Income effects, cost damping and the value of time: theoretical properties embedded within practical travel choice models
Abstract
Mackie et al. (Values of travel time savings in the UK. Report to Department for Transport. Institute for Transport Studies, University of Leeds & John Bates Services, Leeds and Abingdon, 2003) proposed an identity relating the value of time (VoT) for commute and leisure travel to income and travel cost, reporting the prevalence of ‘cost damping’ (i.e. the phenomenon where VoT increases as travel cost increases). This identity (or a variant thereof) has been adopted within official methods for estimating VoT in the UK, Switzerland and The Netherlands. The present paper shows that Mackie et al.’s identity: (i) implies linear preferences, not strictly convex preferences as reported by Mackie et al.; (ii) complies with homogeneity and symmetry by construction; (iii) complies with addingup if and only if VoT is unit elastic with respect to income; (iv) complies with negativity if VoT is unit elastic or greater with respect to income; (v) violates both addingup and negativity in the case of the 2003 UK national VoT study. We propose alternative identities which comply with addingup and homogeneity by construction, and offer comparable fit to Mackie et al.’s identity on the UK VoT dataset. We also find that the imposition of addingup and negativity on Mackie et al.’s identity, through appropriate constraint on model estimation, leads to an increase of around 20% in valuations from the 2003 UK dataset.
Keywords
Value of time Cost damping Addingup Homogeneity NegativityJEL Classification
B41 economic methodology D01 microeconomic behaviour Underlying principlesIntroduction

To consider the theoretical properties inherent within the CIUF (2), with particular reference to the cost damping phenomenon.

To consider the theoretical properties of the conditional demand function associated with (2), and the extent to which these properties are affected by cost damping.

To draw implications for the practical specification of travel choice models when estimating VoT.

To illustrate these practical implications using an official national VoT dataset.
Theoretical basis of VoT

Goods 1 and 2 are in De Serpa’s terms ‘intermediate goods’; more specifically, let us assume that they are travel options.

Prices and travel costs are oneandthesame.
First derivatives from Mackie et al.’s CIUF
This section will consider the manner in which the value of time theory summarised in the "Theoretical basis of VoT" section relates to Mackie et al.’s (2003) empirical CIUF (2). More specifically, this section will derive three relevant quantities, namely the conditional marginal utility of travel cost, the conditional VoT, and the conditional demand for travel.
Conditional marginal utility of travel cost
Conditional value of travel time
Conditional demand for travel

If \(\eta_{y} < 0\) then \(\tilde{x}_{i} \left[ {c_{i} ,y} \right] > 0\)

If \(\eta_{y} > 0\) then \(\tilde{x}_{i} \left[ {c_{i} ,y} \right] < 0\)
Second derivatives from Mackie et al.’s CIUF, and theoretical properties of the indifference map
Building upon the working from the "First derivatives from Mackie et al.’s CIUF" section, the present section will take second derivatives of the CIUF (2), and then interpret these derivatives through reference to the theoretical properties of indifference maps. In this way, we will revisit the assertions of Mackie et al. (2003) and Daly (2010)—reported earlier in the "Introduction" section—concerning the theoretical properties of indifference maps and the consistency of these properties with the cost damping phenomenon.
The UK, Swiss and Dutch national VoT studies cited in the "Introduction" section all found the conditional marginal disutility of design cost (4) to decrease—in absolute terms—as current cost increased. By implication, the conditional VoT for the design journey (5) was reported to increase as current cost increased. Although cost damping is frequently observed empirically (see Daly (2010) for a review of evidence), some researchers have argued that this phenomenon is inconsistent with microeconomic theory. Indeed, Mackie et al. (2003) expressed dissatisfaction with this finding in the UK VoT study, arguing that it violates ‘strict concavity’—a wellbehaved property of indifference curves.
Elasticities of conditional marginal utility of design cost from Mackie et al. (2003)
Elasticity with respect to  Commuting  Other nonwork 

