Linear learning in changing environments

The decision maker receives signals imperfectly correlated with an unobservable state variable and must take actions whose payoffs depend on the state. The state randomly changes over time. In this environment, we examine the performance of simple linear updating rules relative to Bayesian learning. We show that a range of parameters exists for which linear learning results in exactly the same decisions as Bayesian learning, although not in the same beliefs. Outside this parameter range, we use simulations to demonstrate that the consumption level attainable under the optimal linear rule is virtually indistinguishable from the one attainable under Bayes’ rule, although the respective decisions will not always be identical. These results suggest that simple rules of thumb can have an advantage over Bayesian updating when more complex calculations are more costly to perform than less complex ones. We demonstrate the implications of such an advantage in an evolutionary model where agents “learn to learn.”     

Supporting others and the evolution of influence

In this paper we study environments in which agents can transfer influence to others by supporting them. When planning whom to support, they should take into account the future effect of this, since the receiving agent might use this influence to support others in the future. We show that in the presence of a finite horizon there is an essentially unique optimal support behavior which can be characterized in terms of associated marginal value functions. The analysis of these marginal value functions allows us to derive qualitative properties of optimal support strategies under different specific environments and to explicitly compute the optimal support behavior in some numerical examples. We also investigate the case of an infinite horizon. Examples show that multiple equilibria may appear in this setting, some of which sustaining a degree of cooperation that would not be possible under a finite horizon.     

Egalitarianism under population change: Age structure does matter

We study the compatibility of the optimal population size concepts produced by different social welfare functions and egalitarianism meant as “equal consumption for all individuals of all generations”. Social welfare functions are parameterized by an altruism parameter generating the Benthamite and Millian criteria as polar cases. The economy considered is in continuous time and is populated by homogeneous cohorts with a given life span. Production functions are linear in labor, (costly) procreation is the unique way to transfer resources forward in time. First, we show that egalitarianism is optimal whatever the degree of altruism in “perpetual youth” model, that is when lifetime span is finite but age structure does not matter: in this case egalitarianism does not discriminate between the social welfare functions considered. Then we show that, when life span is finite but age structure matters, egalitarianism does not arise systematically as an optimal outcome. In particular, in a growing economy, that is when population growth is optimal in the long-run, this egalitarian rule can only hold when the welfare function is Benthamite. When altruism is impure, egalitarianism is impossible in the context of a growing economy. Either in the Benthamite or impure altruism cases, procreation is never optimal for small enough life spans, leading to finite time extinction and maximal consumption for all existing individuals.     

Adaptive learning,endogenous uncertainty,and asymmetric dynamics

I present a simple model where forecasting confidence affects aggregate demand. It is shown that this model has similar stability properties, under statistical and evolutionary learning, as a model without a confidence affect. From this setup, I introduce “Expectational Business Cycles” where output fluctuates due to learning, heterogeneous forecasting models and random changes in the efficient forecasting model. Agents use one of two forecasting models to forecast future variables while heterogeneity is dictated via an evolutionary process. Increased uncertainty, due to a shock to the structure of the economy, may result in a sudden decrease in output. As agents learn the equilibrium, output slowly increases to its equilibrium value. Expectational business cycles tend to arrive faster, last longer and are more severe as agents possess less information.     

Adaptive learning and monetary exchange

This paper investigates three classic questions in monetary theory: How can an intrinsically worthless asset, such as fiat money, maintain value as a medium of exchange? What are the short-run and long-run effects of a change in the money supply? What is the social cost of inflation? I answer these questions using a microfounded model of monetary exchange that replaces the rational expectations assumption with an adaptive learning rule. First, I show that monetary exchange is a robust arrangement in the sense that agents are able to learn the stationary monetary equilibrium while the non-monetary equilibrium is unstable under learning. Second, an unanticipated monetary injection has real effects in the short-run because learning the value of money takes time. In the long run, agents successfully learn the value of money, hence money is neutral. Third, under a constant money growth policy, an increase in the growth rate of money increases output in the short-run producing a short-run Phillips curve. A ten percent increase in the money growth rate has a social cost of 0.41 percent of output per year. Alternatively, a ten percent decrease in the money growth rate has a social benefit of 0.37 percent of output per year.     

