Random Utility Model
Model Structure​
- Based on the assumption of utility-maximizing 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 Model​
Perspective of the Decision Maker​
- A decision maker, denoted as , faces a choice among alternatives.
- Each alternative provides a certain utility (where j = ).
- The decision maker chooses the alternative offering the greatest utility.
- Formally, decision maker chooses alternative if and only if , .
Econometricians' Perspective​
- As econometricians, certain elements are not observable:
- The actual utility 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 and how each attribute affects it.
Model of Utility​
Decomposition of Utility​
- Each alternative's utility consists of two parts:
- Observed factors: .
- Unobserved factors: .
- Utility equation: .
Representative Utility​
- Defined as .
- : Vector of attributes of the alternative.
- : Vector of attributes of the decision maker.
Unobserved Utility Component​
- Captures factors affecting utility not included in .
- Treated as a random variable.
- : Joint density of the random vector for decision maker .
Representative Utility​
Function of Representative Utility​
- Modeled as a function of:
- : Vector of attributes of the alternative.
- : Vector of attributes of the decision maker.
- : Vector of structural parameters.
- Often specified as a linear function.
- Flexibility includes interactions, squared terms, etc.
Advantages of Linear Function​
- Closely approximates most utility functions.
- Non-linear utility complicates estimation.
Structural Parameters​
Linear Representative Utility​
- Total utility: .
- : Structural parameters connecting observable attributes to unobserved utility.
- Marginal utilities interpretation.
Objective​
- Find structural parameters consistent with observed choices.
Properties of the Random Utility Model​
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.