CAHIERS ECO & MATHS

94.07

MACROECONOMIC PERFORMANCE AND WAGE SETTING LEVEL
IN SYMMETRIC NON-COOPERATIVE GAMES

Pierre CAHUC

MAD
Macroeconomic performance
and wage setting level
in symmetric non-cooperative games

January 1994 (first draft: December 1993)

Pierre Cahuc*

MAD, Université de Paris 1, Panthéon-Sorbonne
90, Rue de Tolbiac, 75634, Paris Cedex 13

Abstract

This paper shows that the relationship between the wage and the wage setting level is necessa-
rily monotonic in symmetric non cooperative games between monopoly Trade Unions, when
the conditions of stability of Nash equilibria are taken into account. Moreover, the Trade
Unions' welfare raises with the degree of centralisation.

**Introduction**

Empirical evidence suggests that there is a hump-shaped relationship between the extent of centralisation of wage setting and macroeconomic performance, mainly measured by the unemployment rate (Calmfors and Driffill 1988, Freeman 1988, Rowthorn 1992). The theoretical explanation of such a relationship rests on the idea that both very centralised and very decentralised wage settings are likely to produce wage moderation and high employment. Actually, as far as I am aware, the contributions that yield microfoundations to this idea use symmetric Nash equilibria in non-cooperative games between identical wage-setters (Strand (1987), Calmfors and Driffill (1988), Hoel (1991), Danthine and Hunt (1992), and Calmfors (1993)).

In this paper, it is argued that such microfoundations are questionable, because the relation-
ship between the extent of centralisation and the wage is necessarily monotonic, when the
stability conditions of the Symmetric Nash Equilibria are taken into account. The model is
presented in section 1 and the main results are derived in section 2.

I. The model

Let us consider a simple model of monopoly Trade Unions that can coordinate their stra-
tegies at different levels. There is a continuum of identical firms, indexed by *i*, which size
is normalised to one. In the labor pool of each firm, a Trade Union sets the nominal wage,
denoted by *w*ᵢ. We are looking for the subgame perfect equilibrium of a two-stage game,
where unions set the wages in the first stage, and firms determine prices and employment in
the second stage.

Let us denote by *P* the consumer price index. We assume that union *i* has a preference ordering over real wage *w*<sub>*i*</sub>/P, and employment in firm *i*, denoted by *l*<sub>*i*</sub>. Hence, we can

<footnote>* I wish to thank Russell Cooper, Giacomo Corneo, Jean-Marc Tallon and André Zylberberg for helpful comments. All remaining errors are mine.</footnote>
represent the preferences of union *i* by the following utility function:

$$U\left(\frac{w_i}{P}, \ell_i\right)$$

$U(\cdot, \cdot)$ is twice differentiable, quasi-concave and increasing with respect to both its arguments. If $p_{-i}$ is the vector of prices of the products sold by firms $j \neq i$, the price of firm $i$ can be written as $p_i = p(w_i, p_{-i})$, and the consumer price index as: $P = P(p_i, p_{-i})$. Firm $i$'s labor demand can be written as: $\ell_i = \ell(w_i/p_i, p_{-i})$. We assume that $p(\cdot), P(\cdot)$, and $\ell(\cdot)$ are twice differentiable functions. Such a specification of prices and labor demands can be interpreted as a reduced form of a macroeconomic model of monopolistic competition, where firms produce substitutable goods (Blanchard and Kiyotaki 1987, Calmfors and Driffill 1988, Hoel 1991). In this case, a rise in $w_j, j \neq i$, has both a positive and a negative effect on union $i$'s utility: on the one hand, a rise in $w_j$ increases the consumer price index, therefore decreasing the utility of union $i$, but on the other hand, a rise in $w_j$ increases firm $i$'s labor demand, because products are substitutable, leading to an increase in union $i$'s welfare. That is the combination of those two effects that can entail a hump-shaped relationship between the extent of centralisation and the wage (cf Calmfors and Driffill (1988), Calmfors (1993)).

