Random Utility Model
Model Structureâ€‹
 Based on the assumption of utilitymaximizing behavior (or profit maximization for firms).
 Components:
 Utility derived from each alternative.
 Dependency on observed and unobserved characteristics.
 Selection of the alternative providing maximum utility.
 Flexibility: RUM can include behavioral and informational parameters deviating from traditional models.
Specifying a Random Utility Modelâ€‹
Perspective of the Decision Makerâ€‹
 A decision maker, denoted as $n$, faces a choice among $J$ alternatives.
 Each alternative $j$ provides a certain utility $U_{nj}$ (where j = $\{1, 2, ..., J\}$).
 The decision maker chooses the alternative offering the greatest utility.
 Formally, decision maker $n$ chooses alternative $i$ if and only if $U_{ni} > U_{nj}$, $\forall j \neq i$.
Econometricians' Perspectiveâ€‹
 As econometricians, certain elements are not observable:
 The actual utility $U_{nj}$ from each alternative is not directly observed.
 Observable data includes:
 The alternative that is chosen.
 Some attributes of each alternative.
 Some attributes of the decision maker.
 The goal is to use this data to infer $U_{nj}$ and how each attribute affects it.
Model of Utilityâ€‹
Decomposition of Utilityâ€‹
 Each alternative's utility $(U_{nj})$ consists of two parts:
 Observed factors: $V_{nj}$.
 Unobserved factors: $\epsilon_{nj}$.
 Utility equation: $U_{nj} = V_{nj} + \epsilon_{nj}$.
Representative Utilityâ€‹
 Defined as $V(x_{nj}, s_n)$.
 $x_{nj}$: Vector of attributes of the alternative.
 $s_n$: Vector of attributes of the decision maker.
Unobserved Utility Componentâ€‹
 Captures factors affecting utility not included in $V_{nj}$.
 Treated as a random variable.
 $f(\epsilon_n)$: Joint density of the random vector $\epsilon_n = \{\epsilon_{n1}, ..., \epsilon_{nJ}\}$ for decision maker $n$.
Representative Utilityâ€‹
Function of Representative Utilityâ€‹
 Modeled as a function of:
 $x_{nj}$: Vector of attributes of the alternative.
 $s_n$: Vector of attributes of the decision maker.
 $\beta$: Vector of structural parameters.
 Often specified as a linear function.
 Flexibility includes interactions, squared terms, etc.
Advantages of Linear Functionâ€‹
 Closely approximates most utility functions.
 Nonlinear utility complicates estimation.
Structural Parametersâ€‹
Linear Representative Utilityâ€‹
 Total utility: $U_{nj} = \beta x_{nj} + \epsilon_{nj}$.
 $\beta$: Structural parameters connecting observable attributes to unobserved utility.
 Marginal utilities interpretation.
Objectiveâ€‹
 Find structural parameters consistent with observed choices.
Properties of the Random Utility Modelâ€‹
General Formula for Choice Probabilities:

This formula reveals two important properties about the Random Utility Model (RUM):
Differences in Utility Matter:
 The focus is not on the absolute level of utility from any alternative, but rather on the differences in utility between alternatives.
 Only parameters that capture these differences can be estimated.
Scale of Utility is Arbitrary:
 Scaling all utilities (e.g., multiplying by a constant) does not change the relative comparison between alternatives.
 Typically, the variance of the error terms is normalized in RUMs.