Point elasticity extremum and reverse analysis of the multinomial logit model

Existing point elasticity theories of the multinomial logit (MNL) model mainly focus on the forward application of the elasticity formula. However, the monotonic and extremum properties of the elasticity formula—critical for exploring the potential to regulate choice probabilities—have received limited attention. Currently, no analytical method exists for calculating attribute variables based on given point elasticity extreme values and thresholds. To address the limitations of reverse point elasticity solutions that depend on numerical methods and the unclear monotonicity of threshold intervals, this study applies differential theory and the properties of the Lambert W function to clarify the monotonic intervals and extreme point properties of the MNL model's point elasticity function. It also proposes generalized closed-form solutions for the reverse solutions of cross and self-point elasticity functions.

The study performs derivative analyses of the cross and self-elasticity functions, introducing the Lambert W function to derive generalized closed forms for the extreme points, determine the monotonic intervals of elasticity functions, and specify the ranges of attribute variables under given elasticity thresholds. First, the derivative of the elasticity function is calculated, revealing that the elasticity functions have a discontinuity at the origin and exhibit a monotonically increasing and then decreasing trend, with only one maximum point. Limit analysis indicates that the elasticity functions have no lower bound. Next, by introducing the main branch theory of the Lambert W function, the generalized closed form of the stationary points is derived, along with a criterion for identifying intervals of positive elasticity based on whether the stationary point exceeds 1. Considering the marginal diminishing effect of adjusting attribute variables on choice probability, the study further calculates the effect's preservation intervals of attribute variables under given elasticity critical thresholds. Because the inverse solution involves double roots, the main and negative branches of the Lambert W function are used to comprehensively obtain the generalized closed analytical form of the double roots and identify the variation intervals of attribute variables with the potential to regulate choice probability. Finally, two transportation mode choice cases—bus fare discounts and bus travel subsidies as attribute independent variables—are analyzed to verify the practicality of the proposed theory in formulating attribute variable adjustment values while accounting for policy costs and marginal diminishing effects.

The theoretical derivation and case analysis of this study indicate that: (1) As attribute variables increase, both the cross and self-point elasticity functions of the MNL model exhibit a monotonic pattern—first increasing and then decreasing, and first negative and then positive—with a single maximum positive point; (2) Considering an alternative associated with attribute variable, when the absolute difference between the system utility of the remaining alternatives and that of the alternative without the attribute variable is not less than 2, the maximum value of the self- or cross-point elasticity is not less than 1; (3) Within the MNL model framework, the marginal effect of bus fare discounts decreases continuously as the discount amount increases, while that of bus ride subsidies first increases and then decreases as the subsidy amount grows. Therefore, to enhance the attractiveness of public transportation, relevant authorities should first perform the reverse quantitative elasticity analysis proposed in this paper to scientifically balance policy implementation costs and effectiveness.

This study provides a theoretical foundation for scientific decision-making on target values of attribute variables by considering adjustment costs and the significance of expected effects. Using the discriminant for the existence of sensitive intervals, decision-makers can assess whether adjustments to specific attribute variables significantly affect choice probabilities and preclude unsuitable variables. Under a given elasticity threshold, the adjustment interval of an attribute variable with a non-decreasing marginal effect can be precisely calculated in reverse, improving the efficiency and accuracy of policymaking.