Survey (NCDS).
Our results show that expected earnings profiles vary according to observed ability and educational choice. Results from the demand model show that expected lifetime earnings have a significant impact on educational choice.
Other socio–demographic factors, particularly social class, also exhibit significant influences on the education decision. We estimate the discount rate to be lower than reported in other studies.     

PRIVATE AND PUBLIC EDUCATION: HUMAN CAPITAL ACCUMULATION UNDER PARENTAL TEACHING*

This paper examines a simple overlapping generations model of human capital accumulation under both the public and private education regimes. Both young individuals and their parents allocate time to human capital accumulation. Under the public education regime, the government collects tax to finance expenditure for education resources. We show that there exists a level of tax which maximizes the speed of human capital accumulation because of parental teaching; and, if the government chooses tax rates adequately, human capital grows faster and welfare levels become higher under the public education regime than under the private.     

The Superiority of Environmental Federalism in the Presence of Lobbying and Prior Tax Distortions

It is generally believed that environmental federalism tends to generate greater pollution emissions than centralized policymaking. This paper demonstrates that the opposite can occur in the presence of lobbying. Although the decentralized regime gives rise to a tax‐interaction effect, which induces policymakers to set lax environmental policies, it may also reduce the political pressure on enlarging allowed emissions. If the latter outweighs the former, then the decentralized regime will generate less pollution than the centralized regime. Moreover, we also show that the decentralized regime can be more efficient than the centralized regime, which provides an alternative theoretical support for the superiority of environmental federalism.     

Self-fulfilling patience

This paper explores macroeconomic implications of investment in patience in a standard neoclassical growth model with Becker-Mulligan endogenous time preferences. The endogenous discount rate acts as a new margin for inter-temporal decisions, in addition to the standard margin that hinges on the marginal return of capital. This time preference margin alters the equilibrium dynamics and stability of the neoclassical growth model substantially. When the discount rate is positive, there may exist multiple steady states that are either saddle-point stable or unstable. When the discount rate is negative, the unique steady state is locally indeterminate due to self-fulfilling patience investment. Interestingly, the existence of the local indeterminacy does not need any externality. When the discount rate is zero, various types of bifurcations can happen, leading to rich equilibrium dynamics such as limit cycles and chaos. We show that opening up the closed economy can cause aggregate instability.     

A model with endogenously determined cycles, discounting and growth

Summary. This article reexamines the role of consumption in growth and emphasises the external effects of aggregate consumption, viewed as consumption standards, as an additional impediment in the growth process. These external effects raise the productivity of the individuals and are positively related to their valuation of the future. Conditions are established under which this results in a marginal value of wealth that is an increasing function of consumption. This brings new types of multiple steady states, local indeterminacies and cyclical motions. Imposing extra homogeneity restrictions, balanced growth solutions with endogenous impatience emerge. The possibility of multiple convergent paths is univocally related to endogenous discount effects. A comparison with a benchmark planning economy indicates an excessive value for the rate of time preference and emphasises its insufficient adaptation to future utility in a stationary setting. Discrepancies along the transition path that rest on endogenous impatience versus fixed discount appear in a non-stationary environment when the competitive balanced growth solution is indeterminate. Received: May 5, 1996; revised version: May 19, 1997     

On recursive utility and the superneutrality of money

This paper investigates whether a change in the growth rate of the money supply enhances the rate of capital accumulation in a cash-in-advance monetary model with recursive utility. Although money is superneutral in the steady state, the effect of the growth rate of money supply on the speed of capital accumulation depends not only on the curvature of the felicity, but also on the slope and curvature of the discount rate function. We find that when the discount rate decreases with consumption and the elasticity of marginal utility is greater than unity, inflation yields a slower speed of capital accumulation.     

