Chapter 3 Static Discrete Choice Models

3.1 In this chapter

  1. Describe static discrete choice models

  2. Derive logit probabilities

  3. Show how to estimate logits and multinomial logits

  4. Cover the IIA property

  5. Discuss expected utility and consumer surplus

3.2 Notation

Let \(d_i\) indicate the choice individual \(i\) makes where \(d_i\in\{1,\cdots, J\}\). Individuals choose \(d\) to maximize their utility, \(U\). \(U\) generally is written as: \[U_{ij}=u_{ij}+\varepsilon_{ij}\] where:

  1. \(u_{ij}\) relates observed factors to the utility individual \(i\) receives from choosing option \(j\)

  2. \(\varepsilon_{ij}\) are unobserved to the econometrician but observed to the individual

  3. \(d_{ij}=1\) if \(u_{ij}+\varepsilon_{ij}>u_{ij'}+\varepsilon_{ij'}\) for all \(j'\neq j\)

3.3 Probabilities

With the \(\varepsilon\)’s unobserved, the probability of individual \(i\) making choice \(j\) is given by: \[\begin{aligned} P_{ij}&=\Pr(u_{ij}+\varepsilon_{ij}>u_{ij'}+\varepsilon_{ij'}\forall j'\neq j)\\ &=\Pr(\varepsilon_{ij'}-\varepsilon_{ij}<u_{ij}-u_{ij'}\forall j'\neq j)\\ &=\int_{\varepsilon}I(\varepsilon_{ij'}-\varepsilon_{ij}<u_{ij}-u_{ij'}\forall j'\neq j)f(\varepsilon)d\varepsilon\end{aligned}\] Note that, regardless of what distributional assumptions are made on the \(\varepsilon\)’s, the probability of choosing a particular option does not change when we:

  1. Add a constant to the utility of all options (utility is relative to one of the options, only differences in utility matter)

  2. Multiply by a positive number (need to scale something, generally the variance of the \(\varepsilon\)’s)

3.4 Variables

Suppose we have: \[\begin{aligned} u_{i1}&=\alpha Male_i+\beta_1 X_i + \gamma Z_1\\ u_{i2}&=\alpha Male_i+\beta_2 X_i+\gamma Z_2\end{aligned}\] Since only differences in utility matter: \[u_{i1}-u_{i2}=(\beta_1-\beta_2)X_i+\gamma (Z_1-Z_2)\] Thus, we cannot tell whether men are happier than women, but can tell whether men have a preference for a particular option over another. We can only obtain differenced coefficient estimates on \(X\)’s, and can obtain an estimate of a coefficient that is constant across choices only if the variable it is hitting varies by choice.