Lecture 4 - Disaggregate Econometric Choice Theory

ENCI707: Engineering Demand and Policy Analysis

Demand Versus Choice

Demand

  • Aggregate
  • Homogenous people
  • Identical behaviour
  • Variables: Price (P) and quantity (Q)
  • Demand function: Q = f(P)
  • Demand = Bundle of choices

Choice

  • Disaggregate
  • Heterogeneous people
  • Random behaviour
  • Variables: Attributes of choice alternatives; attributes of choice maker; context of choices
  • Non-linear choice function

Transportation Choices

Many travel demand related choices are:

  • Discrete: activity type choice, mode choice, route choice, vehicle type choice, time of the day choice, location of trip destination choice etc.
  • Continuous: activity duration, vehicle usage, housing duration, etc.
  • Heterogeneous: Wide variations of choices across the population.
  • Probabilistic/stochastic/random: We can’t defined deterministically.

Choice as Consumption

Choice as Consumption

Disaggregate to Aggregate

  • Choice of private vehicle as mode -> # of vehicle trips
  • Choice of transit as mode -> # of transit trips
  • Choice of shared vehicle as mode -> # of shared vehicle trips

Theory of Choice Behaviour

  • Descriptive (or positive): how human beings behave, not how they should/ought to behave
  • Abstract: that can be formalized in general cases not specific to particular circumstances
  • Operational: can be applied to develop models with variables and parameters that can be observed and estimated. Models can be used for prediction and/or policy evaluation

Framework of Choice Theory

  • Choice as outcome of sequential decision-making:
    1. Definition of the choice problem
    2. Generation of alternatives
    3. Evaluation of attributes of alternatives
    4. Choice making
    5. Executing the choice
  • Define:
    • Decision maker
    • Characteristics of decision maker
    • Alternatives
    • Attributes of alternatives
    • Decision rules to make choice

Decision Maker

  • Disaggregate: Individual or group (household, family, firms, government agency, etc.)
  • Heterogeneous choice makers: Wide variety in choice behaviour across the population
    • Taste variations across the choice/decision makers: Idiosyncrasy
    • Different choice situations for different people
  • Characteristics of heterogeneous choice maker: age, gender, income, education, household/firm size, etc.
  • Choice situations/context: spatial, temporal, economic

Alternatives

  • Choice means choosing an alternative out of a choice set (mutually exclusive and collectively exhaustive countable set of alternatives):
  • Universal choice set: \(C\)
  • Feasible / Individual-specific: \(C_i\)
  • Consideration (awareness) set: \(C_{ci}\)

Example:

\(C\) = { Private car, Ridehail, Taxi, Bus, Bicycle, Walk}

  • No driver’s license, distance > 3 km

\(C_i\) = { Ridehail, Taxi, Bus, Bicycle}

  • No driver’s license, distance > 3 km & not aware of Ridehail

\(C_{ci}\) = { Taxi, Bus, Bicycle}

Alternatives - Attributes

  • Decision maker evaluates alternatives in terms of attribute values: Attribute values can be measured on the scale of attractiveness:
  • Categorical: Binary, ordinal (reliability, safety, convenience, etc.)
  • Cardinal: absolute values (time, cost, etc.)
  • Generic versus alternative-specific
  • Measured versus perceived

Decision Rules

  • Rational behaviour (homo economicus): self-interested economic actor who optimizes own choice outcomes
  • Bounded rationality: Use rules to simplify:
    • Dominance: Rules used to remove inferior alternative
    • Satisfaction: Attributes of the alternatives need to satisfy the expectation level
    • Lexicographic: attributes are rank ordered by their level of importance
    • Utility maximizing: maximize a latent function of different attributes of alternatives in the choice set

Utility Maximization

Utility function:

  • Captures relative attractiveness of alternative in the choice set
  • Measures the satisfaction that the decision maker wants to optimize

Assumptions:

  • Decision maker has full knowledge about the attributes of the alternatives and is able to process information
  • Base on information processing, decision maker associates a utility to each alternative
  • Decision maker has transitive preferences
  • Decision maker is rational and perfect optimizer
  • Decision maker is consistent

Probability in Utility Maximization

Constant utility:

  • Utility is deterministic and cardinal in nature
  • Decision maker does not maximize utility
  • However, human behaviour is inherently random

Random utility:

  • Decision maker is a rational optimizer
  • Modeller does not observe the exact measure of utility used by the decision maker
  • Utility becomes random and ordinal variable
  • Probabilistic choice model because of unobserved heterogeneity (randomness)

