Model I

Earnings

W_t^g(s,x,e) = \omega(s) \, \varepsilon(s) \, \lambda(e) \, x(s)

\varepsilon(s): permanent unobserved sector-specific skill

\omega(s): skill price

x(s): current period human capital for sector s.

\lambda(e): effect of education on human capital.

rewrite:

ln W_t^g(s,x,e) = ln \, \omega(s) \, + ln \, \varepsilon(s) \, + ln \, \lambda(e) + \, ln \, x(s)

equation (4) in IZA_model gives a log-wage equation where the intercept is [ \,ln \omega(s) \, + ln \varepsilon(s) \,], education component is given by some function \lambda(e) and work experience component is x(s).

Questions on earnings:

Need to specify what part of wages are individual- and/or time-specific. For example, compare to a more standard form:

a Ben-Porath framework where earnings in sector s is the product of aggregate skill price [\omega(s)] and the amount of skills the person has in sector s denoted with H(s):

W_{it}(s) = \omega_t(s) \cdot H_{it}(s)

where human capital production technology is a function of education, experience, sector-specific skill endowment, and some idiosyncratic time-varying component:

H_{it}(s) = exp \big( \, \beta_1 \, e_i + \beta_2 \, x_{it}(s) + \varepsilon_i(s) + \nu_{it}(s) \, \big)

ln \, W_{it}(s) = \omega(s) + \beta_1 \, e_i + \beta_2 \, x_{it}(s) + \varepsilon_i(s) + \nu_{it}(s)

which gives the standard Mincer log-wage equation where the intercept \omega(s) + \varepsilon_i(s) consists of sector-specific skill price and individual-specific skill endowment in sector s.

this has the same terms as W(.)=\omega(s) \cdot \varepsilon(s) \cdot \lambda(e) \cdot x(s) , except for the addition of a time-varying idiosyncratic term \nu_{it}(s). need to discuss whether we want such a term.

Note: the intercept \omega(s) + \varepsilon_i(s) should be treated as one parameter i.e. even with panel data, changes in \omega(s) (skill price differences either due to discrimination or due to technology or other things) and differences in \varepsilon_i(s) (skill endowment differences) are observationally equivalent unless we have some exogenous shifters that change skill rental prices (we don’t). so I don’t think we can really talk about discrimination or have a parameter that pertains to it, but this is fine as we can still say something about the differences in the wage opportunities men and women face, stemming from either different price of their skills due to discrimination etc. OR due to differences in their starting endowments. this is not a limitation as it’s a restriction that applies to most other gender wage gap papers.

Singles’ problem

\begin{aligned} V^g_t(s,x;\mathbf{f},e,\mathbb{S}) = {} \max\limits_s \bigg\{ \,\,\, ln\big[ W_t^g(s,x,e) - \kappa(\mathbf{f},s) \big] + \beta \, \mathbb{E} \big[ \sum\limits f(s,e) \, V_{t+1}^g(s',x';\mathbf{f}',e,\mathbb{S}') \big] \bigg\} \end{aligned}

s: sector (1-home production, 2- self-employment, 3-private sector employment, 4-public sector employment)

x: work experience

e: education

\mathbb{S}: sector offer

\kappa(.): childcare cost

f(.): offer probability determined by current sector s and education e

\mathbf{f}: number of kids

Questions on the singles’ problem:

  • notation is a bit confusing though I understand the handout was not meant to be the full exposition of a model. for example, s is both an argument in V and then also the choice variable in the objective function.

  • In addition to \mathbb{S}', need to write out what else the expectation operator \mathbb{E} is defined over.

  • Is \mathbb{S}’ just the index of the sector from which agents receive a job offer in t+1? need to decide whether they receive job offers from every sector each period or do they only receive a job offer from at most one sector each period?

  • the vector of individual-specific sector skills ie \varepsilon = \big( \varepsilon(1),\varepsilon(2),\varepsilon(3),\varepsilon(4) \big) should be one of the state variables. Together with education, it’s part of the initial conditions.

rewrite the above (same model but with slightly different notation):

  • Instead of \mathbf{f}, use n to denote number of kids (to avoid confusing it with the offer probability function f(.))

  • omitting i for now and omitting the gender index g.

