We follow much of the literature on modeling the labour supply of low-income households in approximating the continuous hours of work data by a choice among discrete alternatives. Like Hoynes (1996 and Brewer *et al.* (2007), we allow for unordered labour supply choices, but we do not adopt the logit with mixing to avoid IIA but rather use the multinomial probit. That is, like Keane and Moffitt (1998) we adopt a probit specification, but we do not restrict it to be ordered. Like Hoynes (1996 and Brewer *et al.* (2007) we also allow for unobserved heterogeneity through random parameters. While our Random Utility Model unordered probit approach would not scale up to a larger choice set with the same ease as those based on a multinomial logit framework, it has at least the same degree of flexibility for the problem at hand. Furthermore, we control for the fact that some of those not working would rather be employed – i.e. they are unemployed. This seems particularly important since our data covers a period when there was widespread unemployment: aggregate unemployment rates reached 10% in the mid 1980’s, fell to around 7% in the late 1980’s and started to rise again after that.

There are four elements to our empirical model: the constraint, the specification of preferences, the specification of take-up that embeds assumptions about costs, and how the model combines to allow the identification of fixed and variable costs. We discuss each element in turn.

The income levels associated with each state constitute the constraint which contributes to the determination of the choice of labour market state. Since we only observe the one alternative that is chosen, we need to predict incomes for each state from the income in the observed state. It would be computationally demanding to estimate the incomes associated with each labour market alternative jointly with the choice among alternatives. Since we only require consistent predictions of wages in order to estimate the determinants of each state, we adopt a two-step procedure. In the first step we estimate full-time and part-time wage equations which use a reduced form for labour market status to control for the endogeneity of hours and use these estimates to predict incomes in the part-time and full-time states. ^{10} Income for non-participants is computed from the welfare system and observed unearned (non-transfer) income. For participants, we compute the levels of tax liability and transfer entitlement using these predicted earnings at the specific discrete levels of hours of work.

The budget constraint is approximated by just four discrete labour supply alternatives: non-participation (NP), low hours part-time (LPT), high hours part time (HPT) and full-time (FT) ^{11} in combination with three transfer programmes: Family Credit, Housing Benefit and in-kind transfers to children. We model the choice between 32 alternatives. These are combinations of 4 labour market states and participation in each of three transfer programmes (2*2*2). The 8 possible programme participation combinations, across the 4 labour market states yields the 32 alternatives. Since each alternative is a composite of a labour supply state and a combination of programme participations, we maintain this structure in the decision modeling. ^{12}

In the second step, we estimate the random utility model using the predicted incomes in each state. ^{13} The second element of the model is preferences. Here choices between these alternatives are driven by differences in the utilities attached to them. Consistency with choice theory implies that we determine all 31 utility differences (8 alternatives involve unemployment which we do not regard as a distinct choice driven by utility maximizing considerations). Let *p* index each programme in the set of programmes *P =* {*Housing Benefit, Family Credit, child nutrition programmes*}. Participation in each separate programme is indicated by categorical indicators *T*^{p}, which together compose the complete programme participation vector {\mathbf{\tau}}_{s}^{p}={\left({T}^{\mathit{HB}},{T}^{\mathit{FC}},{T}^{\mathit{CH}}\right)}^{\prime} where HB = Housing Benefit, and FC = Family Credit. Hence labour supply *h*_{
s
} and participation in programmes {\mathbf{\tau}}_{s}^{p} completely characterize a state, *s*.

Let the utility associated with choosing state *s* be {U}^{*}\left({y}_{\mathit{is}}^{0},{h}_{\mathit{is}},{\mathbf{y}}_{\mathit{is}}^{p},{\mathbf{T}}_{\mathit{is}}^{p};\mathbf{X}\right) where {\mathbf{y}}_{\mathit{is}}^{p} is the income associated with the programmes *P*, {y}_{\mathit{is}}^{0} is other income (effectively Income Support and earned income, which we pool because the participation rate in the Income Support programme is so close to unity) ^{14} , and **X** is a vector of individual characteristics. Now consider a statistical specification which allows for random variation in behavior due to an additive disturbance and to variation in tastes, {U}^{*}\left({y}_{\mathit{is}}^{0},{h}_{\mathit{is}},{\mathbf{y}}_{\mathit{is}}^{p},{\mathbf{T}}_{\mathit{is}}^{p};\mathbf{X},{\u03f5}_{\mathit{is}}\right), where {U}_{\mathit{is}}^{*} is unobservable utility of state *s* for individual *i*, and *ε*_{
is
} is an alternative specific random error term. Thus, the utility gain of moving from alternative *s* to *t* is:

{U}_{\mathit{is}}^{*}-{U}_{\mathit{it}}^{*}\equiv {U}^{*}\left({y}_{\mathit{is}}^{0},{h}_{\mathit{is}},{\mathbf{y}}_{\mathit{is}}^{p},{\mathbf{T}}_{\mathit{is}}^{p};\mathbf{X},{\epsilon}_{\mathit{is}}\right)-{U}^{*}\left({y}_{\mathit{it}}^{0},{h}_{\mathit{it}},{\mathbf{y}}_{\mathit{it}}^{p},{\mathbf{T}}_{\mathit{it}}^{p};\mathbf{X},{\epsilon}_{\mathit{it}}\right)

