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The Hensher equation: derivation, interpretation and implications for practical implementation

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Abstract

The ‘Hensher equation’ is a prominent method for valuing the benefits of business travel time savings. This paper derives the equation from first principles, revealing several underpinning assumptions, as follows: (I) production is a function only of labour given fixed capital; (II) the value of the marginal product of labour is equal to the wage; (III) business travel has constant productivity whether it takes place during work or leisure; and (IV) utility is a function of work, leisure and travel time. Informed by this derivation, the paper interprets the features of the resulting valuations. Finally, the paper also derives restricted and extended cases of the Hensher equation, applicable to a range of practical situations where the equation might be implemented.

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Notes

  1. This paper was written by Hensher, as a contribution to both Ian Heggie’s 1976 book on the value of time and the UK Department of the Environment’s 1975 workshop on the value of time.

  2. Whilst (2) focuses upon specific costs and benefits of business travel to the employer and employee, Hensher’s research encompassed additional costs and benefits to both parties as well as to society more generally. Note that, on the basis of the subsequent derivation of (2) from first principles in the “Solving the optimisation problem” section, we will refine Fowkes et al.’s (1986) interpretation of the constituent terms.

  3. See for example the discussion on page 113 of HS2 Ltd. (2013).

  4. Both Carruthers and Hensher (1976) and Hensher (1977) discussed the tax implications of business travel at some length, although taxation does not feature within Fowkes et al.’s (1986) formulation of the Hensher equation (2). In the “Extended cases of the Hensher equation” section of the present paper, we will introduce some notion of taxation.

  5. See Karlström et al.’s equation (13).

  6. Indeed, Hensher (1977) considered not only the benefits and costs of business travel to the employer and employee (p. 82), but also the wider impacts on the ‘community’ through the taxation system (p. 83).

  7. See Kato’s equation (A-28).

  8. To avoid any ambiguities, we will take the liberty of re-defining all terms and tightening definitions where appropriate.

  9. In practice, there may be savings in overheads from travelling instead of spending equivalent time in the workplace, but these are not considered here.

  10. For example, the “Extended cases of the Hensher equation” section of the present paper extends the analysis to consider taxation (see Case 6) and imperfect competition (see Case 7).

  11. On this issue, Hensher (1977) remarked:

    ‘When travel occurs during the employee’s own time the opportunity cost of an average hour of working time to the employer is traditionally assumed to be zero. However, the true net cost or benefit associated with travel in the employee’s own time must reflect the cost of compensation (if any) and the contribution to productivity.’ (p. 76)

  12. Wardman et al.’s (2013a, b) review of empirical evidence concluded that p* is typically less than 0.3, whilst q is typically close to 1.

  13. In the vein of Karlström et al. (2007), an alternative (but equivalent) way of specifying the problem (8) would be to include a budget constraint, but represent the objective statement entirely in terms of the employee, as follows:

    \( \begin{gathered} \mathop {\text{Max}}\limits_{{X,T_{w} ,T_{l} ,t_{w} }} W = U \hfill \\ {\text{s}} . {\text{t}} .\hfill \\ PX = wT_{w} \quad \quad \left( \lambda \right) \hfill \\ T = T_{w} + T_{l} \quad \quad \left( \mu \right) \hfill \\ \end{gathered} \)

    This problem yields exactly the same Lagrangian function as (8), specifically: \( L = U + \lambda \left( {PX - wT_{w} } \right) + \mu \left( {T - T_{w} - T_{l} } \right) \)

  14. This resonates with the argument, advanced by Carruthers and Hensher (1976), that:

    ‘…some of the work done while travelling represents work that would be done in the employee’s own time anyway. If this argument is accepted, then it is strictly not employee’s but employer’s time, and should be included in the total annual hours worked when calculating the hourly productive rate…’ (p. 168)

    In the “Restricted cases of the Hensher equation” section of the present paper, the analysis considers overtime payments (see Case 3).

  15. The “Extended cases of the Hensher equation” section of the present paper considers possible discrepancy between the value of the marginal product of labour and the wage rate (see Case 7).

  16. This is because ∂U/∂T w  = ∂U/∂{·}·∂{·}/∂T w where ∂{·}/∂T w  = 1, and similarly for leisure time. More realistically, we should perhaps expect diminishing marginal utility as ‘effective’ work time increases but ‘contracted’ (i.e. income-earning) hours remain constant, which would call for a non-linear functional form for utility.

  17. In the context of (8), production and consumption are revenue neutral. Recall the assumption that revenue from the sale of goods straightforwardly transfers to the wage-related income of the employee.

  18. For reasons of brevity, subsequent cases will not present full working but simply the final derivation of VBTTS in each case.

  19. Note that r*=0 corresponds with the first outcome in Fig. 2, i.e. ‘effective’ work time is unchanged.

  20. Note that r*=1 corresponds with the second outcome in Fig. 2, i.e. ‘effective’ leisure time is increased.

  21. A premium overtime rate would call for a further adjustment.

  22. The classic example of this case is where work entails travelling per se, such as lorry or bus drivers.

  23. There is a question as to the applicability of the behavioural value (18) to the social value of travel time savings in appraisal, since the latter is concerned with the marginal value of labour productivity and/or the resource value of labour released into the market, but not the marginal value of the trip.

  24. In this case we will adopt the equation (13) rather than (14), since the former explicates the role of wages.

  25. In contrast to the other cases of the Hensher equation considered, it is debatable whether (19) represents the behavioural value of travel time savings, since employer behaviour will not in general be influenced by the tax implications for the employee, and vice versa. Moreover, standard appraisal practice in the UK (and a number of other countries) is to employ the behavioural value of travel time savings, and to consider any tax implications elsewhere in the appraisal.

  26. In contrast to Cases 5 and 6, behavioural and social values of travel time savings will in this case be one-and-the-same.

  27. The sign of the w term in (20) reflects the way in which the optimisation problem (8) is formulated, and is not substantively different from the sign of the corresponding term in Jara-Díaz’s equation.

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Acknowledgments

This paper arises from research commissioned by the UK Department for Transport, under contract reference RGTRAN484855, and published as Wardman et al. (2013a, b). The analysis and interpretation presented herein should, however, be regarded as a statement of the private views of the author, and not as a statement of the Department’s policy. The paper has greatly benefitted from the comments of David Hensher, Hironori Kato, James Laird, Peter Mackie, Mark Wardman and Manuel Ojeda Cabral. Any remaining deficiencies are the sole responsibility of the author.

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    Richard Batley

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Batley, R. The Hensher equation: derivation, interpretation and implications for practical implementation. Transportation 42, 257–275 (2015). https://doi.org/10.1007/s11116-014-9536-3

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