Deriving pass-through for an N-firm Cournot oligopoly

02 Oct 2015

In the previous post, I mentioned the concept of pass-through. Pass-through is the partial derivative of price with respect to marginal cost; it tells us how the equilibrium price will change in response to a change in marginal cost. As we will see here, it is closely related to the curvature of the inverse demand function.

Consider a homogeneous product market with \(N \ge 2\) firms which are Cournot competitors with the same constant marginal cost \(c\). The inverse demand function is \(P(Q)\), and it satisfies \(2P'(Q) + QP''(Q) \lt 0\). The first order condition for firm \(i\) when every firm produces positive output is given by \(P(Q) + q_{i}P'(Q) - c = 0\). Because the firms are symmetric, \(q_i = \frac{Q}{N}\), making firm \(i\)’s first order condition \(P(Q) + \frac{Q}{N}P'(Q) - c = 0\). Call this FOC as \(F\).

Define \(\sigma(Q) = -\frac{QP''(Q)}{P'(Q)}\), the curvature of the inverse demand function.

By the implicit function theorem, we can find \(\frac{\partial Q}{\partial c}\), the quantity pass-through, as

\[\begin{align} \frac{\partial Q}{\partial c} & = - \frac{\frac{\partial F}{\partial c}}{\frac{\partial F}{\partial Q}} \cr & = \frac{N}{(N+1)P'(Q) + QP''(Q)} \cr \end{align}\]

Then, by the chain rule,

\[\begin{align} \frac{\partial P(Q)}{\partial c} & = \frac{\partial P}{\partial Q}\frac{\partial Q}{\partial c} \cr \implies \frac{\partial P(Q)}{\partial c} & = \frac{\partial P}{\partial Q} \frac{N}{(N+1)P'(Q) + QP''(Q)} \cr \end{align}\]

Dividing by \(\frac{\partial P}{\partial Q}\) on top and bottom, we get

\[\begin{align} \frac{\partial P}{\partial c} & = \frac{N}{(N+1) + Q\frac{P''(Q)}{P'(Q)}} \cr & = \frac{N}{(N+1) - \sigma(Q)} \cr \end{align}\]

Thus, we see that the pass-through rate is indeed a function of the curvature of the inverse demand function. If we assume inverse demand is linear, we get

\[P'(c) = \frac{N}{N+1}\]

As \(N \to \infty\), \(P'(c) \to 1\). This makes sense, given that we know as \(N \to \infty\), \(P \to c\).

Pass-through is a pretty useful concept, and it seems to be a popular tool these days in analyses of oligopoly and market power.