## Tito Boeri, Agar Brugiavini, and Lars Calmfors

Print publication date: 2001

Print ISBN-13: 9780199246588

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0199246580.001.0001

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# (p.254) Appendix Unions and Benefits: A Simple Analytical Framework

Source:
The Role of Unions in the Twenty-first Century
Publisher:
Oxford University Press

# A1. A Model of Trade Unions and Long‐Term Benefits

This Appendix presents a formal analysis of the theoretical arguments developed in the non‐technical discussion of the main text of Part II.

We start with a monopoly trade union with ‘right to manage’. There are two periods, 1 and 2. In period 1 the trade union sets optimally, given its objective function, the level of wages and the long‐term benefits to be paid in period 2. The firm then decides its optimal level of employment in order to maximize its intertemporal profit. At the end of period 1 production takes place. In period 2 only a fraction of previously employed workers are still attached to the firm and receive therefore their promised long‐term benefits as chosen by the union in the first period.1

More precisely, consider a union with a total pool of M identical workers. Each worker cares about consumption over the two periods and has the following preferences:

$Display mathematics$
where u i(·) is the intratemporal utility function of period i with standard properties, and is increasing concave with u i(0) = 0; c 1 and c 2 are consumption levels in periods 1 and 2; and δ is the discount rate.

Following Askildsen and Ireland (1997), we assume that there is some basic uncertainty concerning the future attachment of the worker to the firm in period 2. This may reflect the existence of a stochastic element affecting the worker's likelihood to stay in the firm in the next period.2 Given such uncertainty, a worker will be concerned about the expected utility of (p.255) consumption Eu 2(c 2) in period 2. In period 1 each worker who is employed receives a current wage rate w. This same worker gets the promise of a future occupational pension s paid out in period 2. This specification therefore captures in a simple way the idea that workers may be more risk‐averse about uncertain benefits paid in the long term than about current wages paid today when u 2(·) is more concave than u 1(·).

An unemployed worker in period 1 receives an intertemporal utility payoff of V R. Besides occupational benefits, we assume that there is a first‐tier public benefit system through which, independently from the employment status in period 1, an agent receives a transfer T in the second period of his life. This public system is financed in the economy by a payroll or income tax τ. Finally, in period 1 an unemployed worker receives an unemployment benefit b. Formally, we may write down the discounted expected utility of a union member when employed and when unemployed as:

$Display mathematics$
and
$Display mathematics$
where q is the exogenous probability of attachment of the worker to the firm in the second period.

We take the standard view that, given a pool of workers M, the union wants to maximize the expected utility of the representative member, which can be written as

$Display mathematics$
where L is the number of employed workers in period 1.

We consider that the firm sells a product which generates a revenue function in period 1, R(L) increasing concave with R(0) = 0. Hence, given that only a fraction q of workers will remain attached to the firm in period 2, the firm's discounted expected profits can be written as

$Display mathematics$
with 1/(1 + r) the discount factor. Finally, let πR be the reservation level of profits of the firm in order to be part of the relationship with the union.

In the ‘right to manage’ model, the firm takes as given the current wage rate w and the long‐term benefit s and chooses its employment level in order to maximize profits. This provides a demand function for labour L D(W) (p.256) decreasing in W = w + [1/(1 + r)]sq, the total expected discounted cost to hire a worker.

## A1.1. The Determinants of the Trade‐Off Between Current Wages and Deferred Benefits

The problem of the monopoly trade union is then written as

$Display mathematics$

This problem is easily decomposed into two steps. First, we may solve the optimal mix between current wages w and deferred payments s for a typical employed worker, given a fixed expected discounted cost of hiring W for the firm. Second, we may determine the trade union's optimal expected discounted labour cost.

### Optimal Trade‐Off Between Current Wages and Deferred Benefits

The first step can be written as the following problem:

$Display mathematics$
which, after a change of variable w′ = w(1 − τ) and s′ = s + T, is rewritten as
$Display mathematics$
The optimal wage and long‐term benefits are given by the standard marginal condition
$Display mathematics$
(1)
and the budget constraint. From this we find the net wage w′ = w(1 − τ) = w * (q, W, τ, T), total benefits s′ = s + T = s * (q, W, τ, T), and the indirect utility function of an employed worker U(w *, s *) = U *(q, W, τ, T).

