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Efficiency-Wage Competition: What Happens as the Number of Players Increases?

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In this paper, I explore the consequences of extending the number of firms within an efficiency-wage competition setting by showing that the shape of the effort function is crucial in determining key features of the economy. Specifically, when workers are endowed with a concave (sigmoid) effort function, the wage behaviour of firms follows a collusive (competitive) pattern and the symmetric Nash equilibrium is unstable (stable). Moreover, when effort is concave (sigmoid), full employment is characterized by a labour exploitation that increases (decreases) together with the number of productive units required to sustain that allocation. These findings may have intriguing implications for the existence of involuntary unemployment as well as for policies aimed at increasing employment.

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  1. 1.

    The same production function is used by Akerlof (1982) and Alexopulos (2004). None of the results derived in the present paper, however, depend on the specification of the production function.

  2. 2.

    Theoretical efficiency-wage models with concave effort functions can be found in Sparks (1986) and Walsh (1999). Moreover, an attempt to provide an empirical estimation of a concave effort function for a panel of OECD countries is given by de la de la Croix et al. (2000).

  3. 3.

    As it will become clear in the subsequent section, in Eq. (3) the single external wage opportunity is not raised to the power of \(\gamma \) as it happens instead to the internal wage offer since this would be inconsistent with the definition of a well-defined symmetric wage strategy. Further details are available from the author upon request.

  4. 4.

    An additional intriguing features of Eqs. (2) and (3) also shared by the alternatives in which the deviation from the average is taken into account, is the fact that when all the firms change their wage bid by the same amount, individual effort changes accordingly in the same direction. A rationale for that pattern is that workers may have some concern not only for relative wages but also for the labour share of output. Indeed, the higher the wage levels, the lower the profits of the firms and the higher the output share available for workers.

  5. 5.

    In order to avoid the questionable situation in which workers reduce effort provision when a new firm enters the output market by offering a barely positive wage, we can assume that entrance is impeded when entering productive units bid a wage below a certain threshold.

  6. 6.

    On a general equilibrium perspective, a tentative to track down reduced forms for workers’ preferences implied by the concave and sigmoid effort specifications by means of straightforward integration can be found, respectively, in Guerrazzi (2013) and Wu and Ho (2012).

  7. 7.

    The non-monotonicity of the strategic relation among optimal wage bids in a similar efficiency-wage competition framework is addressed by Guerrazzi and Sodini (2018).

  8. 8.

    Using a concave effort function, Guerrazzi (2013) shows that the symmetric Nash equilibrium can be stabilized by assuming that firms adjust their wage bids in the direction of increasing profits by conjecturing—in a myopic manner—a certain degree of substitutability among optimal wage offers.

  9. 9.

    According to Weiss (1991), the full employment allocation is actually achieved even when the symmetric Nash equilibrium implies that firms are rationed in the labour market, i.e., when \(NL^{*}\) is higher than the aggregate labour supply that holds at the prevailing equilibrium wage.

  10. 10.

    When effort is sigmoid, the reduction of \(L^{*}\) induced by an increase of is so strong that \(NL^{*}\) is always a decreasing function of the number of competing firms. Formally speaking, in this case \(\left( \partial L^{*}/\partial N\right) \left( N/L^{*}\right) >1\).


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Correspondence to Marco Guerrazzi.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

I would like to thank Ilham Ksebi, Nicola Meccheri and Paolo Scapparone and two for their valuable suggestions.

Comments from two anonymous referees substantially increased the quality of the paper. The usual disclaimers apply.

Appendix: Derivation of the reaction functions

Appendix: Derivation of the reaction functions

Without any loss of generality, assume that the effort function has a concave shape as conveyed by Eq. (2). In this case, when the participation constraint is not binding and the internal wage offer of each player is substituted for the expression of the relevant incentive compatibility constraint, the problem of the representative firm can be written as

$$\begin{aligned} \underset{e_{i},L_{i}}{\max }\pi _{i}=A\left( e_{i}L_{i}\right) ^{\alpha }-\left( w_{i}^{\min }+e_{i}^{^{\frac{1}{\beta }}}\right) L_{i} \end{aligned}$$

Recalling that \(w_{i}^{\min }\equiv {\textstyle \sum \nolimits _{j\ne i}^{N-1}} w_{j}-\kappa \), the first-order conditions for \(e_{i}\) and \(L_{i}\) are respectively given by

$$\begin{aligned}&A\left( e_{i}L_{i}\right) ^{\alpha -1}-\frac{e_{i}^{^{\frac{1-\beta }{\beta }} }}{\beta }=0 \end{aligned}$$
$$\begin{aligned}&A\left( e_{i}L_{i}\right) ^{\alpha -1}e_{i}- {\displaystyle \sum \limits _{j\ne i}^{N-1}} w_{j}+\kappa -e_{i}^{^{\frac{1}{\beta }}}=0 \end{aligned}$$

Solving (A2) with respect to \(A\left( e_{i} L_{i}\right) ^{\alpha -1}\) and plugging the result into (A3) allows us to find the profit-maximizing effort provision as a function of the external wage bids and the parameters of the effort function. Formally speaking, we find that

$$\begin{aligned} e_{i}=\left( \frac{\beta \left( {\textstyle \sum \nolimits _{j\ne i}^{N-1}} w_{j}-\kappa \right) }{1-\beta }\right) ^{\beta } \end{aligned}$$

Plugging the expression in (A4) into Eq. (2) and solving for \(w_{i}\) returns exactly the reaction function in Eq. (9). A similar procedure can be followed in order to retrieve the reaction function in Eq. (10) that holds in the case of a sigmoid effort function.

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Guerrazzi, M. Efficiency-Wage Competition: What Happens as the Number of Players Increases?. Ital Econ J 6, 13–35 (2020).

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  • Efficiency-wage competition
  • Number of competitors
  • Effort function
  • Nash equilibrium
  • Labour exploitation

JEL Classification

  • C72
  • E12
  • E24
  • J41