Optimal tax theory or the theory of optimal taxation is the study of designing and implementing a tax that maximises a social welfare function subject to economic constraints. The social welfare function used is typically a function of individuals' utilities, most commonly some form of utilitarian function, so the tax system is chosen to maximise the aggregate of individual utilities. Tax revenue is required to fund the provision of public goods and other government services, as well as for redistribution from rich to poor individuals. However, most taxes distort individual behavior, because the activity that is taxed becomes relatively less desirable; for instance, taxes on labour income reduce the incentive to work. The optimization problem involves minimizing the distortions caused by taxation, while achieving desired levels of redistribution and revenue. Some taxes are thought to be less distorting, such as lump-sum taxes (where individuals cannot change their behaviour to reduce their tax burden) and Pigouvian taxes, where the market consumption of a good is inefficient, and a tax brings consumption closer to the efficient level. In the Wealth of Nations, Adam Smith observed that
“Good taxes meet four major criteria. They are (1) proportionate to incomes or abilities to pay (2) certain rather than arbitrary (3) payable at times and in ways convenient to the taxpayers and (4) cheap to administer and collect.”
Tax revenue Generating a sufficient amount of revenue to finance government is arguably the most important purpose of the tax system. Optimal taxation theory attempts to derive the system of taxation that will achieve the desired revenue and income distribution with the least inefficiency—that is, that interferes least with market participants making Pareto optimal exchanges—economic transactions that make both parties better off. Free Market economies use prices to allocate resources to produce the products society wants most. If demand exceeds supply, the price will rise as those who want the product most compete to buy it. The high price induces producers to make more, until supply is adequate to meet demand and the price comes down. If supply exceeds demand, the price falls as producers try to induce more people to buy the product. The low prices then induce producers to make something else, that consumers want more. If the government imposes a tax however, the price the consumer pays is different from the price the producer receives because the government takes its cut. If demand is inelastic—if consumers will pay what they must to get the product at any price, consumers will pay the tax and government will appropriate some of their benefit from the transaction (and hopefully provide useful services like public education in exchange). If supply is inelastic—producers will sell the same amount regardless of price—producers will pay the tax and government will take some of their benefit from the transaction. Note that it does not matter which side actually writes the government's check, the market price will adjust to compensate (see Tax incidence). However, if both supply and demand are elastic—producers will make less at a lower price and consumers will buy less at a higher price—then the equilibrium quantity will decrease. There may be a consumer willing to buy at a price for which a producer is willing to sell, but this Pareto optimal transaction does not occur because neither is willing to pay the government's cut. The consumer then buys something less desirable and the producer makes something less profitable (or simply produces less and enjoys more leisure), so that the economy is no longer producing the optimal mix of products. Moreover, the sale does not occur, so the government never collects the revenue that was the whole reason for the distortion. This is the deadweight loss—the government has not merely taken a cut of the benefits from the exchange, it has destroyed those benefits for all three. These are the results optimal tax theorists seek to avoid.
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