Privately optimal severance pay

A.1 Proofs

Proof of Proposition 1. From the worker’s sequence problem we have the Euler equation for each t≥0. The CARA felicity function satisfies U′(c)=–αU(c), and it follows that For an unemployed worker equation (1) can then be written as and stationarity allows us to drop the time subscript. Thus we have rW^u(a)=U(c^u(a)), and similarly for an employed worker rW^e(σ, a)=U(c^e(σ, a)). Differentiating these equations with respect to a and using the first-order conditions and for the consumption choices, we obtain The dynamic budget identity then ensures that s^u(a) and s^e(σ, a) are independent of a and so equations (14)–(15) hold.

From equations (16)–(17) we see that consumption depends on wealth only through the additively separable term ra. Together with the CARA felicity function this implies that W^e(σ, a)–W^u(a)=e^–αra[W^e(σ, 0)–W^u(0)] for any 〈σ, a〉. Hence maximization of the Nash objective function Φ(σ, a)=e^–αraγΦ(σ, 0) is unaffected by changes in wealth, and so σ*(a) is independent of a.

Finally, substituting for the value functions in equations (8) and (13), using the first-order conditions for consumption optima, and rearranging terms yields

Equations (18)–(19) then follow from the CARA functional form and replacing for consumption using (16)–(17).                       □

Proof of Proposition 2. In text.                      □

Proof of Lemma 1. We have c^e(σ, a)≥c^u(a+F) if and only if U′(c^e(σ, a))≤U′(c^u(a+F)), which by equation (19) is equivalent to s^e(σ)≥0. The equivalence of both

and W^e(σ, a)≥W^u(a+F) then follows from equations (14)–(15).         □

Lemma 2 We have

Furthermore,

and

are each separately equivalent to W^e(σ, a)≥W^u(a+F).

Proof. From equation (7) we have and using these relationships to differentiate (5) leads to (31)–(32). Moreover, differentiating equation (8) with respect to w yields

Since we have that W^e(σ, a)≥W^u(a+F) is equivalent to equation (33), and the equivalence of (34) is established by a similar argument employing .   □

Lemma 3 We have

if and only if w–b+s^u–rF≥0.

Proof. Equations (31)–(32) yield

Moreover, equations (14)–(15) and the CARA specification for U imply that

Thus and and equations (33), (34), and (37) can be combined to show that

is equivalent to W^e(σ, a)≥W^u(a+F). The result then follows from Lemma 1.   □

Lemma 4 Given J^°, the unique Pareto-optimal contract σ satisfying J^e(1, σ)=J^° is such that w–b+s^u–rF=0. Moreover, the Pareto frontier in payoff space is strictly decreasing, strictly concave, and differentiable.

Proof. Pareto optimality of σ ensures that and from Lemma 3 it follows that w–b+s^u–rF=0. Since the locus of contracts satisfying J^e(1, σ)=J^° is downward-sloping in 〈w, F〉-space, these two equations characterize the unique Pareto optimum for fixed J^°. Setting w=b–s^u+rF in equation (7), we obtain

Hence is independent of F for contracts yielding payoff vectors on the Pareto frontier, and so from equation (5) we have dJ^e(1, b–s^u+rF, F)/dF=–1. In view of Lemma 1 and equation (14), we have also that W^e(b–s^u+rF, F, a)=W^u(a+F) is differentiable, strictly increasing, and strictly concave in F; and the desired properties of the Pareto frontier follow as a consequence.   □

Proof of Proposition 3. Since the agreed contract σ* maximizes the objective function in equation (9), the associated payoff vector 〈W^e(σ*, a), J^e(1, σ*)〉 is on the Pareto frontier. Moreover, this frontier is strictly concave by Lemma 4 and the Nash maximand is strictly quasi-concave in the payoffs, so the solution is uniquely determined. Finally, Lemma 4 ensures that w*–b+s^u–rF*=0.   □

Proof of Proposition 4. We have w*–b+s^u–rF*=0 by Proposition 3, and so F*≥F is equivalent to s^u≥–p(θ*)[w*–b]/[p(θ*)+r]. Moreover s^e(σ*)=0 by Lemma 1, and so equation (18) takes the form