Current cost \(\eta _{c}\)  −0.42 (9.08)  −0.31 (11.86) 
Income \(\eta _{y}\)  −0.36 (7.58)  −0.16 (5.49) 
Continuing with our qualification of Mackie et al.’s rationale, the conditional marginal utility of design time within the CIUF (2) is similarly independent of current time. Since \({{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial t_{i} < 0}}} \right. \kern0pt} {\partial t_{i} < 0}}\) and \({{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} < 0}}} \right. \kern0pt} {\partial c_{i} < 0}}\), whilst \({{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial t_{i} }}} \right. \kern0pt} {\partial t_{i} }}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial t_{i} }}} \right. \kern0pt} {\partial t_{i} }}} \right)} {\partial t = 0}}} \right. \kern0pt} {\partial t = 0}}\) and \({{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}} \right)} {\partial c > 0}}} \right. \kern0pt} {\partial c > 0}}\), Mackie et al. in practice found that design time and cost were ‘bads’, and that the conditional marginal disutility of design cost decreased—in absolute terms—relative to the conditional marginal disutility of design time as current cost increased (and current time decreased, again given the twodimensional nature of the indifference map). Therefore, Mackie et al. observed the marginal rate of substitution between design time and cost to be increasing as current cost increased (and current time decreased), and vice versa. In other words, these empirical results contradict Mackie et al.’s theoretical hypothesis noted earlier in this section, and (in terms of the cross partial derivatives of the conditional marginal utility of design cost with respect to current cost, at least) imply strict convexity rather than strict concavity.
Theoretical properties of Mackie et al.’s conditional demand
Addingup
Homogeneity
Negativity

If \( 1 < \eta_{y} \le 0\), such that VoT increases with income but at a decreasing rate (i.e. is inelastic with respect to income), then negativity fails.

If \(\eta_{y} \le  1\), such that VoT increases with income and at an increasing rate (i.e. is elastic with respect to income), then negativity holds.
Symmetry
Alternative specifications of the CIUF

Model I: As a benchmark against which the alternative specifications can be judged, this specification is the ‘normalised’ version of Mackie et al.’s CIUF, i.e. (2).

Model II: This specification adopts a CobbDouglas form for income and design cost.

Model III: This entails a linear ‘residual income’ specification, where residual income is expressed in terms of design cost.

Model IV: This specification expresses design cost as a proportion of income, and introduces a power term on this ratio.

Model V: This employs the same ratio as Model IV, but takes logarithms instead of specifying a power term.^{11}
Some common practical model specifications, and their compliance with addingup and homogeneity
Model I  Model II  Model III  Model IV  Model V  Model VI^{a}  Model VII^{a}  