Discrete Choice

Two alternatives: transit, car \[π‘ˆ_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}=\beta_𝑑 π‘‡π‘–π‘šπ‘’_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}+\beta_𝑐 πΆπ‘œπ‘ π‘‘_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} \text{ (Typically, }\beta_t, \beta_c <0)\]

\[π‘ˆ_{π‘π‘Žπ‘Ÿ}=\beta_𝑑 π‘‡π‘–π‘šπ‘’_{π‘π‘Žπ‘Ÿ}+\beta_𝑐 πΆπ‘œπ‘ π‘‘_{π‘π‘Žπ‘Ÿ}\]

Equivalent specification \[π‘ˆ_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}=(\beta_𝑑/\beta_𝑐)π‘‡π‘–π‘šπ‘’_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}+πΆπ‘œπ‘ π‘‘_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}\]

\[π‘ˆ_{π‘π‘Žπ‘Ÿ}=(\beta_𝑑/\beta_𝑐)π‘‡π‘–π‘šπ‘’_{π‘π‘Žπ‘Ÿ}+πΆπ‘œπ‘ π‘‘_{π‘π‘Žπ‘Ÿ} \]

Choice: car, if \[\beta_𝑑 π‘‡π‘–π‘šπ‘’_{π‘π‘Žπ‘Ÿ}+\beta_𝑐 πΆπ‘œπ‘ π‘‘_{π‘π‘Žπ‘Ÿ} \geq \beta_𝑑 π‘‡π‘–π‘šπ‘’_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} + \beta_𝑐 πΆπ‘œπ‘ π‘‘_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} \]

or \[(\beta_𝑑/\beta_𝑐)π‘‡π‘–π‘šπ‘’_{π‘π‘Žπ‘Ÿ}+πΆπ‘œπ‘ π‘‘_{π‘π‘Žπ‘Ÿ} \leq (\beta_𝑑/\beta_𝑐)π‘‡π‘–π‘šπ‘’_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}+πΆπ‘œπ‘ π‘‘_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} \]

Discrete Choice

Car is the dominant alternative, if \[π‘‡π‘–π‘šπ‘’_{π‘π‘Žπ‘Ÿ} < π‘‡π‘–π‘šπ‘’_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} \text{ & } πΆπ‘œπ‘ π‘‘_{π‘π‘Žπ‘Ÿ} < πΆπ‘œπ‘ π‘‘_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} \therefore π‘ˆ_{π‘π‘Žπ‘Ÿ} > π‘ˆ_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}\]

Car and transit in competition, if \[π‘‡π‘–π‘šπ‘’_{π‘π‘Žπ‘Ÿ} < π‘‡π‘–π‘šπ‘’_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} \text{ & } πΆπ‘œπ‘ π‘‘_{π‘π‘Žπ‘Ÿ} > πΆπ‘œπ‘ π‘‘_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}\]

Is the traveler willing to pay extra \((Cost_{car}-Cost_{transit})\) to save \((Time_{transit} -Time_{car})\)?

If yes (pick car): \(\left(\frac{\beta_𝑑}{\beta_𝑐}\right)π‘‡π‘–π‘šπ‘’_{π‘π‘Žπ‘Ÿ} + πΆπ‘œπ‘ π‘‘_{π‘π‘Žπ‘Ÿ} \leq \left(\frac{\beta_𝑑}{\beta_𝑐}\right)π‘‡π‘–π‘šπ‘’_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} + πΆπ‘œπ‘ π‘‘_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}\)

Willingness-to-pay or Value of Travel Time Savings (VOTS): \[\left(\frac{\beta_𝑑}{\beta_𝑐}\right) \geq \frac{(πΆπ‘œπ‘ π‘‘_{π‘π‘Žπ‘Ÿ}βˆ’πΆπ‘œπ‘ π‘‘_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘})}{(π‘‡π‘–π‘šπ‘’_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘}βˆ’π‘‡π‘–π‘šπ‘’_{π‘π‘Žπ‘Ÿ})}\]

Discrete Choice

Discrete Choice

Stochastic Discrete Choice

  • Choice Probability of an alternative \(j\) for an individual \(i\) with choice set \(C_i\) \[Pr⁑( 𝑗|𝐢_𝑖)=Pr⁑( π‘ˆ_{𝑖𝑗} \geq π‘ˆ_{π‘–π‘˜}) \text{, } π‘˜ \neq 𝑗 \text{ & } π‘˜ \in 𝐢_𝑖\]

  • Random utility: \[π‘ˆ_{𝑖𝑗}=𝑉_{𝑖𝑗}+\epsilon_{𝑖𝑗} \text{, } 𝑗 \in 𝐢_𝑖\]

