Using dynamic programming with adaptive grid scheme for optimal control problems in economics

The study of the solutions of dynamic models with optimizing agents has often been limited by a lack of available analytical techniques to explicitly find the global solution paths. On the other hand, the application of numerical techniques such as dynamic programming to find the solution in interesting regions of the state was restricted by the use of fixed grid size techniques. Following Grüne (Numer. Math. 75 (3) (1997) 319; University of Bayreuth, submitted, 2003), in this paper an adaptive grid scheme is used for finding the global solutions of discrete time Hamilton–Jacobi–Bellman equations. Local error estimates are established and an adapting iteration for the discretization of the state space is developed. The advantage of the use of adaptive grid scheme is demonstrated by computing the solutions of one- and two-dimensional economic models which exhibit steep curvature, complicated dynamics due to multiple equilibria, thresholds (Skiba sets) separating domains of attraction and periodic solutions. We consider deterministic and stochastic model variants. The studied examples are from economic growth, investment theory, environmental and resource economics.     

A review of estimates of the schooling/earnings relationship, with tests for publication bias

In this paper we provide an analytical review of previous estimates of the rate of return on schooling investments and measure how these estimates vary by country, over time, and by estimation method. We find evidence of reporting (or “file drawer”) bias in the estimates and, after due account is taken of this bias, we find that differences due to estimation method are much smaller than is sometimes reported, although some are statistically significant. We also find that estimated returns are higher in the US and they have increased in the last two decades.     

Matching and semi-parametric IV estimation, a distance-based measure of migration, and the wages of young men

Our paper estimates the effect of US internal migration on wage growth for young men between their first and second job. Our analysis of migration extends previous research by: (i) exploiting the distance-based measures of migration in the National Longitudinal Surveys of Youth 1979 (NLSY79); (ii) allowing the effect of migration to differ by schooling level and (iii) using propensity score matching to estimate the average treatment effect on the treated (ATET) for movers and (iv) using local average treatment effect (LATE) estimators with covariates to estimate the average treatment effect (ATE) and ATET for compliers.We believe the Conditional Independence Assumption (CIA) is reasonable for our matching estimators since the NLSY79 provides a relatively rich array of variables on which to match. Our matching methods are based on local linear, local cubic, and local linear ridge regressions. Local linear and local ridge regression matching produce relatively similar point estimates and standard errors, while local cubic regression matching badly over-fits the data and provides very noisy estimates.We use the bootstrap to calculate standard errors. Since the validity of the bootstrap has not been investigated for the matching estimators we use, and has been shown to be invalid for nearest neighbor matching estimators, we conduct a Monte Carlo study on the appropriateness of using the bootstrap to calculate standard errors for local linear regression matching. The data generating processes in our Monte Carlo study are relatively rich and calibrated to match our empirical models or to test the sensitivity of our results to the choice of parameter values. The estimated standard errors from the bootstrap are very close to those from the Monte Carlo experiments, which lends support to our using the bootstrap to calculate standard errors in our setting.From the matching estimators we find a significant positive effect of migration on the wage growth of college graduates, and a marginally significant negative effect for high school dropouts. We do not find any significant effects for other educational groups or for the overall sample. Our results are generally robust to changes in the model specification and changes in our distance-based measure of migration. We find that better data matters; if we use a measure of migration based on moving across county lines, we overstate the number of moves, while if we use a measure based on moving across state lines, we understate the number of moves. Further, using either the county or state measures leads to much less precise estimates.We also consider semi-parametric LATE estimators with covariates (Frölich 2007), using two sets of instrumental variables. We precisely estimate the proportion of compliers in our data, but because we have a small number of compliers, we cannot obtain precise LATE estimates.     

Labour adjustment costs: Estimation of a dynamic discrete choice model using panel data for Greek manufacturing firms

In this paper we estimate a dynamic structural model of employment at firm level. Our dataset consists of a balanced panel of 2790 Greek manufacturing firms. The empirical evidence of this dataset stresses three important stylized facts: (a) there are periods in which firms decide not to change their labour input, (b) there are periods of large employment changes (lumpy nature of labour adjustment) and (c) the commonality is employment spikes to be followed by smooth and low employment growth periods. Following Cooper and Haltiwanger [Cooper, R.W. and Haltiwanger, J. “On the Nature of Capital Adjustment Costs”, Review of Economic Studies, 2006; 73(3); 611–633], we consider a dynamic discrete choice model of a general specification of adjustment costs including convex, non-convex and “disruption of production” components. We use a method of simulated moments procedure to estimate the structural parameters. Our results indicate considerable fixed costs in the Greek employment adjustment.     