\(\tilde{v}_{i}\)  \(\beta \cdot y^{{\eta_{y} }} \cdot c^{{\eta_{c} }} \cdot c_{i} + \alpha \cdot t_{i}\)  \(\beta \cdot y^{{\eta_{y} }} \cdot c_{i}^{{\eta_{c} }} + \alpha \cdot t_{i}\)  \(\beta \cdot \left( {y  c_{i} } \right) + \alpha \cdot t_{i}\)  \(\beta \cdot \left( {\frac{{c_{i} }}{y}} \right)^{{\eta_{c} }} + \alpha \cdot t_{i}\)  \(\beta \cdot \log \left( {\frac{{c_{i} }}{y}} \right) + \alpha \cdot t_{i}\)  \(\beta \left[ {c,y} \right] \cdot c_{i} + \alpha \left[ {t,y} \right] \cdot t_{i}\)  \(\beta \left[ {c,y} \right] \cdot c_{i} + \alpha \cdot t_{i}\) 
\({{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial y}}} \right. \kern0pt} {\partial y}}\)  \(\eta_{y} \cdot \beta \cdot y^{{\eta_{y}{  1} }} \cdot c^{{\eta_{c} }} \cdot c_{i}\)  \(\eta_{y} \cdot \beta \cdot y^{{\eta_{y}{  1} }} \cdot c_{i}^{{\eta_{c} }}\)  β  \( \eta_{c} \cdot \beta \cdot \left( {\frac{{c_{i} }}{y}} \right)^{{\eta_{c} {  1} }} \cdot \left( {\frac{{c_{i} }}{{y^{2} }}} \right)\)  \(\frac{\beta }{{c_{i} }}\)  \({{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot c_{i} + {{\partial \alpha } \mathord{\left/ {\vphantom {{\partial \alpha } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot t_{i}\)  \({{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot c_{i}\) 
\({{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}\)  \(\beta \cdot y^{{\eta_{y} }} \cdot c^{{\eta_{c} }}\)  \(\eta_{c} \cdot \beta \cdot y^{{\eta_{y} }} \cdot c_{i}^{{\eta_{c} {  1} }}\)  −β  \(\eta_{c} \cdot \beta \cdot \left( {\frac{{c_{i} }}{y}} \right)^{{\eta_{c}{  1} }} \cdot \left( {\frac{1}{y}} \right)\)  \( \frac{\beta \cdot y}{{c_{i}^{2} }}\)  \(\beta \left[ {c,y} \right]\)  \(\beta \left[ {c,y} \right]\) 
\(\tilde{x}_{i}\)  \( \frac{y}{{\eta_{y} \cdot c_{i} }}\)  \( \frac{{\eta_{c} }}{{\eta_{y} }} \cdot \frac{y}{{c_{i} }}\)  1  \(\frac{y}{{c_{i} }}\)  \(\frac{y}{{c_{i} }}\)  \( \frac{{\beta \left[ {c,y} \right]}}{{{{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot c_{i} + {{\partial \alpha } \mathord{\left/ {\vphantom {{\partial \alpha } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot t_{i} }}\)  \( \frac{{\beta \left[ {c,y} \right]}}{{{{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot c_{i} }}\) 
\({{\partial \tilde{x}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{x}_{i} } {\partial y}}} \right. \kern0pt} {\partial y}}\)  \( \frac{1}{{\eta_{y} \cdot c_{i} }}\)  \( \frac{{\eta_{c} }}{{\eta_{y} }} \cdot \frac{1}{{c_{i} }}\)  0  \(\frac{1}{{c_{i} }}\)  \(\frac{1}{{c_{i} }}\)  \( \frac{{{{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}}}}{{{{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot c_{i} + {{\partial \alpha } \mathord{\left/ {\vphantom {{\partial \alpha } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot t_{i} }}\)  \( \frac{1}{{c_{i} }}\) 
\({{\partial \tilde{x}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{x}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}\)  \(\frac{y}{{\eta_{y} \cdot c_{i}^{2} }}\)  \(\frac{{\eta_{c} }}{{\eta_{y} }} \cdot \frac{y}{{c_{i}^{2} }}\)  0  \( \frac{y}{{c_{i}^{2} }}\)  \( \frac{y}{{c_{i}^{2} }}\)  \(\frac{{\beta \left[ {c,y} \right] \cdot {{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}}}}{{\left( {{{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot c_{i} + {{\partial \alpha } \mathord{\left/ {\vphantom {{\partial \alpha } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot t_{i} } \right)^{2} }}\)  \(\frac{{\beta \left[ {c,y} \right]}}{{{{\partial \beta } \mathord{\left/ {\vphantom {{\partial \beta } {\partial y}}} \right. \kern0pt} {\partial y}} \cdot c_{i}^{2} }}\) 
Complies with addingup?  Yes, if \(\eta_{y} =  1\)  Yes, if \( \eta_{y} = \eta_{c}\)  No  Yes  Yes  No  Yes 
Complies with homogeneity?  Yes  Yes  Yes  Yes  Yes  No  Yes, if \(\tilde{x}_{i} = {y \mathord{\left/ {\vphantom {y {c_{i} }}} \right. \kern0pt} {c_{i} }}\) 

Model VI: This specifies generalised relationships between each of the conditional marginal utilities that comprise VoT (i.e. α and β) and features of the journey/traveller (i.e. c, t and y). The intention of Model VI is to mimic relationships reported in the most recent Danish national VoT study (Fosgerau et al. 2007), although it should be qualified that the Danish study specified models in VoTspace rather than utilityspace^{12}.

