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.

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Specifying a Random Utility Model

Perspective of the Decision Maker

• 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' Perspective

• 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.

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Model of Utility

Decomposition of Utility

• 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 Utility

• 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 Component

• 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 $n$.

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Representative Utility

Function of Representative Utility

• 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 Function

• Closely approximates most utility functions.
• Non-linear utility complicates estimation.

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Structural Parameters

Linear Representative Utility

• Total utility: $U_{nj} = \beta x_{nj} + \epsilon_{nj}$.
• $\beta$: Structural parameters connecting observable attributes to unobserved utility.
• Marginal utilities interpretation.

Objective

• Find structural parameters consistent with observed choices.

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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.

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