The desirability of workfare in the presence of misreporting

Abstract

In this paper we demonstrate that in addition to its acknowledged screening role, workfare—namely, introducing work (or training) requirements for welfare eligibility in means-tested programs—also serves to mitigate income misreporting by welfare claimants. It achieves this goal by effectively increasing the marginal cost of earning extra income in the shadow economy for claimants who satisfy the work requirement. We show that when misreporting is sufficiently prevalent, supplementing a means-tested transfer system with work requirements is socially desirable.

Appendix: Proofs

Proof that constraint ( 4 ) is non-binding in the optimal solution

Let $\hat{u} = \overline {V}$. By virtue of (3), it follows that $\underline {c}- h(\underline {y}/\underline {w}+\varDelta )\geq \overline {V}$. Thus, $\underline {c}- h(\underline {y}/\overline {w}+\varDelta )\geq \overline {V}$; hence, by virtue of (5), $\overline {c}- h(\overline {y}/\overline {w})> \overline {V}$. It follows that the constraint in (4) is satisfied as a strict inequality. By continuity considerations, the result extends to values of $\hat{u}$ sufficiently close to $\overline {V}$. This completes the proof.

Proof that constraint ( 5 ) is binding in the optimal solution

Consider first the benchmark case in the absence of misreporting; namely, when $\overline {\alpha}\to \infty$. In this case we can ignore constraint (7), as the set of individuals who misreport is of zero measure. Suppose by way of contradiction that constraint (5) is not binding. Thus, as constraint (4) is also non-binding (as shown above), then by continuity considerations, one can slightly reduce the level of $\overline {c}$ without violating any of the constraints (4) and (5). Notice that by reducing the level of $\overline {c}$ none of the other two constraints ((3) and (6)) is violated either. This slight modification will economize on government expenditure and attain the desired contradiction to the presumed optimality. Consider next the case where a non-zero measure of individuals is misreporting. A reduction in $\overline {c}$ would have two effects on government expenditure, a mechanical effect and a behavioral one. As in the previous case with no misreporting, a reduction in $\overline {c}$ would lower the level of government expenditure. However, as can be observed from condition (7), the number of high-ability individuals who misreport would then adjust in equilibrium. In particular, α ₀ will increase (that is the number of individuals who misreport will increase). This will result in a corresponding increase in government expenditure, which may all in all increase overall government expenditure. To see why an increase in α ₀ will increase the government expenditure (other things equal), note that in the optimal solution it is necessarily the case, that $\underline {c} - \underline {y} \geq \overline {c} - \overline {y}$. If it were not the case, one could replace the presumably optimal program with a universal system that would offer all agents a lump-sum transfer equal to $\underline {c} - \underline {y}$, which would trivially satisfy all constraints and reduce total government expenditure. Then it follows from the objective in (2) that when the system is indeed means-tested (that is, $\underline {c} - \underline {y} > \overline {c} - \overline {y}$), an increase in α ₀ does increase total government expenditure.

The overall impact on government expenditure of the combined mechanical and behavioral effect is generally ambiguous. However, our result in the case of no misreporting extends by continuity consideration to the case where the level of misreporting is sufficiently small, that is, $\overline {\alpha}$ is sufficiently large. In this case the mechanical effect would prevail.

We will show below that a sufficient condition for the incentive constraint in (5) to be binding in the optimal solution for the government problem is the following:

$\overline {\alpha} \geq 2 \cdot [[\overline {y}- h(\overline {y}/\overline {w})] -[\underline {y}- h (\underline {y}/\overline {w})]]$, where $\overline {y}$ and $\underline {y}$ denote the levels of income chosen by a high-skill and low-skill individuals, respectively in the absence of a transfer program.

Signing the optimal marginal tax rates

Assuming no workfare in place (Δ=0) we turn to verify the standard properties of the (implicit) marginal tax rates embedded in the welfare system. Substituting for the term $(1- \alpha_{0}/\overline {\alpha})$ from (9) into (10) and re-arranging yields

$$ h' (\overline {y}/\overline {w}) =\overline {w}.$$

(15)

Thus, we obtain the standard ‘efficiency at the top’ result.