  • Random Utility Maximizing (RUM) Choice Model: \[Pr⁑( 𝑗|𝐢_𝑖)=Pr⁑( 𝑉_{𝑖𝑗} + \epsilon_{𝑖𝑗} \ge 𝑉_{π‘–π‘˜} + \epsilon_{π‘–π‘˜}) \text{, } π‘˜ \neq 𝑗 \text{ & } π‘˜ \in 𝐢_𝑖\] Or \[Pr⁑( 𝑗|𝐢_𝑖)=Pr⁑( \epsilon_{ik} - \epsilon_{ij} \leq V_{ij} - V_{ik}) \text{, } π‘˜ \neq 𝑗 \text{ & } π‘˜ \in 𝐢_𝑖\]

RUM-Based Discrete Choice

  • Fully specified choice set \(C_i\)
  • Attributes of alternatives are completely defined
  • Only differences in utility matter
  • Systematic utility function is specified as a function of attributes
  • An attribute may have the same (generic) weighting factors (coefficient) for all alternatives if it varies across the alternatives (e.g., travel time, travel cost, etc.).
  • An attribute may have different weighting factors (coefficient) for different alternatives if it does not vary across the alternatives (age, gender, income, etc.).
  • Assumption on distribution of random utility is necessary to derive choice probability: Stochastic Choice Model

Binary Discrete Choice

Choice Probability of an alternative \(j\) for an individual \(i\) with choice set \(C_i\) of two alternatives: \(j, k\) (dropping \(i\) for sake of simplicity) \[Pr⁑( 𝑗|𝐢_𝑖)=Pr⁑( 𝑉_𝑗+\epsilon_𝑗 \ge 𝑉_π‘˜+\epsilon_π‘˜)\] \[Pr⁑( 𝑗|𝐢_𝑖)=Pr⁑( \epsilon_π‘˜βˆ’\epsilon_𝑗≀𝑉_π‘—βˆ’π‘‰_π‘˜)\] Binary Probit:

  • \(\epsilon_j,\epsilon_k\) are both normally distributed
  • So,(\(\epsilon_k-\epsilon_j\)) is also normally distributed
  • With fully specified alternative specific constants, (\(\epsilon_k-\epsilon_j\)) is \(N(0,\sigma^2)\): \[V-j - V_k = \beta_0 + \sum \beta(x_j - x_k)\]
  • Resulting Choice model: \(Pr(j) = \Phi(V/\sigma)\)
  • \(\sigma\) normalized to 1, giving probit model with unit variance

Binary Discrete Choice

Binary Logit:

  • \(\epsilon_j,\epsilon_k\) are IID Type I EV distributed with scale \(\mu\) and variance \(\pi^2/6\mu^2\)

  • With fully specified alternative specific constants, (\(\epsilon_k-\epsilon_j\)) is logistically distributed with scale \(\mu\) and variance \(\pi^2/3\mu^2\): \[V_j - V_k = V = \beta_0 + \sum \beta (x_j - x_k)\] Resulting Choice model: \[Pr(j) = \frac{1}{1+\exp(-\mu(V_j - V_k))} = \frac{\exp(\mu V_j)}{\exp(\mu V_j)+\exp(\mu V_k)}\]

  • \(\mu\) normalized to 1, giving logit model with variance of \(\pi^2/3\)

Binary Discrete Choice

Binary Probit/Logit Identification

  • Only one alternative specific constant
  • Scale of logit model is not identified as a constant
  • Reference alternative is irrelevant \[𝑉_π‘—βˆ’π‘‰_π‘˜=𝑉=\beta_0+\sum \beta(π‘₯_π‘—βˆ’π‘₯_π‘˜)\] \[𝑉_𝑗=\beta_0+\beta(π‘₯_{π‘”π‘’π‘›π‘’π‘Ÿπ‘–π‘})_𝑗+\beta_{(π‘—βˆ’π‘ π‘π‘’π‘π‘–π‘“π‘–π‘)} π‘₯_{(π‘—βˆ’π‘ π‘π‘’π‘π‘–π‘“π‘–π‘)}\] \[𝑉_π‘˜=0+\beta(π‘₯_{π‘”π‘’π‘›π‘’π‘Ÿπ‘–π‘})_π‘˜ + \beta_{(π‘˜βˆ’π‘ π‘π‘’π‘π‘–π‘“π‘–π‘)} π‘₯_{(π‘˜βˆ’π‘ π‘π‘’π‘π‘–π‘“π‘–π‘)}\] \[𝑉_{π‘π‘Žπ‘Ÿ}=\beta_0+\beta_𝑑 time_{π‘π‘Žπ‘Ÿ} + \beta_𝑐 cost_{π‘π‘Žπ‘Ÿ} +\beta_π‘œ CarOwnership \] \[𝑉_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} = 0 + \beta_𝑑 time_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} + \beta_𝑐 fare_{π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘–π‘‘} + \beta_π‘Ÿ Onβˆ’TimePerformance\]
  • Allows non-linear transformation and normalizations