# Average cost and marginal cost pricing rule

Average cost and marginal cost pricing rule are both regulatory regimes that can be used by governments in attempt to regulate a monopoly.

### Average cost pricing rule

The average cost pricing rule is when the government forces a monopoly to charge a price which coincides with the average cost of production.

### Marginal cost pricing rule

Marginal cost pricing rule is when the government requires a monopoly to charge a price which equates to the marginal cost of production.

Consider the following example:

Determine the equilibrium price and quantity of a monopoly where (a) the monopolists behaves as a profit maximizing firm, (b) the monopolist is required to produce output according to the average cost pricing rule and (c) the monopolist is regulated such that it must produce output according to the marginal cost pricing rule. Suppose that the monopoly faces the inverse demand equation $P + 100 - 2Q$ and the firms total costs are defined $TC = 4Q + 2Q^2$.

To determine the profit maximizing level of 0utput for the monoply, we must equate marginal cost with the marginal revenue of the monopoly. Firstly, to calculate the marginal revenue $(MR)$ function we require the revenue $(R)$ function. The revenue function is defined as the product of the firms quantity and price, such that

$R = 100Q - 2Q^2$

The corresponding marginal revenue function is defined

$MR = 100 - 4Q$

We have been given the total cost equation and thus only need to derive the marginal cost equation. The marginal cost equation is:

$MC = 4 + 4Q$

Equating the marginal cost and marginal revenue yields

$4 + 4Q = 100 - 4Q$

$Q = 12$

Substituting this quantity into our inverse demand equation gives the price as

$P = 100 - 2(12) = 76$

The firms profit is defined as:

Profit = R - TC,

such that

$\text{profit} = 76*12 - [4*(24) + 2*24^2] = 912 - 288 = 624$

In case (b) the monopolist is required to produce output according to the average cost pricing rule. This is achieve by setting the price of output equal to the average cost.

The average cost $(AC)$ is defined as

$AC = \frac{TC}{Q}$,

such that

$AC = \frac{4Q + 2Q^2}{Q}$

$AC = 4 + 2Q$

In this case, the average cost curve is a positive function of the quantity produced which means average costs increase as the monopolies output increases. The monopoly also faces a downward sloping demand curve, which means that the price they can charge decreases the more output that they produce. Thus to find the point where the price and the average cost curve are equal, we must equate the inverse demand equation with the average cost function, such that:

$100 - 2Q = 4 + 2Q$

Rearranging to find for Q, we find that:

$Q = 24$

Substituting this back into the inverse demand equation yields

$P = 100 - 2*(24) = 52$

In this case, the output is higher and the price is lower than if the monopoly operated without regulation. The profit in this case will equal 0 - which you should be able to verify yourself.

In case (c) we are asked to find the equilibrium price and quantity when the marginal cost pricing rule is applied. In case (b) we observed how average costs were an increasing function of the quantity of output produced. Similarly, the marginal cost function is an increasing function of the amount of output produced. Therefore, to determine the equilibrium level of output, we must equate the marginal cost function with the inverse demand function, such that

$4 + 4Q = 100 - 2Q$

Rearranging to solve for Q yields:

$Q = 16$

Substituting this Q back into our inverse demand equation yields

$P = 100 - 2(16) = 68$

Finally, the profit in this scenario can be calculated as:

$\text{Profit} = 16*68 - 4*16 - 2*(16)^2 = 1088 - 576 = 512$

We can see in this case, that the marginal cost pricing rule generates a higher price and lower output compared with the average cost pricing rule. However, it is still less than the monopoly operating in the absence of regulations.