On dynamic programming with unbounded returns

Summary. Some economic models like those of endogenous growth motivate the analysis of a class of recursive models sharing the property that the return function is not bounded along feasible paths. We consider a strategy of proof which allows to deal with many unbounded recursive models exploiting bounds to the rates of growth rather than to the levels.     

Observational equivalence of the overlapping generations and the discounted dynamic programming frameworks for one-sector growth

Two alternative frameworks for one-sector growth are considered: an overlapping generations model with two-period lived, strictly selfish agents and a discounted dynamic programming model with infinitely lived agents. It is shown that under certain assumptions the two models are observationally equivalent, i.e., generate identical timepaths for all aggregate variables. One of the key assumptions is that the steady-state real interest rate in the overlapping generations model be strictly greater than the growth rate. The result is proved for the deterministic case and extension to a stochastic case is indicated.     

On optimal cycles in dynamic programming models with convex return function

Summary. In this paper we study the behavior of optimal paths in dynamic programming models with a strictly convex return function. Such a model has been investigated in Dawid and Kopel (1997) who assume that the growth of a renewable resource is governed by a piecewise linear function. We prove that in their model the optimal cycles undergo the following qualitative changes or bifurcations: a cycle of period n“bifurcates” into a cycle of period n+1 for increasing elasticity of the return function. We also show that under the assumption of a concave differentiable growth function the qualitative properties of the optimal policy remain valid: oscillating behavior is optimal. Furthermore, we demonstrate numerically that the period of a cyclic optimal path increases if the convexity of the return function (measured by the elasticity) increases. Received: January 22, 1997; revised version: October 13, 1997     

On knife-edge conditions with unbounded growth

Imagine an economic process modeled with help of ordinary differential equations with some fixed initial conditions and suppose that a solution to the system under consideration, for some reason, should enjoy desired conditions. It appears that small changes in the system describing the modeled process may have devastating effect on the fulfillment of the desired conditions by solutions of the perturbed system. Such a feature of a system is called a knife-edge condition in the literature on economic growth and it has recently found some deeper interest in that literature. However, the available results seem to be unclear and not correct – this paper shows how to correct and improve these results, creating consistent mathematical foundations of that theory. The main contribution of the paper is that knife-edge conditions are present whenever long-term unbounded growth of an economic variable is required as (or implied by) a desired regularity condition.     

Robust ergodic chaos in discounted dynamic optimization models

Summary We identify a family of discounted dynamic optimization problems in which the immediate return function depends on current consumption, capital input and a taste parameter. The usual monotonicity and concavity assumptions on the return functions and the aggregative production function are verified. It is shown that the optimal transition functions are represented by the quadratic family, well-studied in the literature on chaotic dynamical systems. Hence, Jakobson's theorem can be used to throw light on the issues of robustness of ergodic chaos and sensitive dependence on initial conditions.     

Stochastic convexity in dynamic programming

Summary. This paper explores sufficient conditions for a continuous stationary Markov optimal policy and a concave value function in stochastic dynamic programming problems. Also, the paper addresses conditions needed for the differentiability of the value function. The paper uses conditions such as first order stochastic dominance, second order stochastic dominance and concave stochastic dominance that are widely applied in economics. Received: February 23, 2001; revised version: May 19, 2002 RID="*" ID="*" I am deeply indebted to Prajit Dutta for patient assistance and advice. This paper has benefited from discussions with Tsz Cheong Lai, Tackseung Jun, Werner Stanzl and Satyajit Bose as well as feedback from an anonymous referee.     

On inferior inputs and marginal returns

An input is inferior if and only if an increase in its price raises all marginal productivities. A sufficient condition for input inferiority under quasi-concavity of the production function is then that there are increasing marginal returns with respect to the other input and a non-positive marginal productivity cross derivative. Thus, contrary to widespread opinion, input “competitiveness” is not needed. We discuss these facts and illustrate them by introducing a class of simple production function functional forms. Our results suggest that the existence of inferior inputs is naturally associated with increasing returns, and possibly strengthen the case for inferiority considerably.     

Solving optimal growth models with vintage capital: The dynamic programming approach

This paper deals with an endogenous growth model with vintage capital and, more precisely, with the AK model proposed in [R. Boucekkine, O. Licandro, L.A. Puch, F. del Rio, Vintage capital and the dynamics of the AK model, J. Econ. Theory 120 (1) (2005) 39-72]. In endogenous growth models the introduction of vintage capital allows to explain some growth facts but strongly increases the mathematical difficulties. So far, in this approach, the model is studied by the Maximum Principle; here we develop the Dynamic Programming approach to the same problem by obtaining sharper results and we provide more insight about the economic implications of the model. We explicitly find the value function, the closed loop formula that relates capital and investment, the optimal consumption paths and the long run equilibrium. The short run fluctuations of capital and investment and the relations with the standard AK model are analyzed. Finally the applicability to other models is also discussed.     

On the concepts of returns to scale

We clarify various concepts of returns to scale scattered in the literature by presenting their six different concepts. We find the global relationship among them and apply some of the results to determine the shape of the average cost curves.     

Recursive utility with unbounded aggregators

A new framework is presented for the study of the existence and uniqueness of solutions to the Koopmans’ equation in the unbounded case, that is based on the contraction mapping approach. In the bounded below case with bounded consumption streams, uniqueness of the solution in the whole class of weak-star continuous utility functions is obtained. When the aggregator is unbounded below and/or consumption streams are unbounded, existence of a weak-star continuous solution is shown, and a simple criterium to check the sufficient conditions for existence is provided. Juan Pablo Rincón-Zapatero and Carlos Rodríguez-Palmero gratefully acknowledge financial support by the Ministerio de Educación y Ciencia and FEDER funds under Research Projects MTM2005-06534 and SEJ2005-08709/ECON, respectively, and by Consejería de Educación de la Junta de Castilla y León under Research Projects VA99/04 and VA017B05, respectively. This paper has substantially benefited from the comments of an anonymous referee. Particular and special thanks are due to Robert Becker for his very helpful comments and valuable suggestions.     