Estimates of the return to schooling in Sweden from a large sample of twins

A large sample of twins was used to examine whether conventional estimates of the return to schooling in Sweden are biased because ability is omitted from the earnings–schooling relationship. Ignoring measurement error, the results indicate that omitting ability from the earnings–schooling relationship leads to estimates that are positively biased. However, reasonable estimates of the measurement-error-adjusted returns are both above and below the unadjusted estimates, showing that the results depend crucially on a parameter not known at this time. However, an estimate of the reliability ratio was obtained using two measures on educational attainment. With this estimate of the reliability ratio, the measurement-error-adjusted estimate of the return to schooling in the sample of identical twins indicates that there is at most a slight ability bias in the conventional estimates of the return to schooling. The fundamental assumption of this kind of study is that within-pair differences in educational attainment are randomly determined. This assumption was also tested, but no strong evidence to reject it was found.     

Causality reconsidered: A note

This note extends the recent work of Steinnes by formulating and testing a similar but more complete dynamic reduced form model of intraurban location decisions. The formulation of the model corrects a specification error by Steinnes, and a dummy variable technique is applied to yield consistent estimates. The results include relations between the employment sectors as well as between population and employment and, when they differ from those previously published, they are more plausible. The basic conclusion of the note is that “jobs follow people.”     

Risk aversion and the elasticity of substitution in general dynamic portfolio theory: Consistent planning by forward looking, expected utility maximizing investors

Using the measure of risk aversion suggested by Kihlstrom and Mirman [Kihlstrom, R., Mirman, L., 1974. Risk aversion with many commodities. Journal of Economic Theory 8, 361–388; Kihlstrom, R., Mirman, L., 1981. Constant, increasing and decreasing risk aversion with many commodities. Review of Economic Studies 48, 271–280], we propose a dynamic consumption-savings–portfolio choice model in which the consumer-investor maximizes the expected value of a non-additively separable utility function of current and future consumption. Preferences for consumption streams are CES and the elasticity of substitution can be chosen independently of the risk aversion measure. The additively separable case is a special case. Because choices are not dynamically consistent, we follow the “consistent planning” approach of Strotz [Strotz, R., 1956. Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies 23, 165–180] and also interpret our analysis from the game theoretic perspective taken by Peleg and Yaari [Peleg, B., Yaari, M., 1973. On the existence of a consistent course of action when tastes are changing. Review of Economic Studies 40, 391–401]. The equilibrium of the Lucas asset pricing model with i.i.d. consumption growth is obtained and the equity premium is shown to depend on the elasticity of substitution as well as the risk aversion measure. The nature of the dependence is examined. Our results are contrasted with those of the non-expected utility recursive approach of Epstein–Zin and Weil.     

Household search and health insurance coverage

Health insurance in the United States is typically acquired through an employer-sponsored program. Often employees offered employer-provided health insurance have the option to extend coverage to their spouse and dependents. We investigate the implications of the “publicness” of health insurance coverage for the labor market careers of spouses. The theoretical innovations in the paper are to extend the standard partial–partial equilibrium labor market search model to a multiple searcher setting with the inclusion of multi-attribute job offers, with some of the attributes treated as public goods within the household. The model is estimated using data from the Survey of Income and Program Participation (SIPP) using a Method of Simulated Moments (MSM) estimator. We demonstrate how previous estimates of the marginal willingness to pay (MWP) for health insurance based on cross-sectional linear regression estimators may be seriously biased due to the presence of dynamic selection effects and misspecification of the decision-making unit.     

Convergence to symmetry in dynamic strategic models of R&D: The undiscounted case

Will R&D increase or decrease the asymmetry between firms over time? We examine this issue in the context of a dynamic, alternate-move duopoly model of non-cooperative R&D. The asymmetry we consider is with respect to the initial stocks of technological knowledge which provides one firm a greater potential for current and future profits in the product market. Utilizing a value loss process constructed from the Optimality Equation of dynamic programming, we show that for the undiscounted case the asymmetry between the firms disappears over time. We obtain this convergence-to-symmetry result by imparting a temporal character to R&D through the notion that R&D investment cannot be changed instantaneously, by allowing research externalities, complementarity or substitutability between own and appropriated R&D, and either increasing, decreasing or constant returns to scale in the production of technical knowledge from own and rival R&D.     