(1)

In a discrete choice model the set of alternatives is assumed to be common across individuals. This is a rather general specification since it allows for the possibility that the effect of entitlement on programme participation and the effect of each type of income on labour supply might differ across *X* and *both p* and *s.* Of course, this is far too general to be practical even though some of the types of income are not available in some of the *s* because of the nature of the welfare rules. Thus, we assume (as seems reasonable) that programme participation decisions over *p* should be affected by **X** and by its level of entitlement, *y*^{p}, and not by the entitlement to any other programme or by one’s labour market state (except insofar as entitlement varies across *s*). It also seems reasonable that labour supply choices should depend on **X** and on levels of receipt of each type of income and not on programme participation *per se*.

Thus, we assume that labour supply is a function (which is allowed to vary across hours) of individual characteristics (which are fixed irrespective of hours), and a function (which is fixed across hours, but varies across programmes) of characteristics of alternative combinations of programmes and hours (which vary across hours and programmes). In other words, labour supply is a function of individual-specific characteristics and alternative-specific characteristics. In particular, hours comparisons are a function of demographics and incomes. As usual in this class of model, only the utility *differences* between the number of alternatives minus one can be identified. Since, we are implicitly assuming that the decision to participate in a programme affects welfare in the same way for all comparisons of labour market states, then utility differences between labour supply states can be expressed as

{U}_{\mathit{is}}^{*}-{U}_{\mathit{it}}^{*}\equiv \mathbf{g}\left({\mathbf{y}}_{\mathbf{is}}\mathbf{-}{\mathbf{y}}_{\mathbf{it}}\right){\mathbf{\psi}}_{i}+{X}_{i}{\mathbf{\omega}}_{\mathit{st}}+\left({\epsilon}_{\mathit{is}}-{\epsilon}_{\mathit{it}}\right)

(2)

where here **g**(**y**_{
is
} - **y**_{
it
}) is assumed to be linear ^{15} , and **ψ** = (*ψ*^{HB}, *ψ*^{FC}, *ψ*^{CH}, *ψ*^{0})^{′}, with {\psi}_{i}^{p}={\overline{\psi}}^{\mathit{p}}+{\tilde{\psi}}_{i}^{p}, is a matrix of parameters of functions of differences in the levels of each type of income across pairs of states (from HB = Housing Benefit, FC = Family Credit and CH = child nutrition transfer programmes \left({y}_{\mathit{is}}^{p}\right) and other sources ({y}_{\mathit{is}}^{0}, which is Income Support and other non-transfer income)). That is, the programme participation decisions difference out of this expression. ^{16} We think of the *ψ*^{p}**’** s as capturing the value that individuals attach to variations in differences in the *p*-type of income differences relative to *ψ*^{0} which captures the effects of wage income on the choice of state. In particular, we think of *ψ*^{CH} as capturing the discount that the individual applies to the market value of the in-kind transfers. \overline{\psi} reflects the mean tastes of the sample while {\tilde{\psi}}_{i} is a coefficient which shows how *i* differs from the mean individual, and (*ε*_{
is
} − *ε*_{
it
}) is an additive disturbance assumed to be i.i.d. across *i* but not necessarily across *s.* The *ε*_{
is
} term captures taste variation (or unobserved attributes of alternatives) that is uncorrelated with the income levels – that is purely random variation. The interpretation of the parameters **ω**_{
st
} is the gain (or loss) in utility from comparing the alternative *s* to the alternative *t*, where the latter choice is the reference, when one has the characteristics **X**.

To summarize, from equation (2), the probability of observing *i* in labour market state *s* is given by

\mathrm{Pr}\left[{U}_{\mathit{is}}^{*}>{U}_{\mathit{it}}^{*}\right]=\mathrm{Pr}\left[\mathbf{g}\left({y}_{\mathit{is}}-{y}_{\mathit{it}}\right){\mathbf{\psi}}_{i}+{X}_{i}{\mathbf{\omega}}_{\mathit{st}}>\left({\epsilon}_{\mathit{it}}-{\epsilon}_{\mathit{is}}\right)\right]\phantom{\rule{0.5em}{0ex}}\forall s\ne t

(3)

The third element of our model is programme participation which is assumed to be functions (which do not vary across hours) of individual and programme characteristics: specifically, demographic variables and the levels of entitlement. Consequently programme participation can vary with labour market state, as does entitlement and eligibility. In particular, an individual *i*, in labour market state *s* will participate in transfer programme *p* if it offers a utility gain. This is assumed to be determined by the following latent and observed programme participation (take-up) equations:

{T}_{\mathit{is}}^{p}=\{\begin{array}{c}\hfill 1\phantom{\rule{0.25em}{0ex}}\mathrm{if}\phantom{\rule{0.25em}{0ex}}{T}_{\mathit{is}}^{p*}\equiv {\mathbf{V}}_{i}^{p}{\beta}^{p}{+\mathrm{y}}_{\mathit{is}}^{\mathit{p}}{\gamma}^{p}+{\eta}_{\mathit{is}}^{p}>0\phantom{\rule{0.25em}{0ex}}\hfill \\ \hfill 0\phantom{\rule{0.25em}{0ex}}\mathrm{otherwise}\phantom{\rule{7.5em}{0ex}}\hfill \end{array}