The solution of this problem is illustrated in Figure A1. In the plane (w′, s′) of after‐tax current wages, w′ + w(1 − τ) and total long‐term benefits s′ = s + T, (p.257)

Figure A1

iso‐utility curves u(w′, s′) = u 1(w′) + δ qu 2(s′) = cst are drawn with the usual decreasing convex shape. The ‘budget line’ BB under which the employed worker's expected utility is maximized is given by the expression [1/(1 − τ)]w′ + [1/(1 + r)]sq = W + [1/(1 + r)]qT. It goes through point H(W(1 − τ), T) with a slope [(1 − τ)/(1 + r)]q. The optimal allocation (w*, s*) of after‐tax current wages and total long‐term benefits is given at point A, where the budget line BB is tangent to the optimal iso‐utility curve U * = U(w *, s *) = U *(q, W τ, T).

An increase in q induces a clockwise rotation around point B of the budget line (see Figure A2). At the same time, it makes the iso‐utility curves of the worker steeper. In general, the rotation of the budget line has three effects on (w*, s*). First, there are the traditional substitution and income effects coming from the fact that a change in the probability q can be interpreted as a change in the ‘relative price’ [(1 − τ)/(1 + r)]q of occupational benefits s′. Second, there is an additional ‘wealth’ effect emanating from the term [1/(1 + r)]qT in the budget constraint BB. The substitution effect tends to increase w* and to reduce s*. Under standard ‘normality’ assumptions on current and future consumption, the income effect tends to reduce both w* and s*. The wealth effect, on the contrary, induces an increase in w* and s*. The fact that the utility curves become steeper as a result of an increase in q, on the other hand, induces a decrease in w* and an increase in s* as future consumption is more valued by the worker.

It is easy to see that the substitution effect and steeper utility slope effects cancel out for total benefits and that the rotation around point H of the budget line leads along the locus determined by equation (1) to a decrease in the net (p.258) wage rate w′ and total benefit s′ (see Figure A2). Obviously, the effect is similar for gross wages w and occupational benefits s.

Similarly, an increase in the discounted expected labour cost W or an increase in public benefits T shifts out the budget line and implies an increase in both net and gross wages and total benefits. However, it is easy to see that

$Display mathematics$
Hence occupational benefits s = s′ − T decrease with public transfers T. There is some crowding out of public benefits on occupational benefits.

Finally, an increase in the payroll tax τ has the usual substitution and income effects with a resulting decrease in the net wage rate w′ and a total ambiguous effect on benefits s′ and s.

### The Quasi‐Linear Case

An interesting tractable case is the quasi‐linear case, where u 1(c 1) = c 1 and u 2(c 2) = u(c 2). It is easy to get

$Display mathematics$
and
$Display mathematics$

Figure A2

(p.259) and the indirect utility function of an employed worker U(w *, s *) = U *(q, W, τ, T) is
$Display mathematics$
In the simple quasi‐linear specification, as income and wealth effects fall entirely on w*, there is no net effect of q or W on total benefits s*.

### Employment and Labour Costs

The second step of the union problem can be written as

$Display mathematics$
which collapses to the fairly standard problem of the monopoly trade union. Given that there is a participation constraint of the firm, the total labour cost solution is given by W = W * = Min(W̃(q, τ, T, b), W̃(πR)) where W̃(q, τ, T, b) is the interior solution of the first‐order condition
$Display mathematics$
(2)
and W̃(πR) is the labour cost level which makes the firm's participation constraint binding:
$Display mathematics$
From this, it is straightforward to recover wages, benefits, and employment:
$Display mathematics$

The solution is depicted in Figure A3 in the space (W, L) of total expected labour cost W and employment level L. The iso‐utility curves of the trade union are depicted by the curve V while the labour demand L = L D(W) is the locus of vertical tangents of the iso‐profit curves π and π* of the firm. Given that there is a participation constraint of the firm, the labour cost solution is given by W = W * = Min(W̃(q, τ, T, b), W̃(πR)), where W̃(q, τ, T, b) is associated with the ‘monopoly trade union equilibrium’ point E, at which the optimal (p.260)

Figure A3

iso‐utility curve of the union is tangent to the labour demand curve, and W̄(πR) is the labour cost level at which the firm's participation constraint is binding.

### Comparative Statics

It is interesting to see how, in this simple framework, wages, occupational pensions and employment change with the probability of attachment to the firm (or concern for long‐term benefits) q, unemployment benefits b, reservation profit level πR, public pensions T, and payroll tax rates τ.