The religious factor in private education

We quantify the religious factor in private education in the United States by calibrating a political economy model of school choice—among public, private-nonsectarian and subsidized religious schools—in which parents differ in their incomes and in their preferences for religious education. This shows strong latent demand for religious schooling, conditioned on current subsidized tuition levels in parochial schools, which is suppressed by the need to “pay twice” for private education. Applying the results of the calibration to gauge the effect of means-tested school voucher programs on religious and nonsectarian private enrollment, we find that when the amount of the voucher is relatively small, including subsidized religious schools in such programs is necessary for providing low-income families effective access to private education.     

Stopping with anticipated regret

This paper analyzes a stopping problem where the decision maker is driven by anticipated ex-post regret. There are two sources of potential dynamic inconsistency, one is arrival of information and the other is changing choice opportunities over time—discarding the current stopping option may change how she stops the game in the future.First we consider a naive planner who prescribes a commitment solution, and illustrate the nature of the inconsistency problem. Then we consider a sophisticated planner who plays backward induction against her ‘successive selves’. The resolution of dynamic inconsistency does not in general allow the use of standard dynamic programming technique. We provide, however, a simple characterization of the backward induction strategy, which is given in a recursive formula.We also provide a behavioral implication, that larger indeterminacy of belief may lead to a more aggressive behavior, that is, continuing the gamble longer, which contrasts to the implication of ambiguity aversion.     

Education choice under uncertainty: Implications for public policy

We analyse how progressive taxation and education subsidies affect schooling decisions when the returns to education are stochastic. We use the theory of real options to solve the problem of education choice in a dynamic stochastic model. We show that education attainment will be an increasing function of the risk associated with education. Furthermore, this result holds regardless of the degree of risk aversion. We also show that progressive taxes will tend to increase education attainment.     

Finite sample inference for quantile regression models

Under minimal assumptions, finite sample confidence bands for quantile regression models can be constructed. These confidence bands are based on the “conditional pivotal property” of estimating equations that quantile regression methods solve and provide valid finite sample inference for linear and nonlinear quantile models with endogenous or exogenous covariates. The confidence regions can be computed using Markov Chain Monte Carlo (MCMC) methods. We illustrate the finite sample procedure through two empirical examples: estimating a heterogeneous demand elasticity and estimating heterogeneous returns to schooling. We find pronounced differences between asymptotic and finite sample confidence regions in cases where the usual asymptotics are suspect.     

Is econometrics useful for private policy making? A case study of replacement policy at an auto rental company

The goal of this paper is to illustrate the potential usefulness of econometrics as a tool to assist private policy makers. We provide a case study and detailed econometric analysis of the automobile replacement policy adopted by a large car rental company. Unlike public policy making–where the benefits from using econometric models and “science-based” approaches to policy making are hard to quantify because the outcomes of interest are typically subjective quantities such as “social welfare”–in the case of firms there is an objective, easily quantifiable criterion for judging whether policy A is better than policy B: profits. We introduce and estimate an econometric model of the rental histories of individual cars in the company’s fleet. Via stochastic simulations, we show that the model provides a good approximation to the company’s actual operations. In particular, the econometric model is able to reproduce the extraordinarily high rates of return that the company obtains on its rental cars, with average internal rates of return between purchase and sale of approximately 50%. However, the econometric model can simulate outcomes under a range of counterfactual vehicle replacement policies. We use the econometric model to simulate the profitability of an alternative replacement policy under pessimistic assumptions about the rate maintenance costs would increase and rental rates would have to be decreased if the company were to keep its rental cars longer than it does under the status quo. Depending on the vehicle type, we find that the company’s expected discounted profits would be between 6% to over 140% higher under the suggested alternative operating strategy where vehicles are kept longer and rental rates of older vehicles are discounted to induce customers to rent them. The company found this analysis to be sufficiently convincing that it undertook an experiment to verify the predictions of the econometric model.     