Using the definitions of $\ell(\cdot), P(\cdot)$ and $p(\cdot)$, we can represent the preferences of union $i$, in the first stage of the game, by a twice differentiable function $\sigma(w_i, w_{-i})$, where $w_{-i}$ is the vector of wages set by unions $j \neq i$:

$$U\left(\frac{w_i}{P(p_i, p_{-i})}, \ell(w_i/p_i, p_{-i})\right) \equiv \sigma(w_i, w_{-i})$$

Let us assume that there are $n \ge 1$, identical coalitions of Trade Unions. Within each coalition, the Trade Unions coordinate their strategies (i.e. their nominal wages), and take the wages of the other coalitions as given. The number of coalitions measures the degree of decentralisation of wage setting. A wage setting at the firm level corresponds to an infinite number of coalitions, but if there is one coalition, the wage setting is perfectly centralised. We consider the Symmetric Nash Equilibria with $n \ge 1$ coalitions. Let us denote by $C_s$ a coalition, the optimal wage set by union $i$, which belongs to coalition $C_s$, is the solution to the following program:

$$\max_{w_i} \int_{j \in C_s} \sigma(w_j, w_{-j}) dj \qquad (1)$$

The first order condition of this program is:

$$\frac{\partial \sigma(w_i, w_{-i})}{\partial w_i} + \int_{j \in C_s; j \neq i} \frac{\partial \sigma(w_j, w_{-j})}{\partial w_i} dj = 0$$

Let us denote the payoff to union $i$ of wage $w_i$ when all other unions set a wage $w$, by $V(w_i, w)$. Since we consider only symmetric equilibria, we can rewrite the first order condition as follows:

$$V_1(w_i, w) + \frac{1}{n} V_2(w_i, w) = 0 \qquad (2)$$
where the subscript denotes a partial derivative in the usual manner: $V_1(w_i, w) \equiv \frac{\partial\sigma(w_i, w_{-i})}{\partial w_i}$,
$V_2(w_i, w) \equiv \frac{\partial\sigma(w_i, w_{-i})}{\partial w_j}$; with $w_j = w$, $\forall j \neq i$.

The Symmetric Nash Equilibria with *n* coalitions (nSNE) are defined by:

$$ S_n = \begin{cases} w \in [0, \infty[ / V_1(w, w) + \frac{1}{n}V_2(w, w) = 0; \\ V_{11}(w, w) + \frac{2}{n}V_{12}(w, w) + \frac{1}{n^2}V_{22}(w, w) < 0 \end{cases} \quad (3) $$

We assume that $w$ is positive and finite, i.e.:

$$ \lim_{w \to 0} V_1(w, w) + \frac{1}{n} V_2(w, w) > 0; \quad \lim_{w \to \infty} V_1(w, w) + \frac{1}{n} V_2(w, w) < 0 $$

Let us denote by $w_n^*(\bar{w})$ the optimal response of a union in a coalition, when all other unions that do not belong to its coalition set a wage $\bar{w}$. A nSNE is locally stable if, around equilibrium : $dw_n^*(\bar{w})/d\bar{w} \in ]-1, 1[$ (see for instance Cooper and John 1988). Differentiating equation (2) around equilibrium yields the slope of the optimal response function. Hence, a nSNE is locally stable if and only if, around equilibrium:

$$ \frac{dw_n^*(\bar{w})}{d\bar{w}} = \frac{(n-1)\left(V_{12} + \frac{1}{n}V_{22}\right)}{n\left(V_{11} + \frac{2}{n}V_{12} + \frac{1}{n^2}V_{22}\right)} \in ]-1, 1[ \quad (4) $$

The definition of a locally stable nSNE allows us to study the relationship between the wage, the welfare and the degree of cooperation.

## II. Wage setting level, wage and welfare

The wage setting level influences the wage, and accordingly the employment level and the welfare of the Trade Unions.

### Proposition 1

At a locally stable nSNE, the wage is a monotonic function of the degree of centralisation.

### Proof

From the definition of a nSNE (equation (3)), one gets:

$$ \frac{dw}{dn} = \frac{V_2(w, w)}{n^2 \left( V_{11}(w, w) + \left(\frac{n+1}{n}\right) V_{12}(w, w) + \frac{1}{n} V_{22}(w, w) \right)} \quad (5) $$

It can be easily checked that the stability condition of the nSNE, defined in equation (4), implies that the denominator of the right hand side of equation (5) is necessarily negative. Then $dw/dn$ and $V_2(w, w)$ have an opposite sign. Now, let us assume that $w \in S_n$ is not
a monotonic function of $n$. This implies that there exists $w \in S_n$ such that $V_2(w, w) > 0$ and $V_2(w, w) < 0$, which is impossible, since $V_2(w, w)$ is a function. QED

It is worth noting that proposition 2 does not rely upon a priori restrictions on the form of the payoff functions. Equation (5) shows that if there are, according to Cooper and John's (1988) definition, positive spillovers (i.e. if $V_2(w, w) > 0$), the equilibrium wage is larger at a perfectly centralised wage setting ($n = 1$) than at a completely decentralised wage setting ($n \to \infty$). In this case, a higher degree of centralisation always entails a rise in the equilibrium value of $w$. Such a case can arise in an open economy, where the centralised union does still neglect a large part of the relationship between the nominal wage and the consumer price that is partly exogenously determined. Hence, the main effect of centralisation is a decrease in the elasticity of the labor demand, that increases the unions' monopoly power, and induces the unions to set higher wages in a centralised than in a decentralised wage setting (see e.g.: Layard, Nickell and Jackman (1991)).

