Skip to main content

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 nn, faces a choice among JJ alternatives.
  • Each alternative jj provides a certain utility UnjU_{nj} (where j = {1,2,...,J}\{1, 2, ..., J\}).
  • The decision maker chooses the alternative offering the greatest utility.
  • Formally, decision maker nn chooses alternative ii if and only if Uni>UnjU_{ni} > U_{nj}, ∀j≠i\forall j \neq i.

Econometricians' Perspective​

  • As econometricians, certain elements are not observable:
    • The actual utility UnjU_{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 UnjU_{nj} and how each attribute affects it.

Model of Utility​

Decomposition of Utility​

  • Each alternative's utility (Unj)(U_{nj}) consists of two parts:
    • Observed factors: VnjV_{nj}.
    • Unobserved factors: ϵnj\epsilon_{nj}.
  • Utility equation: Unj=Vnj+ϵnjU_{nj} = V_{nj} + \epsilon_{nj}.

Representative Utility​

  • Defined as V(xnj,sn)V(x_{nj}, s_n).
  • xnjx_{nj}: Vector of attributes of the alternative.
  • sns_n: Vector of attributes of the decision maker.

Unobserved Utility Component​

  • Captures factors affecting utility not included in VnjV_{nj}.
  • Treated as a random variable.
  • f(ϵn)f(\epsilon_n): Joint density of the random vector ϵn={ϵn1,...,ϵnJ}\epsilon_n = \{\epsilon_{n1}, ..., \epsilon_{nJ}\} for decision maker nn.

Representative Utility​

Function of Representative Utility​

  • Modeled as a function of:
    • xnjx_{nj}: Vector of attributes of the alternative.
    • sns_n: Vector of attributes of the decision maker.
    • β\beta: 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: Unj=βxnj+ϵnjU_{nj} = \beta x_{nj} + \epsilon_{nj}.
  • β\beta: 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.