An increase in unemployment benefits b implies, in a standard fashion, an increase in the equilibrium intertemporal labour cost W * (at least, so long as the participation constraint of the firm is not binding). This, in turn, is associated with an outward shift of equilibrium wages w and a reduction in employment L. Similarly, a positive change in the firm's reservation profit level πR implies a decrease in W * (when the firm's participation constraint is binding) with a reduction in w and an increase in employment L.

The impact of a change in q is less straightforward (see Figure A4). Getting back to the definition of the indirect intertemporal utility U *(q, W, τ, T) of an employed worker and the intertemporal payoff V R(b, q, T) of an unemployed worker, it can be seen that an increase in q will induce flatter iso‐utility curves for the trade union. A higher probability of future attachment to the firm increases the worker's value to be employed today by the firm, reducing therefore the union's incentive to choose a high total labour cost W. Under reasonable concavity assumptions, this implies that the new tangency point at E′ between the labour demand curve and the union's iso‐utility is in the (p.261)

Figure A4

north‐west direction of the initial equilibrium point E. This is associated with a lower expected cost of labour W̃(q, τ, T, b) and a higher employment level. Given that the direct impact of q on w is negative, this implies that the total impact of an increase in q is to reduce current wages.

Formally, differentiation of the indirect intertemporal utility of an employed worker and an unemployed worker provides

$Display mathematics$
and
$Display mathematics$
as workers are risk‐averse (i.e. u2(s) < 0). Therefore, when the union problem is well behaved (i.e. ∂2 V/∂ W 2 < 0), differentiation of the first‐order condition provides
$Display mathematics$
(p.262) with
$Display mathematics$
Hence when the condition
$Display mathematics$
is satisfied, W̃(q, τ, T, b) is decreasing in q and the trade union solution W * is decreasing in q. This is obviously satisfied for the quasi‐linear case.

The equilibrium employment level is then increasing in q. Given that the direct impact of q on w is negative, this implies that the total impact of an increase in q is to reduce current wages.

Similarly, it can be seen that an increase in public pensions T will induce steeper trade union iso‐utility curves (see Figure A5). The intuition is the fact that, as public transfers are not conditional on the employment status, they tend to reduce the relative value for a worker to be employed in period 1. This, in turn, increases the union's incentive to choose a high total labour cost W with an associated lower employment level. Hence W̃(q, τ, T, b) is increasing in T, implying that the equilibrium intertemporal labour cost W * is also weakly increasing in T.

Figure A5

(p.263) Formally, the effect of an increase in public pensions T on labour costs W will have the sign of ∂2 V/∂ WT:

$Display mathematics$
But
$Display mathematics$
and
$Display mathematics$
Hence W̃(q, τ, T, b) is increasing in T when
$Display mathematics$
which is again satisfied for the quasi‐linear case. This implies that the equilibrium intertemporal labour cost W * is also weakly increasing in T. Gross and net current wages will consequently also increase with T.

In the general case (when preferences are not quasi‐linear), total pensions s′ are likely to increase. Occupational pensions are partially crowded out by public pensions through the direct effect of T. At the same time, there is an additional income effect coming from the fact that the equilibrium inter‐temporal labour cost W * does increase. The whole effect of public pensions on occupational pensions is therefore on a priori grounds ambiguous. When the impact of public pensions on wages and labour costs is strong enough (resp. weak enough), the second effect is likely to dominate (resp. be dominated) and there is a complementarity (substitutability) between public pensions and occupational pensions.

Finally, let us close this section by investigating the effect of an increase in the payroll tax rate τ. As is straigtforward to see, the indirect utility of an employed worker U *(q, W, τ, T) is obviously negatively related to the payroll tax rate. This implies the standard result (Alesina and Perotti, 1997) that the union passes to the employer some fraction of the fiscal burden and that the union‐determined discounted labour cost W̃(q, τ, T, b) is increasing in τ. Thus, an increase in the payroll tax, everything else being equal, has a negative impact on employment L. The effect on current wages is generally ambiguous. The direct impact of an increase in τ is negative but there is an additional positive income effect coming from the increase in W. For occupational pensions, the positive (p.264) substitution effect and the induced income effect through the increase in W imply that, in general, they are increasing in the payroll tax τ.

