An endogenous monitoring technology

4. An endogenous monitoring technology

Why should employers keep on monitoring their workers when the NSC is not binding? To answer this question, we endogenize the monitoring technology by allowing employers to choose the frequency of inspections. 4.1. The choice of the inspection rate Ž . We introduce a monitoring cost, denoted by c l , which is a strictly convex and increasing function of the inspection rate. The net production flow of each job Ž . is then equal to the worker’s productivity y minus monitoring expenses. y s y y c l . Ž . Ž . The value function of a filled job satisfies a generalized version of Eq. 7 : rW s max y y c l y w l y sW , 20 4 Ž . Ž . Ž . J J q lgR Ž . where w l , the wage paid to workers, depends on the inspection rate. Each employer faces a trade-off between wages and monitoring costs. In general, an employer saves some monitoring costs by decreasing the frequency of inspections but must offer a higher wage to motivate workers. As it has been outlined in Section 3, the wage received by workers is at least Ž . w Ž . Ž .x equal to the efficiency wage. Hence, w l s max w l , w l , with, from Eqs. ˜ ˆ Ž . Ž . 10 and 11 , w l s 1 y b rW q e q b y y c l , 21 Ž . Ž . Ž . Ž . Ž . Ž . ˜ U e w l s r q s q e q rW . 22 Ž . Ž . Ž . ˆ U l For rW given, 6 wages are decreasing in the intensity of the monitoring activity. U Ž . According to Eq. 22 , a higher frequency of inspections reduces the incentive to Ž . shirk; according to Eq. 21 , it raises monitoring expenses and decreases the net production flow. A case that is worth mentioning is when workers have no bargaining power Ž . Ž . Ž . b s 0 . Then, the wage is always an EW. From Eqs. 20 and 22 , the profit-maximizing inspection rate is l which satisfies: r q s X c l s e. Ž . 2 l The marginal cost of the inspection technology is equal to the marginal gain in terms of the wage bill. 6 Workers and firms take W as given. Of course, the value of W in equilibrium also depends on U U the inspection rate. Proposition 3. There are two types of equilibrium: v EW equilibria: the inspection rate is l and w l w l . ˆ ˜ v FNW equilibria: the inspection rate is inferior to l and is such that w l s w l ˆ ˜ Proof. Let l eq 0 denote the value of the inspection rate in equilibrium. There are three possible a priori cases: Ž . Ž eq . Ž eq . i w l - w l . ˆ ˜ Ž . It is impossible. Indeed, from Eq. 21 E c l q w l Ž . Ž . Ž . ˜ X s 1 y b c l 0. Ž . Ž . E l Ž . Each firm bears a fraction 1 y b of the monitoring cost and has an incentive to reduce its inspection rate below l eq . Ž . Ž eq . Ž eq . ii w l w l . ˆ ˜ Then, profits maximization implies l eq s l . Ž . Ž eq . Ž eq . iii w l s w l . ˆ ˜ For profits to be maximized, employers must have no incentive either to increase or to decrease the inspection rate. For values of l slightly below l eq , firms want to Ž . Ž . increase their inspection rate if both w l w l and l - l . This requires ˆ ˜ eq X Ž eq . X Ž eq . l F l , and so c l F yw l . Indeed, ˆ c X l eq F yw X l eq ´bc X l eq - yw X l eq ´w X l eq - w X l eq . Ž . Ž . Ž . Ž . Ž . Ž . ˆ ˆ ˆ ˜ eq Ž . Ž . Under this last condition, if l is slightly above l then w l - w l and, ˆ ˜ Ž . according to case i , firms want to reduce their inspection rate. I Proposition 3 establishes that the wage expression in the two regimes is Ž . identical, and is given by Eq. 22 . When the NSC is not binding, workers have no incentive to cheat. Consequently, employers can relax supervision of employees and save some monitoring costs by decreasing the inspection rate below l . 