Since the LHS and the RHS of this equation are strictly increasing and decreasing in s^u, respectively, it suffices to show that the LHS is no larger than the RHS for the value s^u=–p(θ*)[w*–b]/[p(θ*)+r]. This amounts to the inequality

which is easily seen to hold strictly for w*>b and as an equality for w*=b. It follows that when γ>0 and the income loss w*–b from unemployment is strictly positive, we have F*>F. Alternatively, when γ=0 and w*–b=0 we have F*=F=0.   □

Proof of Proposition 5. Since γ=0 we have W^e(σ*, a)=W^u(a) and thus w*=b+s^e(σ*) by equations (14)–(15) and Proposition 2. Furthermore, since F*=F_m>0 we have that c^u(a+F*)>c^u(a)=c^e(σ*, a), that U′(c^u(a+F*))<U′(c^e(σ*, a)), and that s^e(σ*)<0 by equation (19). Hence w*<b.   □

Proof of Proposition 6. Since γ=0 and F*=F_m>0 we have w*<b by Proposition 5, and moreover s^u=0 by Proposition 2. Hence w*–b+s^u–rF*<0, and thus by Lemma 3 we have This establishes that the contract σ* is inefficient, and nevertheless the worker obtains the same payoff gross of lump-sum taxes as under (efficient) laissez-faire. It follows that the firm’s payoff J^e(1, σ*) is lower; equation (20) then implies that q(θ*) is higher; and since the matching technology M is strictly increasing we have that job creation p(θ*) is lower. Furthermore, writing σ for the laissez-faire equilibrium contract, we have

by equation (7). This yields and so job destruction is lower.   □

A.2 Renegotiation of the severance payment

After a productivity shock (i.e., for y<1) the agreed contract may no longer be efficient. In this section we consider the possibility of renegotiation to capture additional surplus. For simplicity we model only the option to adjust the severance payment, since this is what is important for determining the impact of a government mandate.²⁹

Renegotiation after a shock proceeds as follows. The worker employed under contract σ=〈w, F〉 makes a new severance offer F′ to the firm. The firm can accept this offer and pay F′, reject it and propose that employment continue under the contract σ, or dismiss the worker outright and pay F. Continuation of employment requires the consent of the worker, who remains free to quit and receive no payment. Importantly, the renegotiated quantity F′ is paid on the spot and unconstrained by the government-mandated minimum F_m on the contractual amount.³⁰

Note first that if W^e(σ, a)<W^u(a), then a worker who is not dismissed will choose to quit. Monotonicity of U guarantees that W^e will be strictly increasing in all arguments, and it follows that for each relevant 〈F, a〉 there exists a unique reservation wage w(F, a) that satisfies W^e(w(F, a), F, a)=W^u(a). If w is below w(F, a) then the worker will quit and the continuation values are W^s(y′, σ, a)=W^u(a) for the worker and 0 for the firm.

Renegotiation is relevant when the worker will not quit and the firm will not dismiss. (The condition for dismissal remains as in Section 3.1.) Since the worker makes a take-it-or-leave-it offer, the amount proposed will be the firm’s reservation payoff F′=–J^e(y′, σ), and the firm will be left indifferent between accepting the offer and continuing the match. Moreover, the productivity threshold below which renegotiation occurs is determined by the relation

When y′ is above this threshold the match survives and the continuation payoffs remain 〈W^e(σ, a), J^e(y′, σ〉〉, as in the model without renegotiation. Alternatively, ifthen the two parties separate with a renegotiated transfer and the continuation payoffs are 〈W^u(a–J^e(y′, σ)), J^e(y′, σ)〉. For fixed 〈F, a〉, Figure 3 plots the renegotiation threshold in 〈w, y′〉-space together with the thresholds w(F, a) for quitting and for dismissal.