(4)

where {T}_{\mathit{is}}^{p*} is the latent variable corresponding to observed take-up {T}_{\mathit{is}}^{p} of a transfer programme *p* (Housing Benefit, Family Credit, Chuld nutrition), which we define to be unity if *i* is observed to be participating in the programme *p* and zero otherwise; {\mathbf{V}}_{i}^{p} is a vector of individual characteristics which do not vary across labour market states; *β*^{p} is a corresponding vector of parameters; {y}_{\mathit{is}}^{p} is transfer entitlement which may vary across labour market states; *γ*^{p} is an associated coefficient and {\eta}_{\mathit{is}}^{p} is a random error.

The final element of the model is the relationship between labour supply and programme incomes which is established through the functions, **g**(.,.). Labour market status choices are made on the basis of these income differences, amongst other things. These differences are decomposed into Housing Benefit, Family Credit, child nutrition programmes and other (Income Support and non-transfer) income differences separately. Other income is differenced directly, whereas the differences in programme incomes are the differences in the programme participation indices, which are, in turn, a function of entitlement levels. It is straightforward to show that when comparing labour market states *s* and *t*, the difference in programme participation indices between states turns out to be a function of entitlement differences only. That is,

{T}_{\mathit{is}}^{p*}-{T}_{\mathit{it}}^{p*}\equiv \left({Y}_{\mathit{is}}^{p}-{Y}_{\mathit{it}}^{p}\right){\gamma}^{p}+\left({\eta}_{\mathit{is}}^{\mathit{p}}-{\eta}_{\mathit{it}}^{\mathit{p}}\right)

(5)

It is evident from equation (5) that {T}_{\mathit{is}}^{p*} has the dimension of income, and can be interpreted accordingly. Restricting programme participation to be a function of size of programme entitlements captures the idea of “fixed cost stigma” in the terminology of Moffitt (1983). This fixed cost depends on individual demographic characteristics, {\mathbf{V}}_{i}^{p}, so that “fixed cost stigma” varies with observed characteristics. Because programme participation is discrete it must be the case that programme participation decisions can only identify fixed costs.

However, our model does not require that we impose additive separability between labour supply and programme participation. ^{17} Indeed, imposing the restriction *ψ*^{p} = 0 allows a direct test of separability between labour supply and receipt of each type of programme income. It is this feature that allows us to capture the “variable stigma costs” that arise because, conditional on programme participation, the level of each type of income matters for labour supply and welfare. Allowing for fixed and variable cost stigma with multiple programmes is an innovation of our work. Furthermore, {\tilde{\psi}}_{i}^{p} allows taste heterogeneity to vary across types of income.

The model is complex, so it is useful to summarize the restrictions we have put on the labour supply and programme participation model so as to place these in the context of the literature. We assume that participation in a programme is a function of demographics and income from that programme (but not of income that might come from any other programme). This function does not vary across labour market states and while demographics do not vary across state, programme income does. Hence we obtain a programme participation index which varies across labour market states according to this function of entitlement. Exploiting the nature of the choice set and restricting programme participation functions makes the problem much more tractable without imposing further restrictions on preferences or functional form. For example, Family Credit eligibility is restricted to those in work, and child nutrition programme eligibility is restricted to Income Support recipients and to pre-reform Family Credit recipients.

The relationship between labour supply and programme participation comes through *differences* in incomes and functions of entitlements. We assume multivariate normality of the error terms and allow additional flexibility by estimating random coefficients on income differences. The novelty of our empirical approach is that: we allow taste heterogeneity through random coefficients; we nest additive separability of labour supply and programme participation, but impose only a minimal economic structure on the data. Details concerning stochastic specification and likelihood contributions are available online and on request from the authors.

The labour supply parameters are identified because there are households without eligibility to *any* transfers at *any* employment status: largely because they have high wages and/or unearned (non-transfer) incomes that imply zero entitlements even at low hours of work (recall that we exclude Income Support as a choice on the grounds that the take-up rate is close to 100%). The labour supply choice itself is distinguished from unemployment rationing by the exclusion of the regional unemployment rate from the labour supply functions and through joint normality. Identification of the determinants of participation in the various programmes is achieved through exogenous variation in eligibilities and entitlements. For example, time series variation in real housing costs are important in affecting Housing Benefit entitlement, and the variation in real school lunch and milk prices determine the market value of child nutrition entitlements. For both Family Credit and child nutrition programmes we exploit the fact that the data spans the reform in 1988: Family Credit entitlements were increased and associated in-kind transfers lost. Thus, our method uses both step changes associated with the policy reform and the time series variation in entitlements that using 15 years of data allows. In estimation we pool 15 years of cross-sections, and assume preferences are stable over the period having controlled for the observed drivers of sample composition and behavior described in Table 1, together with year and region effects.