For a person with state variables x, n, e, \varepsilon and offer \mathbb{S}, the alternative-specific value of choosing sector j \in \{ 1,2,3,4 \} is: The contemporaneous utility of working in sector j (given by ln[W(.)-\kappa(.)]) plus the expected continuation value next period given current period choice is j.

\begin{aligned} v^j_t( x_t, \, n_t, \mathbb{S}_t ;\, e, \, \varepsilon ) = {} ln\big[ W_t^j(x_t\, ;\, e, \, \varepsilon) - \kappa(j,n_t) \big] + \beta \, \mathbb{E} \bigg[ \sum\limits_{\mathbb{S}_{t+1}} \,f(\mathbb{S}_{t+1} \mid s_t=j,\,e) \, V_{t+1}(x_{t+1},n_{t+1},\mathbb{S}_{t+1};e,\varepsilon) \bigg] \end{aligned} and V_t(.) is the maximum over alternative-specific value functions v_t(j,...):

\begin{aligned} V_t(x_t, \, n_t, \, \mathbb{S}_t;e, \, \varepsilon) = {} \max\limits_j \,\, v^j_t( x_t, \, n_t, \mathbb{S}_t ;\, e, \, \varepsilon ) \end{aligned} so that V_{t+1}(.) is defined similarly.

f(\mathbb{S}_t+1;\, s_t=j,e) specifies offer probabilities in period t+1, which depends on the sector choice in preceding period (in this case given by choice j) and education level e.

to clarify the structure of uncertainty agents face, write out the rest of the expectation operator:

\begin{aligned} v^j_t( x_t, \, n_t, \mathbb{S}_t ;\, e, \, \varepsilon ) = {} ln\big[ W_t^j(x_t\, ;\, e, \, \varepsilon) - \kappa(j,n_t) \big] + \beta \, \sum\limits_{n_{t+1}} \, \sum\limits_{\mathbb{S}_{t+1}} \, prob(n_{t+1} \mid n_t) \,f(\mathbb{S}_{t+1} \mid s_t=j,\,e) \, V_{t+1}(x_{t+1},n_{t+1},\mathbb{S}_{t+1};e,\varepsilon) \end{aligned}

Notes:

  • Work experience evolves deterministically given preceding period’s x_t and sectoral choice so I don’t include it in the expectation. However, since we don’t really observe work experience in the data (which would require us to be able to observe each person’s employment history from some labor market entry age), I suggest we make work experience evolve stochastically instead. this would reduce the state space (especially if we plan to have sector-specific experience. the notes mention a human capital transfer matrix and we also talked about sector-specific experience. this blows up the state space considerably since then we have to keep track of three different work experience variables. this needs to be discussed.

  • The continuation value also needs to include the probability of marriage if single etc. I omit that for now.

  • If the intention is to have some kind of random time-varying wage component (other than experience or age) such as that \nu_{it}(s), then we also have to integrate over those draws.

Home sector and income

In the above specification, agents’ utility flow for each alternative j is given by

U^j = ln\big[ W_t^j(x_t\, ;\, e, \, \varepsilon) - \kappa(j,n_t) \big] \,\,\,\, j=1,4

This is a problem for the home alternative j=1 since we don’t observe any income.

Given the non-linear utility specification, we need some kind of non-labor income in there anyway. When utility is not well-defined for some parameter values, the model solution/estimation behaves unpredictably.

The addition of non-labor income addresses the problem of non-observable home income problem and also ensures that the utility is well-defined for all wage and child-care cost values.

Non-labor income can either come from the data or we can just estimate it as a parameter that is the same for everyone or that depends on some characteristics. The first option (pinning it down using data) is probably not possible. Even with much better data, non-labor income information is rarely any good. The second option (treating non-labor income as a parameter to be estimated) would look like the following:

U^j = ln\big[ Y + W_t^j(x_t\, ;\, e, \, \varepsilon) - \kappa(j,n_t) \big] \,\,\,\, j=1,4 where Y is a parameter to be estimated and we set W_t^1(.)=0 (home). Y can be allowed to differ by gender and/or marital status and/or education as well.

The model solution will be very sensitive to the values of Y, which is a problem especially if we are not pinning it down using non-labor income data. So it would be identified, but maybe not separately from some of the other channels we are interested in.

For couples, Y would be added to total household earnings for each alternative the same way as singles.

