Wage bargaining and vertical differentiation

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vertically differentiated industries, firms' production costs are evaluated at ex- ... Consider for instance the case of airline companies hiring different types of pilots ... the high-quality firm determines in a one-to-one correspondence the quantity of ..... agents both on the labor and on the product's market invites to use a cooper-.

Wage bargaining and vertical differentiation∗ Emanuele Bacchiega† December 2002

Abstract This article intends to apply the Nash Bargaining solution to wage setting in a vertically differentiated oligopoly and to study its welfare effects. The market outcome crucially depends on the bargaining power attributed to the agents. I show that the resulting wage bargaining structure is likely to lead to another source of distortion that adds to the classical one derived by oligopoly pricing and quality choice. Keywords: Vertical differentiation, workers’ skills, wage bargaining, welfare. JEL Classification: L11, J00



It is often assumed that firms operating in an oligopolistic environment buy their inputs in competitive markets. In particular, in the recent literature analyzing vertically differentiated industries, firms’ production costs are evaluated at exogenously given inputs’ prices, in spite of the oligopolistic context in which they operate on the selling side of the market. Yet, in this context, this assumption is far from being innocuous because high-quality variants of a product often embed ”sophisticated” inputs which can hardly be substituted by alternative ones. The owners of these inputs are accordingly conscious of the role they play in the production process, and are likely not to behave competitively. Conversely, the economic value of these inputs is strongly dependent on the very existence of firms which produce the high-quality variants for the production of which these inputs are indispensable. ∗ I wish to thank professor Jean J.Gabszewicz whose help has been more than necessary in the achievement of this work. I am grateful to professors Xavier Wauthy and Paolo Garella for their very helpful suggestions and comments. I had useful discussions on the issues treated in this work with Geoffroy de Clippel, Armando Dominioni, Giordano Mion and professor Anna Rubinchik-Pessach. Of course all errors are mine. † e-mail: [email protected]; [email protected]


A significant example of these relations is provided by a vertically differentiated industry in which the high-quality variant requires for its production the contribution of highly skilled workers. The owners of such skills are generally few in number, which gives them some market power with respect to those firms which are meant to hire them. Yet, if these skills are ”product-specific”, their economic value crucially depends on the employment decisions of the same firms. Consider for instance the case of airline companies hiring different types of pilots according to the sophistication of the planes in their fleet. Clearly, to pilot a ”Concorde” requires much more skills and specialization than to pilot an ATR. But, without any airline company using Concordes, there would be no employment at that level of competence for those who obtained their qualification to a Concorde and they would still have to pilot ATR’s in spite of their qualification. Obviously, this reasoning works the other way around: no company could fly a Concorde without Concorde-qualified pilots. Both airline companies and Concorde pilots hold market power and it would be unreasonable to assume a price (wage)-taking behavior in this situation. In this paper, I study a simple example inspired both from Gabszewicz and Turrini ((1999) and (2000)) and Brander and Spencer (1988). This example is based on an industry with two firms selling products of different quality. The high-quality variant can be produced only thanks to the contribution of skilled workers; by contrast, the low-quality variant can be produced using either skilled or unskilled workers. I assume that the production function of each firm exhibits constant-returns to scale with respect to the quantity of the input used: the production of each unit of each variant requires exactly one unit of the corresponding factor. The resulting quantities of each variant are sold to consumers with varying willingness to pay for the products, in a demand model ”` a la Mussa-Rosen (1978)”. Each firm selects which quality to produce and how much. In particular, the quantity of the high-quality variant produced by the high-quality firm determines in a one-to-one correspondence the quantity of skilled workers which are hired in the skilled labor market. The wage level in this market is crucial to evaluate the profits of the firms, and thus the economic consequences of their decisions. It is this fundamental role, as well as the intrinsic non-competitive nature of the markets for sophisticated inputs, which have induced me to adopt a non-competitive approach 2

