We use a bargaining model first developed by Manning (1987). We derive the structural equations for determining employment and wages on the assumption that both are sequentially negotiated between an employer and a trade union and that these parties are completely informed. The resulting model will be used as the basis for an empirical analysis in Section 4. The basic idea here is that the employer seeks to maximise profit, given a production technology, and that the union seeks to maximise its members’ utility. How well each party succeeds depends on its bargaining strength. Bargaining strength may, in turn, depend on a number of different things, one of which is the institutional setting. In this paper, we ultimately set out to test whether foreign ownership affects employment and wage-setting, and our hypothesis is that if there is an effect, this effect may be transferred through the trade union’s bargaining power.

In most collective bargaining, there is a two-stage negotiation process where wages are determined prior to employment. In Sweden, such employment contracts often last for a year or two. Partly because of this, trade unions may act differently—and may have a different bargaining power—when negotiating about wages, compared to when they are negotiating about employment. Manning (1987) offers a theoretical explanation for this phenomenon. Firstly, owing to uncertainty and the impossibility of concluding complete contracts that will cover all the possible scenarios and supplements, it may be best to reach agreement on wages first and then determine employment. Secondly, wage negotiations are mostly carried out at a higher level, e.g. nationally, while employment levels are usually negotiated at plant level.

To start, we assume that *supply of labour* is given by a set of workers, where each worker has a unit of labour that they supply with no disutility. Furthermore, the members of the trade union are assumed to have identical preferences that are represented by the following (indirect) utility function:

$$ U=Pu(w)+\left(1-P\right)u(R) $$

(1)

where *w* denotes the real wage, *P* is the probability of being employed in the firm and *R* denotes the expected real income of a worker who loses his/her job in the firm (the reservation wage). *u*(*·*) is strictly increasing and concave. If we normalise the labour force to unity, the union utility function (1) can be written as

$$ U\left(w,\;L,\;R\right)=Lu(w)+\left(1-L\right)\;u(R) $$

(2)

where *L* is the share of the labour force that is employed and (1 − *L*) the unemployed share.

The profit-maximising firm is assumed to use labour, *L*, and capital, *K*, and a strictly concave technology, *f*, to produce its output. Assuming that capital is fixed in the short run, the firm’s short-run profit can be written as

$$ \varPi \left(L,\;K,\;p,\;w\right)=pf\left(L,\;K\right)-wL $$

(3)

where *p* is the price of output.

The first stage in the negotiation process between firm and trade union is where the parties bargain over the wage. In the second stage, the parties bargain over the employment level—subject to the negotiated wage from the first stage. The negotiation process between the union and employer is formalised by the strategic model of bargaining between completely informed players. This can be set up as a Nash product, where the objective is to maximise a weighted product of the union’s utility and the firm’s profit. The weights are determined by the bargaining power, which in turn is a function of a vector of variables, **z**, such as institutional setting, firm characteristics, labour market conditions and type of ownership. The problem can be solved backwards, i.e. by solving for the employment conditional on a given wage, as follows:

$$ \underset{L}{ \max }{\left(U\left(w,\;L,\;R\right)-{U}_0\right)}^{\varPhi_2\left(\mathbf{z}\right)}{\left(\varPi \left(L,\;K,\;p,\;w\right)-{\varPi}_0\right)}^{1-{\varPhi}_2\left(\mathbf{z}\right)} $$

(4)

where *Φ*
_{2} ∈ [0, 1] is the union’s bargaining power over employment. If *Φ*
_{2} = 0, the firm sets the employment level. *U*
_{0} and *Π*
_{0} are the fallback utility and fallback profit, assumed to be given by *U*
_{0} = *u*(*R*) and *Π*
_{0} = 0, respectively.

The first-order condition to this problem can be written as,

$$ {\varPhi}_2\left(\mathbf{z}\right)\frac{pf\left(L,\;K\right)}{L}+\left(1-{\varPhi}_2\left(\mathbf{z}\right)\right)p{f}_L\left(L,\;K\right)=w. $$

(5)

From Eq. (5), we see that the employment level is chosen such that a linear combination of the value of the average productivity of labour and the value of the marginal productivity of labour is equal to the wage. If *Φ*
_{2} = 1, the firm has no influence on determining employment, and the wage will be set equal to the value of the average product. If *Φ*
_{2} = 0, then employment is chosen such that the marginal product of labour equals the wage.

