¹ More technical discussions can be found in the surveys by Rust (1993, 1994), Miller (1997) and Aguirregebaria and Mira (forthcoming), as well as in a number of papers cited throughout this chapter.

² Their use has spread to areas outside of traditional economics, such as marketing, in which it is arguably now the predominant approach to empirical research.

³ Most applications of DCDP models assume that agents, usually individuals or households, solve a finite horizon problem in discrete time. For the most part, we concentrate on that case and defer discussion of infinite horizon models to the discussion of the special case of job search models. We do not discuss continuous time models except in passing.

⁴ The conventional approach assumes that agents have rational expectations. An alternative approach directly elicits subjective expectations (see, e.g., Dominitz and Manski, 1996, 1997; Van der Klaauw, 2000; Manski, 2004).

⁵ A notable omission is the literature on retirement behavior. Although that literature relies heavily on the DCDP approach, the previous Handbook of Labor Economics chapter by Lumsdaine and Mitchell provides an extensive survey up to that time. We decided to concentrate on DCDP literature that to date has not been surveyed in the Handbook.

⁶ As will be seen in the empirical applications we consider, there are a wide range of types of variables that would be included in X. Their common feature is that they are not directly choices of the agent, although they may be affected by prior choices or correlated with choices without being directly affected by them.

⁷ By forward looking, we simply mean that agents take into account the effect of their current actions on future welfare. How exactly they form expectations about the impact of those actions and about future preferences and constraints are specific modeling choices.

⁸ By maintaining the same joint distribution when performing the ceteris paribus change, we are assuming that the change in an observable variable does not induce a change in the joint distribution of unobservables. This assumption is not the same as assuming conditional independence.

⁹ The modern approach to this topic began with Mincer (1962).

¹⁰ We treat the price of child care as parametric in part to illustrate how alternative approaches to estimation are related to achieving goal 3. A more complete model would allow for a choice among alternative types of child care, for example, of varying qualities, which differ in their price and which may vary over time.

¹¹ In this taxonomy, semi-parametric and semi-structural categories fall into the parametric (P) and structural (S) categories.

¹² As before (see footnote 8), we assume that the change in an observable variable does not induce a change in the joint distribution of unobservables.

¹³ Results from Matzkin (1993) apply to the case where all wage offers are observed (regardless of participation). In that case, aside from normalizations, and the joint distribution, are nonparametrically identified.

¹⁴ Pagan and Ullah (1999), Chapter 7, provides a good introduction to semi-parametric estimation of discrete choice models.

¹⁵ The unconventional assumption of normality for the wage distribution (allowing, as it does, for negative wage offers) is adopted in order to obtain a decision rule that is linear and additive in unobservables. We present a more general formulation in later sections.

¹⁶ As we show below, the additive error is convenient in calculating choice probabilities and is maintained for illustrative purposes. However, as we also show below, the additive structure is fragile. It is lost, for example, if the wage function takes a semi-log form or if the utility function is nonlinear in consumption. Note that the linearity and separability of consumption in the utility function implies that husband’s income does not enter and, thus, does not affect the participation decision.

¹⁷ We call a probability, but it is actually a mixed probability for and a density for . Note that the Jacobian of the transformation from the wage density to the wage error density is one.

¹⁸ Given the assumptions of the model, full independence of the joint error distribution with respect to observables is not necessary. See French and Taber (2011) for an extended discussion of identification of selection models.

¹⁹ If all wage offers were observed, it would be possible to achieve all three goals without imposing parametric assumptions or structure. With respect to the policy counterfactual (goal 3), because of the subsidy acts like a wage tax, the effect of the subsidy can be calculated by comparing participation rates of women with a given wage to women with a wage augmented by (see Ichimura and Taber (2002) and Todd and Wolpin (2010)).

²⁰ Another reason for adopting the P-S estimation approach is that separating out preferences from opportunities (wage offers) helps to understand important social and economic phenomena, for example, in assessing how much of the difference in labor market outcomes of black and white women is due to differences in preferences and how much to differences in wage opportunities. Such an assessment could be useful in the design of public policies aimed at ameliorating those differences.