Substituting for the term $\alpha_{0}/\overline {\alpha}$ from (11) into (12) and re-arranging yields

$$ h' (\underline {y}/\underline {w})/\underline {w} = 1- \mu /\lambda \cdot \bigl[1- h' (\underline {y}/\overline {w})/\overline {w}\bigr].$$

(16)

By virtue of the single crossing property and the fact that the incentive constraint of high-skill individual (constraint (5)) is binding, $\overline {y}>\underline {y}$. Hence by virtue of (15) and the convexity of h, $h' (\underline {y}/ \overline {w})/\overline {w}< 1$. It follows that $h' (\underline {y}/ \underline {w})/\underline {w}\leq 1$ (with strict inequality when μ>0). Thus, the (implicit) marginal tax rate levied on the low-skill individual is strictly positive.

Proof of the proposition

The construction of the proof will be in three stages.

Stage I We first turn to simplify the expression in (14), which is reproduced for convenience:

$$ \partial L/ \partial \varDelta |_{\varDelta =0} = \lambda \cdot h'(\underline {y}/\underline {w})- \mu \cdot h'(\underline {y}/\overline {w}) - \eta \cdot \partial \overline{\overline{V}} / \partial \varDelta |_{\varDelta =0}.$$

(17)

Substituting for the term (μ−η) from (9) into (11) and re-arranging yields

$$ \lambda =2.$$

(18)

By the definition of $\overline{\overline{V}}(0)$ and by virtue of (15) it follows that $\overline{\overline{V}}(0) = \overline {y}-h(\overline {y}/\overline {w})$. Substituting into (7) and re-arranging then yields

$$ \alpha_0 = (\underline {c} - \underline {y})- (\overline {c} - \overline {y}).$$

(19)

Substituting into (13) yields

$$ \eta = -\alpha_0 /\overline {\alpha}.$$

(20)

Employing (18) and (20) to simplify (11) yields

$$ \mu = 1- 2 \alpha_0 / \overline {\alpha}.$$

(21)

Differentiating $\overline {\overline {V}}$ with respect to Δ, employing the envelope theorem, yields

$$ \partial \overline {\overline {V}} /\partial \varDelta |_{\varDelta = 0} = -\overline {w}.$$

(22)

Substituting into the expression in (14) yields

$$ \partial L/ \partial \varDelta |_{\varDelta = 0} = \lambda \cdot h' (\underline {y}/\underline {w})-\mu \cdot h' (\underline {y}/\overline {w}) + \eta \cdot \overline {w}.$$

(23)

Finally, by employing conditions (12), (18), (20) and (21), following some algebraic manipulations, one can obtain the following simplified form of the expression in (23):

$$ \partial L/ \partial \varDelta |_{\varDelta = 0} = \overline {w}\cdot (1 + \alpha_0 / \overline {\alpha})-2 \cdot (\overline {w}/\underline {w}-1) \cdot h' (\underline {y}/\underline {w}).$$

(24)

Stage II We next derive two useful properties of the optimal system that will be employed in what follows. In order to prove these properties we make the additional technical assumption that h‴≥0. Notice that when h takes an iso-elastic functional form, the assumption implies that the (constant) elasticity of labor supply is bounded above by unity, which is consistent with existing empirical evidence (see, e.g., Salanie 2003).

Lemma

(i) $\partial (\alpha_{0} /\overline {\alpha})/ \partial \overline {\alpha}<0$, (ii) $\partial \underline {y}/ \partial \overline {\alpha}<0$.

Proof

(i) Substituting for λ, η and μ from (18), (20) and (21) into (12) and re-arranging yields the following simplified expression:

$$ h'(\underline {y}/\underline {w})/\underline {w} = 1 + \frac{(1-2\alpha_0/\overline {\alpha})\cdot [h' (\underline {y}/\overline {w})/\overline {w}-h'(\underline {y}/\underline {w})/\underline {w}]}{(1+ 2\alpha_0/\overline {\alpha})}.$$

(25)

Fully differentiating the expression in (25) with respect to $\overline {\alpha}$ and re-arranging yields

(26)