# A2. Bargaining

The simple model of the previous section can be extended along several dimensions. First, one may generalize the analysis to situations where the union bargains with the firm on wages and benefits, while the firm keeps its ‘right to manage’ on employment decisions. Figure A6 reflects such a situation in the plane (W, L) of discounted expected labour cost and employment. The downward‐sloping demand curve of the firm is represented by L D(W) while typical iso‐profit curves π* and πR and trade union iso‐utility curves V * are also shown. A bargaining game on wages and benefits with ‘right to manage’ will essentially pick a point on the labour demand curve between the reservation level curve of the firm πRπR and the reservation discounted labour cost of the union W R corresponding to its reservation value V R. Once such a solution (W B, L B) is found, it is easy to recover wages and benefits as w B = w *(q, W B) and s B = s *(q, W B). It should be clear by then that all the previous comparative‐static results on b, πR, q τ, T are qualitatively the same as in the monopoly model of the trade union.

In a similar fashion, one can consider the situation of full bargaining between the firm and the union on the three dimensions of current wages, occupational pensions, and employment (w, s, L). This too is represented in Figure A6. The main difference between this and the previous case is simply that now the

Figure A6

(p.265) optimal point has to be picked up on the ‘contract’ curve CC of efficient allocations (W, L) between the union and the firm with an outcome implying more employment L N, lower expected discounted labour costs W N, and lower wages and benefits than in the ‘right to manage’ case. Again, one gets the same qualitative comparative statics as in the case of the monopoly union model.

# A3. Political Economy Considerations Within the Union

So far we have considered a framework in which all workers are identical. Obviously, one of the important features of non‐wage and benefit policies for unions is the fact that they do not affect all workers the same way. In this section we introduce some heterogeneity across workers' preferences with respect to long‐term benefits and employment. Various dimensions may be captured by such differentiation: age, seniority, or degree of attachment to the firm. As long as unions can be viewed as groups taking collective decisions along more or less democratic rules, this introduces a number of interesting political economy issues within the union. What are the points of conflict or convergence between young and old workers inside the union? What kind of preferences of the various workers are represented in the objective function of the union? What is the effect of workers' heterogeneity on wages, pensions, employment, membership? To these questions we now turn.

## A3.1. Introducing Young and Old Workers

We start first by amending our preceding two‐period framework and allowing for the fact that there are young and old workers in the union. Young workers may work in the first period (period 1) of their life and retire in the second period (period 2) (if still alive). Old workers, in fixed number L 0, are retirees and live only in period 1. They do not work and receive benefits s 0 in that period. At the beginning of period 1, the ‘monopoly’ union chooses the current wage rate of young workers w y, their future occupational benefits s y to be paid in period 2. Given this, the firm chooses the employment level L y. In period 2, benefits s y are paid to the workers who were previously employed.

We will first assume that the structure of political representation within the union is such that the union's objective function will be the expected discounted utility of young workers in period 1, given some veto power of the old workers to receive what they were promised to receive, namely their benefits s 0.

Given this, and noting W y = w y + [1/(1 + r)]s y q, the discounted labour cost of a young worker in period 1, the problem of the union can be written as

$Display mathematics$

(p.266)

Figure A7

With our assumptions, the only difference between this and the previous section is the fact that the firm now has to generate a large enough surplus to be sure that benefits to old workers can be paid in period 1. Hence the minimum surplus that has to be left over for the firm and old workers is πR + s 0 L 0. Obviously, the solution of this problem is simply (with the same notation as before) $W y = W y * = Min ( W ~ ( q , τ , T , b ) , W ~ ( π R + s 0 L 0 ) )$.

This is represented in Figure A7. The first quadrant represents the, by now, usual monopoly union equilibrium in terms of the young worker employment level L y and the discounted labour cost W y. The iso‐utility curve of the young worker EV y is tangent to the demand function at the point W̃(q, τ, T, b). The second quadrant plots the labour cost W̄(πR + s 0 L 0) such that the participation constraint of the firm is binding, as a function of the size L 0 of old workers in the union. This is a downward‐sloping relationship, as a larger current surplus and a lower associated labour cost W is necessary to accommodate higher expenditures of the firm on occupational pensions of old workers. The equilibrium‐discounted labour cost of young workers $W y *$ is then also depicted by a thicker line. When the size of old workers L 0 increases, the minimum ‘reservation surplus’ to be satisfied, πR + s 0 L 0, moves up the labour demand curve L D(W y) from point E 0 to some point E′. It follows immediately from this that, as long as the surplus constraint is not binding, the equilibrium labour cost of young workers $W y *$ is first fixed at the level W̃(q, τ, T, b); then, after some threshold L̄0, the surplus constraint binds, and the equilibrium labour cost is driven by W̄(πR + s 0 L 0) and is consequently decreasing with L 0.

