Home > Micro concepts, Monopoly and market power > Taxing a monopoly firm

## Taxing a monopoly firm

September 29, 2011

The model of a monopoly firm I made had a simple demand function of Q = 500 – P and a marginal cost of MC = 150 with no fixed costs.

This firm was producing output of 175 and selling at price 325, bringing in profits of 30265.

Now we can look at what would happen if the government introduced a tax on the monopoly. There are two common types of taxes that are analysed in microeconomics – specific taxes (also known as sales taxes or quantity taxes) and ad valorem taxes.

Specific taxes are charged per unit of output. This means that the price that the supplier (the firm) receives is going to be different from the price the demander (the consumer) pays.

Consumers will not change their view on how they value the product just because there is a tax on it, they will still demand the same amount of product at the same price (to them) as before. So the consumer demand function remains the same, $P_D = 500 - Q$.

However, the price the supplier receives will now be less, as the supplier has to pay $\tau$, the specific tax, on each unit, to the government. So the effective demand function to the supplier is $P_S = 500 - Q - \tau$.

So lets see what happens if the government introduces a specific tax of 50 per unit. The effective demand function to the supplier is $P_S = 500 - Q - 50 \Rightarrow P_S = 450 - Q$.

The total revenue to the supplier is $TR_S = 450Q - Q^2$ so marginal revenue is $MR_S = \frac{d(TR_S)}{dq} = 450 - 2Q$.

Setting MR = MC when MC is 150 means $450 - 2Q = 150 \Rightarrow 150 = Q$.

Now if the supplier produces output of 150, we can look back at the original consumer demand curve to see the price at which consumers would demand 150 – this is 500-150 = 350.

So the supplier produces output of 150 and sells at a price of 350. However of the price of 350 for each item, 50 goes to the government and 300 goes to the supplier, the tax drives a wedge between the price paid by the consumer and the price received by the supplier.

On the graph D1 shows the original consumer demand curve, and D2 shows the effective demand curve faced by the firm once the tax is taken into account. MR2 shows the marginal revenue curve faced by the supplier once the tax is taken into account.

Pm1 and Qm1 give the original price and quantity combinations for the monopoly firm before the tax was introduced. Pm2 and Qm2 give the new price and quantity combinations when the tax is in place. Ps2 is the price received by the supplier when the tax is in place.

The green shaded area represents the profits for the government (the tax revenue). The red shaded area shows the profits of the firm.

Remember that originally without the tax, the firm made profits of 30265. Now, the firm’s profits are $\pi = TR - TC = PQ - (MC)Q = 300(150) - (150)(150) = 22500$. Governments profits are $\tau Q = 50(150) = 7500$. If you add the two together you get total profits of 30000, which is lower than the firm’s profits in the situation without the tax, so there is some deadweight loss here.

Ad valorem taxes are charged as a proportion of the price of the product. VAT is an example of an ad valorem tax, you pay an additional 20% of the price to the producer, in tax.

Again the price that the supplier (the firm) receives is going to be different from the price the demander (the consumer) pays. The supplier has to pay $\alpha$, the ad valorem tax, as a proportion of the price on each unit, to the government. The relationship between the price the demander pays and the price the supplier gets is $P_D = P_S (1+\alpha)$

So here the effective demand function to the supplier is $P_S (1+\alpha) = 500 - Q$.

We can look at what would happen if the government introduced an ad valorem tax of 33.333% of the price received by the supplier. The effective demand function to the supplier is $1.333 P_S = 500 - Q \Rightarrow P_S = 375 - 0.75Q$.

The total revenue to the supplier is $TR_S = 375Q - 0.75Q^2$ so marginal revenue is $MR_S = \frac{d(TR_S)}{dq} = 375 - 1.5Q$.

Setting MR = MC when MC is 150 means $375 - 1.5Q = 150 \Rightarrow 150 = Q$.

Again looking back at the original consumer demand curve, if the supplier produces output of 150, the price at which consumers will demand this amount is 350. The reason I chose the ad valorem rate of 33.333% was to fit with the other example – this level of ad valorem tax has the same impact on consumers as the specific tax of 50 per unit, it means the quantity produced falls to 150 and the price rises to 350.

But this time the wedge between the price paid by consumers and price received by suppliers is different. As $P_D = P_S (1+\alpha)$, $350 = 1.333 P_S \Rightarrow P_S = 262.5$, which means that of the price of 350 for each item, 262.5 goes to the supplier and 87.5 goes to the government. The government is making more revenue from each unit this time.

This time the green shaded area representing the profits for the government (the tax revenue) is bigger, and the red shaded area showing the profits of the firm, is smaller.

Under this ad valorem tax, the firm’s profits are $\pi = TR - TC = PQ - (MC)Q = 262.5(150) - (150)(150) = 16875$. Governments profits are $\alpha P Q = 0.333(262.5)(150) = 13125$. If you add the two together you get total profits again of 30000, but there has been more redistribution of profits away from the firm towards the government.

Generally when a firm has market power, the government will make more revenue by charging an ad valorem tax as it takes advantage of the firm’s power to charge a price higher than marginal cost. Monopolists generally produce less output than the competitive level, and use their market power to charge a higher price for less production. A specific tax will not be as much of a revenue earner when output is cut as it is a constant tax charged per unit sold, but the value of an ad valorem tax increases as the firm increases the price. So ‘pound-for-pound’ as it were, when a specific tax and ad valorem tax on a monopolist have the same effect on the price faced by the consumer, the ad valorem tax will be better for government and worse for the firm.