Computational Implementation#

We will present the computational implementation of the model.

To Explanation

A list of the parameter-covariates combination as implemented in respy is presented in Parameterization

We follow individuals over their working life from young adulthood at age 16 to retirement at age 65. The decision period \(t = 16, \dots, 65\) is a school year and individuals decide \(a\in\mathcal{A}\) whether to work in a blue-collar or white-collar occupation (\(a = 1, 2\)), to serve in the military \((a = 3)\), to attend school \((a = 4)\), or to stay at home \((a = 5)\).

../_images/event_tree.svg

Individuals are initially heterogeneous. They differ with respect to their initial level of completed schooling \(h_{16}\) and have one of four different \(\mathcal{J} = \{1, \dots, 4\}\) alternative-specific skill endowments \(\boldsymbol{e} = \left(e_{j,a}\right)_{\mathcal{J} \times \mathcal{A}}\).

To How-to guide

An explanation how to set and change initial conditions in respy is provided in Initial Conditions

The per-period utility \(u_a(\cdot)\) of each alternative consists of a non-pecuniary utility \(\zeta_a(\cdot)\) and, at least for the working alternatives, an additional wage component \(w_a(\cdot)\). Both depend on the level of human capital as measured by their alternative-specific skill endowment \(\boldsymbol{e}\), years of completed schooling \(h_t\), and occupation-specific work experience \(\boldsymbol{k_t} = \left(k_{a,t}\right)_{a\in\{1, 2, 3\}}\). The immediate utility functions are influenced by last-period choices \(a_{t -1}\) and alternative-specific productivity shocks \(\boldsymbol{\epsilon_t} = \left(\epsilon_{a,t}\right)_{a\in\mathcal{A}}\) as well. Their general form is given by:

\[\begin{split}u_a(\cdot) = \begin{cases} \zeta_a(\boldsymbol{k_t}, h_t, t, a_{t -1}) + w_a(\boldsymbol{k_t}, h_t, t, a_{t -1}, e_{j, a}, \epsilon_{a,t}) & \text{if}\, a \in \{1, 2, 3\} \\ \zeta_a(\boldsymbol{k_t}, h_t, t, a_{t-1}, e_{j,a}, \epsilon_{a,t}) & \text{if}\, a \in \{4, 5\}. \end{cases}\end{split}\]

Work experience \(\boldsymbol{k_t}\) and years of completed schooling \(h_t\) evolve deterministically.

\[\begin{split}k_{a,t+1} = k_{a,t} + \mathbb{1}[a_t = a] &\qquad \text{if}\, a \in \{1, 2, 3\} \\ h_{t + 1\phantom{,a}} = h_{t\phantom{,a}} + \mathbb{1}[a_t = 4] &\qquad\end{split}\]

The productivity shocks are uncorrelated across time and follow a multivariate normal distribution with mean \(\boldsymbol{0}\) and covariance matrix \(\boldsymbol{\Sigma}\). Given the structure of the utility functions and the distribution of the shocks, the state at time \(t\) is summarized by the object \(s_t = \{\boldsymbol{k_t}, h_t, t, a_{t -1}, \boldsymbol{e},\boldsymbol{\epsilon_t}\}\).

To Reference guide

A sophisticated account of the complete state space is integrated in respy. The particular implementation is described in respy.state_space

Empirical and theoretical research from specialized disciplines within economics informs the exact specification of \(u_a(\cdot)\). We now discuss each of its components in detail.

Non-pecuniary utility#

We present the parameterization of the non-pecuniary utility for all five alternatives.

Blue-collar#

Equation (1) shows the parameterization of the non-pecuniary utility from working in a blue-collar occupation.

(1)#\[\begin{split}\zeta_{1}(\boldsymbol{k_t}, h_t, a_{t-1}) = \alpha_1 &+ c_{1,1} \cdot \mathbb{1}[a_{t-1} \neq 1] + c_{1,2} \cdot \mathbb{1}[k_{1,t} = 0] \\ & + \vartheta_1 \cdot \mathbb{1}[h_t \geq 12] + \vartheta_2 \cdot \mathbb{1}[h_t \geq 16] + \vartheta_3 \cdot \mathbb{1}[k_{3,t} = 1]\end{split}\]

A constant \(\alpha_1\) captures the net monetary-equivalent of on the job amenities. The non-pecuniary utility includes mobility and search costs \(c_{1,1}\), which are higher for individuals who never worked in a blue-collar occupation before \(c_{1,2}\). The non-pecuniary utilities capture returns from a high school \(\vartheta_1\) and a college \(\vartheta_2\) degree. Additionally, there is a detrimental effect of leaving the military early after one year \(\vartheta_3\).

White-collar#

The non-pecuniary utility from working in a white-collar occupation is specified analogously. Equation (2) shows its parameterization.

