Monitoring Intensity and Technology Choice in a Model of Unemployment

Abstract

The interaction among a firm’s choices of output, technology, and monitoring intensity is studied in a general equilibrium model. Firms engage in oligopolistic competition, and unemployment is a result of the existence of efficiency wages. The following results are derived analytically. First, an increase in the cost of exerting effort leads a firm to choose a more advanced technology and a lower level of monitoring intensity. Second, an increase in the discount rate does not change a firm’s choices of technology and monitoring intensity. Third, an increase in the elasticity of substitution among goods leads a firm to choose higher levels of monitoring intensity and technology. In a model in which the level of monitoring is exogenously given, there is a negative relationship between the wage rate and the monitoring intensity. In this model with endogenously chosen monitoring intensity, the wage rate and the monitoring intensity can move either in the same direction or in opposite directions.

Appendix

The system of Eqs. (8)–(16) defining the steady state can be reduced alternatively into the following system of three equations defining three variables $w$, $n$, and $q$ as functions of exogenous parameters:

$$\varOmega_{1} \equiv f + \frac{\theta }{2}q^{2} - \left( {1 - \beta w} \right)\sigma K = 0,$$

(21)

$$\varOmega_{2} \equiv - f^{\prime } - \beta^{\prime } w\sigma K = 0,$$

(22)

$$\varOmega_{3} \equiv \theta q^{2} - \beta \sigma K\left( {w - e} \right) = 0.$$

(23)

The derivation of Eqs. (21)–(23) is as follows. First, Eq. (21) is the same as Eq. (18). Second, Eq. (22) is derived by plugging the value of $r$ from Eqs. (16) and (18) into Eq. (10). Third, Eq. (23) is derived by plugging the value of $u$ from Eq. (17) into Eq. (19a).

Partial differentiation of Eqs. (21)–(23) yields

$$\left( {\begin{array}{*{20}c} {\frac{{\partial \varOmega_{1} }}{\partial w} 0 \frac{{\partial \varOmega_{1} }}{\partial q}} \\ {\frac{{\partial \varOmega_{2} }}{\partial w} \frac{{\partial \varOmega_{2} }}{\partial n} 0 } \\ {\frac{{\partial \varOmega_{3} }}{\partial w} \frac{{\partial \varOmega_{3} }}{\partial n} \frac{{\partial \varOmega_{3} }}{\partial q}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {{\text{d}}w} \\ {{\text{d}}n} \\ {{\text{d}}q} \\ \end{array} } \right) = - \left( {\begin{array}{*{20}c} {\frac{{\partial \varOmega_{1} }}{\partial K}} \\ {\frac{{\partial \varOmega_{2} }}{\partial K}} \\ {\frac{{\partial \varOmega_{3} }}{\partial K}} \\ \end{array} } \right){\text{d}}K - \left( {\begin{array}{*{20}c} {\frac{{\partial \varOmega_{1} }}{\partial \theta }} \\ 0 \\ {\frac{{\partial \varOmega_{3} }}{\partial \theta }} \\ \end{array} } \right){\text{d}}\theta - \left( {\begin{array}{*{20}c} 0 \\ 0 \\ {\frac{{\partial \varOmega_{3} }}{\partial e}} \\ \end{array} } \right){\text{d}}e - \left( {\begin{array}{*{20}c} {\frac{{\partial \varOmega_{1} }}{\partial \sigma }} \\ {\frac{{\partial \varOmega_{2} }}{\partial \sigma }} \\ {\frac{{\partial \varOmega_{3} }}{\partial \sigma }} \\ \end{array} } \right){\text{d}}\sigma .$$

(24)

Let $\Delta_{\varOmega }$ denote the determinant of the coefficient matrix of (24). For stability, it is assumed that $\Delta_{\varOmega } < 0$. Comparative statics results from (24) are the same as those from (20). For example, applying Cramer’s rule to (24) yields $\frac{{{\text{d}}w}}{{{\text{d}}e}} = \frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial e}/\Delta_{\varOmega } > 0$. This result is the same as that in Proposition 4. However, the system (24) can be used to derive the following additional results. Those results are not available from (20).

Partial differentiation of (21) and (23) reveals that $\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{3} }}{\partial K} - \frac{{\partial \varOmega_{1} }}{\partial K}\frac{{\partial \varOmega_{3} }}{\partial q} = \frac{2qf\theta }{K} > 0$. With $\frac{{\partial \varOmega_{2} }}{\partial n} < 0$ and $\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial K}\frac{{\partial \varOmega_{3} }}{\partial n} > 0$, it is clear that $\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial K} - \frac{{\partial \varOmega_{1} }}{\partial K}\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial q} - \frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial K}\frac{{\partial \varOmega_{3} }}{\partial n} < 0$. Partial differentiation of Eqs. (21) and (23) reveals that $\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{3} }}{\partial \sigma } - \frac{{\partial \varOmega_{1} }}{\partial \sigma }\frac{{\partial \varOmega_{3} }}{\partial q} = \frac{2fq\theta }{\sigma } > 0$. With $\frac{{\partial \varOmega_{2} }}{\partial n} < 0$ and $\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial \sigma }\frac{{\partial \varOmega_{3} }}{\partial n} > 0$, $\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial \sigma } - \frac{{\partial \varOmega_{1} }}{\partial \sigma }\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial q} - \frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial \sigma }\frac{{\partial \varOmega_{3} }}{\partial n} < 0$. Applying Cramer’s rule to (24) yields

$$\begin{aligned} \frac{{{\text{d}}w}}{{{\text{d}}K}} & = \left( {\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial K} - \frac{{\partial \varOmega_{1} }}{\partial K}\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial q} - \frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial K}\frac{{\partial \varOmega_{3} }}{\partial n}} \right)\big/\Delta_{\varOmega } > 0, \\ \frac{{{\text{d}}w}}{{{\text{d}}\sigma }} & = \left( {\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial \sigma } - \frac{{\partial \varOmega_{1} }}{\partial \sigma }\frac{{\partial \varOmega_{2} }}{\partial n}\frac{{\partial \varOmega_{3} }}{\partial q} - \frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{2} }}{\partial \sigma }\frac{{\partial \varOmega_{3} }}{\partial n}} \right)\big/\Delta_{\varOmega } > 0. \\ \end{aligned}$$

Partial differentiation of Eqs. (21) and (23) reveals that $\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{3} }}{\partial \theta } - \frac{{\partial \varOmega_{1} }}{\partial \theta }\frac{{\partial \varOmega_{3} }}{\partial q} = \theta q^{3} - \theta q^{3} = 0$. Applying Cramer’s rule to (24) yields

$$\begin{aligned} \frac{{{\text{d}}w}}{{{\text{d}}\theta }} & = \frac{{\partial \varOmega_{2} }}{\partial n}\left( {\frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{3} }}{\partial \theta } - \frac{{\partial \varOmega_{1} }}{\partial \theta }\frac{{\partial \varOmega_{3} }}{\partial q}} \right)/\Delta_{\varOmega } = 0, \\ \frac{{{\text{d}}n}}{{{\text{d}}\theta }} & = \frac{{\partial \varOmega_{2} }}{\partial w}\left( {\frac{{\partial \varOmega_{1} }}{\partial \theta }\frac{{\partial \varOmega_{3} }}{\partial q} - \frac{{\partial \varOmega_{1} }}{\partial q}\frac{{\partial \varOmega_{3} }}{\partial \theta }} \right)/\Delta_{\varOmega } = 0. \\ \end{aligned}$$