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This paper studies the behaviour of a labour union in a model with employment and membership dynamics. It is known from the literature that the relative level of employment between intertemporal and static union models is affected by the structure of the union density function. We show that this result does not hold in general but rather that it depends on the union objective function. If the union maximises the wage bill of their members, employment in the intertemporal model will differ from employment in the static model depending on the union density function. However, if the union maximises the rent from unionisation, the intertemporal model will yield the same level of employment as the static model regardless of the union density.

Collective bargaining and labour unions that represent the interests of different workers are important institutional factors in many labour markets. Although union membership has declined over the last several years, the proportion of workers covered by collective agreements measured in terms of the bargaining coverage is about 50% in the OECD [

The economic aspects of unions have been broadly discussed in the literature [

Considering the evolution of union membership over time, [

A more plausible union density function is proposed by [

Examining dynamic models of union wage-setting is an important step to better understand what unions do. The studies mentioned above, however, do not focus on the role of the union objective function. Our paper aims to fill this gap. We analyse the impact of the particular forms of the objective function (wage bill maximizing or rent maximizing) on employment and membership dynamics and on the steady state results. We show that the static model and the intertemporal model yield the same level of employment when unions maximise the rent collected from unionisation. If unions maximise their members’ wage bill, however, employment in the intertemporal model may differ from the static model depending on the functional form of the union density. Our results indicate that a precise definition of the empirically relevant union objective function is needed.

This paper proceeds as follows. Section 2 sets up the intertemporal labour market framework. Section 3 includes an analysis of employment and membership dynamics. The steady state results are discussed in Section 4. Finally, Section 5 offers conclusions.

The firm’s production function is given by f ( n t ) , with f ′ ( n t ) > 0 and f ″ ( n t ) < 0 , where n t denotes employment in period t. The firm’s profit is Π t = p f ( n t ) − w t n t , where p is the price of the output good and w t is the worker’s nominal wage. The firm’s labour demand results from the marginal productivity condition and is implicitly given by

f ′ ( n t ) = w t p . (1)

We assume that the wage is set by a monopoly union. The union chooses the wage that maximises its objective function. Given this wage, the firm then chooses the level of employment according to its labour demand (1). The objective function of the monopoly union is given by

U t = n t ( w t − b t ) + ρ m t b t , (2)

where m t denotes the number of union members, b t is the unemployment benefit, and ρ ∈ [ 0,1 ] is a measure of the weight of unemployed members. Thus, Equation (2) is a general objective function which captures the specific functional forms discussed above. When ρ = 0 , this indicates that the union maximises the difference between the wage and the unemployment benefit, i.e., the rent from unionisation. When ρ = 1 , the union maximises the total wage bill of its members.

This formulation reflects the seminal discussion of the appropriate union objective function. On the one hand, [

Regarding the union membership dynamics, we assume that unemployed members leave the union [

m ˙ = σ ( w ) n − m , (3)

where σ ( w ) , with σ ′ ( w ) > 0 and σ ″ ( w ) < 0 , is the proportion of employed workers who join the union.

We now examine the dynamic aspects of employment and membership determination. Our starting point is the intertemporal formulation provided by [

max n ∫ 0 ∞ { n ( p f ′ ( n ) − b ) + ρ m b } e − r t d t s . t . m ˙ = σ [ p f ′ ( n ) ] n − m , (4)

where r denotes the rate of time preference. The current-value Hamiltonian appropriate to the optimization problem (4) can be expressed by

H = n [ p f ′ ( n ) − b ] + ρ m b + λ [ σ ( p f ′ ( n ) ) n − m ] (5)

where λ is the co-state variable, m is the state variable, and n is the control variable. The first-order conditions are

H ′ n = β ( n ) − b + λ α ( n ) = 0 (6)

H ′ m = ρ b − λ = − λ ˙ + r λ , (7)

with β ( n ) = p f ′ ( n ) + n p f ″ ( n ) , β ′ ( n ) < 0 , and α ( n ) = σ ( p f ′ ( n ) ) + σ ′ ( p f ′ ( n ) ) p f ″ ( n ) n . From (6) we obtain

λ = − β ( n ) + b α ( n ) . (8)

Differentiating (8) with respect to t gives

λ ˙ = n ˙ [ − β ′ ( n ) α ( n ) − ( − β ( n ) + b ) α ′ ( n ) ] α ( n ) 2 . (9)

Finally, substituting equations (8) and (9) into (7) yields the following equation, which implicitly defines the time path for employment:

n ˙ [ − β ′ ( n ) α ( n ) − ( − β ( n ) + b ) α ′ ( n ) ] α ( n ) = ( 1 + r ) [ − β ( n ) + b ] − ρ b α ( n ) (10)

The employment path (10) and the evolution of union membership (3) describe the solution path for the union’s intertemporal maximisation problem. In the following discussion, however, we focus only on employment and membership in the steady state, where n ˙ = m ˙ = 0 hold. From (3) and (10), we derive the following steady state conditions:

β ( n * ) = b − ρ α ( n * ) 1 + r b (11)

m * = σ ( p f ′ ( n * ) ) n * . (12)

We first compare steady state employment given by Equation (11) with the result of the static model where employment n s is given by

β ( n s ) = b . (13)

The structure of the union objective function clearly does not affect the static result. That is, Equation (13) is independent of ρ . Employment—and thus, the union-set wage—is the same under a wage-bill-maximising union as it is under a rent-maximising union. This result might explain why both objective functions are treated very similarly in most union models.

However, this independency result does not hold if we model union behaviour in an intertemporal framework. If the union maximises the wage bill ( ρ = 1 ), the results derived for employment and membership in the steady state confirm the findings of [

For ρ = 1 , our result is also in line with the findings of [

If the union maximises the rent from unionisation, however, we have ρ = 0 . In this case, the steady-state employment in Equation (11) reduces to β ( n * ) = b , which is the same result that is derived from Equation (13) for the static model. That is, neither the time preference rate nor the union density affect employment in the steady state.

This result has some interesting implications. If the union’s objective is characterised by maximizing the rent from unionisation, then membership dynamics do not affect optimal employment. Unlike in static models, the two objective functions should not be treated similarly because they lead to different results.

For cases where 0 < ρ < 1 , Equation (11) indicates that the comparison between static and dynamic employment is driven by three effects. Two of these are conflicting effects that are discussed above and which stem from α ( n * ) ≷ 0 . The third effect that determines whether employment in the dynamic model is greater than in the static model pertains to the weighting factor 0 < ρ < 1 .

We studied the behaviour of a labour union in a model with employment and membership dynamics. Our results suggest that the structure of the union objective function plays a decisive role in an intertemporal union model. Contrary to a static model, in which maximising the wage bill leads to the same level of employment as maximising the wage surplus, the union objective function affects the level of employment in an intertemporal framework with membership dynamics. If the union maximises the wage surplus, the intertemporal model yields the same result as the static model. If the union maximises the wage bill, however, employment in the intertemporal model differs from that of the static model. In this case, the question whether the intertemporal model overstates or understates the distortions caused by unions depends on the form of the union density function.

The primary conclusion of this research is that the form of the union objective function plays a more important role than that implied by conventional static models. This result may be important in gaining a better understanding of the behaviour of labour unions. Future research could focus on empirically testing the union objective function and the determinants of membership dynamics.

The author declares no conflicts of interest regarding the publication of this paper.

Dittrich, M. (2019) Union Membership and Employment Dynamics: The Role of the Union Objective Function. Theoretical Economics Letters, 9, 1473-1479. https://doi.org/10.4236/tel.2019.95094