Model in Keane and Wolpin (1994)#

The explanations section of this documentation gives a detailed outline of the economic modeling components and the mathematical framework in Eckstein-Keane-Wolpin models using the example of Keane and Wolpin (1997, [13]). In the documentation, you will often another model specification rooted in the publication Keane and Wolpin (1994, [15]). This model constitutes a similar but simpler version of the model. We give a brief overview of the reward functions and components distinctive to this specification here. Note that the underlying economic and mathematical framework remains the same.

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Find the economic model and mathematical framework in the Explanations

The model from Keane and Wolpin (1994, [15]) is characterized by four distinct choices. At each point in time \(t \in \{0, ...,39\}\) individuals decide between \(a \in \{1,2,3,4\}\) mutually exclusive alternatives: working in occupation A, working in occupation B ($a=1,2$), investing in education ($a=3$), or staying home ($a=4$). The alternatives are associated with the rewards:

\[\begin{split}\text{Occupation A: } R_1(t) &= w_{1t} = r_{1}exp\{\alpha_{1} + \beta_{1,1}h_{t} + \gamma_{1,1}k_{1t} + \gamma_{1,2}k^2_{1t} + \gamma_{1,7}k_{2t} + \gamma_{2,8}k^2_{2t} + \epsilon_{1t}\} \nonumber \\ \text{Occupation B: } R_2(t) &= w_{2t} = r_{2}exp\{\alpha_{2} + \beta_{2,1}h_{t} + \gamma_{2,1}k_{2t} + \gamma_{2,2}k^2_{2t} + \gamma_{2,7}k_{1t} + \gamma_{2,8}k^2_{1t} + \epsilon_{2t}\} \nonumber \\ \text{School: }R_3(t) &= \alpha_3 + \beta_{tc}I(h_t \geq 12) + \beta_{rc}(1-d_3(t-1)) + \epsilon_{3t}, \nonumber \\ \text{Home: }R_4(t) &= \alpha_4 + \epsilon_{4t}\end{split}\]

These rewards enter the alternative specific value functions of individuals. In these equations \(h(t)\) denotes schooling in period \(t\) and \(k_{at}\) denotes work experience from sector \(A\) or \(B\) (\(a=1,2\)). The reward for schooling includes an indicator \(I(h_t \geq 12)\) which is connected to the cost of schooling after 12 periods (i.e. post-secondary schooling costs) and component that captures costs of returning to school when the choice in the previous period was something else. Aside from the parameters connected to these various components, each reward function also contains a constant and an alternative specific shock. The skill price in occupations is denoted by \(r_{a}\), it is set to 1 in this model. [1]

The model from Keane and Wolpin (1994, [15]) is not a complete subset of the model outlined in the explanations. The most important deviations are:

  • The model includes an additional squared experience term with parameter \(\gamma_{2,8}\) for experience in the other occupation.

  • It also does not include unobserved heterogeneity i.e. types. Here we thus define \(\alpha_{a}\) as the constant for alternative \(a\).

  • We do not distinguish between different levels of post-secondary education. The parameters \(\beta_{tc}\) and \(\beta_{tr}\) are thus not enumerated.

Footnotes