Long horizon versus short horizon planning in dynamic optimization problems with incomplete information

Summary. This paper compares the implications of short and long horizon planning in dynamic optimization problems with the structure of a standard one-sector growth model if agents have incomplete knowledge about the production function. Agents know the output and rate of return at the current capital stock and use an estimation of the production function based on this knowledge to determine current consumption. For standard utility functions without wealth-effects both long and short planning horizons yield convergence to the steady state - however at a faster rate than optimal -, or fluctuations around the steady state, and in both cases, long horizon planning yields a policy which locally at the steady state is closer to the optimal one than short horizon planning. On the other hand, for preferences with wealth effects where the intertemporal optimal path exhibits fluctuations, long horizon planning destabilizes the path and short horizon planning can generate paths which are qualitatively closer to the optimal one and yield higher discounted utility.Received: 5 April 2001, Revised: 15 September 2003, JEL Classification Numbers: C61, D83, D90.Herbert Dawid: The author would like to thank Richard Day for numerous stimulating discussions which led to this article and an anonymous referee for helpful comments     

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.     

Optimal Two Sector Growth Models with Three Factors

The paper is devoted to construction of optimal trajectories in the model, which balances growth trends of investments in capital and labor efficiency. The model is constructed within the framework of classical approaches of the growth theory. It is based on three production factors: capital, educated labor and useful work. GDP level is described by a production function of the Cobb–Douglas type. The utility function of the growth process is given by an integral consumption index discounted on the infinite horizon. The optimal control problem is posed to balance investments in capital and labor efficiency. The problem is solved on the basis of dynamic programming principles. A novelty of the solution consists in constructing nonlinear stabilizers constructed on the feedback principle, which leads the system from any current position to a steady state. Growth and decline trends of the simulated trajectories are studied for all components included in the model.     

Consumption Externalities,Habit Formation and Equilibrium Efficiency*

We analyze the welfare properties of the competitive equilibrium in a capital accumulation model where individual preferences are subject to both habit formation and consumption spillovers. Using an additive specification for preferences, according to which the argument in the utility function is a linear combination of present and past values of own consumption and consumption spillovers, we analyze the circumstances under which these spillovers are a source of inefficiency. It is shown that consumption externalities have to interact with habits in order to generate an inefficient dynamic equilibrium. Finally, we characterize optimal tax policies aimed at restoring efficient decentralized paths.     

On income fluctuations and capital gains

We consider the infinite time horizon problem of asymptotically maximizing the expected accumulated discounted utility in a one-good production economy. The available capital in a given period is given by the production of the previous period plus a random variable. The product of the discount and interest factors is either (1) greater than or (2) equal to one. Under (1) the optimal policy exists under certain conditions and always under (2). The optimal capital sequence almost surely goes to infinity. Under (1) with conditions on the utility one almost surely reaches a capital level above which the sequence is increasing.     

Does exchange rate variability matter for welfare? A quantitative investigation of stabilization policies

This paper studies what degree of exchange rate stabilization is optimal for several types of open economies. This is accomplished through a quantitative evaluation of optimal monetary policy rules in a two-country sticky-price model. First, a calibrated benchmark model with incomplete asset markets supports past conclusions from simpler models, emphasizing inflation stabilization rather than exchange rate stabilization. It also highlights that the utility gains from optimal stabilization policy are small. Second, while an economy extended to include consumer habits implies greater sensitivity by households to consumption variability, it has only minor effects on the benchmark conclusions and benefits. Finally, these conclusions are altered under an alternative environment where international asset markets exhibit asymmetry in the form of “original sin.” Such countries can benefit from policies that aggressively stabilize the exchange rate, with utility gains larger than the previous cases.     

Dynamic Tax Evasion with Habit Formation in Consumption*

We model the optimal intertemporal decision of an agent who chooses tax evasion and consumption, over an infinite lifetime horizon, where consumption is driven by habits. We find the following: (i) tax evaders reduce consumption in the early stages of habit accumulation and increase it over time; (ii) habit formation has a dampening effect on tax evasion; (iii) neglecting tax evasion can lead to habit overestimation; (iv) the effect of the tax rate on tax evasion is ambiguous; (v) heavy fines are more efficient than frequent controls in reducing tax evasion.     

