Economic Model#

To illustrate the usage of respy in solving finite-horizon discrete choice dynamic programming (DCDP) problems we will present the basic setup of the human capital model as conceptualized in Keane and Wolpin (1997, [13]).


The promise: respy will become your preferred tool to develop, solve, estimate, and explore models within the EKW framework


At time t = 1,…,T each individual observes the state of the economic environment \(s_{t} \in \mathcal{S}_t\) and chooses an action \(a_t\) from the set of admissible actions \(\mathcal{A}\). The decision has two consequences:

  • Individuals receive per-period utility \(u_a(\cdot)\) associated with the chosen alternative \(a_t \in \mathcal{A}\). [1]

  • The economy evolves from \(s_{t}\) to \(s_{t+1}\).

Individuals are forward-looking and so do not simply choose the alternative with the highest immediate per-period utility. Instead, they take future consequences of their actions into account and implement a policy \(\pi \equiv (a_1^{\pi}(s_1), \dots, a_T^{\pi}(s_T))\). A policy is a collection of decision rules \(a_t^{\pi}(s_t)\) that prescribe an action for any possible state \(s_t\). The implementation of a policy generates a sequence of per-period utilities that depends on the objective transition probability distribution \(p_t(s_t, a_t)\) for the evolution of state \(s_{t}\) to \(s_{t+1}\) induced by the model. Individuals entertain rational expectations (Muth, 1961, [18]) so their subjective beliefs about the future agree with the objective transition probabilities of the model.

Timing of events depicts the timing of events in the model for two generic periods. At the beginning of period t, an individual fully learns about the immediate per-period utility of each alternative, chooses one of them, and receives its immediate utility. Then the state evolves from \(s_t\) to \(s_{t+1}\) and the process is repeated in \(t+1\).

Illustration timing of events

Timing of events#

Individuals face uncertainty (Gilboa, 2009, [8]) and discount future per-period utilities by an exponential discount factor \(0 < \delta < 1\) that parameterizes their time preference. Per-period utilities are time-separable (Samuelson, 1937, [20]). Given an initial state \(s_1\) individuals implement the policy \(\pi\) from the set of all feasible policies \(\Pi\) that maximizes the expected total discounted utilities over all \(T\) periods.

\[\underset{\pi \in \Pi}{\max} \, \mathbb{E}_{p^{\pi}} \left[ \sum_{t = 1}^T \delta^{t - 1} u(s_t, a_t^{\pi}(s_t)) \right],\]

where \(\mathbb{E}_{p^{\pi}}[\cdot]\) denotes the expectation operator under the probability measure \(p^{\pi}\). The decision problem is dynamic in the sense that expected inter-temporal per-period utilities at a certain period \(t\) are influenced by past choices.

To Explanation

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

Footnotes