<blockquote>
  <p>As a result we have <span class="math-container">$$ z\{p(t)\}\beta(t) = F\{K(t),L(t)\}. $$</span> Solving
  for the output price we obtain <span class="math-container">$$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$</span></p>
</blockquote>

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

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

CC BY-SA 4.0
2020-01-29T20:40:25.530Z
2020-01-29T20:40:25.530Z
Giskard
<p>I'm looking through <em>The Investment Decisions of Firms</em> by S.J. Nickell, and I've come across some notation that I don't quite understand. Any clarification would be very welcome.</p>

<p>Nickell first assumes that firms face a downward sloping demand curve:</p>

<p><span class="math-container">$$
z\{p(t)\}\beta(t) \textrm{ where } z_p&lt;0.
$$</span>
He says: </p>

<blockquote>
  <p>The demand curve itself thus has a constant shape defined by
  <span class="math-container">$z\{p(t)\}$</span> and is shifted up and down by the time function
  <span class="math-container">$\beta(t)$</span>.</p>
</blockquote>

<p>So <span class="math-container">$z$</span> is some kind of functional form whose first derivative in <span class="math-container">$p$</span> is negative.</p>

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

<blockquote>
  <p>As a result we have <span class="math-container">$$ z\{p(t)\}\beta(t) = F\{K(t),L(t)\}. $$</span> Solving
  for the output price we obtain <span class="math-container">$$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$</span></p>
</blockquote>

<p>How did he get this? <span class="math-container">$F$</span> equals quantity demanded. Fine, it's the assumption. But why is <span class="math-container">$p$</span> the inverse of <span class="math-container">$z$</span>?</p>

<p>Then he derives these partial derivatives, which are also confusing me because he brings <span class="math-container">$z$</span> back in:</p>

<blockquote>
  <p><span class="math-container">$$
\frac{\partial p}{\partial K} = \frac{F_K}{z_p \beta(t)} \quad\textrm{and}\quad 
\frac{\partial p}{\partial L} = \frac{F_L}{z_p \beta(t)}.$$</span></p>
</blockquote>

<p>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.</p>

<p>Thanks.</p>

CC BY-SA 4.0
2020-01-26T20:34:54.503Z
Is it possible that it is just a typo?
Walrasian Auctioneer
<microeconomics><mathematical-economics>
Notation? Inverse of demand function
Lou
# Notation? Inverse of demand function

(Asked by: username_0 on 2020-01-25)

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.

**Tags:** microeconomics, mathematical-economics
> **Comments:**
> **username_1**: Is it possible that it is just a typo?  

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## 1 answers

### Answer by username_2 on 2020-01-29. Score: 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.


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