(2)#\[\begin{split}\zeta_{2}( \boldsymbol{k_t}, h_t, a_{t-1} ) = \,\alpha_2 & + c_{2,1} \cdot \mathbb{1}[a_{t-1} \neq 2] + c_{2,2} \cdot \mathbb{1}[k_{2,t} = 0]\\ & + \vartheta_1 \cdot \mathbb{1}[h_t \geq 12] + \vartheta_2 \cdot \mathbb{1}[h_t \geq 16] + \vartheta_3 \cdot \mathbb{1}[k_{3,t} = 1]\end{split}\]

Military#

Equation (3) shows the parameterization of the non-pecuniary utility from working in the military.

(3)#\[\zeta_{3}( k_{3.t}, h_t) = c_{3,2} \cdot \mathbb{1}[k_{3,t} = 0] + \vartheta_1 \cdot \mathbb{1}[h_t \geq 12] + \vartheta_2 \cdot \mathbb{1}[h_t \geq 16]\]

Search costs \(c_{3, 1} = 0\) are absent but there is a mobility cost if an individual has never served in the military before \(c_{3,2}\). Individuals still experience a non-pecuniary utility from finishing high-school \(\vartheta_1\) and college \(\vartheta_2\).

School#

Equation (4) shows the parameterization of the non-pecuniary utility from schooling.

(4)#\[\begin{split}\zeta_4(k_{3,t}, h_t, t, a_{t-1}, e_{j,4}, \epsilon_{4,t}) = e_{j,4} & + \beta_{tc_1} \cdot \mathbb{1}[h_t \geq 12] + \beta_{tc_2} \cdot \mathbb{1}[h_t \geq 16] \\\nonumber & + \beta_{rc_1} \cdot \mathbb{1}[a_{t-1} \neq 4, h_t < 12] + \beta_{rc_2} \cdot \mathbb{1}[a_{t-1} \neq 4, h_t \geq 12] \\\nonumber & + \gamma_{4,4} \cdot t + \gamma_{4,5} \cdot \mathbb{1}[t < 18] \\\nonumber & + \vartheta_1 \cdot \mathbb{1}[h_t \geq 12] + \vartheta_2 \cdot \mathbb{1}[h_t \geq 16] + \vartheta_3 \cdot \mathbb{1}[k_{3,t} = 1]\\ & + \epsilon_{4,t}\end{split}\]

There is a direct cost of attending school such as tuition for continuing education after high school \(\beta_{tc_1}\) and college \(\beta_{tc_2}\). The decision to leave school is reversible, but entails adjustment costs that differ by schooling category (\(\beta_{rc_1}, \beta_{rc_2}\)). Schooling is defined as time spent in school and not by formal credentials acquired. Once individuals reach a certain amount of schooling, they acquire a degree. There is no uncertainty about grade completion (Altonji, 1993, [2]) and no part-time enrollment. Individuals value the completion of high-school and graduate school (\(\vartheta_1, \vartheta_2\)).

Home#

Equation (5) shows the parameterization of the non-pecuniary utility from staying at home.

(5)#\[\begin{split}\zeta_5(k_{3,t}, h_t, t, e_{j,5}, \epsilon_{5,1}) = e_{j,5} & + \gamma_{5,4} \cdot \mathbb{1}[18 \leq t \leq 20] + \gamma_{5,5} \cdot \mathbb{1}[t \geq 21] \\ \nonumber & +\vartheta_{1} \cdot \mathbb{1}[h_t \geq 12] + \vartheta_{2} \cdot \mathbb{1}[h_t \geq 16] + \vartheta_3 \cdot \mathbb{1}[k_{3,t} = 1] \\ & + \epsilon_{5,t}\end{split}\]

Staying at home as a young adult \(\gamma_{5, 4}\) is less stigmatic as doing so while already being an adult \(\gamma_{5,5}\). Additionally, possessing a degree \((\vartheta_1, \vartheta_2)\) or leaving the military prematurely \(\vartheta_3\) influences the immediate utility.

Wage component#

The wage component \(w_{a}(\cdot)\) for the working alternatives is given by the product of the market-equilibrium rental price \(r_{a}\) and an occupation-specific skill level \(x_{a}(\cdot)\). The latter is determined by the overall level of human capital.

\[w_{a}(\cdot) = r_{a} \, x_{a}(\cdot)\]

This specification leads to a standard logarithmic wage equation in which the c onstant term is the skill rental price \(\ln(r_{a})\) and wages follow a log-normal distribution.