Now suppose by way of contradiction that $\partial (\alpha_{0}/\overline {\alpha})/\partial \overline {\alpha}\geq 0$. We will examine separately the case where $\partial (\alpha_{0}/\overline {\alpha})/\partial \overline {\alpha}> 0$ and the one in which $\partial (\alpha_{0}/\overline {\alpha})/\partial \overline {\alpha}= 0$. Consider first the case where $\partial (\alpha_{0}/\overline {\alpha})/\partial \overline {\alpha}> 0$. As h is convex, h‴≥0, by assumption, and $\alpha_{0}/\overline {\alpha} \leq 1/2$, by our earlier derivations (21), it follows that $\partial \underline {y}/ \partial \overline {\alpha}>0$, otherwise, it is straightforward to verify that the right-hand side of the expression in (26) is positively signed, whereas the left-hand side is negatively signed. Now consider the illustrative figure below, which depicts the optimal solution for the government program in the net income-gross income (c,y) space. We denote by U(w,c,y)≡c−h(y/w), the utility derived by an individual of type w with gross income y and net income c. Note first that by the convexity of h, the single crossing property holds and in particular, the indifference curve of the low-ability type is steeper than that of the high-ability type (the slope of the indifference curve is given by MRS(w)=h′(y/w)/w). By virtue of our earlier derivations, conditions (3) and (5) are binding in the optimal solution. Fixing the initial level of $\overline {\alpha}$, the equilibrium is given by the two bundles depicted as triangles in the figure. Now consider a downward shift in $\overline {\alpha}$; namely $\overline {\alpha}' < \overline {\alpha}$. By virtue of our presumption, $\underline {y}(\overline {\alpha}') < \underline {y}(\overline {\alpha})$ and $\alpha_{0}'/ \overline {\alpha}' < \alpha_{0}/\overline {\alpha}$, hence, $\alpha_{0}' < \alpha_{0}$. By virtue of (15) the gross income level chosen by the high-ability type remains unchanged (at the efficient level). The new equilibrium is then given by the two bundles depicted as squares in the Fig. 1.

Fig. 1

The optimal solution for the government problem

By virtue of our earlier derivations, $h' (\underline {y}, \underline {w})/\underline {w}\geq 1$, thus the slope of the indifference curve of the low-ability type in the initial equilibrium (the triangle lying on the steeper indifference curve) is (weakly) lower than unity. Thus, it follows that $\underline {c}(\overline {\alpha}')- \underline {y}(\overline {\alpha}')\geq \underline {c}(\overline {\alpha}) -\underline {y}(\overline {\alpha})$. As can be straightforwardly observed from the figure, $\overline {c}(\overline {\alpha}')- \overline {y}(\overline {\alpha}')< \overline {c}(\overline {\alpha}) -\overline {y}(\overline {\alpha})$. We thus conclude

$$\bigl[\underline {c}(\overline {\alpha}')- \underline {y}(\overline {\alpha}')\bigr]-\bigl[ \overline {c}(\overline {\alpha}') - \overline {y}(\overline {\alpha}')\bigr]>\bigl[\underline {c}(\overline {\alpha}) - \underline {y}(\overline {\alpha})\bigr]-\bigl[ \overline {c}(\overline {\alpha}) - \overline {y}(\overline {\alpha})\bigr].$$

However, by virtue of (19) the last inequality implies that $\alpha_{0}' > \alpha_{0}$. We thus obtain the desired contradiction.

We turn next to the other case, where $\partial (\alpha_{0}/\overline {\alpha})/ \partial \overline {\alpha} = 0$. As h is convex, h‴≥0, by assumption, and $\alpha_{0}/\overline {\alpha}\leq 1/2$, by our earlier derivations (21), it follows that $\partial \underline {y}/\partial \overline {\alpha}= 0$, by virtue of (26), by a similar reasoning to the case examined above. Now consider again a downward shift in $\overline {\alpha}$; namely $\overline {\alpha}'< \overline {\alpha}$. By virtue of our presumption, $\underline {y}(\overline {\alpha}') = \underline {y}(\overline {\alpha})$ and $\alpha_{0}'/\overline {\alpha}'= \alpha_{0}/\overline {\alpha}$, hence, $\alpha_{0}'< \alpha_{0}$. In this case the triangles and the corresponding squares depicted in the figure coincide; that is the two equilibria associated with $\overline {\alpha}'$ and $\overline {\alpha}$ are identical. It then follows that

$$\bigl[\underline {c}(\overline {\alpha}') - \underline {y}(\overline {\alpha}')\bigr]- \bigl[\overline {c}(\overline {\alpha}') - \overline {y}(\overline {\alpha}')\bigr]=\bigl[\underline {c}(\overline {\alpha}) - \underline {y}(\overline {\alpha})\bigr]- \bigl[\overline {c}(\overline {\alpha}) - \overline {y}(\overline {\alpha})\bigr].$$

By virtue of (19) the last equality implies that $\alpha_{0}' = \alpha_{0}$. We thus obtain the desired contradiction, as our presumption implies that $\alpha_{0}' < \alpha_{0}$.