Thus, Eq. (5) then gives us the employment level as a function of the given real wage, the producer price, the capital stock and the trade union’s bargaining power:

$$ L=L\left(w,\;p,\;K,\;{\varPhi}_2\left(\mathbf{z}\right)\right). $$

(6)

The first stage in the negotiation process is where the union and the employer negotiate over wages, given the employment level from the second stage, i.e.

$$ \underset{w}{ \max }{\left(U\left(w,\;R,\;L\left(w,\;p,\;K,\;{\varPhi}_2\left(\mathbf{z}\right)\right)\right)-{U}_0\right)}^{\varPhi_1\left(\mathbf{z}\right)}{\left(\varPi \left(L\left(w,\;p,\;K,\;{\varPhi}_2\left(\mathbf{z}\right)\right),\;K,\;w\right)-{\varPi}_0\right)}^{1-{\varPhi}_1\left(\mathbf{z}\right)} $$

(7)

where *Φ*
_{1} ∈ [0, 1] is the trade union’s bargaining power in the wage-setting stage. If *Φ*
_{1} = 1, the trade union sets the wage, which in combination with *Φ*
_{2} = 0 gives us the monopoly model (see e.g. Dunlop 1944).^{Footnote 1} If *Φ*
_{1} and *Φ*
_{2} are equal, we have the efficient bargaining model of McDonald and Solow (1981); Manning (1987) notes that in practice, *Φ*
_{1} > *Φ*
_{2} in many real-world unionised labour markets and that different bargaining strengths on different issues are a key source of economic inefficiencies associated with unionisation.

The first-order condition to Eq. (7) then gives us the equilibrium wage-setting rule, which can be written as,

$$ {w}^{\ast }=w\left(L\left(w,\;p,\;K,\;{\varPhi}_2\left(\mathbf{z}\right)\right),\;R,\;{\varPhi}_1\left(\mathbf{z}\right)\right). $$

(8)

Equation (8) indicates that the wage-setting rule, in general, is a function of the employment level (and, thus, of bargaining power over employment), of the reservation wage and of bargaining power over the wage.^{Footnote 2} Substituting Eq. (8) into (6) then gives us the equilibrium level of employment, i.e.

$$ {L}^{\ast }=L\left({w}^{\ast },\;p,\;K,\;{\varPhi}_2\left(\mathbf{z}\right)\right)=L\left(p,\;K,\;R,\;{\varPhi}_2\left(\mathbf{z}\right),\;{\varPhi}_1\left(\mathbf{z}\right)\right). $$

(9)

Following Manning (1987), by assuming that *u*(*w*) *= w*, *u*(*R*) = *R* and *U*
_{0} = *R*, it can be shown that

$$ \frac{\partial {L}^{\ast }}{\partial {\varPhi}_1}\le 0,\;\frac{\partial {L}^{\ast }}{\partial {\varPhi}_2}\ge 0,\;\frac{\partial {w}^{\ast }}{\partial {\varPhi}_1}\ge 0,\;\frac{\partial {w}^{\ast }}{\partial {\varPhi}_2}?0. $$

(10)

Perhaps surprisingly, we cannot say how the wage is affected by a change in bargaining power over employment without having a more explicit model specification that lets us determine the wage-setting rule (e.g. if the production function is of a Cobb-Douglas type, it can be shown that ∂*w*/∂*Φ*
_{2} = 0).

Equations (8) and (9) constitute a system of equations describing equilibrium employment and equilibrium wage determination that can, in principle, be estimated. A problem, however, is that the bargaining parameters, *Φ*
_{1} and *Φ*
_{2}, are not directly observable. To get around this problem, we will follow the empirical approach developed by Alogoskoufis and Manning (1991), Doiron (1992) and Vannetelbosch (1996), in which it is assumed that bargaining power can be expressed as a function of a vector of exogenous variables, **z**.

In particular, we are interested in investigating how differences in ownership affect bargaining power and, ultimately, employment and wage-setting. If the belief that foreign ownership reduces unions’ bargaining power is correct, we would expect the bargaining parameters to be lower for foreign-owned firms. However, even if there is an impact on unions’ bargaining power, it is not clear which of the two bargaining parameters will be affected the most. Thus, the actual impact of foreign ownership on the two bargaining parameters is a matter for empirical analysis.