I mark all my own exam papers (whereas many other lecturers out-source this to tutors). I think this gives me a better feel for where students are going wrong and usually I can figure out why. This helps me to improve my teaching for future semesters, especially where it is some passing comment that students have adopted in their answer. Sometimes though, I'm sure it isn't something I've said that has led the students astray. One case of the latter happened in the A Semester ECON100 exam, when I asked a question about the options government has to restrict the market power of monopolies. Many students answered that an appropriate way to restrict market power was to tax the monopoly. Here's why that's not a correct answer.
A monopoly is a sole seller of its product. This gives it market power - the ability to choose a price that will maximise its profit. Consider the diagram below (which assumes a constant-cost firm). The monopoly will operate at the profit-maximising quantity (where marginal revenue (MR) is equal to marginal cost (MC)), which is QM. To sell that profit-maximising quantity they will set a price of PM (because at the price PM consumers will demand exactly QM units of the product).
With that price and quantity, the consumer surplus (the difference between the amount that consumers are willing to pay (shown by the demand curve), and the amount they actually pay (the price)) is the triangle area ABF. The producer surplus (the difference between the amount the monopoly producer receives (the price), and their costs (which are shown by the marginal cost curve)) is the rectangle area FBDG. Total welfare is the combination of consumer and producer surplus, i.e. the area ABDG.
If this was a perfectly competitive market, the market would operate at the point where supply is equal to demand (and note that the supply curve is the marginal cost curve for these constant-cost firms). The perfectly competitive market would operate at the quantity QC and price PC. Consumer surplus would be the triangle AEG, while producer surplus would be zero (because the price PC is equal to the cost of every unit produced), so total welfare is also the triangle AEG.
Now, comparing the monopoly with perfect competition, we see that market power (exercised by the monopoly firm) leads to a higher price (PM) than under perfect competition (PC). That in itself isn't a problem though. What is a problem is the loss of total welfare, which is ABDG with a monopoly firm, but AEG with perfect competition. The difference is the area BED - the deadweight loss of monopoly. It is this deadweight loss that is the main reason why governments might prefer to restrict market power.
So, what happens if you tax the monopoly firm? Consider the diagram below. The tax would likely be levied on the seller since this is administratively easier, so this is like increasing their costs. We represent the tax as a new curve S + tax (or MC + tax in this case for a monopoly). These 'higher' costs for the monopoly move the profit-maximising quantity down to QT (the quantity where MC + tax is equal to marginal revenue), and lead to an even higher price PT.
Consumer surplus with the taxed monopoly is the triangle area AJH, and producer surplus is the area HJKL. The government receives tax revenue equal to the area LKNG (this is part of total welfare for society because the government can use that revenue to pay for roads, schools, etc.). So total welfare with the taxed monopoly is the area AJNG. Notice that the combination of monopoly plus tax has increased the deadweight loss from the area BED to the area JEN. Taxing the monopoly makes the problem of lost welfare worse not better.
And that is why you can't use taxes to restrict the market power of a monopoly. The monopoly can still use its market power to set the price, and in response to the tax it will set an even higher price, increasing the size of the deadweight loss. A better response is to use a price control (a price ceiling - a maximum price that is lower than PM), government ownership (so the government can choose any price it wants, including a price below PM), or using anti-trust laws to prevent firms from merging into larger firms with market power in the first place.