²¹ The assumption that the woman’s initial work experience at the time marriage is zero, which is undoubtedly in many cases untrue, is made for ease of exposition. We discuss in a later section the complications introduced by accounting for the fact that work experience is accumulated prior to marriage and varies across women.

²² The finite horizon assumption is immaterial for the points we wish to make. If the current period utility is bounded at all and the discount factor is less than one, then the solution to the infinite horizon problem can be approximated arbitrarily closely by the solution to a long but finite horizon problem. The essential difference between a finite and infinite horizon model in terms of the predictions about behavior is that in the finite horizon case there are implications for age patterns in behavior.

²³ The terminal period of the model would be at the termination of the marriage or the retirement of the wife. Accounting for divorce, even taking it to be exogenous, would unduly complicate the model. For illustrative purposes, then, we assume that the wife retires at . The value function at is normalized to zero, although a more complete formulation would make the retirement decision of both spouses a choice and would, at the least, specify the determination of post-retirement income through the social security system.

²⁴ Given the lack of separate identification of and , we set to reduce notation.

²⁵ Note that if preference or wage shocks were serially correlated, the observable and unobservable state variables would not generally be additively separable as in the second equality. The additive separability arises because, with serial independence, , which does not include or can replace in the future component of the value functions,.that is . We discuss the case of serially correlated errors below.

²⁶ Because of the linearity and additive separability of consumption in utility, husband’s income does not affect the participation decision. We therefore do not need to specify what is known about future husband’s income (see below). Again, this assumption is made so that the solution method can be illustrated most effectively.

²⁷ Later, we introduce stochastic fertility, allowing for the decision model to begin at the time of marriage, when we consider an extension of the model to a multinomial choice setting.

²⁸ Suppose we define a young child as a child under the age of six (that is, not of school age). Consider a couple who at the start of the woman’s infecund period has a 3 year old child and thus for whom . Then, for that couple, and .

²⁹ This expression uses the fact that for any two random variables and ,

³⁰ Although would surely be zero at some point, we carry it along to emphasize its perfect foresight property.

³¹ In solving for the latent variable functions, we could thus set (or any other arbitrary value) for all .

³² If the structure does not yield an additive (composite) error, the latent variable function becomes . Calculating the joint regions of , that determine the probabilities that enter the likelihood function and that are used to calculate the function must, in that case, be done numerically. We address this more general case below.

³³ As in the static case, the Jacobian of the transformation from the density of the wage offer to the density of is one.

³⁴ In the current example, couples are assumed to know the full structure of the model and to use it in forming their forecasts of future wage offers and their future preferences.

³⁵ It is possible that in some models additional parameters might enter , say through the transition functions of state variables (see below for an example). While the same heuristic argument would apply, its validity would be less apparent.

³⁶ More generally, if agents have beliefs about future policies (or policy changes), such beliefs should be incorporated into the solution and estimation of the decision model.

³⁷ We ignore the possibility that the husband’s type also affects his earnings because, in the model as specified, his earnings has no effect on the participation decision. In a more general specification, one would probably add this source of heterogeneity.

³⁸ There are obviously restrictions across the husband and wife individual type probabilities and couple type probabilities.

³⁹ We could combine the permanent-transitory scheme with the AR(1) scheme by allowing the and shocks in (41) and (42) to be AR(1).

⁴⁰ Alternatively as noted, we could assume, unrealistically as well, that the experience that women have at the start of marriage is exogenous with respect to future participation decisions.

⁴¹ We would also need to include any other initial conditions that affect wage offers (’s) or preferences (’s), for example, completed schooling.

⁴² If there is both unobserved permanent heterogeneity and serial correlation, and letting be the exogenous initial conditions at the time of marriage, then in the likelihood function (43), would be replaced with . Note that must contain a variable other than and in order to identify the effect of on a couple’s type.