(ii) This part follows immediately from the expression in (26) and part (i). □

Stage III Our final step would be to provide sufficient conditions for the expression in (24) to be negative, that is, for imposing workfare to be socially desirable as it economizes on government costs.

By virtue of (21) and as the incentive constraint of the high-skill individual is binding, it follows that $\alpha_{0}/ \overline {\alpha} \leq 1/2$ (ensuring that the Lagrange multiplier associated with the high-skill incentive constraint is non-negative). Employing this inequality condition, it follows that a sufficient condition for the expression in (24) to be negative is

$$ 3/2 \cdot \overline {w} -2 (\overline {w}/ \underline {w}-1)\cdot h' (\underline {y}/\underline {w}) <0.$$

(27)

By part (i) of the lemma, the term $\alpha_{0}/\overline {\alpha}$ is decreasing with respect to $\overline {\alpha}$. Suppose that $\overline {\alpha}$ is sufficiently small such that the term $\alpha_{0}/\overline {\alpha}$ attains its upper-bound; namely, $\alpha_{0}/\overline {\alpha} =1/2$ (μ=0). For later purposes denote the level of $\overline {\alpha}$ for which $\alpha_{0}/\overline {\alpha} =1/2$ by $\overline {\overline {\alpha}}$. As, in this case, the multiplier associated with the high-ability type’s incentive compatibility constraint is equal to zero, it follows from (25) that $h' (\underline {y}/\underline {w}) = \underline {w}$. Substituting into (27) then yields

$$ 3/2 \cdot \overline {w} - 2 \cdot (\overline {w} - \underline {w})< 0\quad \Longleftrightarrow\quad \overline {w}/\underline {w}>4.$$

(28)

It follows that within the range where the incentive constraint of the high-skill individual is binding (namely, $\overline {\alpha} \geq \overline {\overline {\alpha}}$, implying indeed by virtue of part (i) of the lemma that $\alpha_{0} /\overline {\alpha} \leq 1/2$, hence, μ≥0), by continuity considerations, for sufficiently small values of $\overline {\alpha}$ and sufficiently high wage (skill) ratios, the expression in (27) is negative; hence, the expression in (24) is negative.

Taking the other limiting case, by letting $\overline {\alpha}\to \infty$, that is, assuming no misreporting, it follows from (18), (20) and (21) that μ=1 and η=0 (and evidently, λ=2). By the convexity of h, $h' (\underline {y}/\underline {w}) > h' (\underline {y}/\overline {w})$. It thus follows from (23) that ∂L/∂Δ| _Δ=0>0. That is, imposing a workfare requirement in the case of no misreporting is undesirable. We thus replicate the result obtained by Besley and Coate (1995). We conclude that the expression in (24), which is a simplified form of (23), is positive. Hence, as $\alpha_{0}/\overline {\alpha}\leq 1/2$, it follows that in the limiting case where $\overline {\alpha}\to \infty$, hence, by continuity considerations, for sufficiently large values of $\overline {\alpha}$:

$$ 3/2 \cdot \overline {w} -2 (\overline {w}/\underline {w}-1) \cdot h' (\underline {y}/\underline {w}) >0.$$

(29)

Combining the two (strict) inequality conditions given in (27) and (29), and provided that $\overline {w}/\underline {w} > 4$, by continuity, it follows by virtue of the intermediate-value theorem that there exists some value of $\overline {\alpha}$, denoted by $\overline {\alpha}_{0}$, for which:

$$ 3/2 \cdot \overline {w} - 2 (\overline {w}/\underline {w}-1) \cdot h' (\underline {y}/\underline {w}) =0.$$

(30)

By virtue of part (ii) of the lemma, the expression on the left-hand-side of (30) is strictly increasing with respect to $\overline {\alpha}$, hence, $\overline {\alpha}_{0}$ is uniquely defined. Moreover, it follows by part (ii) of the lemma that when the moral costs entailed by misreporting are sufficiently small $(\overline {\overline {\alpha}} \leq \overline {\alpha} < \overline {\alpha}_{0})$, and when the difference between the skill levels is large enough $(\overline {w}/\underline {w}> w_{0} = 4)$, the sufficient condition in (27) holds; hence, imposing workfare is socially desirable as it economizes on government expenditure. This completes the proof.