Model VII: This is a simplified version of Model VI which specifies relationships between the β parameter and features of the journey/traveller, but excludes similar relationships from the α parameter. This model might also be seen as a generalised version of the relationships inherent within Models I–V.
Considering first the explicit specifications (i.e. Models I–V), note that Model III fails addingup by construction, whilst Models IV and V comply with both addingup and homogeneity by construction. Models I and II are distinct in that theory confirms compliance with homogeneity, but compliance with addingup is inconclusive and can be confirmed only through empirical analysis.^{13} Turning to the generalised specifications, Model VI fails both addingup and homogeneity, whilst Model VII complies with addingup by construction and with homogeneity provided budget is exhausted (i.e. \(\tilde{x}_{i} = {y \mathord{\left/ {\vphantom {y {c_{i} }}} \right. \kern0pt} {c_{i} }}\) for \(i = 1,2\)).
Empirical analysis
 1.
Empirical analysis will reveal whether, in the case of certain models, addingup holds in practice.
 2.
Empirical analysis will reveal whether compliance with addingup and homogeneity engenders models which better explain the empirical data visàvis models which violate either or both of these properties.
Summary of parameter estimates and goodness of fit
Model I  Model II  Model III  Model IV  Model V  Model I (Constrained)  Model II (Constrained)  

Est.  t  Est.  t  Est.  t  Est.  t  Est.  t  Est.  t  Est.  t  
β  −0.876  −4.14  −0.586  −3.73  0.0163  19.30  −2.04  −7.84  −2.08  −17.22  −3.98  −3.58  −0.306  −17.07 
α  −0.0932  −14.94  −0.0867  −14.12  −0.0755  −12.58  −0.0771  −13.35  −0.0393  −7.87  −0.0594  −10.82  −0.0541  −9.95 
η _{c}  −0.474  −11.85  0.671  18.86  –  –  0.563  18.29  –  –  −0.448  −8.57  1.000  – 
η _{y}  −0.359  −8.02  −0.382  −8.16  –  –  –  –  –  –  −1.000  –  −1.000  – 
γ  0.924  18.70  0.973  19.63  0.911  19.04  0.982  19.89  0.999  20.88  0.912  18.98  0.913  19.19 
Null loglikelihood  −3283.438  −3283.438  −3283.438  −3283.438  −3283.438  −3283.438  −3283.438  
Final loglikelihood  −2705.165  −2733.621  −2813.415  −2746.202  −2869.293  −2793.097  −2832.740  
Parameters  5  5  3  4  3  4  3  
Adjusted ρ ^{2}  0.175  0.166  0.142  0.162  0.125  0.148  0.136  
Implied VoT at reference values^{a} (p/min, 1994 prices and incomes)  3.38  5.60  4.63  6.01  5.67  4.11  6.19 
Theoretical interpretation of the empirical analysis
With reference to Table 3, note that Model I—which is ostensibly the model recommended by Mackie et al. (2003)^{14}—gives the best explanatory fit (in terms of the adjusted \(\rho^{2}\) statistic) of the models considered, followed closely by Models II and IV. Models III and V give reduced fit, and this is possibly because the former excludes cost damping whilst the latter admits a restrictive form of cost damping,^{15} when the evidence from other models suggests that a more complex form of this phenomenon was prevalent in the dataset. Across all models, the estimated parameters carry their expected signs. Turning to the VoTs derived in terms of the design journey, Model I recommended by Mackie et al. yields the lowest valuation (3.38p/min, in 1994 prices and incomes), whilst Model IV yields the highest (6.01p/min).

Model I: Since η _{ y } is significantly different from minus one, we can infer that addingup is violated.^{16}

Model II: Since η _{ y } is significantly different from η _{c}, we can similarly infer that addingup is violated.

