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# 3 Measuring welfare changes

This section briefly describes the method used to compute welfare changes for each individual and household. Subsection 3.1 discusses the approach in a random utility framework. Subsection 3.2 examines the calculation of marginal welfare costs for individuals and groups, paying particular attention to the treatment of cases where the net tax paid by some individuals actually declines when income tax rates are increased, as a result of large labour supply responses.

## 3.1  Welfare changes in the random utility model

It was mentioned above that, in the discrete hours approach, direct utility functions are assumed to consist of a deterministic plus a random component, implying that each individual has a probability distribution over the available hours levels. Simulation involves the use of ‘calibration', which ensures that each individual's optimal labour supply under the actual tax structure corresponds to the observed (discretised) hours level. This means that the post-reform distribution of hours worked, and of welfare changes, is effectively a conditional distribution.[14]

The calibration approach proceeds as follows.[15] If a set of random utility terms (one for each hours level) is drawn from the distribution of the stochastic component of utility for an individual, the resulting utility levels can be computed.[16] These combine point estimates of the utility parameters and observed characteristics to calculate the deterministic utility components, to which are added the random components. Comparing the utility values at each labour supply point, optimal labour supply can be determined. In the calibration approach, only those sets of error terms which result in the implied optimal hours being equal to the observed labour supply in the pre-reform tax system are retained.[17]

A new tax and transfer system gives rise to a new set of net incomes for each hours level, and each retained set of random draws produces a single optimal post-reform hours level. Pre-reform hours are always equal to observed hours but, using the retained sets of error terms, a frequency distribution of post-reform hours arises. The calibration approach is preferred in this case as it uses important information in the sample about each individuals' actual labour supply in a given base tax structure. The resulting conditional expected welfare change is also easily interpreted, as starting from a single hours level. It is thus relatively straightforward to embed the calculation of welfare changes in this process of simulating labour supply responses arising from policy reforms.

The compensating variation, for any set of random values, is obtained, as in Creedy, Herault and Kalb (2011), as follows. Utility for working hours level hi, giving rise to net income of ci under tax structure Tk, is equal to a deterministic component plus a random component vi, so that total utility is:[18]

The usual indifference curve diagram used to consider welfare changes is no longer helpful because the indifference curves - obtained from the deterministic component of utility - through any particular combination of hours of leisure and net income for different values of vi are identical in a two-dimensional graph, although they represent different levels of total utility. Hence indifference curves which are located further away from the origin (zero leisure and consumption) no longer necessarily represent higher total utility than indifference curves located closer to the origin, depending on the random components for each discrete hours level. A three-dimensional graph would be needed to present all relevant information. However, it is straightforward to explain the approach in terms of equation (1).

Suppose there are two discrete hours levels, and consider a single set of random utility draws, giving v1 and v2. With tax structure 0, hours level 2 is chosen if . Suppose the tax structure changes to structure 1. Using the calibration approach, the random terms are the same before and after the change. Hours level 2 continues to be chosen if , but hours level 1 is chosen if the inequality is reversed. In the case of the compensating variation, for this set of draws, conditional on hours level 2 being chosen under the initial tax structure and irrespective of the hours chosen after the policy change, the compensating variation is the smaller of two values of CV, obtained by solving each of the following two equations:

If, under the initial tax structure, hours level 1 had been chosen instead of level 2, the compensating variation would be the smaller of two values of CV, obtained by solving two equations similar to (2) and (3), but with the right-hand side of each replaced by . Thus in general, where there are H hours levels, suppose that level hm is chosen in the base tax structure, 0. After the shift to tax structure 1, calculate the H values of CVj Where:

and the CV is given by . When these calculations are repeated for all sets of draws, a probability distribution is obtained, from which the expected value of CV for this individual can be calculated by averaging across all sets of draws. In the case of the equivalent variation, equation (4) is replaced by:

and EV is given by .

In carrying out the calculations, it is necessary to obtain the net income corresponding to a specified hours level and for a given utility level (computed from income and hours at an optimal position). In the present context, as mentioned above, quadratic utility functions are used, so that the well-known expression for the roots of a quadratic can be employed. The appropriate root is obvious, as this lies on the ‘upward sloping' part of the utility function.[19]

In this random utility context, the application of this process to each set of random draws producing observed (discretised) hours as the optimal working hours, a conditional probability distribution of welfare changes can be obtained for each individual or family. The resulting arithmetic mean value is used where appropriate.

### Notes

• [14]An alternative method would be to use the analytical expression for the unconditional distribution of hours worked and a corresponding analytical method to derive expected welfare changes (the numerical approach could be used to obtain the unconditional distribution as well). Preston and Walker (1999) and Dagsvik and Karlstrom (2005) use the unconditional distribution. The present approach has the advantage of simplicity combined with the benefit of being able to exploit observed behaviour in placing individuals at their pre-reform observed hours. In addition, the approach works equally well using the unconditional pre-reform expected labour supply.
• [15]For a detailed introduction to modelling and estimation of labour supply, and the use of these models in behavioural microsimulation, see Creedy and Kalb (2005b).
• [16]In the TaxWell-B simulations reported here, individuals and partnered women have the choice of 11 hours levels (0, 5, 10, … , 50 hours). Partnered men have the choice of six hours levels (0, 10, 20, … ,50). Hence partners jointly have a choice of 66 discrete hours levels.
• [17]In the simulations reported here, 100 sets of such draws are retained for each individual or couple.
• [18]The Extreme Value Type I distribution is used for this random component.
• [19]A check is made at each hours level to ensure that utility increases when net income increases. In the empirical application below, there were no cases of this kind for couples and single men. For single parents, 0.7 per cent of equations did not satisfy this condition (remembering that 100 sets of draws are used for 11 hours levels for each individual). For single women, only 0.1 per cent of equations contained no feasible solution.
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