Figure 3

Post-shock outcomes allowing for renegotiation. Given 〈F, a〉, the consequences of a productivity shock are shown in 〈w, y′〉-space. The worker quits (case Q) if the wage is below w(F, a). The firm dismisses the worker (case D) if productivity is below Employment continues (case C) if productivity exceeds both and And the two parties separate with a renegotiated severance payment (case R) if productivity is between and In each region the first component of the payoff vector accrues to the worker and the second to the firm.

When W^e(σ, a)≥W^u(a) and so the worker does not quit, we can now express the job destruction threshold and the severance payment after renegotiation as

With these definitions, equation (3) can be written as

Allowing for renegotiation requires the following modifications to Proposition 1. First, in part A we show also that w, and y^d are independent of wealth. Second, in part C we replace equation (19) by

And third, we show in addition that the job-destruction threshold is given by

(This implies that separation is consensual for the marginal job loser – the job destruction threshold is above the firing threshold – as long as workers strictly prefer being fired to continuing employment under the existing contract.) Moreover, equation (22) must be replaced by

Renegotiation does not affect Lemma 1 or Propositions 2–4, which continue to hold as stated. In Proposition 5 we can show also that establishing that in the binding-constraint equilibrium the severance payment is renegotiated with positive probability. The optimality of separation, together with the functional form for the value function, also implies that the consumption of the marginal job loser is not higher than it would be in case of continuation.

As for Proposition 6, when renegotiation is allowed this result is replaced by the following.

Proposition 7 If γ=0 and F*=F_m>0, then unemployment u* and job destruction λG(y^d(σ*)) are higher while job creation p(θ*) and workers’ welfare W^e(σ*, a)=W^u(a) are lower than under laissez-faire.

Figure 4 is the counterpart of Figure 2 above for the case of renegotiation. Note that, in line with our theoretical result, binding mandates increase rather than decrease unemployment in the presence of renegotiation.

Figure 4

Effects of severance-pay mandates for a range of benefit replacement rates, with optimal, ex-post renegotiation of the severance payment.

A.3 Data and data sources

Table 3 contains the data used to construct the country-specific upper bounds on the excess of mandated over privately-optimal severance pay used in Sections 5–6. These bounds, reported in the last column of the table, amount to the maximum (over white and blue collar workers) difference between the legislated payment f_m and the bound on the privately-optimal payment in equation (25).

The monthly exit rates from unemployment p(θ) are from the OECD unemployment duration database. The benefit replacement rates ρ come from Nickell (1997), except for the Italian rate which has been updated using Office of Policy (2002). The interest rate is set at 4% annually.

Legislated dismissal costs are constructed as the maximum over blue-collar and white-collar workers of the sum of notice-period and severance pay. The latter quantities are obtained by applying the formulas for legislated payments to the average completed job tenures (ACJT) in the third column of the table (from the data set in Nickell et al. 2002) averaged over each country’s sample period. The formulas for European countries come from Grubb and Wells (1993); with the exception of those for Austria, Finland, Norway, and Sweden, which are derived from IRS (Industrial Relations Service) (1989). The size of the legislated severance payment for Italy includes damages workers are entitled to if their dismissal is deemed unfair (5 months) plus the amount they receive if they give up their right to reinstatement (15 months). Our value is consistent with the estimates in Ichino (1996).³¹ The data for Portugal and New Zealand come from European Foundation (2002) and CCH New Zealand Ltd (2002), respectively. The data for legislation in Australia, Canada, and the US are from Bertola et al. (1999).

A.4 Positive utility from leisure

As a further robustness exercise, this section presents an alternative calibration of the model which allows for a positive utility from leisure and perfect substitutability of consumption and leisure.³² The felicity function becomes

where leisure l equals zero for an employed and one for an unemployed worker.

Calibration now requires an additional moment to pin down the extra parameter ψ. Following Mortensen and Pissarides (1999), we set the cost of posting a vacancy m so that the average cost per hired worker m/q(θ)=0.33. This corresponds to the average cost of recruiting and hiring a worker in the US, according to the survey results in Hamermesh (1993).³³ Matching the Portuguese average unemployment duration and unemployment rate requires setting ψ=0.316 and λ=0.143. All other parameters are the same as in the main calibration.