Model I (Constrained): \(\eta_{y} = 1\), thereby imposing addingup.^{17}

Model II (Constrained): \( \eta_{y} = \eta_{c} = 1\), again imposing addingup.^{18}
Finally, it is worth reflecting that Model IV offers superior fit to Models I and II (Constrained), accommodates cost damping in the form of second own partial derivatives of the CIUF with respect to design cost, and complies with addingup and homogeneity by construction. This model would therefore seem to represent a credible alternative to Models I and II.
Summary and conclusions
A popular interest in transport demand modelling is to estimate the value of travel time (VoT), by implementing De Serpa’s (1971) theory through empirical models from the random utility model (RUM) class. Within this framework, Mackie et al.’s (2003) conditional indirect utility function (CIUF) has proved influential, since this identity (or a variant thereof) has formed the basis of official national estimates of VoT in the UK, Switzerland and The Netherlands. A key feature of this CIUF, which was introduced as equation (1) in the present paper, is that it specifies utility as a function of changes in ‘design’ travel time and cost (i.e. for a given policy scenario such as an infrastructure scheme) from ‘current’ travel time and cost (i.e. in the absence of the scheme). However, when estimating as RUM, the property of ‘translational invariance’ means that the CIUF (1) is not identifiably different from one that specifies utility as a function of the absolute levels of design travel time and cost; the latter CIUF was introduced as equation (2) in the present paper. Estimation of (2) has commonly found that \({{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} < 0}}} \right. \kern0pt} {\partial c_{i} < 0}}\) and \({{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}} \right)} {\partial c_{n} > 0}}} \right. \kern0pt} {\partial c_{n} > 0}}\), such that the conditional marginal disutility of design cost decreases—in absolute terms—as current cost increases; this phenomenon is referred to as ‘cost damping’ in the literature.

When estimating (2) in the course of the 2003 UK VoT study, Mackie et al. reported the implied indifference map to be strictly convex. We have argued that this result arises from an unconventional definition of the elasticity of VoT with respect to travel cost. Using a more conventional definition, the implied indifference map was shown to be linear rather than strictly convex.

In an attempt to clarify the theoretical validity of the cost damping phenomenon, we assessed the compliance of the conditional demand implied by (2) with the theoretical properties of demand functions. We found that homogeneity and symmetry are imposed by construction, but addingup holds only where the VoT is unit elastic with respect to income, and negativity holds where the same elasticity is unit elastic or greater.

Relating these requirements to Mackie et al.’s empirical results, it was concluded that—in the case of the 2003 UK VoT study—the conditional demand violates both addingup and negativity.

The sign of \({{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}} \right)} {\partial c_{n} }}} \right. \kern0pt} {\partial c_{n} }}\) had no bearing on the aforementioned violation of addingup and homogeneity. In fact, the determining factor was the sign (and magnitude) of \({{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial c_{i} }}} \right. \kern0pt} {\partial c_{i} }}} \right)} {\partial y}}} \right. \kern0pt} {\partial y}}\), i.e. the prevalence of cost damping with respect to income (rather than to travel cost).

Repeating the same analysis with alternative specifications of the CIUF, we found that specifications which a) express design cost as a proportion of income, and b) introduce nonlinearity through logarithmic transformation or through a power term, comply with the theoretical properties of demand functions. Comparing against (2) on Mackie et al.’s data, these specifications offered comparable explanatory power, whilst complying with addingup and homogeneity. Furthermore, these specifications elicited higher valuations of travel time for the design journey.

