Models of travel demand with endogenous preference change and heterogeneous agents

Abstract

In the literatures of regional science, urban economics, and urban development planning, a working assumption is that individuals respond to incentives and regulations, given their preferences. Models for planning and policy analyses are used to consider what might occur if the incentives or regulations were different. In these models, however, preferences are usually assumed to be given and stable, and agents are usually assumed to be homogeneous. This paper focuses on the implications of making preferences in models of policy implementation endogenously determined and time varying heterogeneous agents. We consider first the recent literature on intertemporal choice and preference change, which cuts across many disciplines, and more briefly the literature on norm-regarding behavior. We then elaborate a simple model of transportation demand—from a static to a dynamic orientation, from fixed and exogenously given preferences of strictly self-regarding agents to endogenously determined and policy-induced preferences of heterogeneous agents—and illustrate its characteristics with simple numerical examples.

Technical appendix: properties of the MPIGLOG demand system

To discuss the regularity properties of Eq. 2, which we reprise here as

$$ U(c,p) = \ln (c/P1)\,(c^{\eta } /P2), $$

(16)

it is helpful to recall the set of conditions associated with regular indirect utility functions that are dual to cost functions. From standard duality theory (Diewert 1982; McFadden 1978), the indirect utility functions should satisfy the following conditions:

• U1: U is HD0 in (c, p),

• U2: U is non-decreasing in c,

• U3: U is non-increasing in p,

• U4: U is quasi-convex in p.

If P1 and P2 are positive functions, HD1 and HDη, respectively, and η ≥ 0, (2) will satisfy U1 and U2. If P1 and P2 are also non-decreasing in prices, then U3 will be satisfied over the region where c ≥ 0. Finally, concavity of P1 and P2, when c ≥ P1, ensures quasi-convexity of U in p (Greenberg and Perskalla 1971), although concavity of P2 also requires that 0 ≤ η ≤ 1.

As noted above, applying Roy’s identity to (16) yields expenditure share equations of the form

$$ s_{j} = [\varepsilon_{1j} + \varepsilon_{2j} \,\ln (c/P1)]/[1 + \eta \,\ln (c/P1)], $$

(17)

where $s_{j} = p_{j} q_{j} /c, \, \varepsilon_{ij} = \partial \ln \,P_{i} /\partial \ln \,p_{j} , \, \sum\nolimits_{j} {\varepsilon_{1j} = 1.0{\text{ and }}\sum\nolimits_{j} {\varepsilon_{2j} = \eta .} }$

In the region c ≥ P1, ln(c/P1) will be non-negative. Thus restricting the price elasticities of P1 and P2, $\varepsilon_{1j} {\text{ and }}\varepsilon_{2j} ,$ to be non-negative will ensure that $0 \le s_{j} \le 1.$ Following Deaton and Muellbauer (1980), Cooper and Mclaren 1992 take the MPIGLOG share equations to indicate that, for given prices and rising income, the share s _j moves monotonically from ɛ _1j , the level of the subsistence expenditure share for the ‘poor’, toward $(\varepsilon_{2j} /\eta )$, the asymptote share level for the ‘rich.’

Some insight into (17) can be gained by defining the following expression:

$$ Z = \eta \,{ \ln }(c/P1)/[1 + \eta \,\ln (c/P1)]. $$

(18)

Then, the share equations can be rewritten in terms of Z as

$$ s_{j} = \varepsilon_{1j} (1 - Z) + (\varepsilon_{2j} /\eta )Z, $$

(19)

that is, as weighted averages of the shares of the ‘rich’, $(\varepsilon_{2j} /\eta )$ and ‘poor,’ ɛ _1j .

It is apparent from the form of the share equations that the potentially unbounded variable, ‘real expenditures’, (c/P1), enters the right-hand side only through the bounded variable, Z, and the potentially unbounded price term enters through the HD0 elasticity terms ɛ _1j and ɛ _2j . From the perspective of co-integration accounting, then, the budget shares have an appropriate set of explanatory variables.

Other properties of the MPIGLOG system can also now be easily derived in terms of Z. The expenditure elasticities of demand for each type of service j, E _j , satisfy

$$ E_{j} = 1 + (\varepsilon_{2j} /s_{j} - \eta )\,(1 - Z), $$

(20)

while the price elasticities of demand, M _ij , satisfy

$$ M_{ij} = [\varepsilon_{1j} (1 - Z) + (\varepsilon_{2j} /\eta )Z - \varepsilon_{2i} \varepsilon_{1j} (1 - Z)]/[s_{i} - \delta_{ij} + \eta \varepsilon_{1j} (1 - Z)], $$

(21)

in which δ _ij is the Kronecker delta—i.e., assumes the value of 1.0 when i = j, and is zero everywhere else. A typical term of the Slutsky matrix is

$$ S_{ij} = (c/p_{i} p_{j} )\,[s_{i} M_{ij} + s_{i} s_{j} E_{i} ]. $$

(22)