Couples’ problem

The couples’ problem looks the same as above but with the addition of spousal state variables and spousal employment decisions:

\begin{aligned} M^h_t(x_t, \, \mathbb{S}_t \, \tilde{x}_t, \, \tilde{\mathbb{S}}_t, \,n_t; e, \, \varepsilon, \, \tilde{e}, \, \tilde{\varepsilon}) &= {} \max\limits_{j,\tilde{j}} \,\, m^h_t(j,\tilde{j}, \, x_t, \, \mathbb{S}_t, \, \tilde{x}_t, \, \tilde{\mathbb{S}}_t , \,n_t; e, \, \varepsilon,\, \tilde{e}, \, \tilde{\varepsilon}) + m^w_t(j,\tilde{j}, \, x_t, \, \mathbb{S}_t, \, \tilde{x}_t, \, \tilde{\mathbb{S}}_t , \,n_t; e, \, \varepsilon,\, \tilde{e}, \, \tilde{\varepsilon}) \\ M^w_t(x_t, \, \mathbb{S}_t \, \tilde{x}_t, \, \tilde{\mathbb{S}}_t, \,n_t; e, \, \varepsilon, \, \tilde{e}, \, \tilde{\varepsilon}) &= {} \max\limits_{j,\tilde{j}} \,\, m^h_t(j,\tilde{j}, \, x_t, \, \mathbb{S}_t, \, \tilde{x}_t, \, \tilde{\mathbb{S}}_t , \,n_t; e, \, \varepsilon,\, \tilde{e}, \, \tilde{\varepsilon}) + m^w_t(j,\tilde{j}, \, x_t, \, \mathbb{S}_t, \, \tilde{x}_t, \, \tilde{\mathbb{S}}_t , \,n_t; e, \, \varepsilon,\, \tilde{e}, \, \tilde{\varepsilon}) \end{aligned}

  • M(.) and m^j(.) are the married counterparts of the V(.) and v^j functions defined before. I label them differently to highlight that the marriage value functions are different objects since their state space as well as the uncertainty agents face is different.

  • if we go with Model I, one way to reduce the dimensionality of this problem would be to get rid of work experience altogether and just have age to fit wage profiles. this is a choice between whether we want to put more structure on human capital accumulation (sectoral differences, transferrability of human capital across sectors etc.) vs. household stuff. we can discuss this.

Model II

Earnings defined the same way as Model I.

ln \, W_{it}^{s,g} = \beta_0^{s,g} + \beta_1^{s,g} \, e_i + \beta_2^{s,g} \, x_{it} + \varepsilon_{i}^s

where individuals’ vector of initial sector-specific skills \varepsilon_i = (\varepsilon_i^s,\,s=1,4 ) follow a gender-specific distribution: \varepsilon_i \sim F^g(.) with normalized mean 0 (or alternatively, we can also get rid of \beta_0 and estimate the mean for \varepsilon for each gender for reasons discussed above.)

I omit a time-varying idiosyncratic component \nu because it was not in Model I, but we can add it on or leave it.

For married and single agents, the alternative-specific utility flows for each sector is specified as:

\begin{aligned} U_{it}^{j,g} &= W_{it}^{j,g}(\cdot) + \pi^{j,g}(n_{it}, m_{it}, e_i) \,\,\,\,\,\,\,\,\, j=1,4 \end{aligned} where

\begin{aligned} &\pi^j(.) = \pi_{0}^{j,g} \, m_{it} + \pi_{1}^{j,g} \, n_{it} + \pi_{2}^{j,g} \, e_i + interaction \,\,\,\,terms\,\,\, \,\,\,\,\,\,\,\,\,\,\, j=1,4 \end{aligned}

Alternative-specific value functions defined the same way as before:

\begin{aligned} v^j_t( m_t,x_t, \, n_t, \mathbb{S}_t ;\, e, \, \varepsilon ) = {} \pi^j(.) + \beta \, \sum\limits_{m_{t+1}} \sum\limits_{n_{t+1}} \, \sum\limits_{\mathbb{S}_{t+1}} \, prob(m_{t+1} \mid m_t) \, \cdot prob(n_{t+1} \mid n_t) \,\cdot f(\mathbb{S}_{t+1} \mid s_t=j,\,e) \, \cdot V_{t+1}(m_{t+1},x_{t+1},n_{t+1},\mathbb{S}_{t+1};e,\varepsilon) \end{aligned} where the g and i indices are omitted for clarity but everything is indexed by gender.