Empirical applications of discrete choice dynamic programming models

The development over the past 25 years of methods for the estimation of discrete choice dynamic programming (DCDP) models opened up new frontiers for empirical research in a host of areas, including labor economics, industrial organization, economic demography, health economics, development economics, political economy and marketing. In this paper, we first describe the development of the DCDP framework, showing how it was a natural extension of static discrete choice modeling. We then summarize six papers that adopt the DCDP paradigm that address substantively important social and economic questions. Finally, we consider the issue of the credibility of empirical findings based on the structural estimation of DCDP models.     

On optimal steady states of n-sector growth models when utility is discounted

This paper contains results on local and global stability of n-sector growth models when utility is discounted mostly for small rates of discount. It is well known that when future utility is not discounted one can prove precise results about optimal steady states (OSS's) under fairly general assumptions. In particular, existence, uniqueness, and turnpike properties have been established by several authors. The counter examples presented by Kurz, Sutherland, and Weitzman, however, show that when utility is discounted, additional assumptions are required to obtain turnpike results. In general, it would be interesting to know how the submanifolds of stability change as δ changes. One hopes that certain conditions on the utility function would be sufficient to “classify” the submanifolds of stability and instability. Such a question is apparently very difficult to answer, but we think that the results obtained here will help in this task.The proof that the turnpike theorem holds for discount factors near one is divided in two parts. First, we prove that optimal paths “visit” neighborhoods of the modified OSS's. Then, we prove that local stability holds for such neighborhoods.In order to show this fact, we must prove that the local “stable manifold” varies continuously with the discount factor. This roundabout method is necessary since our problem is similar to proving uniform continuity with respect to a parameter of solutions of a differential equation in a noncompact interval of time.Other problems analyzed here include uniqueness and continuity of OSS's. We also discuss the relation between the saddle-point property and the local stability of infinite horizon optimal growth paths.     

Computing equilibria with increasing returns

This paper shows how standard algorithms for solving economic general equilibrium models may be modified so as to solve models in which some or all industries have increasing returns to scale. The modification requires that some quantities as well as prices be quoted by the algorithmic auctioneer. It is applicable to a wide range of models in which firms pricing behavior follows simple rules. As an illustration of the technique a small general equilibrium model with increasing returns is presented and solved numerically.     

On the correlation between stocks and art market returns

We estimate the correlation between the returns of an S&P 500-based portfolio and Renoir paintings. Unlike previous studies that relied on single-point estimates of the correlation to explore the merits of adding art assets to a portfolio of stocks, we rely on a wild bootstrap algorithm to determine confidence intervals for the correlation estimates. We find that these confidence intervals are so wide (a situation not peculiar to our example) that it seems impossible to make absolute remarks about the merits of adding art-related assets to stocks portfolios. Moreover, our results suggest that previous conclusions regarding the correlation between art and stocks should be taken with some scepticism.     

On the effect of R&D in returns to experience

Using a recent macroeconomic database, we empirically estimate an ‘erosion effect’: we show that Total Factor Productivity (TFP) growth decreases returns to experience. The ‘erosion effect’ is typically stronger within the rich and the most experienced countries in the world. This article complements the scarce microeconomic literature on the issue and the theoretical macroeconomic literature recently coming to the conclusion that this effect has important implications for the interaction between human capital accumulation and TFP.     

The multiplant firm with increasing returns to scale

The paper analyzes allocation among plants whose production is characterized by “nicely convex-concave production functions” (intended to fulfill the property of initial increasing returns to scale followed by subsequent decreasing returns to scale). Necessary conditions for optimal allocation are derived. The complete solution is given for the simplified case of identical plants, and examples show the various possibilities that may arise. The relative difficulty of solution for even the simplest case emphasizes the problem of applying our economic theories to the real world.     

Estimating tax noncompliance with evidence from unaudited tax returns

This article estimates the degree of tax noncompliance using evidence from unaudited tax returns. Measurements of noncompliance are derived from the relationship between reported charitable contributions and reported income from wages and salary as compared to alternative reported income sources such as self-employment, farm and other small business income. Assuming that the source of one's income is unrelated to one's charitable inclinations and that the ratio of true income to taxable income does not vary by income source, any difference in the relationship between charitable contributions and the source of income can be attributed to (relative) underreporting by the individual. We find that the implied amount of noncompliance is significant and that it varies by source of income, as well as between positive and negative values of each type of income.     

On the relationship between energy returns and trading volume: a multifractal analysis

Forecasting and modelling commodities price movements and the activity of energy markets are of real interest to investors and policymakers, especially during turbulent times. This study investigates the volume–returns relationship for two major energy markets (oil and gas) during the recent global financial crisis. Unlike previous studies, we examine this relationship by applying an original fractal approach to intraday data, which has the advantage of accounting for further non-normality, nonstationarity, and fat-tailedness properties. Our study provides two interesting findings. First, we find a significant multifractal relationship between returns and volume in both markets and across all timescales, suggesting nonlinearity in the cross-correlation between returns and volume and rejecting the efficiency assumption. Second, the measure of multifractality in this relationship shows that the magnitude of the fluctuations during bearish and bullish trends affects the volume–return relationship differently, and that the oil market exhibits higher volatility than does the gas market.