Workforce or Workfare?

for this activity. The utility level for a person that is unemployed — and, therefore, par-
ticipating in workfare — is given by u(b, kr).⁵ Individuals are endowed with an ordered
triple of characteristics (m, n, k). The joint distribution of these three characteristics is
denoted F (m, n, k) and the associated density, f (m, n, k). Apart from continuity, we
make no assumptions on the joint distribution function. Thus, the model admits an ar-
bitrary correlation structure among individual characteristics. For ease of notation only,
we assume that the support of the distribution is a cube [m, m] × [n, n] × [k, k]. Individ-
uals that are indifferent between market work and public welfare benefits are described
by the equation

u(c(n), m) = u(b, kr)                                  (9)

For a fixed skill level n and tax-benefit system, (9) determines an upward-sloping locus
in (k, m)-space. We parameterize this locus by m = φ(k, r). We suppress the dependence
of this locus on the parameters of the tax-transfer system c(n) and b, but, given our focus
on workfare, make explicit the dependence of this locus on r.⁶ Individuals above and to
the left of this locus remain out of the labor market, because these people either find
market work relatively more costly in terms of utility or find workfare relatively less
onerous. The mass of workers of skill type n that choose market work is given by⁷

Ï [ ^ψ(,) f (m,n,k)dmdk.
k k m∏

An increase in r shifts the locus m = φ(k,r) upward, leading to increased participation
in market work. To see this, applying the Implicit Function Theorem to (9) yields

⁵The model of Section 2 corresponds to the special case of k = 1 for all individuals.

⁶By the Implicit Function Theorem, ^d÷ = rul(^b,rk) > 0.

∂k     ul (c(n),m)

⁷We are implicitly assuming that the locus φ(k, r) intersects the line k = k above m = m, so that for
each skill type there is some sufficiently low value of m that induces market work. If this is not the case,
we can reformulate our analysis by parameterizing the locus as k = η(m, r) and reversing the order of
integration in (10) and all subsequent integrals. Our results do not change.

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