A sufficient condition for the incentive constraint in ( 5 ) to bind

In the proof of the proposition it is assumed that the incentive constraint in (5) is binding. We turn next to establish a sufficient condition for the incentive constraint in (5) to be binding. We then turn to verify that the range of parameters for which workfare is a desirable supplement to means testing (characterized in stage III of the proof of the proposition) is indeed well defined, by verifying that the incentive constraint of the high-skill individual is binding.

Claim

The following condition suffices for the incentive constraint in (5) to be binding:

$$ \overline {\alpha} \geq 2 \cdot \bigl[ \bigl[\overline {y}-h (\overline {y}/\overline {w})\bigr]- \bigl[\underline {y}-h (\underline {y}/\overline {w})\bigr] \bigr],$$

(31)

where $\overline {y}$ and $\underline {y}$ denote the levels of income chosen by a high-skill and low-skill individuals, respectively, in the absence of a transfer program.

Proof

Suppose that there is no workfare requirement in place (as in the statement of the proposition) and suppose further, by way of contradiction, that the condition in (31) holds but the incentive constraint in (5) does not bind. Hence, the Lagrange multiplier associated with the incentive constraint in (5) is equal to zero.

Repeating the steps in Stage I of the proof of the Proposition (see above for details) it follows by virtue of (21) that

$$ \mu = 1 -2 \alpha_0 / \overline {\alpha}.$$

(32)

Hence,

$$ \mu = 0 \quad \Longleftrightarrow\quad \alpha_0/ \overline {\alpha} = 1/2.$$

(33)

By virtue of (15) and (16) it follows that

(34)

(35)

Hence, $\overline {y}$ denote $\underline {y}$ the levels of income chosen by high-skill and low-skill individuals, respectively, in the absence of a transfer program.

By virtue of (19) it follows that

$$ \alpha_0 = (\underline {c} -\underline {y}) - (\overline {c} -\overline {y}).$$

(36)

By virtue of the (non-binding, by presumption) incentive constraint in (5) it follows that

$$ \overline {c} - h(\overline {y}/ \overline {w}) > \underline {c} - h \bigl[(\underline {y}/\overline {w})\bigr].$$

(37)

It hence follows that

$$ h \bigl[(\underline {y}/\overline {w})\bigr] - h(\overline {y}/ \overline {w})>\underline {c}-\overline {c}.$$

(38)

Substituting into (36), employing (33) and re-arranging yields

$$ \overline {\alpha} < 2 \cdot \bigl[ \bigl[\overline {y}-h (\overline {y}/\overline {w})\bigr] -\bigl[\underline {y} - h (\underline {y}/ \overline {w})\bigr] \bigr].$$

(39)

As (39) violates (31), we obtain the desired contradiction. This completes the proof. □

Finally, we turn to show that the range of parameters for which workfare is a desirable supplement to means testing is indeed well defined. Recall that in Stage III of the proof of the proposition we have established that for values of $\overline {\alpha}$ which lie within the range $\overline {\overline {\alpha}} \leq \overline {\alpha} \leq \overline {\alpha}_{0}$ and for wage ratios that satisfy the condition, $\overline {w}/\underline {w}>4$, imposing workfare does economize on government costs and is hence socially desirable. Recall further that $\overline {\overline {\alpha}}$ was defined as the level of $\overline {{\alpha}}$ for which the incentive constraint of the high-skill individual is binding and μ=0 (hence, $\alpha_{0}/\overline {\alpha} =1/2$). When the incentive constraint of the high-skill individual is binding, it follows that

$$ h\bigl[(\underline {y}/\overline {w})\bigr] - h(\overline {y}/\overline {w}) = \underline {c}- \overline {c}.$$

(40)

Substituting for the term $\underline {c}- \overline {c}$ from (40) into (36) yields:

$$ \alpha_0 = \bigl[\overline {y}- h(\overline {y}/\overline {w})\bigr]- \bigl[\underline {y}- h(\underline {y}/\overline {w})\bigr].$$

(41)

As $\alpha_{0} / \overline {\overline {\alpha}} = 1/2$, it follows that:

$$ \overline {\overline {\alpha}} = 2 \cdot \bigl[ \bigl[ \overline {y}- h(\overline {y}/\overline {w})\bigr]- \bigl[ \underline {y}- h(\underline {y}/\overline {w})\bigr] \bigr].$$

(42)

Thus, we have established that for the range of parameters for which introducing workfare is socially desirable the incentive constraint of the high-skill individual (given in condition (5)) is indeed binding.