⁴³ Note that we do not discuss methods like Hotz and Miller (1993) here. They propose a method to circumvent having to obtain a full solution of the DP problem while still obtaining parameter estimates, not a method for solving the DP problem (see below).

⁴⁴ There is no formal proof of this proposition, though, as noted, Keane and Wolpin (1994) provide Monte Carlo evidence for a particular model that supports the intuition.

⁴⁵ Geweke and Keane (2001) give an example where the curse of dimensionality is broken. This is when the Emax can be expressed as a function of the expected value of each alternative. (That is, these expected values are a sufficient statistic for all the state variables that determine them.) The size of this set of variables remains fixed at , where is the number of alternatives, even as the state space grows larger.

⁴⁶ Technically this is not quite enough, as convergence must be uniform and not just pointwise.

⁴⁷ Because is now an unobserved component of the state space, estimation of must be carried out jointly. This would require a distributional assumption for and raises issues of the separate identification of and of the effect of on wages.

⁴⁸ This is analogous to the fact that the asymptotic properties of competing estimators (under the hypothetical scenario of increasing sample size) do not reveal which will perform best given finite samples.

⁴⁹ Stinebrickner (2000) compares several approximation methods in the context of a DCDP model with serially correlated shocks.

⁵⁰ The Ben-Porath (1967) model of human capital accumulation leads to a semi-log form and Heckman and Polachek (1974) show using a Box-Cox transformation that a semi-log form is not rejected by the data.

⁵¹ Approximations to DCDP model decision rules were first discussed in Heckman (1981) and Wolpin (1984). For an empirical application in the labor economics literature, see Keane and Wolpin (2001).

⁵² In theory, the period length should correspond to the frequency of decision-making, which, in principle, may differ among choice variables. Like the specification of the model structure (including assumptions about expectations formation and optimization), the discrete time framework is adopted as an approximation. A continuous time framework would be more general, but would require assumptions about the joint process generating decision times for the choice variables.

⁵³ To the extent that variations in hours worked within those categories represents differences in the choice of optimal hours, the discretization of hours induces measurement error. In the data, the mean and standard deviation of hours based on the categorization (where the categories are assigned 0, 1000, 2000 and 3000 hours) are almost identical to that based on actual annual hours worked. The standard deviation of hours within the categories is 145, 286, 224 and 429.

⁵⁴ We have not, however, in this exploratory stage allowed for serially correlated unobservables either through permanent unobserved heterogeneity or serially correlated shocks.

⁵⁵ The constant term in the contraceptive cost function, say cannot be separately identified from , that is, the goods cost of a newborn (a child age 0-1). Note that .

⁵⁶ One could instead allow for a choice of whether to contracept or not with pregnancy being an uncertain outcome. We ignore this extension for ease of presentation.

⁵⁷ In Ben-Porath’s (1967) model of the production of human capital, an individual’s wage was given by the product of a human capital per-unit rental price times the individual’s human capital stock. Griliches (1977) operationalized the human capital production function as depending on arguments such as schooling, work experience and ability.

⁵⁸ Husband’s are assumed to work full-time, which implies that, given schooling, age and work experience are isomorphic.

⁵⁹ For convenience, is in 1000 hour units.

⁶⁰ is obtained after substituting for the wife’s wage and the husband’s earnings in the budget constraint and then substituting for consumption in the utility function.

⁶¹ Keane and Wolpin (1994) discuss various specifications of the regression function.

⁶² In the labor force participation model, the total number of potential state points increases in as feasible work experience and numbers of children increase. A researcher might, as the notation indicates, vary the number of randomly drawn state points with .

⁶³ Interpolating functions should be chosen with great care. To avoid overfitting, it is useful to solve the model at more state points than used in estimating the interpolating function and use the additional points for cross-validation. For example, we might solve the model at 4000 state points, estimate the interpolating function on 2000 points and check the fit, say the , using the other 2000 points.

⁶⁴ Note that the type-specific parameters, ’s, are essentially the constant terms in the production function and cannot be separately identified from the skill rental prices.

