Chapter 2 Simple Static Monopsony A good starting point for the discussion of spatial monopsony is a simple static monopsony model in the fashion of Robinson (1969, pp. 211 231), i.e., considering a spaceless market for homogenous labour with a single non-discriminating firm producing a homogenous commodity from its labour input. 1 This single firm has not to bother with other firms decisions and is unable to pay different wages to its employees. Let L s WD L.w/ (2.1) denote the firm s labour supply. Assume that the wage w is the firm s only instrument to affect the quantity of labour supplied to it and that L is twice continuously differentiable with L 0.w/ >0for all w >0. 2 Since the firm is the single buyer in this labour market, the firm s labour supply is identical to market labour supply, so that we have both upward-sloping labour supply at the level of the firm and the market. Let in the following w.l/ denote the inverse of L.w/. We assume that labour is the firm s only factor of production. Let C denote the firm s production costs which are C.L/ WD wl C f; (2.2) where f denotes the firm s fixed costs and wl its labour costs. Hence, there are economies of scale because of strictly decreasing average costs. The firm is assumed to produce a homogenous single commodity from its labour input L with a constant marginal revenue product of labour. 3 Thus, its revenue is given by 1 For a textbook treatment of simple static monopsony, see Blair and Harrison (1993, pp. 36 42), Manning (2003a, pp. 30/31), or Cahuc and Zylberberg (2004, pp. 257 261). 2 Manning (2006) considers the more general case where the monopsonist is also able to raise her labour supply by increasing her expenditures on recruitment. 3 It is straightforward to generalise this setting to the case with a second factor of production, say capital, and a constant returns to scale production technology. In this case, we get a constant marginal revenue product of labour for each ratio of the output price and the capital rental rate due to the firm s optimal adjustment of the capital stock employed (e.g., Bhaskar and To, 1999; B. Hirsch, Monopsonistic Labour Markets and the Gender Pay Gap, Lecture Notes in Economics and Mathematical Systems 639, DOI 10.1007/978-3-642-10409-1_2, c Springer-Verlag Berlin Heidelberg 2010 11
12 2 Simple Static Monopsony Y.L/ WD L: (2.3) Next, the firm is assumed to maximise its profits by choosing the amount of labour employed. 4 The firm s profits are its revenue net of production costs.l/ WD Y.L/ C.L/ D Œ w.l/ L f; (2.4) where the firm s labour input decision is constrained by market-level inverse labour supply w.l/. Since the firm gains a profit of w from every unit of labour employed, its labour demand L d is infinite if w >0, zero if w <0,and indefinite if w D 0, i.e., 8 < 1 if >w L d D Œ0; 1/ if D w (2.5) : 0 if <w. 5 Maximising profits as given by (2.4) yields the first-order condition 0.L/ Š D 0. In the optimum, firm s labour demand behaviour is described by Y 0.L m / D C 0.L m /, D w.l m / C w 0.L m /L m : 6 (2.6) This condition says that the marginal revenue product has to be equal to the marginal cost of labour, where a diagrammatic representation with linear market-level labour supply for expositional convenience is given by Figure 2.1 on the following page. Since the non-discriminating monopsonist has to increase her wage paid to all her existing employees in order to raise her labour supply, she does not hire additional workers until the wage equals the marginal revenue product of labour. But instead, she stops hiring at a lower level L m, so that this other than under perfect competition yields a monopsony wage w m below workers marginal revenue 2003). Moreover, Bradfield (1990) shows that the (long-run) marginal revenue product of labour is constant if the firm s production technology inhibits constant returns of scale and if there is perfect competition as well on all other factor markets than the labour market as on the monopsonist s output market (see also Hicks, 1963, pp. 242 246). Hence, we can think of labour as one of the factors of production, where the other factors are thought of as set at values that maximise the firm s profits given its labour input. 4 Of course, this is equivalent to maximising profits by choosing the wage offered to workers. Using employment as the choice variable, however, leads to intuitively more appealing results. 5 Note that, though the fixed costs f affect overall profits, they do not affect marginal decisionmaking and therefore are irrelevant for the firm s labour demand behaviour. 6 We assume that >w.l/ C w 0.L/L for some L>0in order to avoid a corner solution with L m D 0. Furthermore, we assume that this unique global maximum of profits actually exists. This is guaranteed if 0.L/ > 0 for all L<L m and 0.L/ < 0 for all L>L m. This holds, for example, if profits are strictly concave in w, which in turn is implied if, for instance, market-level labour supply is concave in w.
2 Simple Static Monopsony 13 Fig. 2.1 Wage and employment chosen by a non-discriminating monopsonist product. Some minor algebraic manipulations on (2.6) reveal that the monopsonist s wage is implicitly given by w m D e.w m/ ; (2.7) 1 C e.w m / where e.w/ WD L 0.w/w=L.w/ is the elasticity of the firm s labour supply at wage w. Hence, the percentage gap between workers marginal revenue product and the monopsony wage is represented by E m WD w m w m D 1 e.w m / : (2.8) E m represents the classic measure of exploitation introduced by Pigou (1932, pp. 813/814) and is an analogue to the Lerner index utilised in the industrial organisation literature (e.g., Tirole, 1988, p. 66; Blair and Harrison, 1993, pp. 48/49). In our case with a constant marginal revenue product of labour (and thus a horizontal labour demand curve), E m also gives the percentage gap between the competitive wage w c D and the monopsony wage w m Pigou s (1932, ibid.) unfairness. It therefore serves as a direct measure of the welfare loss caused by the monopsonist which is given by. w m /.L c L m /. 7 Obviously, the case of perfect competition is nested in this model. If labour supply becomes infinitely elastic, i.e., e.w m /!1, the monopsonist pays the workers their marginal revenue product. We have w m! and, equivalently, E m! 0 as 7 This does not hold if labour demand is not horizontal but decreasing. In this case, w c D Y 0.L c /< Y 0.L m / holds, so that E m overstates the departure of the monopsony wage from the competitive wage and the associated welfare loss (e.g., Boal and Ransom, 1997).
14 2 Simple Static Monopsony can be seen from (2.7) and (2.8). Moreover, (2.7) shows that the more elastic is the firm s labour supply the higher is the monopsony wage chosen. 8 It is straightforward to extend this framework by considering more than one employer (though we shall not do so here). For instance, one could consider a Cournot model or a collusive model with more than just one employer. In the collusive model, things are virtually the same as under simple monopsony (cf. Blair and Harison, 1993, pp. 42 46), while the general message of the Cournot model is that, other things being equal, employers market power decreases with the number of firms competing for workers (cf. Boal and Ransom, 1997). In these models of classic monopsony, labour markets therefore have to be thin in the literal sense of consisting of few employers on the demand side to generate substantial monopsony power. Things work differently in models of new monopsony. One strand of these models, which we refer to as spatial monopsony and the discussion of which we will begin in the next chapter, gives rise to monopsony power due to mobility costs and/or workers heterogenous preferences over non-wage job characteristics. The implication of economic space which can be thought of as geographical space or as job characteristics space is that agents incur significant travel/transportation cost, for costless space is not economic space. (Ohta, 1988, p. 5) On account of the fixed costs, however, firms are not set up at any location possible, so that labour markets are thin to some extent even if there are many employers in economic space. 8 This follows at once from (2.7) because @w m =@e.w m / D =Œ1 C e.w m / 2 >0for all w m >0.
http://www.springer.com/978-3-642-10408-4