Ergodic chaos in optimal growth models with low discount rates

Summary We show that for every discount factor(0,1) one can find infinitely many strictly concave discrete-time optimal growth models in reduced form which have optimal policy functions exhibiting ergodic chaos. These reduced form models are interpreted in a two-sector optimal growth setting with utility functions depending on consumption as well as on capital.We thank Luigi Montrucchio for pointing out an error in a previous version of the paper.     

A qualitative approach to Markovian equilibrium in infinite horizon economies with capital

Using lattice programming and order theoretic fixpoint theory, we develop a new class of monotone iterative methods that provide a qualitative theory of Markovian equilibrium decision processes for a large class of infinite horizon economies with capital. The class of economies includes models with public policy, valued fiat money, monopolistic competition, production externalities, and various other nonconvexities in the production sets. The results can be adapted to construct symmetric Markov equilibrium in models with many agents and market incompleteness. As the methods are constructive, they provide the foundations for a rigorous analysis of numerical approximation schemes that study extremal Markovian equilibrium. Equilibrium comparative statics results relative to the space of economies are available. Of independent interest, we provide new conditions for preserving complementarity under maximization, and new generalized envelope theorems for nonconcave dynamic programming problems. Our fixed point algorithms are sharp, and are able to distinguish sufficient conditions under which Markovian equilibrium form a complete lattice of Lipschitz continuous, uniformly continuous and semicontinuous monotone functions as well as unique continuously differentiable equilibrium.     

Seigniorage and public good in an OLG model with cash-in-advance constraints

This paper is a contribution to the theory of the optimal monetary policy in overlapping generations models with a cash-in-advance constraint applying on old age consumption purchases. We are particularly interested in the study of seigniorage when the latter is used to finance public goods valued by consumers. Assuming that a public good enters the utility function and the existence of cash-in-advance constraints, we find a Laffer curve of seigniorage at steady-state. We also analyze the dynamic optimal monetary policy when the government maximizes an intertemporal social welfare function and can only resort to seigniorage to finance the purchases of the public good. We show that the optimal rate of money growth may be strictly higher than the one which maximizes steady-state seigniorage: the optimal amount of seigniorage may lie on the decreasing part of the Laffer curve of seigniorage. We finally suppose that the government can use lump-sum taxes in addition to seigniorage. Then, the Friedman rule prevails and public expenditures are only financed through lump-sum taxes.     

Dynamic efficiency of soil erosion and phosphor reduction policies combining economic and biophysical models

In this paper, we propose the use of the metamodeling approach to determine the optimal intertemporal management of soil and phosphorus losses from agricultural land. This approach enables to find a common equilibrium of the economic and biophysical systems. In contrast to the existing literature, the model takes into account nonlinear biophysical relationships and land-use choices. As a solution to the mathematical problems arising from this complex setup, we propose and employ a modified Cobb Douglas function in the empirical part of the paper. Most importantly, we allow for the comparison of different soil erosion and phosphorus reduction policies. The results show that an indirect policy in the form of soil protection scores (SPS) is highly inefficient, while another indirect policy in the form of land-use taxes is nearly as efficient as a direct policy.     

INTERTEMPORAL COMPLEMENTARITY AND OPTIMALITY: A STUDY OF A TWO‐DIMENSIONAL DYNAMICAL SYSTEM*

We study the underlying structure of the two‐dimensional dynamical system generated by a class of dynamic optimization models that allow for intertemporal complementarity between adjacent periods, but preserve the time‐additively separable framework of Ramsey models. Specifically, we identify conditions under which the results of the traditional Ramsey‐type theory are preserved even when the intertemporal independence assumption is relaxed. Local analysis of this theme has been presented by Samuelson (Western Economic Journal 9 (1971), 21–26). We establish global convergence results and relate them to the local analysis, by using the mathematical theory of two‐dimensional dynamical systems. We also relate the local stability property of the stationary optimal stock to the differentiability of the optimal policy function near the stationary optimal stock, by using the Stable Manifold Theorem.     

Optimal chaos,nonlinearity and feasibility conditions

Summary This study constructs a class of dynamic models in which optimal paths are generated by nonlinear transition functions similar to a tent map. We provide a sufficient condition under which such a transition function is a chaotic map. This characterization provides a way to construct complex nonlinear dynamics in a broad range of dynamic economic models.We would like to thank Michele Boldrin, William Brock, Richard Day, Mukul Majumdar, Lionel McKenzie, Tapan Mitra, Luigi Montrucchio, Jose Scheinkman, Rangarajan Sundaram and the referees for comments and discussions.     