⁶⁵ Note that in writing (82) we are implicitly assuming that the state space evolves deterministically, conditional on the current state and current choice. Otherwise (82) would require a double sum, where the inner sum is over states that could potentially be reached from given the choice . Norets (2009) handles the stochastically evolving state space case. See also Ching et al. (2010).

⁶⁶ Of course, providing such a distribution is possible without adopting a Bayesian approach, although it can be computationally burdensome.

⁶⁷ The main insight in the multinomial setting is the same and the extension is straightforward.

⁶⁸ Lee and Wolpin (2006); Lentz (2009) allow for (equilibrium) skill prices to change with calendar time due to technical change.

⁶⁹ In recent work, Arcidiacono and Miller (2008) have developed methods for extending the HM approach to allow for unobserved state variables. However, there has as yet been no empirical implementation of that approach to a model as rich as those found in the literature.

⁷⁰ Indeed, to our knowledge the only paper that has attempted to do so is Keane and Wolpin (2001). That paper models the labor supply and human capital investment decisions of young men, who often have low participation rates.

⁷¹ For men, strict application of the DCDP approach would require discretization of hours as an approximation to the choice set. In that case, the parallel to the multinomial choice problem considered above is exact. However, the main insight of the DCDP approach to estimation applies as well to continuous choices and to discrete-continuous choices in which the underlying dynamic programming problem is solved based on first-order conditions or Kuhn-Tucker conditions. That insight was simply the observation that because the continuation value (the max function) is a deterministic function of state variables, the static model and the dynamic programming model have a common empirical structure.

⁷² Their approach builds on the seminal work of MaCurdy (1981) on the labor supply of men.

⁷³ is set at 8760 hours.

⁷⁴ If a woman never works, the likelihood of that event is maximized by setting the fixed effect to . Adjustments for this sample selection made little difference to the estimates.

⁷⁵ Although the effect of the wife’s age may be interpreted as an estimate of , it may also reflect changing preferences for leisure with age.

⁷⁶ Note that this is still a rather large increase, consistent with a Marshallian elasticity of 13. 6/15 = 0.90.

⁷⁷ The only exceptions we have come across are Van Soest et al. (1990) and Keane and Moffitt (1998). Both papers note that it is rare to observe people working very low levels of hours (the former paper looking at men, the latter looking at single mothers). Van Soest et al. (1990) capture this by building in a job offer distribution where few jobs with low levels of hours are available. Keane and Moffitt (1998) build in actual measures of fixed costs of working (e.g., estimates of child care costs).

⁷⁸ See Altug and Miller (1998), Eqs. (6.8) and (6.9), which give the final simple expressions for the labor supply and participation equations. Hotz et al. (1994) develop a simulation method for implementing the Hotz and Miller (1993) conditional choice probability approach.

⁷⁹ Note it is important not to include the aggregate prices and in these regressions. Agents are assumed not to know the future realizations of these variables and so cannot condition on them when forming expected future payoffs.

⁸⁰ An alternative computational approach to taking out group and time means is to regress the group mean of hours on the group mean of wages and a complete set of time and group dummies. Then the wage effect is identified purely from the wage variation not explained by time or group. The advantage of the more involved two-step procedure is that the coefficient on the residual provides a test of exogeneity of wages.

⁸¹ Annual earnings if the woman works are assumed to equal 2000 times the hourly wage rate, regardless of how many hours the woman actually works. This is necessitated by the 1/0 nature of the work decision.

⁸² Eckstein and Wolpin (1989b) also assume that husband’s earnings is a deterministic function of husband’s age, a fixed effect, and a schooling/age interaction. If there were taste shocks or shocks to husband’s earnings they would have to be integrated out in solving the DP problem.

⁸³ Note that the measurement error in wages cannot be estimated using wage data alone. But joint estimation of a wage equation and a labor supply model does allow measurement error to be estimated, as true wage variation affects behavior while measurement error does not. Of course, any estimate of the extent of measurement error so obtained will be contingent on the behavioral model.