Consumption externality, efficiency and optimal taxation in one-sector growth model

In an economy with consumption externalities, existing studies find that a competitive equilibrium is efficient in the long run and remains efficient in transitions if preferences are homothetic. This paper revisits the efficiency issue in an otherwise standard one-sector growth model where consumption externalities affect a utility via their effects on the time preference. We find that even if preferences are homothetic, the externality changes the marginal rate of substitution between now and future and leads to a disparity in the intertemporal elasticity of substitution between the centrally planned economy and a decentralized economy. As a result, a competitive equilibrium is inefficient in transition dynamics. We characterize an optimal tax/subsidy structure that enables the allocation in a decentralized economy to replicate the social optimum.     

Risk Aversion and the Investment Horizon: A New Perspective on the Time Diversification Debate

Investment managers generally subscribe to the principle of time diversification. This implies that a larger portion of the portfolio should be devoted to risky assets as the investment horizon increases. In contrast, academics have shown that for investors with utility functions characterized by constant relative risk aversion, the optimal asset-allocation strategy is independent of the investment horizon. The relative risk aversion in these studies is assumed to be constant both with respect to wealth as well as investment horizon. We suggest a utility function that explicitly captures the notion that individuals are more risk tolerant when the investment horizon is long, thereby validating the intuitively appealing time diversification argument.     

Optimal Intertemporal Consumption under Uncertainty

We analyze the optimal consumption program of an infinitely lived consumer who maximizes the discounted sum of utilities subject to a sequence of budget constraints where both the interest rate and his income are stochastic. We show that if the income and interest rate processes are sufficiently stochastic and the long run average rate of interest is greater than or equal to the discount rate, then consumption eventually grows without bound with probability one. We also establish conditions under which the borrowing constraints must be binding and examine how the income process affects the optimal consumption program. Journal of Economic Literature Classification Number: D91.     

Conditions for Sustainable Optimal Economic Development

This paper shows that, for dynamic optimizing economies with different types of natural resource, environmental, and human‐made capital stocks, a necessary and sufficient condition for permanently sustaining an optimal utility/consumption level is the stationarity of the current‐value Hamiltonian. For economies whose development is not exogenously and directly affected by time (i.e., time‐autonomous economies), this stationarity condition generalizes Dixit et al.’s (1980 ) “zero‐net‐aggregate‐investment” rule of sustain‐ability, which in turn generalizes Solow‐Hartwick’s sustainability rule. For non‐autonomous economies, the stationarity condition is not generally fulfilled, and the current‐value Hamiltonian under (over) estimates the true welfare level by an amount equal to the discounted value of the net “pure time effect.” For the non‐autonomous case of a time‐dependent utility discount rate, a general condition on the discount rate function (of which the hyperbolic discount rate function is a special case) upholds the results obtained for autonomous cases. The paper concludes with a discussion of policies that promote both optimality and sustainability objectives.     

On Optimal Commodity Taxes When Consumption Is Time Consuming

This paper studies the problem of optimal taxation of commodities when consumption is a time‐consuming activity. This is done under two distinct preference separability assumptions: between goods and labor supply, and between goods and leisure. It argues that with the labor separability, the traditional uniform taxation results of optimal tax theory continue to hold. With leisure separability, on the other hand, consumption time is a major ingredient of optimal tax rates. However, the relationship between consumption time and optimal tax rates depends crucially on the representation of the economy. In representative consumer economies, time differences determine the pattern of optimal tax rates so that goods whose consumption take more time are subjected to higher tax rates. When individuals have different earning abilities, redistributive, incentive, and efficiency considerations also come into play resulting in a complex relationship. The paper derives formulas for optimal commodity taxes in this case on the basis of three different tax structures: linear commodity taxes in combination with linear and nonlinear income taxes, and nonlinear commodity taxes in combination with nonlinear income taxes.     

An income fluctuation problem

A consumer has a t period planning horizon problem, at each period, he gets additionals incomes that are i.i.d. random variables, and he must decide how much of his income will be spent in consumption, yielding some utility, and how much will be saved to maximize the total expected utility. No borrowing is allowed, and for the amounts saved no interest is paid. Using the concepts of competitive prices it is shown that as t → ∞ the corresponding limit of the consumption policy is strictly bounded above by the expected value of the random income.     

A simple explanation of some key time preference anomalies

Abstract Individual time discounting behaviour experimentally exhibits important anomalies that are inconsistent with the standard discounted utility model. These include the time preference reversal characteristic of hyperbolic discounting, the magnitude effect and the extreme sign effect. I propose a simple explanation of discounting that accounts for these three anomalies simultaneously, within the context of the expected utility model with uncertainty, risk aversion, and preference for precautionary saving.     