The intertemporal substitution effect of government purchases on the interest rate: Empirical estimates for the Netherlands

In an intertemporal equilibrium setting, temporary government purchases may affect the real interest rate. A temporary change in the level of government spending does not affect the optimal intertemporal allocation of rational consumers, who will therefore try to maintain their consumption plans. Assuming no change in the supply of goods, there is an excess demand for goods and in a closed economy the real interest rate has to rise to maintain equilibrium at the goods market. In this paper it is examined whether predictions of intertemporal substitution models hold up to the experience of the Netherlands. We extend Barro's empirical work in three ways. First, we construct a new measure of temporary government outlays. Second, we take the time series behaviour of the data concerned into account and employ a less restrictive specification for the equation to be estimated. Third, we examine the intertemporal substitution effect of government purchases onreal interest rates. Our results provide only mixed support for the hypothesis that temporary government purchases raise real interest rates. It turns out that only temporary military outlays affect the nominal and real interest rate.     

On capital accumulation paths in a neoclassical stochastic growth model

Summary. Boldrin and Montrucchio [2] showed that any twice continuously differentiable function could be obtained as the optimal policy function for some value of the discount parameter in a deterministic neoclassical growth model. I extend their result to the stochastic growth model with non-degenerate shocks to preferences or technology. This indicates that one can obtain complex dynamics endogenously in a wide variety of economic models, both under certainty and uncertainty. Further, this result motivates the analysis of convergence of adaptive learning mechanisms to rational expectations in economic models with (potentially) complicated dynamics. Received: June 21, 1996; revised version: October 31, 1996     

Robust chaos in dynamic optimization models

The purpose of this paper is to investigate the (theoretical) importance of chaos as a phenomenon occurring in dynamic optimization problems. The intertemporal models we focus on are specified by a standard aggregative production function, an immediate return function depending on current consumption, capital input and a taste parameter, and a discount factor.We interpret “chaos” as a situation in which the Liapounov exponent of the relevant dynamical system is positive. This notion of chaos is related to the concept of “unpredictability” as measured by the Kolmogorov-Sinai entropy.In the family of intertemporal models, indexed by the taste parameter (with values lying in a closed interval), chaos is considered to be an “unimportant” phenomenon, if the set of parameter values for which chaos occurs is of Lebesgue measure zero.We identify a family of dynamic optimization models, for which the optimal transition functions are represented by the quadratic family of maps. Relying on the mathematical literature on the robustness of chaos for this family of maps, we conclude that chaos cannot be considered to be an unimportant phenomenon in dynamic optimization models.     

Optimal trade policy in tariff games with inside money

We construct a bilateral trade model incorporating two physical goods and a financial asset (inside money) to discuss the optimal trade policy that countries would choose to maximize their respective utilities. In this Nash tariff game, the trade of physical commodities only occurs geographically across countries, and the trade of inside money allows for intertemporal allocation of consumptions. When the preferences, present and future endowments for each country are given, according to our numerical analysis, trade surplus or deficit (inside money) and optimal tariff rates are endogenously determined when general equilibrium conditions hold. One country may purchase inside money to shift current consumption to the future, and the other may be willing to issue inside money for smoothing its consumptions in two periods. This imbalance trade contradicts traditional trade models which imply a balanced trade policy. We further find that the price of inside money as an implied interest rate also is determined by the trade intervention policies.     

Modeling intertemporal general equilibrium: An application to Austrian commercial policy

We present a multi-sector CGE model featuring forward looking investiment and savings behavior within an intertemporal optimization framework. Thus, the model captures several of the intertemporal effects of commercial policy that have been stressed by recent literature on current account adjustment. We argue that pursuing a simulation approach in addressing these issues is warranted by certain limitations and ambiguities of the analytical literature. In addition to presenting the details of the model structure, the paper addresses calibration issues relating to intertemporal parameters. The model is calibrated to a microconsistent data set for the Austrian economy. Finally, the paper features an application of the model to a simple tariff liberalization scenario.     

The Relative Dynamics of Investment and the Current Account in the G7-Economies

This paper contributes to the empirics of the intertemporal approach to the current account. We use a cointegrated VAR framework to identify permanent and transitory components of country-specific and global shocks. Our approach allows us to investigate empirically the sensitivity to persistence implied by many forward-looking models and our results shed new light on the excess volatility of investment encountered by Glick and Rogoff (1995). In G7 data, we find the relative current-account and investment response to be in line with the intertemporal approach.     