⁸⁴ This method is not necessarily more parsimonious than modeling a variable as a choice, trading off an additional choice variable (whether to have a child in this case with the corresponding utility and cost parameters ) against additional parameters governing the stochastic outcome (the probability of having a child). A limitation of this method is that it does not allow for effects of contemporaneous shocks, for example a high wage draw for the female, on the probability of having a child.

⁸⁵ An alternative approach would be to assume the four errors follow a generalized extreme value distribution (see Arcidiacono (2005)).

⁸⁶ The estimates imply that a married woman who works receives 34% of husband income. Unfortunately, the share if she does not work is not identified. As can be seen from (118), if a married woman does not work her utility from consumption is times her share of husband income. Only this product is identified in the model.

⁸⁷ Van der Klaauw (1996) simulates that a $1000 (or 5%) increase in husband offer wages would reduce average duration to first marriage by 1 year, increase average years of marriage (by age 35) by 2.3 years, and reduce average years of work by 2.6 years, or 27%. These are very large income effects, but they are not comparable to standard income effect measures, as they refer to changes in husband offer wages as opposed to changes in actual husband wages (or changes in some other type of non-labor income). Furthermore, it is not clear how much credence we can give to these figures since, as noted earlier, all permanent differences in husband income in the model are generated by differences in the wife’s own characteristics.

⁸⁸ There are separate part- and full-time wage functions.

⁸⁹ This is a sub-sample of a group of 1,783 women who were married at least once during the period (the larger sample including women who leave a partner during the sample period).

⁹⁰ Neither model captures the sharp decline in participation in their 60’s due to retirement. But to be fair neither model incorporates any features designed to explain retirement behavior (such as pensions or Social Security).

⁹¹ The choice set differs across women for a number of reasons. For instance, only unmarried women with children under 18 have the option to participate in welfare, and working while on welfare is not an option if the offer wage rate is high enough that income would exceed the eligibility level. Also, girls under 16 cannot choose marriage.

⁹² Childcare time is, in turn, a weighted sum of time required to care for children in different age ranges.

⁹³ The utility function includes some miscellaneous additional terms that were added to capture some specific features of the data. Full and part-time work are interacted with school to capture the fact that people who work while in school tend to have a strong preference for part-time over full-time work. Work variables are also interacted with a school less than 12 dummy to capture that part-time work is far more prevalent among high school students. Pregnancy is interacted with school to capture that women rarely go to school while pregnant. Tastes for school, marriage and pregnancy are also allowed to shift at certain key ages (16, 18 and 21). And there is a linear time trend (across cohorts) in tastes for marriage.

⁹⁴ Keane and Wolpin (2002), which presents a nonstructural analysis of the same data, provides a more detailed description.

⁹⁵ Note that this is an elasticity for hours conditional on working. It is unfortunate that Cogan does not report a participation elasticity, as, given his estimates, this would presumably have been much larger.

⁹⁶ For recent attempts see Mazzocco and Yamaguchi (2006) and Tartari (2007).

⁹⁷ Imai and Keane (2004) actually assume a much more complex process, designed to capture patterns of complementarity between human capital and hours of work in the human capital production function. But use of this simpler form helps to clarify the key points.

⁹⁸ Although Imai and Keane assume a constant rental price, allowing for time varying rental rates is fairly straightforward.

⁹⁹ Adding a bequest motive to the model, as in Imai and Keane (2004) is straightforward. This extension can be accommodated by adding a terminal value function, say to .

¹⁰⁰ These equations may have multiple solutions. If there are, then one would need to check second order conditions or calculate .

¹⁰¹ Keane and Wolpin (2001) first developed this approach to forming the likelihood in DCDP models. Keane and Sauer (2009) extended the approach to nonstructural panel data models.

¹⁰² Van der Klaauw and Wolpin (2008) estimate a collective model of the joint labor supply decisions of a married couple nearing retirement They allow for savings and human capital accumulation, incorporating as well a detailed representation of US social security system.rules. As noted in the introduction, we do not review the DCDP retirement literature in this chapter.