to the process of wage-setting in this market as, for example, in Brander and Spencer (1988) and Bughin (1996). This should be contrasted with Gabszewicz and Turrini ((1999) and (2000)) who determine the skilled workers’ wage through the skilled labor market clearing condition. In their study, both the high-quality firm and the workers behave as price takers on the skilled labor market. This assumption, despite the ease it creates when solving their model, has two important drawbacks. The first is a technical one: the market clearing condition hypothesis, combined with a fixed supply of skilled labor, generates, due to the linearity of the production function, a fixed supply of the high-quality product. The second is that their approach does not take into account the non-competitiveness of the markets for ”sophisticated” production factors. In the present paper I assume that the unit production cost for the skilled labor is derived through the generalized (or weighted) Nash bargaining solution. This approach is not new in our context since it has been extensively studied and used in the theoretical literature related to wage setting in a non-competitive environment (see, for instance, Clark (1984), Svejnar (1986) , Brander and Spencer (1988), Dowrick (1989), Bughin (1996), Coles and Smith (1998), Coles and Hildreth (2000)). In order to apply this concept to the present context, I suppose that all the skilled workers that can be potentially hired in the industry are represented by a union, whose aim is to maximize the skilled workers’ expected wage. The firm that decides to produce the high-quality variant of the good, which requires skilled labor as input, bargains with the union the extent of the skilled workers’ wage. First, I explicitly compute the Nash solution as a function of the respective weights of both the union and the firm and of the payoffs they obtain at their outside options. Then I show that the resulting wage is always inefficient when efficiency is measured through total surplus, except when no weight is given to the union: it turns out, indeed, that total surplus is a decreasing function of this weight. Thus welfare maximization is not compatible with a positive weight assigned to the union, so that its members get at the Nash solution the payoff of their outside option, which is simply the unskilled workers’ wage.

Obviously, this result is obtained under the assumption that

skilled workers are already endowed with their skills. If the cost of skill acqui3

sition would be taken into account, it would not be acceptable for the skilled workers to be paid as the unskilled ones. This would constrain the expected bargained wage to be at least as high as this cost, further augmented by the remuneration the candidate to skill-acquisition could obtain in an alternative job. Thus welfare maximization should be accompanied by a transfer policy from the high-quality firm to the skilled workers. Notice, moreover, that even the case of no costly skill acquisition and zero bargaining weight to the union, the welfare distortion derived by the final market’s oligopolistic structure remains. In the following, section 2 describes the model. In section 3 the equilibrium analysis is developed, section 4 studies the welfare implications, while the related literature is commented in section 5. The last section proposes some extensions and provides a short conclusion.


The Model

I consider an industry embodying two firms producing each one variant of a given product: high-quality or low-quality. The game I analyze has three stages; in the first each firm decides the quality level of its product. Due to a Bertrand argument they will never specialize in the same variant at a subgame perfect equilibrium. Accordingly, for the two following stages, each firm is specialized on a different quality level. In the second the firm which has decided to produce the high-quality variant bargains with the union the skilled workers’ wage; finally, in the third stage, the high-quality firm and the low-quality firm compete in prices on the product market. Each firm is endowed with a linear technology: to produce one unit of the good, one unit of labor is required irrespective of its quality. However, one unit of high-quality requires one unit of high-skill labor, while one unit of lowquality requires one unit of unskilled, or skilled, labor. Thus I assume perfect substitutability between skilled and unskilled labor in the production of the low-quality variant of the good, while the contrary does not hold, in the fashion of the Concorde-ATR pilots’ example. Moreover both skilled and unskilled workers perform equally well outside the industry I consider, in other words human capital is purely specific. Thus the production function is constantreturns-to-scale when relating the quantity of input to the quantity of output;