FIRST‐BEST FISCAL POLICY WITH SOCIAL STATUS*

This paper examines the first‐best fiscal policy in a stochastic, infinite‐horizon representative agent model that exhibits consumption‐enhanced as well as wealth‐enhanced social status in the household utility. We show that the first‐best labour tax rate is a positive constant that is used to correct negative consumption externalities. The first‐best tax rate on capital income is also positive in order to overturn agents' status‐seeking capital over‐accumulation. Moreover, we find that in sharp contrast to a conventional automatic stabilizer, the first‐best capital tax rate moves in the opposite direction with shocks to firms' production technology, thus exacerbating the business cycle.     

Consumption adaptation,anticipation‐bias,and optimal income taxation

Adaptation is omnipresent but people systematically fail to correctly anticipate the degree to which they adapt, leading them to make irrational intertemporal decisions. This paper concerns optimal income taxation to correct for such anticipation‐biases in a framework where consumers adapt to earlier consumption levels. The analysis is based on a general equilibrium OLG model with endogenous labor supply and savings where each consumer lives for three periods. The results show how a paternalistic government may correct for the effects of anticipation‐bias through a combination of time‐variant marginal labor income taxes and savings subsidies/taxes. The optimal policy mix remains the same, irrespective of whether consumers commit to their original life time plan for work hours and consumption or reoptimize later on when realizing that they have already adapted more than expected.     

Consumption, Wealth, and Indebtedness in the Context of Uncertainty: The Consumption Function Meets Portfolio Theory

The objective of this paper is to provide a sound theoretical framework for the empirical analysis of consumer indebtedness, by integrating Portfolio theory with the Life-Cycle hypothesis (LCH) model of consumption. Modern versions of this LCH theory almost always assume that utility is additive over time, but in this study, the multiplicative Cobb–Douglas function is used. The new synthesis also explains the stochastic properties of consumption more fully and clearly than previous studies, in particular the uncertainty arising from rates of return on risky assets. The new theory will also help to improve the explanation of the surprise changes in consumption because these sources of risk are incorporated explicitly into the analysis.     

Intertemporal Consumption Smoothing and Capital Mobility: Evidence from Australia

This paper examines the optimality of international capital flows to Australia, a persistent net importer of capital, during its post‐capital controls period 1984–99. The evolution of Australia’s current account balance is compared against a benchmark derived from an optimising model of intertemporal consumption smoothing. The consumption‐smoothing approach to the determination of the current account implies that international capital flows act as a buffer to smooth aggregate consumption in the face of temporary shocks to the economic fundamentals: changes in national cash flow (that is, changes in the level of output, investment or government spending). It is found that in the early 1990s a structural break occurred in the relationship between consumption and national cash flow, which coincides with a switch from debt‐financing to equity‐financing of the current account deficit. In the decade of the 1990s following this structural break (and unlike the decade of the 1980s which preceded this break), international capital flows to Australia implied a path for consumption which was broadly consistent with expected‐utility maximisation under the consumption‐smoothing model of the current account.     

Rational inattention,long-run consumption risk,and portfolio choice

This paper explores how the introduction of rational inattention (RI) – that agents process information subject to finite channel capacity – affects optimal consumption and investment decisions in an otherwise standard intertemporal model of portfolio choice. We first explicitly derive optimal consumption and portfolio rules under RI and then show that introducing RI reduces the optimal share of savings invested in the risky asset because inattentive investors face greater long-run consumption risk. We also show that the investment horizon matters for portfolio allocation in the presence of RI, even if investment opportunities are constant and the utility function of investors is constant relative risk aversion. Second, after aggregating across investors, we show that introducing RI can better explain the observed joint dynamics of aggregate consumption and the equity return. Finally, we show that RI increases the implied equity premium because investors under RI face greater long-run consumption risk and thus require higher compensation in equilibrium.     

Conspicuous Leisure: Optimal Income Taxation When Both Relative Consumption and Relative Leisure Matter*

In previous studies on public policy under relative‐consumption concerns, leisure comparisons have been ignored. In this paper, we consider a two‐type optimal non‐linear income tax model, in which people care about both their relative consumption and their relative leisure. Increased consumption positionality typically implies higher marginal income tax rates for both ability types, whereas leisure positionality has an offsetting role. However, this offsetting role is not symmetric; concern about relative leisure implies a progressive income tax component (i.e., a component that is larger for the high‐ability type than for the low‐ability type). Leisure positionality does not modify the policy rule for public‐good provision.