The occupation-specific skill level \(x_{a}(\cdot)\) is determined by a skill production function, which includes a deterministic component \(\Gamma_a(\cdot)\) and a multiplicative stochastic productivity shock \(\epsilon_{a,t}\).

\[x_{a}(\boldsymbol{k_t}, h_t, t, a_{t-1}, e_{j, a}, \epsilon_{a,t}) = \exp \big( \Gamma_{a}(\boldsymbol{k_t}, h_t, t, a_{t-1}, e_{j,a}) \cdot \epsilon_{a,t} \big)\]

Blue-collar#

Equation (6) shows the parameterization of the deterministic component of the skill production function.

(6)#\[\begin{split} \Gamma_1(\boldsymbol{k_t}, h_t, t, a_{t-1}, e_{j, 1}) = e_{j,1} & + \beta_{1,1} \cdot h_t + \beta_{1, 2} \cdot \mathbb{1}[h_t \geq 12] + \beta_{1,3} \cdot \mathbb{1}[h_t\geq 16]\\ & + \gamma_{1, 1} \cdot k_{1,t} + \gamma_{1,2} \cdot (k_{1,t})^2 + \gamma_{1,3} \cdot \mathbb{1}[k_{1,t} > 0] \\ & + \gamma_{1,4} \cdot t + \gamma_{1,5} \cdot \mathbb{1}[t < 18]\\ & + \gamma_{1,6} \cdot \mathbb{1}[a_{t-1} = 1] + \gamma_{1,7} \cdot k_{2,t} + \gamma_{1,8} \cdot k_{3,t}\end{split}\]

There are several notable features. The first part of the skill production function is motivated by Mincer (1958, [16]) and Mincer and Polachek (1974, [17]) and hence linear in years of completed schooling \(\beta_{1,1}\), quadratic in experience (\(\gamma_{1,1}, \gamma_{1,2}\)), and separable between the two of them. There are so-called sheep-skin effects (Spence, 1973, [22], Jaeger and Page, 1996, [10]) associated with completing a high school \(\beta_{1,2}\) and graduate \(\beta_{1,3}\) education that capture the impact of completing a degree beyond just the associated years of schooling. Also, skills depreciate when not employed in a blue-collar occupation in the preceding period \(\gamma_{1,6}\). Other work experience (\(\gamma_{1,7}, \gamma_{1,8}\)) is transferable.

White-collar#

The wage component from working in a white-collar occupation is specified analogously. Equation (7) shows the parameterization of the deterministic component of the skill production function.

(7)#\[\begin{split} \Gamma_2(\boldsymbol{k_t}, h_t, t, a_{t-1}, e_{j,2}) = e_{j,2} & + \beta_{2,1} \cdot h_t + \beta_{2, 2} \cdot \mathbb{1}[h_t \geq 12] + \beta_{2,3} \cdot \mathbb{1}[h_t\geq 16] \\ & + \gamma_{2, 1} \cdot k_{2,t} + \gamma_{2,2} \cdot (k_{2,t})^2 + \gamma_{2,3} \cdot \mathbb{1}[k_{2,t} > 0] \\ & + \gamma_{2,4} \cdot t + \gamma_{2,5} \cdot \mathbb{1}[t < 18] \\ & + \gamma_{2,6} \cdot \mathbb{1}[a_{t-1} = 2] + \gamma_{2,7} \cdot k_{1,t} + \gamma_{2,8} \cdot k_{3,t}\end{split}\]

Military#

Equation (8) shows the parameterization of the deterministic component of the skill production function.

(8)#\[\begin{split}\Gamma_3( k_{3,t}, h_t, t, e_{j,3}) = e_{j,3} & + \beta_{3,1} \cdot h_t \\ \nonumber &+ \gamma_{3,1} \cdot k_{3,t} + \gamma_{3,2} \cdot (k_{3,t})^2 + \gamma_{3,3} \cdot \mathbb{1}[k_{3,t} > 0]\\ \nonumber& + \gamma_{3,4} \cdot t + \gamma_{3,5} \cdot \mathbb{1}[t < 18]\end{split}\]

Contrary to the civilian sector there are no sheep-skin effects from graduation (\(\beta_{3,2} = \beta_{3,3}= 0\)). The previous occupational choice has no influence (\(\gamma_{3,6}= 0\)) and any experience other than military is non-transferable (\(\gamma_{3,7} = \gamma_{3,8} = 0\)).

Remark: Our parameterization for the immediate utility of serving in the military differs from Keane and Wolpin (1997, [13]) as we remain unsure about their exact specification. The authors state in Footnote 31 (p.498) that the constant for the non-pecuniary utility \(\alpha_{3,t}\) depends on age. However, we are unable to determine the precise nature of the relationship. Equation (C3) (p.521) also indicates no productivity shock \(\epsilon_{a,t}\) in the wage component. Table 7 (p.500) reports such estimates.

To Explanation

The operationalization of the model allows to proceed with the calibration as described in Calibration