Seniority bias in a tournament

The paper presents a model where the probability of promotion tends to increase with seniority (overall labor market experience) without relying on the accumulation of general human capital. To this end, we consider the optimal design of a tournament (a relative compensation scheme) between two agents with different time horizon, the young and the old, in an overlapping generations framework. When the principal can only imperfectly monitor each agent’s effort level, the difference in time horizon leads to the ex post difference in the marginal value of effort between the two agents. In this case, the optimal tournament necessarily involves a bias towards the old agent. Within this framework, we also examine the relationship between: (1) the monitoring accuracy and the optimal bias; and (2) the value of outside options and the optimal bias.     

Incomplete markets, ambiguity, and irreversible investment

The problem of irreversible investment with idiosyncratic risk is studied by interpreting market incompleteness as a source of ambiguity over the appropriate no-arbitrage discount factor. The maxmin utility over multiple priors framework is used to model and solve the irreversible investment problem. Multiple priors are modeled using the notion of κ‐ignorance. This set-up is used to analyze finitely lived options. For infinitely lived options the notion of constant κ‐ignorance is introduced. For these sets of density generators the corresponding optimal stopping problem is solved for general (in-)finite horizon optimal stopping problems driven by geometric Brownian motion. It is argued that an increase in the set of priors delays investment, whereas an increase in the degree of market completeness can have a non-monotonic effect on investment.     

Ratchet consumption over finite and infinite planning horizons

Ratchet consumers want their spending to always increase and never decrease. We find an optimal consumption rule for ratchet consumers by maximizing an expected utility that eschews spending declines, yet permits a range of choices for felicity and time preference functions. This solution can be tailored to fit both retirees with finite planning horizons and endowments with infinite planning horizons. We assume complete markets modeled by a pricing kernel generated by a Lévy process. When the kernel is log-normal, we obtain closed-form solutions for both finite and infinite horizons.     

Consumer misperceptions,uncertain fundamentals,and the business cycle

This paper estimates the importance of shocks to consumer misperceptions “noise shocks” for U.S. business cycle fluctuations. I embed imperfect information as in Lorenzoni (2009) into a Smets and Wouters (2007)-type DSGE model. Agents only observe aggregate productivity and a signal about the permanent component contaminated with noise. Based on this information agents form beliefs about the temporary and the permanent component of productivity. Shocks to the signal (noise shocks) trigger aggregate fluctuations unrelated to changes in productivity. Bayesian estimation shows that noise shocks explain up to 14 percent of output and up to 25 percent of consumption fluctuations. Nominal rigidities and the specification of the monetary policy rule are crucial for the importance of noise shocks. These features help to resolve conflicting results in the previous literature.     

一类变质性物品的最优生产-库存策略

讨论了需求是时间的连续函数、允许缺货且缺货完全回补、变质率为常数、补货率有限的变质性物品在有限计划期内的生产-库存策略。同时证明了最优生产-库存策略的存在和唯一性,并给出了求最优策略的算法和算例。     