Controlling a biological invasion: a non-classical dynamic economic model

This paper analyzes the optimal intertemporal control of a biological invasion. The invasion growth function is non-convex and control costs depend on the invasion size, resulting in a non-classical dynamic optimization problem. We characterize the long run dynamic behavior of an optimally controlled invasion and the corresponding implications for public policy. Both control and the next-period invasion size may be non-monotone functions of the current invasion size; the related optimal time paths may not be monotone or convergent. We provide conditions under which eradication, maintenance control, and no control are optimal policies.      

Labor income,borrowing constraints,and equilibrium asset prices

Summary We develop a duality approach to study an individual's optimal consumption and portfolio policy when the individual has limited opportunities to borrow against future labor income and cannot totally insure the risk of income fluctuations. The individual's intertemporal consumption and portfolio problem is cast in a continuous-time setting under uncertainty. We transform the individual's intertemporal problem into a dual shadow prices problem that solves the shadow prices for the individual's optimal consumption plan or equivalently the individual's intertemporal marginal rates of substitution. We show that the shadow prices process can be expressed as a product of a martingale and a decreasing process (normalized by the bond price). The existence of an optimal solution to the individual's intertemporal consumption and portfolio problem is established via duality. The duality approach also allows us to characterize in a sample way the individual's optimal consumption and portfolio policy in the presence of labor income and borrowing constraints. Equilibrium implications of borrowing constraints on asset prices are also discussed in the paper.This is a revised version of an earlier paper, entitled Consumption and Portfolio Decisions with Labor Income and Borrowing Constraints. We thank George Constandinides, Ayman Hindy, and Chi-fu Huang for helpful comments. We also thank two anonymous referees for their helpful comments and suggestions. Financial support from the Batterymarch Fellowship Program (for Hua He) is gratefully acknowledged.     

Optimal monetary and fiscal policies in a search theoretic model of monetary exchange

Search models of monetary exchange commonly assume that terms of trade in anonymous markets are determined via Nash bargaining, which generally causes monetary equilibrium to be inefficient. Bargaining frictions add to the classical intertemporal distortion present in most monetary models, whereby agents work today to obtain cash that can be used only in future transactions. In this paper, we study the properties of optimal fiscal and monetary policy within the framework of Lagos and Wright (2005). We show that fiscal policy can be implemented to alleviate underproduction while money is still essential. If lump sum monetary transfers are available, a production subsidy can restore the efficiency of monetary equilibria. The Friedman rule belongs to the optimal policy set, but higher inflation rates are also possible. When lump-sum monetary transfers are not available, equilibrium allocations are generally not first-best. Nevertheless, fiscal policy still results in substantial welfare gains. Money can be extracted from circulation via a sales tax on decentralized market activities, and the Friedman rule is only optimal if the buyer has relatively low bargaining power.     

The Dynamics of Temporary Policies in a Small Open Economy

The paper corrects a subtle, but crucial, conceptual flaw in a solution procedure initially proposed in 1990 by Sen and Turnovsky to analyze anticipated regime changes in small open economies based on the intertemporal optimization of rational forward–looking agents. The problem is its failure to consider the intertemporal solvency of the economy consistently. The paper focuses on temporary shocks, although the procedure also applies to announced future permanent policy changes. Since the issue is generic and relevant to a large class of policy changes, it is important for the intertemporal solvency aspect to be incorporated consistently. The authors show that the seriousness of the error in the previous solution procedure depends upon the specific shock, and two contrasting examples are discussed.     

External shocks,consumption-smoothing and capital mobility in India: evidence from an intertemporal optimization approach

We examine the solvency of India’s current account (CA) in the post-liberalization period using intertemporal optimization approach to the CA. Using quarterly data ranging from 1996Q1 to 2014Q2, we estimate a benchmark consumption-smoothing model and an extended model that incorporates external shocks. Overall, we find that the predicted optimal CA in both the models can track the actual CA movements and the extended model performs better over the benchmark model. Further, we also find that the optimal CA is more volatile than the actual CA which implies that the capital flows have been less than optimal and thus makes an interesting case for further liberalization of the capital account. Our findings suggest that policies aimed at further liberalization of capital flows will allow larger CA deficits to achieve higher economic growth since it will help agents to further smoothen their consumption without worrying about risks associated with insolvency.     

On the indeterminacy of capital accumulation paths

In neoclassical optimal growth models the stability of the accumulation paths depends on the discount parameter. We prove that, for discount factors small enough, the policy function which describes an optimal path can be of any type. The result is achieved using the notion of α-concavity. We adopt a constructive approach. Given any twice differentiable map we show how to construct an optimal growth problem which produces that map as the optimal policy function. A consequence is that “chaos” can appear in these models. We also provide bounds on the values of the discount parameter for which “indeterminacy” is possible.