¹⁰³ The estimate of in Van der Klaauw and Wolpin (2008) is −0.6, which is also in line with other estimates from the retirement literature. They also include liquidity constraints.

¹⁰⁴ Wages would grow deterministically if contains age or job tenure.

¹⁰⁵ A number of these earlier papers appeared in a 1977 symposium volume of the Industrial and Labor Relations Review. Most relevant in that volume are papers by Classen, and the comment on them by Welch (1977).

¹⁰⁶ Burdett (1978) extended the standard unemployment search model to allow for search on the job.

¹⁰⁷ It is a common theme in the structural literature to build upon and extend the theoretical literature in developing estimable models. This is the case with Rendon’s (2006) paper, which builds on the earlier work of Danforth (1979). Lentz (2009) also structurally estimates a sequential search model with savings. Unlike the standard model, the wage offer distribution is taken to be degenerate and agents choose their search intensity, which affects the rate at which job offers are received. Lentz uses the model to empirically determine the optimal unemployment insurance scheme.

¹⁰⁸ Examples of papers based on wage posting models include Eckstein and Wolpin (1990), Kiefer and Neumann (1993), Van den Berg and Ridder (1998), Bontemps et al. (1999, 2000) and Postel-Vinay and Robin (2002). Those based on search-matching-bargaining.models include, among others, Eckstein and Wolpin (1995), Cahuc et al. (2006) and Flinn (2006).

¹⁰⁹ See Mortensen (1986) for the continuous time case.

¹¹⁰ The LHS is linearly increasing in and passes through the origin. The RHS is monotonically decreasing in until it reaches , and is then constant. There will be a unique intersection, and a unique , as long as .

¹¹¹ In a continuous time model in which the arrival of offers follows a Poisson process with parameter , the implicit reservation wage equation is identical except that the instantaneous arrival rate replaces the offer probability, .

¹¹² Given a distributional assumption for , the solution for the reservation wage involves numerically (if, as is for most distributions the case, there is no closed form solution) solving a nonlinear equation. An alternative solution method would be to start from the reservation wage for the final period of a finite horizon problem (see below) and iterate on the reservation wage until it converges. Convergence is assured because the value function is a contraction mapping (see Sargent (1987)).

¹¹³ Wolpin (1995) provides a proof.

¹¹⁴ The distribution of accepted wages is the truncated distribution of wage offers, namely, .

¹¹⁵ An alternative would be to assume that once the terminal period is reached, the individual accepts the next offer that arrives, in which case the reservation wage at is zero.

¹¹⁶ See Wolpin (1987), for the particular case in which is either normal or log normal.

¹¹⁷ There are statistical issues better handled by specifying the hazard function, such as dealing with incomplete spells and time-varying regressors. See Meyer (1990) for an example of this approach. The issues we raise, however, are easier to demonstrate in a regression framework, but hold in the hazard framework as well.

¹¹⁸ The assumption that new unemployment spells are renewal processes rules out any structural connection between spells; for example, it rules out that the benefit level depends on the pre-unemployment wage.

¹¹⁹ As we noted, it is also usual to include some aggregate labor market statistic like the local unemployment rate. The idea is that the aggregate statistic reflects labor market demand and so will affect the offer rate or the wage offer distribution. As shown in Wolpin (1995), because the aggregate statistic is simply the aggregation of the search decisions over the unemployed population, it does not reflect solely demand conditions and estimates of UI benefit effects suffer from proxy variable bias.

¹²⁰ This section follows the development in Flinn and Heckman (1982) and Wolpin (1995).

¹²¹ The minimum observed wage is a superconsistent estimator of the reservation wage in that it converges at rate . This leads to nonstandard asymptotics in the likelihood estimation of the search model (see Flinn and Heckman (1982) and Christensen and Kiefer (1991)).

¹²² Setting and yields the infinite horizon implicit reservation wage function.

¹²³ The implicit reservation wage Eq. (138) would no longer hold in this case. In particular, the integration would have be taken also over the distribution of the unobserved cost of search, recognizing that the reservation wage would be a function of that cost.