Optimal growth models as Economic Planning Tools

Conclusion In the present paper, I have carried out an investigation into the form of models which would be suitable for economy-wide planning. Optimal growth models have usually been constructed in an abstract framework. Therefore, the economies for which optimal growth models would be suitable are not identified. I have shown that, because these models only embody purely physical constraints, they will, in general, only be suitable for totally controlled economies. The conclusion, establishing the general inapplicability of optimal growth models to non-controlled economies, is dependent on the view of economic planning which regards the economic structure, particularly the scope of government policy, as fixed prior to plan construction.The second half of the paper elucidates the nature of the planning process when planners take into account the economic structure of the economy which they are planning. I have deliberately chosen to show the planning process in an abstract context for a simple economy, because the object is not to describe how to plan a particular type of economy, but to describe a methodology of planning. In that methodology, there is a place for optimal growth methods but only after the planners have given utmost consideration to the particular features of the economy they are planning. In using that methodology, one finds that problems must be confronted which are hidden when optimal growth models are formulated in the usual manner. Thus, planning for an infinite horizon may be the planners' preferred method on some basic philosophical grounds but in fact in order to construct an implementable plan they may need to follow the uncontrolled sector and adopt a pragmatic finite horizon approach.This paper is a revised version of a chapter of my Ph.D. thesis at the University of Pennsylvania. I would like to thank my adviser, Don Green, for many helpful comments.     

The modified golden rule of a multi-sector economy

We consider an n-good model of optimal accumulation determined by a technology, a utility function, and a discount factor. A technology is δ-productive if it contains an input-output pair such that the discounted output vector strictly dominates the input vector. We show that a δ-productive technology has a non-trivial modified golden rule. We also report a counter- example of David Starrett showing that the modified golden rule need not have turnpike properties, and that there may exist non-trivial periodic optimal consumption plans even when there is no non-trivial modified golden rule.     

Necessary conditions for infinite-horizon discounted two-stage optimal control problems

In many situations in Economics, one would like to analyse the optimal timing of switching between alternative and consecutive regimes. A natural framework for such an analysis seems to be one of multi-stage optimal control problems, where the switching instants between regimes are endogenously determined. The existing literature considers only the case of two-stage optimal control problems with finite horizon. However, in many cases, it seems more appropriate to consider a problem with an infinite horizon. By deriving the appropriate necessary conditions for such a problem we generalise the methodology in analysing endogenous regime switches.     

Historical resources, uncertainty and preservation values: An application of option and optimal stopping models

State agencies and private historical organizations frequently acquire historical sites with unknown characteristics. In this paper, we provide two approaches to evaluating the preservation decision. In the first approach, we show that a historical site which is not permanently preserved provides citizens with a certain flexibility whose value can be measured as an option on the maximum between the current real estate value and the preservation value. In the second approach, we assume that the organization has an infinite planning horizon and chooses the optimal sale date. Using a contingent valuation estimate of the public's willingness to pay for preservation of a specific historical site and the real estate price, we provide simulation values of the preservation option value and the optimal stopping rule.     

Existence and transversality conditions for a general ‘unbounded-horizon’ model of the mining firm

This paper derives conditions under which optimal programs exist for a general model of the competitive or monopolistic mining firm. The model significantly extends and synthesizes previous formulations by including both resource extraction and discoveries of new reserves, with stock-dependent resource exploitation costs and incomplete resource exhaustion, endogenous technical innovation, and state-dependent resource demand. In addition, the analysis addresses complications arising from having optimal programs defined over an ‘unbounded horizon’, with the possibility of either a finite or infinite terminal time and a non-zero terminal (‘salvage’) valuation if the terminal time is finite. The necessity of the infinite-horizon transversality conditions also is established. The paper illustrates a strongly intuitive approach to existence questions for infinite-horizon or unbounded-horizon control problems, with weaker concavity and interiority assumptions than are often encountered in the literature.     

Competitive equilibrium cycles with endogenous labor

In this paper, we study a two-sector optimal growth model with elastic labor supply. We show that the modified golden rule is saddle-point stable when the investment good is capital intensive. To characterize stability with a capital intensive consumption good, we focus on either additively separable or homothetic preferences. In the first specification, we show that optimal oscillations require the elasticity of intertemporal substitution in consumption to be high enough while the elasticity of labor needs to be low enough. At the same time, we prove that with a linear utility in leisure the modified golden rule is always saddle-point stable. In the second specification for preferences, we show that the local dynamic properties of the optimal path depend instead on the shares of consumption and leisure into total utility. We prove that endogenous fluctuations are even more likely with homothetic preferences.     

