SECOND-BEST WORLDS
4.10
Monopoly model: the mathematics II
Now that you have seen your first monopoly model, let’s use the setting of our perfect competition example to analyse how such a model can be designed as an optimization model and as an equilibrium model and learn a bit on the economics of monopolies on the way.
The setup is the same with one change: we now assume monopolistic competition.
So instead of many firms,
(1)
But, given that it is the sole supplier on the market, the firm knows that its output decision will have an impact on the market price. That is why, contrary to a perfectly competitive market, the income side is not only price times quantity,
Now, how does the firm know how
Recall the benefit function
For example the demand-price relation we used in the last model exercises was given by:
or, rearranging the equation,
As a last step, you now need to consider that the monopoly is the only supplier in this market. Thus, the normal demand balance is now simply
Again, you will need to derive the first order conditions of the model with respect to the choice variable of the monopoly,
(2)
Note that the first term in the objective function,
Let’s switch to the equilibrium formulation. Similar to the perfect competitive version, we have three elements: the monopoly maximising its profit, consumers maximising their benefit, and a market that brings the two sides together. Thus, we actually only need to change the firm block and can keep the other elements as before.
Consumers are still described by their zero-profit logic:
(3)
And in the market clearing we just need to replace the output of all firms with the single output of our monopoly:
(4)
The zero-profit condition for the monopoly is a bit more tricky. You can either derive this logic by formulating the maximisation problem for the monopoly and make the first order derivation and implement this as the zero-profit condition; so basically what we did above for the optimization model. Or, you know the economic equilibrium condition for the behaviour of a monopoly: marginal revenue equals marginal costs.
In a perfectly competitive world, a firm simply sells its output on the market assuming that its own decision has no impact on the price. Thus, the marginal revenue a competitive firm can obtain is equal to the market price. Consequently the equilibrium condition was: market price equals marginal costs.
But a monopoly knows that its own output decision has an impact on the price. And since it is the only one selling this also has a feedback effect on the total sale revenue. Increasing the output by one unit increases the income by the market price,
(5)
The left hand side is again the marginal costs, the right hand side is the marginal revenue. A monopoly will produce
In other words, the optimization and equilibrium model are again providing the same result: a monopoly will produce up to the point where its profit is maximised and this point is defined where the additional revenue it can obtain is equal to the cost it has for this additional output.
This results in a significant shift between consumer and producer rent, a lower