¹²⁴ In some instances, wage rates that are directly reported in hours or weeks are inaccurate. In other cases, wage rates are derived from a division of earnings, reported over a longer period, say annually, and hours worked reported over that period. The inaccuracy arises from a seeming mismatch in the time period between earnings and hours.

¹²⁵ Theoretical models in which job searchers faced borrowing constraints appeared much earlier, starting with Danforth (1979). However, formal empirical implementation did not become feasible until the development of estimation methods for DCDP models.

¹²⁶ The search model does not have to incorporate savings for it to be optimal for individuals to quit into unemployment. It is sufficient that there be a finite horizon (retirement) and either that the offer rate be higher in unemployment or that there be a wage return to general experience (Wolpin, 1992).

¹²⁷ This procedure for reducing the size of the state space follows Wolpin (1992).

¹²⁸ Assets are only observed annually. The data are from the NLSY79.

¹²⁹ Note that the cost of search in his model is isomorphic to the probability of receiving an offer.

¹³⁰ The reservation wage, mean accepted wage and hazard rate are all functions of . They can be estimated as nonparametric functions of Paserman also models search after the exhaustion of benefits. In that case, it is assumed that individuals solve an infinite horizon problem.

¹³¹ The model was estimated both for a normal and log normal wage offer distribution. As found by others (for example,Wolpin (1987), the normal distribution assumption proved problematic. Paserman also allows for measurement error for the reasons previously discussed.

¹³² Ferrall also estimates a model for the US, but, because the UI system varies from state to state and is much less generous than the Canadian system, he does not incorporate UI benefits into the analysis. We focus on the Canadian data to highlight the fact that the DCDP approach allows for a detailed representation of UI policy.

¹³³ Actually, this waiting period is only for those who are unemployed through layoff. Those who quit or were fired had a waiting period of five weeks. Ferrall assumes the waiting period to be two weeks independent of the reason for the unemployment spell.

¹³⁴ As previously noted, the lower bound of the support for the Pareto distribution cannot be identified. Ferrall fixes that value. Christensen and Kiefer (2009) also use the Pareto distribution and impose, based on the wage posting model of Burdett and Mortensen (1998), the individual’s reservation wage as the lower bound.

¹³⁵ UI system parameters depend on region and some of the structural parameters are allowed to differ by education.

¹³⁶ Walsh (1935) is cited by Becker (1964), which is perhaps the most influential work in the development of the human capital literature.

¹³⁷ The sources include a survey of 15,000 former members of the Alpha Kappa Psi fraternity on the education and earnings, a survey of Land Grant colleges made by the US Department of Interior, published figures on the earnings of physician and doctors, and a survey of Harvard Law graduates.

¹³⁸ Heckman and Honore (1990) derive additional implications for earnings distributions and provide identification results in the case of non-normal distributions.

¹³⁹ Much of the empirical literature prior to Willis and Rosen (1979) was concerned with estimating rates of return to schooling. A considerable amount of effort was (and still is) devoted to accounting for bias in the schooling coefficient due to omitted ability in a Mincer-style earnings function. See Card (2001) and Wolpin (2003) for an assessment of that literature.

¹⁴⁰ Specifically, pay grades and promotions are assumed to be probabilistic functions of observable state variables.

¹⁴¹ A related study by Daula and Moffitt (1995) uses a similar DCDP model to analyze the effect of financial incentives on Army infantry reenlistments.

¹⁴² The model is solved using the Gittens index, a simplification in the solution of the dynamic programming problem that arises because tenure in one job does not affect the rate of learning in other jobs.

¹⁴³ Work experience, .

¹⁴⁴ Keane and Wolpin (2001) also find that, on average, youths receive a transfer from their parents in excess of what is received when not attending college, sufficient to fully subsidize college tuition costs. The subsidy ranges from about one-half of the tuition cost for youths whose parents are the least educated (neither a high school graduate) to almost twice the tuition cost for youths whose parents are the most educated (at least one parent a college graduate). It might appear that it is because of the largesse of parents that relaxing borrowing constraints has only a minimal impact on college attendance. However, simulating the impact of relaxing the borrowing constraint in a regime where parents are assumed to provide no additional transfers to children who attend college leads to the same result. Transfers do, however, have a non-negligible effect on school attendance. If transfers are equalized across children, high SES children go to school less, but low SES children do not increase their attendance by much.

