Microeconometrics and MATLAB: An Introduction by Adams Abi & Clarke Damian & Quinn Simon

Author:Adams, Abi & Clarke, Damian & Quinn, Simon [Adams, Abi]
Language: eng
Format: epub
ISBN: 9780198754503
Publisher: OUP Oxford
Published: 2016-01-13T22:00:00+00:00

6.4 Stochastic Dynamic Programming

Time for some shocks! Returning to our household microenterprise, let’s imagine that the evolution of capital is subject to a stochastic shock. The transition equation now becomes:

(6.14)

While the decision maker does not know the precise value of εt+1 at time t, we assume that she does know its distribution.

Rather than deciding between consumption now and consumption in the future, the household must now choose between consumption now and expected future consumption. This requires a rewriting of the dynamic programming problem:

(6.15)

where kt+1 is given as in Equation 6.14, and the expectation is taken over the distribution of εt+1.

To illustrate, let’s assume a very simple distribution for the ε term. Specifically, we assume that ε takes two possible states: low (ε = −2) and high (ε = 2). Let each state occur with a probability of 0.5.8 The stochastic portion of this problem is then characterized by two vectors: a vector of shocks (ε = [−2,2]) and a vector of probabilities (π = [0.5,0.5]). Vt+1 now depends on future capital and the realization of future shocks.

The code below computes the value function at each time period, for each possible (optimal) capital–consumption pair. nextKl refers to the future capital stock (kt+1) associated with a low realization of the shock (i.e. εt+1 = −2), and nextKh for a high realization.9 Line 26 is Matlab speak for ‘replace nextKl with zero if nextKl is less than zero’. This is a technological requirement: in this model, capital can never be negative.

The expected future value is given by EnextV. To calculate this, we find the value function associated with the low shock and with the high shock; we then weight by their respective probabilities. To ensure that the predicted kt+1 lies in the discretized state space for capital, nextK is rounded to the closest value in our capital grid. In dynamic problems with continuous state variables, discretization steps such as this are necessary for computation.

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