Table 4 is the counterpart of Table 2 for the case of positive utility from leisure, and reports equilibrium quantities for different levels of severance pay in the calibrated economy. The third column corresponds to the benchmark calibration, and the first column to its laissez-faire counterpart. Note that the optimal severance payment in the latter equals 1 month, rather then 6 months as in the original calibration. Because of the perfect substitutability of leisure and consumption and the calibrated value of the rate of substitution ψ, the utility from leisure compensates job losers for a substantial fraction of their loss of income.

For comparison, the second and fourth columns of Table 4 report equilibrium quantities for mandated severance pay that exceeds the laissez-faire quantity by 11 and 18 months, respectively (the same differences as in columns two and three of Table 2). It is clear that the allocational and welfare implication of these deviations are both small and very similar under the two calibrations. However, a mandate of 24 months, shown in the fifth column, generates sizeable efficiency and welfare losses in the new calibration.

Figure 5 is the counterpart of Figure 2 for the case of positive utility from leisure. Here, for different values of the benefit replacement rate, we plot the optimal severance payment and the employment, efficiency, and welfare changes associated with a mandate 11 months higher than the private optimum. In this case an equilibrium with positive employment does not exist for ρ exceeding 0.65.

Figure 5

Effects of severance-pay mandates for a range of benefit replacement rates, with utility from leisure.

A.5 Calibration choices

In this section we discuss choices involved in the calibration of the model’s parameters. The two moments used in the calibration are the value of the exit rate from unemployment (or equivalently its inverse, the unemployment duration) and the value of the job destruction rate (or equivalently the unemployment rate, given the flow equilibrium condition in equation (22)). Let p* and s* denote, respectively, the target values for the unemployment exit and job destruction rates.

Consider first calibration of the parameters that determine the job destruction rate, assuming that some other parameters can be adjusted to keep the unemployment duration at its target level. That is to say, consider how the targeted job destruction rate identifies the shock arrival rate and the distribution of shocks via the moment condition

Given the assumption of a uniform distribution of shocks with upper support normalized to 1, this condition involves two parameters: the arrival rate λ and the lower support y_l of the distribution. Blanchard and Portugal (2000) choose y_l to match the coefficient of variation for output in Portugal while we set y_l=0. In either case, λ is calibrated to satisfy the moment condition. Note that, since the uniform distribution has a flat density, the decomposition of the separation rate between the arrival rate λ and the conditional probability of separation is of little relevance provided the lower bound of the support is not binding (i.e., ) in equilibrium. Mortensen and coauthors normally set λ to some arbitrary value (typically 0.1 for the US) and then choose y_l to satisfy the moment condition in equation (53). That is to say, in this sort of model λ and y_l are not separately identified without an additional “normalization.” Normalizing y_l to zero seems the least objectionable choice.

We turn now to the calibration of the parameters that affect the job creation margin, assuming that (for example) λ is adjusted to keep the job destruction rate at its target value s*. Note that given the matching function M(u, v)=A(uv)^0.5 we have

which implies that the equilibrium vacancy filling cost equals q*=A²/p* in the calibrated economy. Substituting for q* in the job creation equation (21) then yields

Given the equilibrium value of J^e(1, σ*) implied by the targeted moments, it follows that there is an infinite set of 〈m, A〉-pairs consistent with the targeted p*. In other words, m and A are also not separately identified without an additional normalization. The main calibration normalizes A to one and calibrates m to satisfy equation (55).³⁴

The calibration reported in Appendix A.4 introduces an extra parameter, the marginal utility of leisure ψ, leaving us short one moment condition. In the absence of a comparable number for Portugal we have again followed Mortensen and set m such that the average cost of posting a vacancy m/q(θ), on the left hand side of equation (55), equals its estimate of 0.33 for the US. We then choose ψ so that J^e(1, σ*) satisfies the moment condition in equation (55).

The effect of mandates is small under all of these parameterizations, which we take as evidence that this result is driven by the economics of the model and not any particular parameter choice.