(p.267) From this it is easy to see that, after the threshold level L̄0, employment of young workers L y is positively related to the number of old workers in the union. The intuition is simple. A larger volume of occupational pensions paid to old workers induces the union to moderate its demand on labour costs W for the next generation of workers in order to generate a high enough firm's surplus. This, in turn, is associated with a larger fraction of employed workers in period 1.

Nevertheless, it should be noted that such a situation is not in the interest of the young workers. As can be seen in Figure A7, when L 0 becomes larger than L̄0, the equilibrium expected discounted utility $EV y T$ of a young worker is decreasing with the number of old workers in the union. Obviously, this aspect has interesting implications for the dynamics of membership inside the union. Consider for instance that trade union membership M is now endogenous and that young workers will enter the union when their expected discounted utility in the union is larger than some reservation level V̄. The preceding discussion implies that young workers will be less likely to enter the union, the larger is the number of retired old workers.

## A3.2. Voting on Benefits and Wages

In the preceding framework, the decision‐making process within the union on current wages and occupational pensions was quite simple. The objective function of the union represented the preferences of the young workers under the constraint of veto power of the old retired union members. Following Askildsen and Ireland (1997), we assume now that workers inside the union are continuously differentiated according to their relative attachment to the firm or, more broadly, their relative evaluation between current wages and long‐term benefits, and that they vote to decide the union position on wages, benefits, and employment.

Indeed, suppose that the parameter q is distributed in the union along a distribution, f(·) with mean q̄ and median q m. We abstract, for simplicity, from taxes and denote public benefits to the previously unemployed retirees as T b. Also, we restrict ourselves to the case of quasi‐linear preferences:

$Display mathematics$
The expected utility of a worker with attachment q can then be written as: (L/M) U(w, s, q) + (1 − L/M)V R(b, q). The firms's average profit in such a
$Display mathematics$
implying an average discounted labour cost for each hired worker of W = w + [1/(1 + r)]sq.

(p.268) As we now have some degree of heterogeneity across workers, we need to model the political mechanism by which collective decisions will be taken within the union. Let us consider first the case where wages and benefits are decided by simple majority voting. A typical technical problem in terms of the determination of the political equilibrium is the fact that voting has to be on two dimensions (w, s). In order to alleviate this issue, assume further that there is sequential voting. First, union members vote on the discounted labour cost W faced by firms (or, equivalently, on the employment level L = L D(W) in the ‘right to manage’ specification) and then they vote on how to allocate this cost W between current wages and future benefits. We solve the game, as usual, by backward induction.

Let us look therefore at the second stage of this collective decision mechanism. For a given L = L D(W) and W = w + [1/(1 + r)]sq̄, the typical problem of a worker of type q is now written as

$Display mathematics$

The solution of this problem is illustrated in Figure A8 at the tangency point of the iso‐utility curve UU q of a typical worker and the budget line BBq̄, providing the optimal current wage w = w *(q/q̄, q̄, W) and benefits s = s *(q/q̄) and the indirect utility function of an employed worker U(w *, s *, q) = U *(q, q̄, W) (see Section A5). An increase in q induces steeper indifferences curves for the worker without changing the budget line. This in

Figure A8

(p.269) turn leads to a lower current wage w * and a higher longer‐term benefit s * at the optimal tangency point. Obviously, the indirect utility of an employed worker of type q is increasing in q. An increase in the average value q̄ induces a clockwise rotation of the budget line around point W. Because of conflicting substitution and income effects, the impact of such a change on current wages w is ambiguous. In the quasi‐linear specification, an increase in q̄ has a negative substitution effect only on benefits s. The indirect utility of an employed worker of type q is clearly decreasing in q̄. As preferences are single‐peaked, the voting equilibrium will be the preferred allocation of the median w m = w *(q m/q̄, q̄, W), and s m = s *(q m/q̄). The indirect utility level of an employed worker at this political equilibrium outcome can be written as U *(q, q̄, W, q m) = w m + δ qu(s m).