On the efficiency of ex ante and ex post pricing institutions

Summary This paper compares the performance of three pricing institutions in a decentralized matching model in which random matching occurs. In the first, sellers make public ex ante commitments to trade at pre-specified prices before matching occurs. In the second, the buyer and seller engage in an alternating offer bargaining game once a match has occurred. In the final pricing institution, one party is chosen at random to make a take it or leave it price offer. The steady state equilibrium with ex ante pricing pareto dominates the steady state equilibrium with ex post take it or leave it offers, which in turn pareto dominates the steady state equilibrium with alternating offers. As the discount factor goes to one, the equilibrium with ex ante pricing generates participation rates that tend toward the pareto efficient level. The ex post pricing institutions generate participation rates that are bounded away from efficient levels. The surplus loss associated with switching from ex ante pricing to take it or leave it offers is small when the discount factor is close to one. Contrary to many models of decentralized trade, ex post pricing institutions encourage too much search by buyers.The financial support of the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. The research for this paper was completed while the author was visiting the University of California at San Diego. The author thanks Douglas Gale for comments.     

Endogenous Discounting and Climate Policy

Under risk of abrupt climate change, the occurrence hazard is added to the social discount rate. As a result, the social discount rate (1) increases and (2) turns endogenous to the global warming policy. The second effect bears profound policy implications that are magnified by economic growth. In particular, we find that greenhouse gases (GHG) emission should be terminated at a finite time so that the ensuing occurrence risk will vanish in the long run. Due to the public bad nature of the catastrophic risk, the second effect is ignored in a competitive allocation and unregulated economic growth will give rise to excessive emissions. In the long term, the GHG emission paths under the optimal and competitive growth regimes lie at the extreme ends of the range of feasible emissions.     

Growth and distribution in an AK-model with endogenous impatience

This paper combines the classical approach to inequality, based on the division of society into classes with different saving propensities, and the social conflict approach, in which inequality inflicts costs to growth. We assume that each consumer’s discount factor is endogenously determined through two channels: (1) it is positively related to the consumer’s relative wealth, and (2) negatively affected by an aggregate measure of social conflict. Unlike in models with exogenous discount rates, steady state equilibria are indeterminate and the set of equilibria is a continuum parameterized by an index of income inequality. Under reasonable assumptions, the relationship between growth and inequality has an inverted-U shape.     

It's a Sin—Contraceptive Use,Religious Beliefs,and Long‐run Economic Development

This study presents a novel theory on the interaction of social norms, fertility, education, and their joint impact on long‐run economic development. The theory takes into account that sexual intercourse is utility enhancing and that the use of modern contraceptives potentially conflicts with prevailing social norms (religious beliefs). The theory motivates the existence of two steady states. At the traditional steady state, the economy stagnates, fertility is high, education is minimal, and the population sustains a norm according to which modern contraceptives are not used. At the modern steady state, the population has abandoned traditional beliefs, modern contraceptives are used, fertility is low and education and economic growth are high. Social dynamics explain why both equilibria are separated by a saddlepoint‐equilibrium (a separatrix), i.e. why it is so hard to transit from the traditional regime to the modern regime.     

Fertility,human capital and the wealth of families

Summary An overlapping generations model with parental altruism is examined. The existence of the optimal value function in a model with an endogenous discount rate is proven. Two development regimes are produced: a high fertility, low income and no growth steady state, and a perpetual growth equilibrium with low fertility and rising income.This paper is adapted from my dissertation. I would like to thank the members of my dissertation committee for helpful comments and suggestions, Messrs, Gary S. Becker, Robert E. Lucas, Jr., Kevin M. Murphy and Sherwin Rosen. I'd like to thank Brooks Pierce, Paul Romer, Ken Judd, Beth Ingram, Ed Prescott and Fernando Alvarez. I also thank the workshop participants of the University of Chicago, University of Pennsylvania, University of Toronto, University of Rochester, University of Washington, Penn State University, University at Buffalo, SUNY, Columbia University and University of Iowa.     

A uniqueness proof for monetary steady state

The framework in Lagos and Wright (2005) [20] combining decentralized and centralized markets is used extensively in monetary economics. Much is known about that model, but there is a loose end: only under special assumptions about bargaining power or decentralized market preferences has it been shown that the monetary steady state is unique. For general decentralized market utility and bargaining, I prove uniqueness for generic parameters with fiat money, and for all parameters with commodity money. As a corollary, I get monotone comparative statics.     