Learning curves and lot sizing for independent and dependent demand

This paper explores the effect of learning curve cost behavior, as opposed to linear, on lot sizing. The first portion of the paper develops optimizing models for the independent demand situation. The second portion examines lot sizing for dependent demand, developing a lot sizing rule similar to Part Period Balancing.After examining the shortcomings of previous attempts at the independent demand lot sizing problem, two models are derived. Excluding material costs (for an assembly operation, the cost of all components), the optimal lot size is seen to vary linearly with demand and inversely with the carrying cost rate.When material costs are included a smaller optimal lot size is derived. The difference between the two, expressed as a fraction of the smaller lot size, equals the material/labor ratio of the last unit produced in the smaller lot size. For dependent demand, the incremental model developed by Freeland and Colley as an improvement on Part Period Balancing is used as a beginning concept. An analogous model, called Assembly Period Balancing, is developed for learning curve cost behavior. The decision rule for combining lots is expressed as a comparison of the material/labor ratio of the lot considered for combining with another expression involving the carrying cost rate, relative lot size and the learning curve exponent.Finally, cost data from an electronics manufacturer are used to examine the cost penalties of failing to recognize learning curve cost behavior. It is shown that optimal lot sizes for learning curve costs can be much larger than those obtained assuming linear costs. It is also shown that much larger lots can be economically combined in the dependent demand case when costs follow a learning curve.     

The uncertainty multiplier and business cycles

I study a business cycle model where agents learn about the fundamentals by accumulating capital. During recessions, agents invest less, and this generates noisier estimates of macroeconomic conditions and an increase in uncertainty. The endogenous increase in aggregate uncertainty further reduces economic activity and thus gives rise to a multiplier effect that amplifies aggregate fluctuations. To discipline learning dynamics, I parametrize the model so that it matches not only standard business cycle moments but also survey data on macroeconomic forecasts. I find that the uncertainty multiplier amplifies output standard deviation by 16%.     

The implications of financial frictions and imperfect knowledge in the estimated DSGE model of the U.S. economy

In this paper, I study how alternative assumptions about expectation formation can modify the implications of financial frictions for the real economy. I incorporate a financial accelerator mechanism into a version of the Smets and Wouters (2007) DSGE framework and explore the properties of the model assuming, on the one hand, complete rationality of expectations and, alternatively, several learning algorithms that differ in terms of the information set used by agents to produce the forecasts. I show that the implications of the financial accelerator for the business cycle may vary depending on the approach to modeling the expectations. The results suggest that the learning scheme based on small forecasting functions is able to amplify the effects of financial frictions relative to the model with Rational Expectations. Specifically, I show that the dynamics of real variables under learning is driven to a significant extent by the time variation of agents’ beliefs about financial sector variables. During periods when agents perceive asset prices as being relatively more persistent, financial shocks lead to more pronounced macroeconomic outcomes. The amplification effect rises as financial frictions become more severe. At the same time, a learning specification in which agents use more information to generate predictions produces very different asset price and investment dynamics. In such a framework, learning cannot significantly alter the real effects of financial frictions implied by the Rational Expectations model.     

On time-consistent policy rules for heterogeneous discounting programs

This article considers a new concept of social optimum for an economy populated by agents with heterogeneous discount factors. It is based upon an approach that constrains decision rules to be temporally consistent: these are stationary and unequivocally ruled by the state variable. For agents who differ only in their discount factors and have equal weights in the planner’s objective, the temporally-consistent optimal solution produces identical consumption for the agents at all time periods. In the long run, the capital stock is determined by a modified golden rule that corresponds to an average-like summation of all discount factors. The general argument is illustrated by various two-agent examples that allow for an explicit determination of the temporally consistent decision rules. Interestingly, this temporally consistent solution can be simply recovered from the characterization of a social planner’s problem with variable discounting and can also be decentralized as a competitive equilibrium through the use of various instruments.