furthermore, it specifies the input(s)’s quality(ies) required to obtain a specific output’s quality1 . The firms’ objective is profit maximization. Skilled workers’ interests are entrusted to a union whose objective is to maximize the skilled workers expected wage. Furthermore, I suppose that the union is composed by N skilled workers. No bargaining between individual skilled workers and the high-quality firm is allowed, so that the skilled workers’ wage -denoted by wis fully determined through the union-firm bargaining process. As for the unskilled workers, I suppose that they are sufficiently numerous in order for their wage to be independent of the number of them hired in the industry. Accordingly, I suppose this wage to be equal to their marginal productivity outside the industry, which is exogenously given and equal to r. These assumptions imply that, if hired in the production of the low-quality variant inside the industry or hired outside the industry, the skilled workers are paid a wage equal to the unskilled workers’ one. Finally, the high-quality firm and the union bargain the skilled workers’ remuneration. Demand for the products are derived from a continuum of consumers differing in terms of their intensity of preferences for quality. They buy one unit of the good only, either of the low-quality variant or of the high-quality one. Denoting by ui , i ∈ {h, l}, uh > ul , the quality level of product i, the utility of the consumer characterized by taste for quality θ is given by:  0 U + θui − pi when buying U (θ, ui ) = 0 when not buying. I assume U 0 > 0; furthermore, the parameter θ is assumed to be uniformly distributed with density 1 on the interval [0, ¯θ], pi denotes the price of one unit 1 This specification is a particular case of Kremer’s (1993) ”O-Ring production function”. Formally, denoting by yh (resp. yl ) the output of the high-quality (resp. low-quality) firm, and by Lh (resp. Ll ) the number of skilled (resp. unskilled) workers, this specification writes as:



Lh ,




for the quantities produced by the high (resp. low) quality firm and qh






This production function can be viewed as an extremely simplified specification of Kremer’s (1993) production function where there is only one task, there is not capital and the quality of the product is not defined by the expected value of the production.


of variant i. I suppose also that the parameter U 0 is high enough to guarantee that the market is covered. Using a standard approach, the market demands can be derived as a function of the prices charged by the firms. The consumer θh,l , indifferent between consuming the high-quality variant at price ph and the low-quality one at price pl obtains as θh,l =

ph − pl . uh − ul


Accordingly, the demands for the high-quality variant and the low-one write as2 : Dh (ph , pl ) = ¯θ − θh,l ,


Dl (ph , pl ) = θh,l ,


respectively. Due to constant returns to scale, firms’ profits write as: π h (ph , pl ) = Dh (.)(ph − w),


π l (ph , pl ) = Dl (.)(pl − r).


In order to solve the model, I proceed backwards from the market game on. First, I determine the prices (and so profits and demands) stemming out from oligopolistic competition on the market for the vertically differentiated good, assuming as given the skilled workers’ wage w. Then I solve the bargaining stage in order to determine the skilled workers’ wage itself; finally I discuss the issue of quality choice.

3 3.1

Equilibrium analysis The price game

It is easy to determine the third-stage price equilibrium, conditional on the quality choice and the wage bargaining outcome. 2 Remember that the market is covered and each consumer purchases one unit of the good only.


Simultaneous maximization of the firm’s profits with respect to price gives the equilibrium prices on the vertically differentiated market, namely I solve the system 

maxph [π h (ph , pl )] . maxpl [π h (ph , pl )]

From the first order conditions for a maximum I obtain:  ¯ h −ul )] p∗h (w) = r+2[w+θ(u 3 . ¯ h −ul ) p∗l (w) = 2r+w+θ(u 3


Notice that these prices can be plugged into (2) to (5) in order to derive the expressions for demands and profits at the price equilibrium on the vertically differentiated market. In particular, I denote by Dh (w) and π h (w) the values of the demand for the high-quality variant of the good and the corresponding profits evaluated at the equilibrium prices p∗h (w) and p∗l (w) These magnitudes are all conditional on the skilled workers’ wage.


