---
tags: micro,
---
$\rho$-concavity and monopoly
===
$$
% My definitions
\def\ve{{\varepsilon}}
\def\dd{{\text{ d}}}
\newcommand{\dif}[2]{\frac{d #1}{d #2}} % for derivatives
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} % for partial derivatives
$$
## Monopolist's price
Suppose a monopoly firm in a market with demand function $q(p)$ , which is twice differentiable. The marginal cost is constant $c$ and the fixed cost is zero.

The firm's problem

$$
\max_p \quad (p-c)q(p)
$$

The first-order condition,

$$
q(p) + (p-c) q'(p)=0 \implies p = c - \frac{q(p)}{q'(p)}.
$$

## Pass-through
The pass-through is $\dif{p}{c}$. By the chain rule,

\begin{align}
\dif{p}{c} &= 1 - (d \frac{q(p)}{q'(p)}/ dp) \dif{p}{c}\\
\dif{p}{c} &= 1 - (\frac{q'(p)q'(p) - q''(p)q(p)}{q'(p)q'(p)}) \dif{p}{c}
\end{align}

To simplify the notation, we use $q'$ for $q'(p)$,

\begin{align}
&\dif{p}{c}(1 + \frac{q'^2-q'' q}{q'^2}) = 1 = \dif{p}{c}( \frac{2q'^2-q'' q}{q'^2})\\
&\dif{p}{c} = \frac{q'^2}{2q'^2-q'' q}
\end{align}

## $\rho$-concavity demand function

If $q(p)$ is a $\rho$-concavity demand function, then $(q(p))^\rho$ is a concave function, and its second derivative is non-positive. We have

\begin{align}
&\frac{d^2(q(p))^\rho}{dp dp} = \rho (\rho -1) q(p)^{\rho-2}q'(p) + \rho q(p)^{\rho-1} q''(p) \le 0 \\
&\implies (\rho -1) q'(p) + q(p) q''(p) \le 0
\end{align}

## Relations

Suppose $\dif{p}{c} \le \frac{1}{k}$, we have

$$
\dif{p}{c} = \frac{q'^2}{2q'^2-q'' q} \le \frac{1}{k} \implies k q'^2 \le 2q'^2-q'' q\\
q'^2(1-2k) \le -q'' qk \implies \left( \frac{1-2k}{k} \right) q'^2 +q'' q \le0
$$

Therefore, if the demand function is $\rho$-concavity, we have

$$
\rho-1 = \frac{1-2k}{k} \implies k = \frac{1}{\rho +1},
$$

and the pass-through is less or equal to $1/(\rho +1)$.

If the pass-through is less or equal to $1/k$ and the quantity is finite, we have

$$
\rho-1 = \frac{1-2k}{k} \implies \rho = \frac{1-k}{k},
$$

and the demand function is $\rho$-concavity.