.

Appendix B - Equations and Computations

Included in this appendix is the mathematics behind the equations summarized in Section IV and V of this paper. Those in the text should provide an adequate understanding of the dynamics of the model, but these are included for reference if anything is unclear. A profit maximizing firm will optimize according to,

$$H = \frac{\partial \pi}{\partial w} = 0,$$ (7)

$$K = \frac{\partial \pi}{\partial l} = 0.$$ (8)

Hence by the implicit function theorem,

$$\frac{d w}{d l} = \frac{-\partial H / \partial l}{\partial H / \partial w}.$$ (9)

Thus, substituting in H,

$$\frac{d w}{d l} = \frac{-\partial^2 \pi / \partial w \partial l}{\partial^2 \pi / \partial w^2}.$$ (10)

Similarly, by solving and substituting in K,

$$\frac{d l}{d w} = \frac{-\partial^2 \pi / \partial w \partial l}{\partial^2 \pi / \partial l^2}.$$ (11)

The production function can be defined as follows,

$$f(w,l) = \left\{ \begin{array}{ll} g(l) & \textrm{if $\epsilon = e$}\\ 0 & \textrm{if $\epsilon = 0.$} \end{array} \right.$$ (12)

Utility variables:

$$\begin{array}{llllllll} V^N &=& \textrm{The level of utili... ...e level of utility gained from not working.}\\ \end{array}$$

The equations for $V^N$, $V^S$, and $V^A$, in terms of the other variables defined thus far:

$$V^N = w - e + \frac{(1-q)V^N}{(1+r)}+\frac{qV^A}{(1+r).}$$ (13)

$$V^S = w + \frac{(1-q)(1-D)V^S}{(1+r)}+\frac{[1-(1-q)(1-D)]V^A}{(1+r).}$$ (14)

$$V^A = \bar{w} + \frac{sV^N + (1-s)V^A}{(1+r)}.$$ (15)

The new production function,

$$f(w,l) = \left\{ \begin{array}{ll} g(l) & \textrm{if $w \geq w_{ns}$}\\ 0 & \textrm{if $w < w_{ns}$}. \end{array} \right.$$

Taking the derivative of profits with respect to labor,

$$\frac{d\pi}{dl} = g'(l) - [w+lw'(l)] = 0.$$ (16)

We obtain this via the chain rule because the wage is now a function of the amount of labor employed at a firm. This was one of our assumptions: $D$ the probability of detecting a shirker is a function of the size of the labor force. Since $D$ is in our equation for $w_{ns}$, $w_{ns}$ is a function of $l$. To determine $w'(l)$, take the derivative of equation 2 with respect to $l$, yielding:

$$w'(l) = -\frac{e(r+s+q)D'(l)}{D^2(1-q)}>0.$$ (17)

Similiarly,

$$l'(w) = -\frac{D^2(1-q)}{e(r+s+q)D'(l)}>0.$$ (18)

Long Run Analysis.
Optimize profit function such that,

$$\partial \pi / \partial l = pg'(l) - [w + lw'(l)].$$ (19)

Setting this equal to zero and solving for w,

$$w = pg'(l) - lw'(l).$$ (20)

Since this is a long run analysis, we can assume the economic profits are 0, or $wl = pg(l) - R$. Thus,

$$w = \frac{pg(l) - R}{l}.$$ (21)

Setting the zero profit criteria equal to the optimization criteria,

$$\frac{pg(l)}{l} - \frac{R}{l} = pg'(l) - lw'(l).$$ (22)

$$p(\frac{g(l)}{l}-g'(l)) = \frac{R}{l} - lw'(l).$$ (23)

$$p(AP_L - MP_L) = \frac{R}{l} - lw'(l).$$ (24)