On the contrary, if there are negative spillovers (i.e. if $V_2(w, w) < 0$), the equilibrium wage decreases with the degree of centralisation. Actually, in this case, the effects of the internalisation of the consumer price index dominates the increase in the unions' monopoly power when the degree of centralisation raises (see e.g.: Calmfors 1993).

But, proposition 1 underscores that the so-called hump-shaped relationship between the wage and the degree of centralisation of wage setting cannot be explained by comparative statics in a symmetric non-cooperative game with stable equilibria. Indeed, if the stability condition is not imposed, it is obvious to show, from equation (5), that there is a hump-shaped relationship between $w$ and $n$ if there is a sufficiently large strategic complementarity (i.e. if $V_{12}(w, w)$ is positive and large enough), because in this case, the denominator of the right hand side of equation (5) can become positive for some values of $n$. But such a large strategic complementarity does not fulfill the stability condition.

**Proposition 2**

At a locally stable nSNE, a rise in the degree of centralisation raises the Trade Unions' welfare.

**Proof**

At a nSNE, equation (2) together with the envelop theorem implies:

$$ \frac{dV(w, w)}{dn} = \frac{dV(w_n^*(\bar{w}), w)}{dn} = \left(1 - \frac{1}{n}\right) \frac{dw}{dn} V_2(w, w) \quad (6) $$

equations (5) and (6) yield, at equilibrium:

$$ \frac{dV(w, w)}{dn} = \frac{(n-1)(V_2(w, w))^2}{n^3 \left(V_{11}(w, w) + \left(\frac{n+1}{n}\right) V_{12}(w, w) + \frac{1}{n} V_{22}(w, w)\right)} \quad (7) $$

the sign of $\frac{dV(w, w)}{dn}$ is the same as the sign of the denominator of the right hand side of equation (7), which is negative (see the stability condition defined in equation (4)) QED.
Therefore unions have always interest in more cooperation, even at intermediate levels. It
does not seem possible to claim that intermediate wage setting levels are the worst situation
for unions with arguments concerning externalities in symmetric non cooperative games.

Conclusion

This paper has shown that there is a monotonic relationship, either positive or negative,
between the wage and the wage setting level in symmetric non cooperative games, when
one takes into account the conditions of stability of Nash equilibria. The results have been
obtained in a monopoly union framework. But it is obvious that they also apply to wage
bargaining models (such as those considered by Calmfors (1993) for instance). Accordingly,
the microfoundations of the hump-shaped relationship between the extent of centralisation
and the wage should likely rely upon asymmetric games.

Moreover, in symmetric games, players' welfare always increases with the extent of coope-
ration. This result is also general and can shed light on many conflictual situations such as
arm race, coordination of monetary, fiscal and trade policies...

References

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Calmfors, L. (1993) "Centralization of wage bargaining and macroeconomic performance, a survey", *Institute for International Economic Studies, Stockholm University, Seminar paper n°536*.

Calmfors, L. and Driffill, J. (1988) "Bargaining Structure, Corporatism and Macroeconomic performance," *Economic Policy*, vol 6, 12-61.

Cooper, R. and John, A. (1988) "Coordinating coordination failures in keynesian models," *Quarterly Journal of Economics*, vol 103, 441-63.

Danthine, J.P. and Hunt, J. (1992) "Wage bargaining structure, employment and economic integration," Mimeo, University of Lausanne and Yale University.

Freeman, H.B. (1988) "Labour market institution and Economic performance". *Economic Policy*, vol 6, 54-80.

Hoel, M. (1991) "Union wage policy: the importance of labour mobility and the degree of centralisation," *Economica*, vol 56, 139-53.

Layard, R. Nickell, S. and Jackman, R. (1991) *Unemployment*, Oxford University Press.

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Strand, J. (1987) "Oligopoly with monopoly unions", Mimeo, FIEF, Stockholm.