Oligopoly model: the mathematics

Let’s examine how the oligopoly model is designed. It closely follows the monopoly model with respect to its general setup: we assume that coal-based producers have market power and the other firms behave competitively.

However, as we can have more than one coal-producer the market clearing is now given by:

[d=Q^{fringe}+sumlimits_{i=1}^{N}{q_i^{coal}}]

You could choose how many ((N)) coal firms, (i), are active in the market.

The remainder of the market interaction between supply and demand is unchanged with the downward sloping demand-price relation again defined as:

[d=bar{Q}-eta cdot p]

Using those two components one can again derive the price relation independent of the coal firm’s output choice. Contrary to the monopoly case the choice of any single firm, (i), is only a part of the total supply on the market. All other firms (termed (–i) or ‘not i’) are also supplying the market and have an impact on the resulting market price.

The coal producers take this relation into account when maximizing their profits. For each firm, (i), the respective profit function is given by:

(1)   (max limits_{q_i^{coal},a_i^{coal}≥0} enspace p(q_i^{coal},q_{-i}^{coal})cdot q_i^{coal} – c_{coal} (q_i^{coal},a_i^{coal} ) – tcdot e_{coal} (q_i^{coal} ,a_i^{coal} ))

The model follows the Cournot-competition logic: firms choose their output and simultaneously take the choice of the other firms as given. In other words, each firm (i) optimizes its own production decision ((q_i^{coal})) while assuming all other firms make their own respective optimal output decision ((q_{-i}^{coal})).

For the model formulation used in the exercise we make one further simplifying assumption: all firms are symmetric. But the formulation also holds if different firm characteristics are assumed (ie, different cost structures (c_i^{coal}) for the different oligopoly firms instead of the generic (c_{coal})).

The model is then again solved numerically with regard to (q_i^{coal},a_i^{coal}) and (p) for the number of firms, (N), you have chosen.