Getting back to the first stage of the voting game, the preferred labour cost value W of a worker of type q will be the solution of the following problem:

$Display mathematics$
Assuming that the participation constraint of the firm is always non‐binding, we easily get the solution for the optimal labour cost W̃(q, q̄, q m) at the tangency point E m of the iso‐utility of the union member and the labour demand curve L D(W) (Figure A9). An increase in q induces flatter iso‐utility curves of the trade union member as long as public benefits after unemployment T b are

Figure A9

(p.270) less than occupational benefits s. Indeed, in such a situation the relative value of being employed for a worker increases with his type q. Hence he is more ready to trade off a lower expected labour cost W for a higher probability of being employed. Consequently the cost of labour W̃(q, q̄, q m) and the employment level preferred by a union member of type q are, respectively, decreasing and increasing in q.

From this and the single‐peakedness of the expected utility function of a typical union member in W, it follows that the majority voting equilibrium in the union in the first stage will again be the one chosen by the median voter of the union q m. The equilibrium labour cost is W m = W̃(q m, q̄, q m), out of which we deduce the employment level L m = L D(W m), the current wage w m = w *(q m/q̄, q̄, W m), and benefits s m = s *(q m/q̄).

Out of this analysis, one may investigate how changes, within the union, in the distribution of concerns for benefits F(q) affect the equilibrium values of current wages, benefits, and employment. Consider first that on average unions' members are more attached to the firm (i.e. q̄ increases). As the value of being employed decreases with q̄, the slope of the iso‐utility curves of the median member in Figure A9 becomes steeper, leading to a higher expected cost of labour W m picked up by the union and lower employment L m. Holding q m constant, the impact on benefits s = s m is unambiguously negative, for occupational benefits are now, ceteris paribus, more costly to the firm. The effect on current wages w m is a priori ambiguous but is likely to be positive if the income effect through W m is large enough.

Consider now an increase in the pivotal agent q m. This may be interpreted as due to the fact that the distribution F(·) is more skewed towards concerns for benefits or that people more attached to the firm get more political leverage within the union. In that case, the pivotal agent in the union obviously wants higher long‐term benefits s m. At the same time, as in the present framework occupational benefits are attached to employment, the pivotal agent also cares more about employment. Consequently, his preferred level of labour costs W m is reduced, current wages w m are reduced, and employment L m is increased.

## A3.3. Insiders, Last‐In–First‐Out Rules, and Soft Landing Plans

So far, an important aspect of the analysis is the fact that members of the union all face the same probability of being unemployed. In reality, this does not hold. Some workers within the union may typically enjoy insider positions within the firm and therefore face very little risk of being dismissed under bad external conditions. Also, very often firms and unions subscribe to seniority rules and last‐in–first‐out (LIFO) conventions. These mechanisms discriminate between young workers and more senior workers in terms of their risks of being made unemployed and their access to occupational benefits. Another dimension of (p.271) implicit employment discrimination across workers is the use of ‘soft landing’ plans, such as early retirement plans, long‐term unemployment insurance, and disability plans, to mitigate unemployment problems. As these plans are applied mainly to senior and older workers, they induce a differential outside payoff according to age or length of service should the firm have to dismiss a worker. This in turn affects differentially the preferred labour package that the worker would wish to vote for in the union.

One can capture the above features in our framework in the following way. First, consider that, for a given employment cost W, the probability of a worker of type q being employed is simply a function Φ (q, W), such that

$Display mathematics$
and satisfying the following conditions:
$Display mathematics$
Condition (i) simply states that the probability of being employed decreases with the expected labour cost of the firm. Condition (ii) captures the idea of seniority rules and last‐in–first‐out (LIFO) conventions in the sense that longer‐serving workers (or senior workers) have a higher probability of keeping their jobs than more mobile or younger workers. Finally condition (iii) states that the sensitivity to labour costs of being employed decreases (in absolute value) with the length of attachment of the worker to the firm. This captures in a certain way the idea that longer serving or older workers are more likely to be insiders; therefore their employment status is less sensitive to wages than that of junior (mobile) workers. Obviously, the specification includes the special case of a uniform probability of employment L D(W)/M when Φ (q, W) is independent of q.

The ‘soft landing’ idea applied to senior workers can be captured by simply supposing that the intertemporal payoff of an unemployed worker V R is designed to be a fraction of what an employed worker of type q would receive in equilibrium V *(q) (i.e., V R = k(q)V *(q) where the fraction k(q) is increasing in q). The cost difference between V R and what the unemployed worker would receive without a ‘soft landing’ is ‘externalized’, i.e. financed outside the firm by general taxation on the rest of the economy.