Human Fallibility, Complementarity, and Fiscal Decentralization

This paper examines economic growth properties under alternative fiscal organizations when a bureau's decisions are fallible. A country consists of J jurisdictions, which need a public service. In a centralized government, one authority decides on services in every jurisdiction. In a decentralized government, J authorities are in charge of each public service. An authority can have high ability or low ability, and an authority with high ability draws a good project with higher probability. We first show that the decentralized government provides the same average quality of public services, with lower variance, than does the centralized government. We then apply this result to an economic growth model where the value of the Solow residual is a constant elasticity of substitution (CES) function of public services. We show that there is a critical value of the degree of complementarity below which fiscal decentralization is more desirable than fiscal centralization for an expected economic growth, and the decentralized government has a lower variance of GDP growth.     

Optimal Experimentation in a Changing Environment

This paper studies optimal experimentation by a monopolist who faces an unknown demand curve subject to random changes, and who maximizes profits over an infinite horizon in continuous time. We show that there are two qualitatively very different regimes, determined by the discount rate and the intensities of demand curve switching, and the dependence of the optimal policy on these parameters is discontinuous. One regime is characterized by extreme experimentation and good tracking of the prevailing demand curve, the other by moderate experimentation and poor tracking. Moreover, in the latter regime the agent eventually becomes "trapped" into taking actions in a strict subset of the feasible set.     

A model of the optimal allocation of government expenditures

Government expenditures can be used for various socioeconomic objectives, including public education, consumption of public goods and services, and social protection. This paper analyzes the optimal allocation of public expenditures among these competing functions. We establish an overlapping generations model with heterogeneous individuals in which the government optimally chooses income tax, transfer payment, educational spending, and public consumption. Our model characterizes the transitional dynamics and the steady state of each function with and without a pay‐as‐you‐go intergenerational contract. We also conduct a simulation illustrating that the presence of an intergenerational contract may raise public consumption and social welfare in the steady state.     

An optimal monetary policy in an aggregate neoclassical model of economic growth

This paper shows that the usual Ramsey-Cass-Koopmans optimal-growth theory is applicable to decentralized monetary economies and illustrates, with a simple model, how optimal growth can be achieved via a simple monetary policy. Securities and the endogeneity aspect of the money supply are explicitly introduced. This paper shows that the steady state under optimal growth is a saddle point, that the dynamic behavior of the capital-labor ratio and real per capita consumption is identical to that found in the usual literature in which money is not introduced, and that the optimal monetary policy is “counter-cyclical.”     

Endogenous fertility and growth in a model with old age support

Summary. This paper examines the interrelationship between capital accumulation, fertility, and growth by introducing an endogenous fertility decision into Diamond's (1965) neoclassical growth model. Under the assumptions that children provide old age support and that individuals incur a variable time cost of raising children, it investigates the potential for cyclical fluctuations in the capital-labor ratio and fertility, as well as for development trap phenomena to be observed. It is shown that when capital and labor are highly substitutable in production, there is a unique steady state equilibrium, and either damped or undamped oscillations in fertility and the capital-labor ratio may occur. However, when the elasticity of substitution between capital and labor is less than one, two steady state equilibria may exist; one with a high capital-labor ratio and a high rate of population growth, and the other with a lower capital-labor ratio as well as a lower population growth rate. The former is a saddle, while the latter may be either a source or a sink. In the latter case development traps are possible. Received: June 16, 1997; revised version: December 18, 1997     

Dynamic optimization with a nonsmooth, nonconvex technology: the case of a linear objective function

This paper studies a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semicontinuous. The model also allows for a general form of irreversible investment. We show that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards. We establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existence of a critical capital stock below which extinction is possible and above which survival is ensured. These conditions generalize those known for the case of S-shaped production functions. We also show that as the discount factor approaches one, optimal paths converge to a small neighborhood of the capital stock that maximizes sustainable consumption.This paper is dedicated to Professor Mukul Majumdar on his 60th birthday. His research with various co-authors in the late 70s and the 80s pioneered innovative techniques for the analysis of nonconvex dynamic optimization models – both deterministic and stochastic. Roy considers himself particularly fortunate for having had the opportunity to learn economic theory and mathematical economics from Professor Majumdar. This paper has benefited from helpful comments and suggestions by an anonymous referee. Financial support from the 21st Century COE Program at GSE and RIEB, Kobe University, is gratefully acknowledged.