Notation? Inverse of demand function

Question

I'm looking through The Investment Decisions of Firms by S.J. Nickell, and I've come across some notation that I don't quite understand. Any clarification would be very welcome.

Nickell first assumes that firms face a downward sloping demand curve:

$$ z\{p(t)\}\beta(t) \textrm{ where } z_p<0. $$ He says:

The demand curve itself thus has a constant shape defined by $z\{p(t)\}$ and is shifted up and down by the time function $\beta(t)$.

So $z$ is some kind of functional form whose first derivative in $p$ is negative.

Now comes the part that confuses me. He assumes that firms adjust output according to demand.

As a result we have $$ z\{p(t)\}\beta(t) = F\{K(t),L(t)\}. $$ Solving for the output price we obtain $$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$

How did he get this? $F$ equals quantity demanded. Fine, it's the assumption. But why is $p$ the inverse of $z$?

Then he derives these partial derivatives, which are also confusing me because he brings $z$ back in:

$$ \frac{\partial p}{\partial K} = \frac{F_K}{z_p \beta(t)} \quad\textrm{and}\quad \frac{\partial p}{\partial L} = \frac{F_L}{z_p \beta(t)}.$$

Like I said, any help is most welcome. I think what's mostly confusing me is the inverse function bit. I can see how he gets the partial derivatives otherwise.

Thanks.

Answer 1

As a result we have $$ z\{p(t)\}\beta(t) = F\{K(t),L(t)\}. $$ Solving for the output price we obtain $$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$

I am pretty sure that what is meant here is that there is a $p(t)$ trajectory for which $$ z\{p(t)\} = F\{K(t),L(t)\} / \beta(t) $$ (see first equation)

Given the function $z$, this depends only on the right hand side of the equation, hence let us denote the trajectory $p(t)$ as a function of this, i.e. $$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$ A different notation would be $$ p(t) = z^{-1}\left( F\{K(t), L(t)\}/\beta(t) \right). $$ This second notation is useful because all the partial derivatives follow from this notation and the chain rule.