Given this setting, one may look once more at the voting equilibrium on wages, benefits, and employment inside the union. Again, one can solve the problem in two stages. The second stage provides, as before, for a given labour cost W, the optimal mix preferred by the pivotal agent q m between current wages w m = w *(q m/q̄,q̄,W) and benefits s m = s *(q m/q̄). In the first stage, the determination of the preferred labour cost W for a worker of type q is given by (p.272) the solution of the amended following maximization problem:

$Display mathematics$

Assuming again that the participation constraint of the firm never binds, we easily get the labour cost W̃(q, q̄, q m) of a worker of type q.

Interestingly now, this labour cost W̃(q, q̄, q m) need not be decreasing in q. On the contrary, because of seniority rules, more attached workers (with a higher q) have a higher probability of being employed and therefore value an increase in W more than without such rules. Since their probability of employment is less sensitive to W, they are also less concerned about the impact of an increase in the labour cost W on their change of employment status. Finally, because of the possibility of ‘soft landing’ plans, their reservation payoff level in the contingency of unemployment is increased, inducing them to demand higher total labour compensations.

For all these reasons, it is very likely that workers more attached to the firm (larger q) will have higher preferred costs of labour W̃(q, q̄, q m). Under single‐peakedness of the expected utility function of a union member in W, the majority voting equilibrium will be again the one chosen by the median voter of the union q m and the equilibrium discounted labour cost will be W m = W̃(q m, q̄, q m) associated with an employment level L m = L D(W m), current wage w m = w *(q m/q̄, q̄, W m) and benefits s m = s *(q m/q̄).

The main difference between this and the previous section is in terms of the impact of a change in the distribution of characteristics of union workers. Consider for instance an increase in the pivotal agent q m. As the pivotal agent in the union becomes more attached to the firm, he also gets more protection against the risk of unemployment through the seniority rules, FILO conventions, and ‘soft landing’ plans. Hence his preferred labour cost W m = W̃(q m, q̄, q m) increases and the employment level L m = L D(W m) decreases. Current wages w m and benefits are now both increased. The cost of such a strategy is obviously paid by junior workers, who have a disproportionately high probability of unemployment.

From the previous discussion, it follows that the more likely seniority rules, LIFO conventions, and ‘soft landing’ plans are to be implemented in the economy, the more likely will ageing and shifts in the distribution of characteristics of unions towards senior workers imply higher labour costs, higher unemployment, and larger occupational benefits. Going one step further, we might also expect the seniority rules, FILO conventions, and ‘soft landing’ plans themselves to be partly endogenous and to be influenced by the unions' activities. As these mechanisms are generally protecting the old or long‐serving (p.273) workers rather than the young workers, one may suspect that the same shift in the distribution towards seniority will make these rules more likely to be implemented, thereby reinforcing the conditions under which one will obtain high labour costs, high unemployment (especially among the young or less attached workers), and greater occupational benefits.

# A4. Endogenous Membership

## A4.1. Endogenous Membership and Political Economy Considerations Within Unions

Another interesting issue concerns the membership evolution of the union. So far, this has been fixed to a given size, M. Suppose now that it is endogenous. Workers are indexed by the characteristic q distributed uniformly on [0, 1] and decide to join the union, comparing their expected payoff inside the union with the reservation payoff V 0 which they might get in some other non‐unionized sector of the economy with V 0 > V R(0). Clearly, in order to decide whether or not to join the union, each worker has to anticipate the political equilibrium (W m, L m, w m, s m) within the union. This obviously depends on the position of the pivotal agent within the union, which in turn is determined by the type of individuals who decide to join the union (Booth, 1984). Hence an equilibrium with endogenous membership should determine jointly the political equilibrium decided within the union, the size of the union, and the nature of the pivotal agent within the union.

