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Monopolies don't maximize welfare because they set prices above the equilibrium price, leading to dead-weight loss. It is possible for the government to provide a per unit subsidy to a monopoly until the producer's marginal cost equals the consumer's marginal benefit at the monopoly's chosen quantity, which would maximize welfare. This change would increase producer surplus and consumer surplus in this market.

To a monopolist producer, a per unit subsidy is essentially equivalent to shifting the demand curve up by the value of the subsidy. It would be possible to shift the demand curve up until the optimal quantity produced is the higher equilibrium quantity. This requires that the subsidy provider knows the complete demand curve to set the optimal subsidy.

The subsidy must come from somewhere, like a tax on a different market. It seems plausible that there would be situations where even if all tax incidence fell on consumers, consumer surplus would still be greater after the subsidization. If consumers and producers are overall better off after a subsidy, why don't consumers do this without government intervention?

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    $\begingroup$ A monopoly's marginal cost does equal it's marginal benefit if it is maximizing profits $\endgroup$ – DornerA Nov 26 '15 at 17:24
  • $\begingroup$ I meant that the consumer's marginal benefit equals the producer's marginal cost at the equilibrium quantity. $\endgroup$ – InquisitivePerson Nov 26 '15 at 17:34
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    $\begingroup$ @HerrK. I think he means a marginal subsidy (x\$ / unit) which would incentivize the monopolist. $\endgroup$ – Giskard Nov 26 '15 at 18:02
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    $\begingroup$ @denesp I don't see why that is necessarily true. For example, consider a demand function D(p) where D(p)=0 at p>3, D(p)=1 at 3>p>1, D(p)=2 at 1>p>0, and D(p)=3.5 at p<=0, Assume marginal cost is always 0. Without a subsidy, the price is set to 3, resulting in a producer surplus of 3. Consumer surplus is 0. A per-unit subsidy of 1 is provided. Now the optimal price is 0. Producer surplus is 3.5. Consumer surplus is 4. The increase consumer surplus (4) is greater than the subsidy cost (3.5). $\endgroup$ – InquisitivePerson Nov 26 '15 at 19:40
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    $\begingroup$ @user2662680 Taxing the market would cause the monopoly to set its price higher. This is because the producer's marginal revenue falls, while marginal costs remain the same, lowering quantity at the intersection point. This means a lower total welfare, since quantity sold is now even farther from the equilibrium quantity. $\endgroup$ – InquisitivePerson Nov 28 '15 at 19:57
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As you pointed out, the problem of inefficient supply by a profit maximizing monopolist can be solved via subsidizing the monopolist to increase his marginal revenue. The subsidy can be paid by the consumers or by a central government.

Consumers unfortunately perceive the problem as a collective action problem: The marginal utility gain from subsidizing the monopoly is much lower than the marginal welfare gain of society, making individuals reluctant to subsidize the monopoly.

Governments, however, can solve the collective action problem by raising a tax and thus financing the subsidy. However, governments usually do not only pursue efficiency but also interpersonal equity. If the government cannot transfer the monopolist's earnings lump sum to the consumers, then this may conflict with the goals of interpersonal equity.

To give a simple example: Suppose rich individual A owns a monopoly on water. There are many consumers of water who are all poor. A sets the profit maximizing price/quantity. If the government maximizes total utility and all consumers have identical concave utility functions, the government would like to impose the efficient subsidy and tax the rich individual A lump sum to finance the subsidy and possibly transfers to the poor. However, if the lump sum tax on A is not available, we are in what is called a second-best solution. In this case, it may be optimal for the government to permit some inefficiency from the monopoly rather than taxing the poor to pay for the subsidy.

As an aside: Most interestingly, if there is an omniscient government which maximizes social welfare and can make lump-sum transfers, then under some regularity conditions the collective action problem disappears without the government imposing the subsidy! Thus, I would have an interest to subsidize the monopoly on my own. This is because the utility possibility frontier is pushed outwards by the subsidy and the government will reimburse me lump sum for my cost.

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The people at large view monopolies as unfair market actors, and the idea of subsidizing such unfairness is the reason that real people in the current world don't want to subsidize monopolies.

If you're asking why economists don't advocate subsidizing monopolies, I'm actually not sure. The standard treatment of monopolies seems to lend itself to that advocation.

But the real reason monopoly subsidies aren't appropriate is because they're simply not necessary (and if used would almost definitely lead to more inefficiency). Price discrimination eliminates most if not all problems of inefficient allocation, since if the monopoly can price discriminate, it can set different prices such that most if not all potentially mutually-profitable trades will be made. And the monopoly has an incentive to do this because it increases their profits above setting a single price.

For more on this, economics PhD Stephen Shmanske wrote a paper entitled The Monopoly Nonproblem

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