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# Monopoly model: the mathematics I

In this article, Frank Krysiak explains the mathematics that underlie the monopoly model.

Let’s examine the model in somewhat more detail. In this model, we assume that one firm has market power (the coal-based producer), whereas the other firms behave competitively.

They form a so-called competitive fringe. The monopolist takes the behaviour of all other actors in this model into account when making his decisions. To model this, we do the following in the underlying model.

First (for simplicity), we take out nuclear power as an option (we only want to have a single actor with market power and nuclear power stations are usually at least as large as coal-fired power stations).

Second, we leave all parts of the model as they have been in the preceding models, but do not solve the optimization problem of the coal-fired power plant yet. Rather, we calculate the production of the competitive fringe as:

[Q^{fringe} = sumlimits_{i=PV,Wind,Coal,Gas}{q_i}]

Demand is again given by

[d=bar{Q} -η cdot p]

Recall that (bar{Q}) is the maximum demand when (p=0).

The market clearing conditions is now

(1)   (d=Q^{fringe}+q_{coal})

This condition implies that production of the competitive fringe plus production of the non-competitive coal-fired power plant has to equal demand. Note that (Q^{fringe}), that is, the total production of the competitive fringe is already fully specified as a function of the market clearing price, (p), and the policy variables.

The coal-fired power plant wants to maximize its profit taking into account all this:

[max limits_{q_{coal},a_{coal}geq 0}enspace ⁡p cdot q_{coal} – c_{coal} (q_{coal},a_{coal} ) – t cdot e_{coal} (q_{coal} ,a_{coal} )]

The optimization model is then solved numerically with regard to (p_{coal}, a_{coal}), and (p) under the constraint given by equation (1). This yields the market outcome that is most desirable for the non-competitive producer, taking into account the behaviour of all other actors.