# Notation? Inverse of demand function

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.

• Is it possible that it is just a typo? – Walrasian Auctioneer Jan 26 '20 at 20:34

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.