The bargaining game

Let me now analyze the bargaining problem, and determine the skilled workers’ wage as a function of the bargaining power of the union and the firm, and conditional on the quality decisions made in the first stage. To this end, I use the concept of weighted Nash bargaining solution. Define by the notation µ the weight of the union; 1 − µ is the weight of the firm. The payoffs of the high-quality firm as a function of the bargaining outcome w is π h (w). To define the payoffs of the union we need to recall our assumption according to which the number N of skilled workers exceeds the largest possible value for the demand of the high-quality variant, i.e.3 . N > max Dh (w) w

3 This

maximum exists because Dh (w) < ¯ θ. At equilibrium this condition can be specified

as N >

2¯ θ = Dh (w)|µ=0 3


Accordingly, the probability Ph (w) for a skilled worker to be hired by the highquality firm is equal to

Dh (w) N .

We assume that the union takes as payoffs the

expected wage E(w) defined by E(w) = Ph (w)w + (1 − Ph (w))r.


The definition of the weighted Nash bargaining solution also requires, as a prerequisite, to identify the outside options of the agents, namely the payoffs the agents could obtain, should the bargaining fail to reach an agreement. In the case of the high-quality firm, this failure constrains it either to produce the low-quality good or to exit from the industry. In both cases its profit is zero. In conclusion the outside option’s payoff for the firm is equal to zero. As for the skilled workers’ union, if no agreement is reached with the high-quality firm, it can only guarantee to its members the unskilled workers’ wage r. By definition, the weighted Nash bargaining solution is the value of w which solves max B(w) = [E(w) − r]µ [π h (w)]1−µ .


w r


Notice that when µ = 0, this problem reduces to maxw>r [π h (w)]. When µ = 1 the problem rewrites as maxw>r [E(w) − r] = maxw>r [ DhN(w) (w − r)]. I explicitly derive the Nash bargaining solution. Proposition 1 The Nash bargaining solution is given by w∗ = r + µ¯θ(uh − ul ).


Proof. First suppose that µ ∈]0, 1[. Since w∗ solves (8) it must cancel the first order derivative

∂B(w) ∂w

= 0. There are three values of w that satisfy this

property, namely w1

= r + 2¯θ(uh − ul ),


= r,



= w∗ .

It is evident that B(w1 ) = 0 and B(w2 ) = 0; by direct substitution it can be checked that B(w3 ) = B(w∗ ) is strictly positive. Consequently, whenever µ ∈ ]0, 1[, B(w) reaches its maximum at w = w∗ . Then suppose µ = 0; remind that in that case the maximization problem boils down to maxw>r [π h (w)], whose 8

solution is given by w∗ = r. Finally, suppose µ = 1; then the maximization problem reduces to maxw>r [ DhN(w) (w − r)]. Differentiating this expression with respect to w, we get ∂[ DhN(w) (w − r)] 2[r − w + ¯θ(uh − ul )] = = 0 ⇔ w = w∗ . ∂w 3N (uh − ul ) First notice that the Nash bargaining wage crucially depends on the weights assigned to the union and the high-quality firm, respectively. Obviously, the higher the union’s weight, the higher the wage. Second, this wage is a linear combination of the unskilled workers’ wage and of an expression which represents the competitive advantage, as determined by preferences, that the high-quality producer has with respect to the low-quality one on the last consumer on the right of the consumers’ distribution. Clearly, the larger the market and the higher this differential, the higher the Nash bargaining solution for the skilled workers’ wage. Plugging the value of the wage as indicated in (9) into the expression of profits, I obtain: π l (µ)


π h (µ)


¯θ2 (1 + µ)2 (uh − ul ) , 9 ¯θ2 (2 − µ)2 (uh − ul ) , 9

(10) (11)

It can be observed that both profits increase with the quality differential between the two variants as well as the individual evaluation for quality that characterizes the last consumer on the extreme right of the distribution. More interestingly, the high-quality firm’s profits decrease with µ, while the low-quality’s ones increase with it. In order to understand this observation, remember that the firms’ profits are equal to the product between the demand addressed to the firms and the mark-up. Each of these two functions depend on µ. In particular: 0

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