V_t(.) is defined as before:

\begin{aligned} V_t(m_t,x_t, \, n_t, \, \mathbb{S}_t;e, \, \varepsilon) = {} \max\limits_j \,\, v^j_t( m_t,x_t, \, n_t, \mathbb{S}_t ;\, e, \, \varepsilon ) \end{aligned}

Note: Now agents have the same value function regardless of marital status because it just enters as an argument in the state space. In Model I, married and single agents solve a different optimization problem (married people maximize the sum of their total utilities with total income and continuation values) so just having marital status as an argument does not cut it.

The transition probabilities that govern the evolution of the vector of state variables comprise of the transition probabilities for fertility and marriage. These processes are independent from each other.

Fertility:

prob_{n}\!\left( n_{t+1} \mid n_{t} \right) = \begin{cases} p_n & \quad \text{for } n_{t+1} = n_{t}+1 \,\text{and } n_{t} \leq \bar{n}, \\ 1-p_{n} & \quad \text{for } n_{t+1}=n_t \end{cases}

Marriage:

The probability of getting married is determined by agent’s gender, age and education and is given by Prob \big( m_{t+1}=1 \mid m_t=0 \big) = \frac{\exp\!\left( \, \delta_0 \, g + \delta_1 \, a + \delta_2 \, a^2 + \delta_3 \, e \, \right)} {1+ \exp\!\left( \, \delta_0 \, g + \delta_1 \, a + \delta_2 \, a^2 + \delta_3 \, e \, \right)}

so that the relationship between marriage rates, age, and education is captured through the \delta coefficients and the dependence on age/education is imposed rather than generated by behavior. These are estimated using the marriage patterns in the data so the estimation is able to incorporate any changing employment incentives that arise due to changing marital status over the life-cycle - in a way that is informed by the data. \delta’s stand in for various underlying forces so that they absorb a lot of structural channels (income effects, match gains, horizon effects, etc.) and are likely not policy-invariant.

These exogenous processes can be estimated outside the model.

Notes:

  • In Model II, the utility specifications don’t include spousal income nor spousal characteristics nor an explicit household budget constraint. Any effect of spousal or marriage-related factors on alternative-specific utilities works through the \pi parameters, which vary by gender and sector s. This allows us to capture the different gender-specific effects of fertility, marriage, and education on the utility of sectoral employment without taking a stance on the underlying mechanism ie without having to deal with spousal characteristics in the state space and without modeling household decisions or the marriage market directly.

  • A richer structural model, such as Model I, would generate these effects explicitly. For example, in Model I, educated women who are married to educated men with higher earnings would have a lower marginal utility of additional income due to the log-linearity of preferences. This might potentially decrease their valuation of sectors that offer higher earnings, especially if they have children and cost of children is higher in those sectors. In Model II, these margins would only be captured through the \pi coefficients on children/marriage/education.

  • For our purposes (eg understanding gender differences in sectoral employment and sector gender wage gaps), Model II does the job but it does limit the kind of counterfactual experiments we can do (for example anything that impacts the marriage market, marriage matching, intra-household time allocations etc. could be problematic though still doable. some arguments can always be made about how some of these parameters are not sensitive to short-run changes and are more governed by social norms, etc.).

  • Model II specifies utility in a mechanical way, but the model solution still allows us to use the theoretical restrictions from the dynamic optimization problem on choice probabilities and therefore handles dynamic selection through the joint optimization of wage parameters using the model solution. In other words, the estimation uses the conditional means of unobserved skills implied by the dynamic model solution ( the model simulated earnings moments include E(\varepsilon \mid \text{decision rules that take into account continuation values/dynamics}) compared to a static selection model where these control functions are usually specified as an ad hoc function of individual characteristics ).

  • the added structure in Model I, especially the non-linearity in the utility function, helps with identification, but here we lose that added benefit.

  • the state space and the choice set are smaller in Model II.

there are tradeoffs involved in either approach and we can discuss more. The good thing is that we can easily alternate between the two. The program for the two are somewhat similar especially for the individual search problem. For this reason, just to get things moving along, singles’ employment decisions could be a good starting point for the code.