Random Utility Theory
3.6 Direct and Cross-elasticities of Random Utility Models *
Analogously,cross-elasticityis defined as the percentage variation in the demand for a certain commodityj divided by the percentage variation in the value of an attributekof another commodityh,Xkh:
Ep[jkh ]=p[j] p[j]
Xkh
Xkh
In the above definitions, the variations in the values of attributes and demand are assumed to be finite. This case defines the arc elasticity, which is calculated as the ratio of incremental ratios over an “arc” of the demand curve. Point elasticities are defined for infinitesimal variations and can be expressed analytically.
The point direct elasticity of the choice probability for alternativej with respect to an infinitesimal variation in thekth attribute Xkj of its own utility function is defined as
Ekjp[j]=∂p[j](X)
∂Xkj Xkj
p[j]=∂lnp[j](X)
∂lnXkj (3.6.1)
whereXincludes the vectors of attributes for all alternatives.
Similarly the point cross-elasticity of the choice probability of alternativej with respect to an infinitesimal variation of thekth attribute,Xkh, of the utility function of alternativehis defined as
Ekhp[j]=∂p[j](X)
∂Xkh Xkh
p[j]=∂lnp[j](X)
∂lnXkh (3.6.2)
Both direct and cross-elasticities24are useful measures of the model’s sensitivity to variations in the attributes. It is evident from (3.6.1) and (3.6.2) that elasticities depend on the functional form of the model as well as on the values of attributes and parameters in the systematic utilities.
Analytic and compact expressions for direct and cross-elasticities (3.6.1) and (3.6.2) can be obtained for the multinomial logit model with a linear systematic
24The elasticities discussed in this section are disaggregate, i.e. related to variations in the prob- abilities of a single decision maker or of a group of decision makers sharing the same attribute values. Aggregate elasticities refer to variations in the average choice fraction:
¯ p(j )=
n
i=1
pi(j )
of a group of decision makers with different attributes. Variations are computed with respect to a uniform infinitesimal variation of a given attribute. In this case, it is possible to express the aggregate elasticity as a weighted average of individual elasticities. For instance the direct point elasticity is:
Ekjp[j¯ ]= n
i=1pi[j]Ekhpi[j]
n i=1pi[j]
3.6 Direct and Cross-elasticities of Random Utility Models 145 utility functionVj=βTXj. In this case:
Ekjp[j]=
1−p[j]
βkXkj/θ (3.6.3)
Ekhp[j]= −p[k]βkXkh/θ (3.6.4) From (3.6.3) it can be deduced that the direct elasticity is positive if attribute Xkj is positive (as is usually the case) and if its coefficientβk is positive. In other words, the choice probability of an alternative increases if the value of an attribute that contributes to its utility (β positive) increases.25 The increase will be higher for higher values of coefficientβk and attribute Xkj, and for lower values of the alternativej choice probability. Thus, in a mode choice model, direct elasticities of the probability of choosing a car with respect to travel time and cost will be negative because the coefficientsβkof these attributes are negative; these elasticities will be larger, in absolute terms, for an origin–destination pair with relatively large time and cost values. Lastly, if the probability of choosing the car is low, its elasticity will be larger, for given values of parameterβk and attributeXkj.
Similar considerations, although with inverted signs, hold for cross-elasticities, which will be positive ifβkorXkhare negative, and will be larger for larger absolute values ofβk, Xk, andp[h]. Continuing with the above example, the cross-elasticities of the probability of using a car with respect to the travel time and cost of another mode will be positive (becauseβk<0).
Qualitatively similar conclusions apply to elasticities of random utility models other thanMNL.
Note that the cross-elasticity (3.6.4) of the multinomial logit model is identi- cal for all alternatives because a variation in the value of one alternative’s attribute produces the same percentage variation in the choice probabilities of all other alter- natives. This result can be considered as a different manifestation of the logit model independence from irrelevant alternatives property described in Sect.3.3.1.
Expressions (3.6.3) and (3.6.4) also show that, for given values of coefficients and attributes, direct and cross-elasticities are higher in absolute terms when the variance of the random residuals (directly related to the scale parameterθ )is lower.
Conversely, as the random residual variances tend to infinity, the elasticities tend to zero. Figure 3.15shows the values of direct and cross-elasticities with respect to a generic attribute in a multinomial logit model.
For more complex random utility models it is not easy, or even possible, to de- rive analytic expressions for direct and cross-elasticities. However, it is useful to discuss elasticities for a single-level hierarchical logit model inasmuch as they pro- vide some insight into the influence of random residual covariances on direct and cross-elasticities.
25The result that multinomial logit choice probabilities increase monotonically with respect to systematic utilities is obtained again. It holds, more generally, for all invariant models described in previous sections.