4.2. Steady-state equilibrium Let us turn to the characterization of the steady-state equilibrium. Given that Ž . W y W s erl, and from Eq. 6 , the permanent income of unemployed workers E U is: e rW s b q u q u . 23 Ž . Ž . U l Ž . Ž . By replacing rW by its expression given by Eq. 23 into Eq. 22 , we obtain a U Ž . wage equation identical to Eq. 15 . Consequently, the labour market tightness in Ž . equilibrium obeys Eq. 16 . Ž Definition 4. A steady-state equilibrium is a sextuplet W , W , W , u , w, U E J . 6q Ž . Ž . Ž . Ž . Ž . l g R that satisfies: Eqs. 2 , 6 , 20 , 8 , 22 and l s l if the NSC is Ž . Ž . Ž . Ž . Ž . Ž . Ž . binding; Eqs. 2 , 6 , 20 , 8 , 22 and w l s w l if the NSC is not binding ˆ ˜ Let us characterize the two types of steady-state equilibrium. For both types of Ž . Ž . equilibria, the labour market tightness u obeys Eq. 16 , that is: e g y y c l y b y r q s q l q u q u s r q s . 24 4 Ž . Ž . Ž . Ž . l q u Ž . v Ž . FNW equilibria: The inspection rate satisfies 14 at equality. b g e s . 25 Ž . 1 y b q u l Ž . v EW equilibria: The inspection rate satisfies: r q s X c l s e. 26 Ž . Ž . 2 l Ž . Ž . Ž . Ž . From Eqs. 21 – 24 , the condition w l w l can be rewritten as: ˆ ˜ e b g . 27 Ž . l 1 y b q u Ž . Ž Ž .. In Fig. 3, the vacancy supply condition Eq. 24 is represented by a curve Ž . labelled VS. The inspection rate schedule in the FNW regime, given by Eq. 25 , is represented by a curve labelled IR. Finally, the inspection rate schedule in the Fig. 3. Equilibrium with endogenous monitoring technology. Ž . EW regime, which obeys Eq. 26 , is represented by the curve IR. The curves VS, IR and IR are respectively l -shaped, downward-sloping and vertical. Further- more, it can be verified that VS reaches a maximum for a value of the inspection rate superior to l . 7 Ž . The equilibrium solution for the pair l, u lies at the unique intersection of VS and IR, or VS and IR. The equilibrium is represented for three different values Ž . of the productivity y - y - y . For instance, when y s y the equilibrium pair 1 1 Ž . Ž . l, u lies at the unique intersection of VS y and IR. 1 Ž . Ž . Ž . Let y , u denote the pair of values y, u which satisfies Eqs. 24 and Ž . 25 when l s l . Proposition 5. If y - y , the unique equilibrium is an EW equilibrium and l s l . If y G y , the unique equilibrium is an FNW equilibrium and l F l . Ž . Proof. According to Eq. 24 , if y - y and l s l then u - u and b g e - . 1 y b q u l Ž . Ž . Thus, the condition for an EW equilibrium is satisfied. Furthermore, Eqs. 24 and Ž . 25 for a FNW equilibrium can only be satisfied for a value of the inspection rate above l , which is impossible. Conversely, if y G y and l s l , then u G u Ž . and the condition 27 for an EW equilibrium is not satisfied. The equilibrium lies at the unique intersection of VS and IR, and l F l . I Proposition 5 confirms and generalizes results from the preceding section. EW equilibria occur when productivity is low and unemployment is high; the inspec- tion rate is then equal to l . Conversely, in tight labour markets firms reduce their inspection rate below l because they gain nothing by monitoring their workers more intensively than what is required to prevent shirking. Again, our model generates a result which challenges the traditional view of EW models. Whereas in EW models the unemployment acts as a discipline device, our model predicts that employers’ monitoring is more intensive in depressed labour markets.

5. Heterogeneity and wage formation