I have a question about a derivation in Abraham (1987)'s simple job market matching model (equations 3 through 7):
She begins by writing down tautologies:
J - V = L - U = E
where J is the number of jobs, V is the number of vacancies, L is the total labor force, U is the total unemployed, and E is the total employed. Consider a probability function that a particular vacancy will be filled:
p(V/U) where $p_u > 0, p_v <0, p_{uu} < 0, p_{vv} > 0$.
Now, letting the total quits and firings from the current stock of employees be sE we must have: \begin{equation} p(V/U)V - sE =0 \end{equation} Letting U/E = u, V/E = v, and dividing through by E we get:
\begin{equation} p(v/u)v = s \end{equation}
Now things get tricky. She writes "along the set of points satisfying (the above)" we have:
\begin{equation} \frac{dv}{du} = - \frac{p_u}{p+p_v v} \end{equation}
My question is: how? Suppose I treat this as an implicit differentiation problem. Write $F(v,u) = p(v/u)v-s = 0$ then we know we know:
\begin{equation} \frac{d v}{d u} = - \frac{\partial F}{du} / \frac{\partial F}{dv} \end{equation}
I obtain $F_v = p + (v/u)p_v $ and $F_u = - (\frac{v}{u})^2 p_{u}$ which yields:
\begin{equation} \frac{dv}{du} = \frac{(\frac{v}{u})^2 p_{u}}{p + (v/u)p_v} \end{equation}
This doesn't look very close. Am I missing something? What's going on here?
EDIT:
Inspired by the answer below, I agree it is as simple as totally differentiating P(u,v) = p(v/u).
\begin{equation} p(v/u) = P(v,u) \rightarrow P(v,u)v - s = 0 \end{equation}
Now, let's totally differentiate this expression with respect to v,u, and s.
\begin{equation} \left[P_v(v,u)v + P\right] dv + v P_u du - ds = 0 \end{equation}
Consider $ds = 0 $ e.g. no change in the separation rate. This yields:
\begin{equation} \frac{dv}{du} = - \frac{vP_u}{vP_v + P} \end{equation}
Substituting $p(v/u)$ for $P(v,u)$ gives the expression off by a factor v.