More precisely, an individual of type q may expect to get a payoff

$Display mathematics$
by joining the union. For given q̄, and q m, this payoff is increasing in q. Hence, ceteris paribus, if an individual of type q decides to join the union, all individuals of type q′ > q will also join the union. Define therefore by q min the lowest level such that all workers with a characteristic q > q min decide to join the union. Hence, for given q̄, W m, q m, q min is determined by
$Display mathematics$

When seniority rules, LIFO conventions, and ‘soft landing’ plans are applied within the union, it is quite likely that q min is increasing in q̄ and q m. The more, on average, the union is constituted of workers with a high attachment or seniority characteristic q (i.e. a high q̄) and reflects the political interest of senior workers (i.e. has a high q m), the less interesting it is for a mobile or young worker with a low q to join that union and the higher will be the threshold level q min above which workers will decide to be members of the (p.274)

Figure A10

union. (See Section A5 for a formal derivation.) At the same time, given that q is uniformly distributed on [0, 1], we have
$Display mathematics$

The resulting union equilibrium with endogenous membership is then described in Figure A10 at the intersection of the curve QQ, which describes the positive relationship between q min and q m, and the line MM, which describes the relationship q m = (1 + q min)/2. As is plotted on the diagram, the two curves may intersect more than once, implying that there may be multiple equilibria in union membership, wages, and benefits. The intuition for such a possibility is quite simple. When the union is composed mainly of workers with a low q, the political equilibrium within the union is more likely to reflect the preferences of these workers with high employment, relatively low wages, and low benefits. Anticipating this, more workers of these types will join the union in the first place. At the same time, however, one may have another equilibrium, at which the union is composed mainly of senior workers with high values of q and protected from unemployment by seniority and LIFO rules. In this case the political pivotal worker within the union is more likely to reflect the preferences of these workers with high wages and benefits, at the cost of a high probability of unemployment for workers with a low q. This, in turn, obviously discourages union membership of the latter and supports a structure of union membership biased towards senior workers.

# (p.275) A5. Proofs

## A5.1. Political Equilibrium Inside the Union

Consider first the second voting stage of this collective decision mechanism. For a given L = L D(W) and W = w + [1/(1 + r)]sq̄, the typical problem of a worker of type q is now written as

$Display mathematics$
The optimal wage and long‐term benefit are given by the standard marginal condition
$Display mathematics$
(3)
and the budget constraint. From this we find the current wage
$Display mathematics$
and benefits
$Display mathematics$
and the indirect utility function of an employed worker U(w *, s *, q) = U *(q, q̄, W). An increase in q induces a downward shift in the first‐order condition (3). This in turn leads to a lower current wage w * and a higher longer‐term benefit s *. Obviously, the indirect utility of an employed worker of type q is increasing in q. An increase in the average value q̄ has an ambiguous effect on the current wage and a negative impact on benefits s. The indirect utility of an employed worker of type q is then decreasing in q̄.

As preferences are single‐peaked, the voting equilibrium will be the preferred allocation of the median, namely

$Display mathematics$
and
$Display mathematics$
(p.276) The indirect utility level of a employed worker at this political equilibrium outcome can be written as
$Display mathematics$
Getting back to the first stage of the voting game, the preferred labour cost value W of a worker of type q will be the solution of the following problem:
$Display mathematics$
Assuming that the participation constraint of the firm never binds, we get the solution for the optimal labour cost W̃(q, q̄, q m) for a worker of type q by the following first‐order condition:
$Display mathematics$
(4)

Simple differentiation of this condition gives that W̃(q, q̄, q m) is decreasing in q as long as public benefits after unemployment T b are less than occupational benefits s. It is increasing in q̄ and increasing (resp. decreasing) in q m when qq m (resp. qq m). From this and the single‐peakedness of the expected utility function of a typical union member in W, the majority voting equilibrium in the union in the first stage is again the one chosen by the median voter of the union q m, and the equilibrium discounted labour cost is W m = W̃(q m, q̄, q m), from which we deduce the employment level L m = L D(W m), and then the current wage

$Display mathematics$
and benefits
$Display mathematics$

## A5.2. Endogenous Membership

q min is determined by:

$Display mathematics$
(p.277) Differentiation provides
$Display mathematics$
At the same time,
$Display mathematics$
Also, as q min < q m and W̃(q min, q̄, q m) < W̃(q m, q̄, q m),
$Display mathematics$
From this and the fact that under seniority rules W m = W̃(q m, q̃, q m) is increasing in q m and q̄, we have
$Display mathematics$

## Notes:

(1) Hence benefits are of a defined benefits (DB) type rather than a defined contribution (DC) type.

(2) Alternatively, if we consider in this two‐period setting that workers work in period 1 and retire in period 2, the second‐period uncertainty may capture the uncertain longevity of the retired worker.