Fig. 3.15 Direct and cross-elasticities for a multinomial logit model
XkA XkB XkC XkD Ep[A] 0.75 −0.25 −0.25 −0.25 Ep[B] −0.25 0.75 −0.25 −0.25 Ep[C] −0.25 −0.25 0.75 −0.25 Ep[D] −0.25 −0.25 −0.25 0.75
Consider the single-level hierarchical logit model in Fig.3.16; it contains one nest whose only component is the elementary alternativeA, and another nest G containing elementary alternativesB, C, andD.
It is possible to obtain in closed form the elasticities of the choice probability of alternativeAwith respect to a generic attributeXkthat is included in the systematic utility of all the alternatives. Applying the definitions of elasticity (3.6.1) and (3.6.2) to the single-level hierarchical logit model in expression (3.3.19) with parameter θo=1, the direct elasticity (variation of attributeXkA) and the cross-elasticity with respect to alternativeB (variation of attributeXkB)are, respectively,
EkAp[A]=
1−p[A]
βkXkA/θ (3.6.5)
EkBp[A]= −p[B]βkXkB/θ (3.6.6) The elasticities in this case are completely analogous to those obtained for the multinomial logit model, expressed by (3.6.3) and (3.6.4). Things are different, however, for the choice probability elasticities of alternativeBin nestG. Its direct elasticity (variation of attributeXkB)is
EkBp[B]=
1−p[G]
·p[B/G] +
1−p[B/G]
/θ
βkXkB (3.6.7) If the hierarchical logit were reduced to a multinomial logit model, that is, ifθ=1, the direct elasticity (3.6.7) would become analogous to (3.6.3) or (3.6.5). On the other hand, ifθis less than one, the hierarchical logit elasticity is larger than that of a multinomial logit model with the same parameters, attributes, and residual variance.
The cross-elasticities ofp[B]with respect to variations in attributeXkA of the
“isolated” alternativeA, and in attributeXkC of alternativeC in the same nestG are, respectively,
Ep[B]X
kA = −p[A]βkXkA/θ (3.6.8) Ep[B]X
kC = −
p[C] +1−θ
θ p[C/G]
βkXkC (3.6.9)
Equation (3.6.8) shows that the cross-elasticity of B’s choice probability with respect to an attribute of alternativeAnot belonging to B’s nest Gis equivalent
3.6DirectandCross-elasticitiesofRandomUtilityModels147
VA=VB=VC=VD βk=1 ∀k Xkj=1 ∀k, j
θ=1 θ=0.8 θ=0.4 θ=0.1
XkA XkB XkC XkD XkA XkB XkC XkD XkA XkB XkC XkD XkA XkB XkC XkD
Ep[A] 0.75 −0.25 −0.25 −0.25 0.71 −0.24 −0.24 −0.24 0.61 −0.20 −0.20 −0.20 0.52 −0.18 −0.18 −0.18 Ep[B] −0.25 0.75 −0.25 −0.25 −0.29 0.93 −0.32 −0.32 −0.39 1.80 −0.70 −0.70 −0.47 6.82 −3.18 −3.18 Ep[C] −0.25 −0.25 0.75 −0.25 −0.29 −0.32 0.93 −0.32 −0.39 −0.70 1.80 −0.70 −0.47 −3.18 6.82 −3.18 Ep[D] −0.25 −0.25 −0.25 0.75 −0.29 −0.32 −0.32 0.93 −0.39 −0.70 −0.70 1.80 −0.47 −3.18 −3.18 6.82
Fig. 3.16 Direct and cross-elasticities for a hierarchical logit model
to that of the corresponding multinomial logit model. On the other hand, the cross- elasticity with respect to an attribute of an alternative belonging toB’s nestG(and so correlated withB) is larger for smaller values of parameterθ, that is, for larger covariance between the two alternatives. If two alternatives are perceived as being very similar (i.e., their respective random residuals are highly correlated), the prob- ability of choosing one of them is very sensitive to variations of the attributes of the other. From (3.6.9) it also follows that ifθ=1 the hierarchical logit model becomes a multinomial logit model and the cross-elasticity is analogous to (3.6.8).
Direct and cross-elasticities of the hierarchical logit model, for different values of parameterθ, are shown in Fig.3.16. Forθ=1, the elasticities reported in Fig.3.15 are obtained.
The general conclusion from the above example is that, given equal attributes and coefficients, the more an alternative is perceived as “similar” to other alternatives, the higher are its direct and cross-elasticities. Thus, for any random utility model, variations in the attributes of an alternative will have the greatest effects on the choice probabilities of alternatives that are perceived as close substitutes to it.