Differences in education and earnings across ethnic groups in Guatemala

Most of the research on ethnicity in Central America looks at two groups: indigenous and nonindigenous populations. While indigenous people are disadvantaged relative to the nonindigenous group, the many groups that make up the indigenous population are often times not very similar in terms of personal and group characteristics. But the differences within these groups are important for public policy. Ethnic groups' experience depends on many factors besides education and skill levels. These additional factors may include what has come to be known as “ethnic capital.” That is, the socioeconomic performance of today's workers depends not only on parental skills, but also on the average skills of the ethnic group in the parent's generation. In the United States, when data on ethnic groups is broken down, the black-white comparison may be too simple a characterization. It is argued here that the indigenous-nonindigenous comparison in Guatemala is too simple. Guatemala's household survey, which utilizes two questions to determine an individual's ethnicity (self-perception and language), is used here to examine the question of ethnicity in terms of schooling attainment, earnings and the returns to schooling. The following hypothesis is tested: do higher levels of schooling correspond to higher returns to schooling? Dividing the indigenous sample into the main ethnic groups produces widely different and dramatic results in terms of schooling, earnings and the returns to schooling. Higher levels of schooling correspond to higher returns to schooling by ethnic group. Thus there may be a need to target interventions appropriately to the specific ethnic group in question.     

Can risk aversion explain schooling attainments? Evidence from Italy

Using unique Italian panel data, in which individual differences in behavior toward risk are measured from answers to a lottery question, we investigate if (and to what extent) risk aversion can explain differences in schooling attainments. We formulate the schooling decision process as a reduced-form dynamic discrete choice, which we estimate flexibly. We analyze how grade transition from one level to the next varies with preference heterogeneity (risk aversion), parental human capital, socioeconomic variables and persistent unobserved (to the econometrician) heterogeneity. We find that differences in attitudes toward risk account for a modest portion of the probability of entering higher education.     

Average and Marginal Returns to Upper Secondary Schooling in Indonesia

This paper estimates average and marginal returns to schooling in Indonesia using a semiparametric selection model. Identification of the model is given by geographic variation in access to upper secondary schools. We find that the return to upper secondary schooling varies widely across individuals: it can be as high as 50% per year of schooling for those very likely to enroll in upper secondary schooling, or as low as ?10% for those very unlikely to do so. Average returns for the student at the margin are substantial, but they are also well below those for the average student attending upper secondary schooling. Copyright © 2016 John Wiley & Sons, Ltd.     

Lower Bound Restrictions on Intensities in Data Envelopment Analysis

We propose an extension to the basic DEA models that guarantees that if an intensity is positive then it must be at least as large as a pre-defined lower bound. This requirement adds an integer programming constraint known within Operations Research as a Fixed-Charge (FC) type of constraint. Accordingly, we term the new model DEA_FC. The proposed model lies between the DEA models that allow units to be scaled arbitrarily low, and the Free Disposal Hull model that allows no scaling. We analyze 18 datasets from the literature to demonstrate that sufficiently low intensities—those for which the scaled Decision-Making Unit (DMU) has inputs and outputs that lie below the minimum values observed—are pervasive, and that the new model ensures fairer comparisons without sacrificing the required discriminating power. We explain why the low-intensity phenomenon exists. In sharp contrast to standard DEA models we demonstrate via examples that an inefficient DMU may play a pivotal role in determining the technology. We also propose a goal programming model that determines how deviations from the lower bounds affect efficiency, which we term the trade-off between the deviation gap and the efficiency gap.     

Identification issues in the public/private wage gap,with an application to Italy

This paper reviews some of the standard assumptions that are imposed in order to estimate the average public/private wage gap and that are mainly related to the possible selection of the sector. There are two contributions to the existing public/private wage gap literature. One is a better understanding of the identified parameters: standard estimators identify a local effect (LATE), which in general cannot be generalized to the entire population, as instead is almost always done. The other is the partial identification of the population average treatment effect, with an instrumental variable. To the best of my knowledge, this is the first paper in this literature that employs bounds. The technique is applied to male workers in Italy. For compliers, LATE estimates a wage advantage from working in the public sector greater than 30%. This return is within the narrowest bounds on the population average treatment effect that are consistent even with a much smaller gap (about 15% or more).     

Dynamic programming for non-additive stochastic objectives

We derive the existence of an optimum and the techniques of dynamic programming for non-additive stochastic objectives. Our key assumption for non-negative objectives is that asymptotic impatience exceeds asymptotic ‘mean’ growth, where ‘mean’ growth is derived not only from intertemporal inelasticity and the random return on investment but also from the curvature of the non-additive stochastic aggregator (i.e. the ‘certainty equivalent’). We provide broad families of new, interesting, and tractable examples. They illustrate that ‘mean’ growth can exist even when the distribution of returns has unbounded support, that power discounting often implies infinite asymptotic impatience, and that non-positive objectives are easily handled with few restrictions on growth.