¹⁴⁵ The approach taken by Cameron and Heckman (1999) can be interpreted as estimating the approximate decision rules from a DCDP model.

¹⁴⁶ All individuals are assumed to complete at least six years.

¹⁴⁷ In the model, there is also a probability of experiencing a so-called interruption, which is a decision period when no decision is made and the stock of accumulated human capital remains fixed, intended to capture an event such as illness or academic failure that lasts one period.

¹⁴⁸ See Belzil and Hansen (2002) Table VIII. The previously described Keane and Wolpin (1997) model did not allow for nonlinearities in returns to education but did allow the return to differ between white and blue collar occupations and found a much higher return in the white collar occupation, which could be viewed as consistent with Belzil and Hansen’s (2002) finding of high returns at higher education levels.

¹⁴⁹ The model is life cycle rather than dynamic in the sense that new information is not revealed to the agent in each decision period.

¹⁵⁰ An earlier paper by Lee and Wolpin (2006) develops a similar model with a focus on examining the relative importance of labor demand and supply factors in explaining the expansion of service sector employment

¹⁵¹ The general rise in inequality and the college premium have often been linked (for example, Murphy and Welch (1992)), but not together with the rise in female-male wages and the growth of the service sector. The growth in the service sector has also been linked with the rise in female employment Fuchs, 1980; Welch, 2000 draws a link between the rise in wage inequality among men and the reduction in the gender wage gap.

¹⁵² For a review of the larger literature, see Katz and Autor (1999). The papers they survey include Bound and Johnson (1992), Gottschalf and Moffitt (1994), Juhn et al. (1993), Katz and Murphy (1992), Krusell et al. (2000) and Murphy and Welch (1992, 1993). Recent contributions to this literature include Baldwin and Cain (2000), Eckstein and Nagypal (2004), Hornstein et al. (2005) and Welch (2000).

¹⁵³ There is a large labor economics literature on interindustry wage differentials among otherwise observably identical workers. Frictions to switching sectors are sometimes proposed to explain these differentials.

¹⁵⁴ The maximum likelihood approach, developed by Keane and Wolpin (2001) and extended in Keane and Sauer (2009) allows for classification error and for missing state variables.

¹⁵⁵ These results are similar to those Keane and Wolpin (2001) obtain from simulations of the effect of reducing borrowing constraints on enrollment.

¹⁵⁶ See Greenberger and Steinberg (1986).

¹⁵⁷ Recent work that specifies and estimates equilibrium models of the college market include Epple et al. (2006) and Fu (2009).

¹⁵⁸ Estimation is based on the EM algorithm developed in Dempster et al. (1977) and adapted to DCDP models by Arcidiacono and Jones (2003).

¹⁵⁹ The DCDP schooling models described previously take endowments at college entry ages as given. It is still an open question whether college subsidies would induce an increase in parental investments at younger ages and thus affect the endowments.

¹⁶⁰ Of course, robustness by itself cannot be conclusive; all of the models could give similarly biased results.

¹⁶¹ Such prejudices are revealed by the contrast between the structure of the DCDP model that Todd and Wolpin (2006) used to evaluate a conditional cash transfer program in Mexico and the model used by Attanasio et al. (2005). As another example, there are several applications of DCDP models applied to traditional topics that take a behavioral economics view. As seen, Paserman (2008) studies a job search model. In addition, Fang and Silverman (2009) study a model of women’s welfare participation assuming that agents use hyperbolic discounting.

¹⁶² Similarly, Lise et al. (2003) used data from a Canadian experiment designed to move people off of welfare and into work to validate a calibrated search-matching model of labor market behavior. Bajari and Hortacsu (2005) employ a similar validation methodology in the case of a laboratory auction experiment.