A final strand of analysis was to impose addingup (and negativity) on Mackie et al.’s specification, through an appropriate constraint on model estimation. Whilst this constraint compromised explanatory fit to some degree, the implied VoT was found to increase by around 20%.
Footnotes
 1.
In practice, Mackie et al. adjusted the specification of travel time in (1) to account for ‘size and sign’ effects and ‘inertia’ effects; the latter phenomenon will be considered in the "Alternative specifications of the CIUF" section of the present paper, whilst the former falls outside the present scope.
 2.
In what follows, \(\left[ \cdot \right]\) will be used to represent the arguments of a function and \(\left( \cdot \right)\) to represent the collection of terms.
 3.
In what follows, we will adopt some brevity in notation by suppressing the traveller index \(n = 1, \ldots ,N\) and focussing upon a single individual. This should not be seen as a substantive loss of generality. Indeed, in the "Alternative specifications of the CIUF" section, we apply the model to a sample of travellers with different incomes and different travel times and costs for the current travel option.
 4.
Mackie et al. (2003) noted that c _{0} and y _{0} are ‘arbitrarily defined base or reference values which do not affect the estimation of the elasticities’ (p. 30).
 5.
Alternatively, and more faithful to De Serpa (1971), we could derive VoT for the design journey in terms of the conditional marginal utility of income rather than the conditional marginal utility of design cost:
\(\tilde{\upsilon }_{i} \left[ {t_{i} ,y} \right] = \frac{{{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial t_{i} }}} \right. \kern0pt} {\partial t_{i} }}}}{{{{\partial \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial \tilde{v}_{i} } {\partial y}}} \right. \kern0pt} {\partial y}}}} =  \frac{\alpha \cdot y}{{\eta_{y} \cdot \beta \cdot y^{{\eta_{y} }} \cdot c^{{\eta_{c} }} \cdot c_{i} }}\quad{\text{for}}\;i = 1,2\quad ({\text{f}}1)\)
This is directly proportional to Mackie et al.’s VoT (5), in the following manner:
\(\tilde{\upsilon }_{i} \left[ {t_{i} ,y} \right] = \tilde{\upsilon }_{i} \left[ {t_{i} ,c_{i} } \right] \cdot \frac{y}{{\eta_{y} \cdot c_{i} }} = \tilde{\upsilon }_{i} \left[ {t_{i} ,c_{i} } \right] \cdot \tilde{x}_{i} \left[ {c_{i} ,y} \right]\quad{\text{for}}\;i = 1,2\quad ({\text{f2}})\)
The substantive distinction between the two measures is that (5) refers to a single trip, whereas (f2) refers to all trips by an individual. Also, the two measures have different signs since the denominator of (5) refers to a ‘bad’ (i.e. travel cost) whilst the denominator of (f2) refers to a ‘good’ (i.e. income).
 6.
Derivation and interpretation of relevant elasticities from (2) will follow in the "Second derivatives from Mackie et al.’s CIUF, and theoretical properties of the indifference map" section.
 7.
That is to say: \({{\partial^{2} \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial^{2} \tilde{v}_{i} } {\partial t_{i}^{2}}}} \right. \,\, \kern0pt} {\partial t_{i}^{2} ,}}{{\partial^{2} \tilde{v}_{i} } \mathord{\left/ {\vphantom {{\partial^{2} \tilde{v}_{i} } {\partial c_{i}^{2} \le 0}}} \right. \kern0pt} {\partial c_{i}^{2} \le 0}}\;{\text{for}}\;i = 1,2\).
 8.
That is to say, if the Marshallian demand exhibits these properties then there exists a direct utility function which is maximised subject to a budget constraint. For fuller discussion of these properties in the context of continuous demand see Deaton and Muellbauer (1980), or in the context of discrete choice demand see Batley and Ibáñez (2013a).
 9.
In the case of the conditional demand, however, there is no need to ‘aggregate’ as such.
 10.
Whilst theory and evidence point to a temporal elasticity of VoT with respect to income of one (e.g. Fosgerau 2008), crosssectional income elasticities estimated on SP data are usually somewhat less than one. In particular, Wardman and Abrantes (2011) reported an average income elasticity from metadata of 0.5.
 11.
This specification corresponds to the nonlinearincost but linearinparameters form promoted by Rich and Mabit (2016).
 12.
Of course, models specified in VoTspace would not in practice identify the conditional marginal utilities of travel time and cost, and would not therefore lend themselves to testing in this manner.
 13.
See the "Addingup" and "Homogeneity" sections for derivation of these theoretical properties for Model I, and the "Empirical analysis" and "Theoretical interpretation of the empirical analysis" sections for empirical testing of these properties for both Models I and II.
 14.
 15.
In Model V, cost is damped in fixed proportion to design cost.
 16.
Based on the discussion within the "Negativity" section, it can also be inferred that negativity is violated.
 17.
Following from the "Negativity" section and footnote 16, this constraint also imposed negativity.
 18.
We also tried imposing the slightly weaker constraint \( \eta_{y} = \eta_{c}\), but this model could not be estimated.
Notes
Acknowledgements
Thanks are due to Dr Manuel Ojeda Cabral, who estimated the models reported in Tables 2 and 3. An earlier version of this paper was presented at the International Association of Travel Behaviour Research Conference (Batley 2015), and the present version has benefitted from the comments of delegates at that conference, especially John Bates and Andrew Daly. Thanks are also due to two anonymous peer reviewers appointed by this journal, whose comments greatly improved the paper. Responsibility for any remaining faults or limitations rest entirely with the author.
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