Optimal Taxation

Optimal Taxation and Health Taxes

Optimal taxation is the part of welfare economics that asks which taxes a government should levy to raise the revenue it needs at the lowest cost in social welfare. The theory treats tax design as the maximisation of a social welfare function, given that people respond to taxes and that the government cannot see each person's earning ability. Health economists meet it most often in taxes on tobacco, alcohol and sugary drinks, where one tax raises revenue and corrects harm while falling hardest on poorer households. It also bears on appraisal, through the extra cost of raising public money with distorting taxes.

The planner's problem in optimal tax theory

Mankiw, Weinzierl and Yagan (2009) summarise the standard theory. A tax system should be chosen to maximise a social welfare function subject to a set of constraints. They trace the modern field to Ramsey (1927) and Mirrlees (1971).

With identical taxpayers the best tax would be a lump-sum tax, a fixed charge that no choice can change, so it distorts nothing. Mankiw and colleagues explain that such taxes are rarely used because they fall equally on rich and poor and so place a greater relative burden on the poor, citing the UK community charge, introduced from 1989 and quickly revoked.

Mirrlees reframed the problem around information. People differ in their innate ability to earn, and the planner can observe income but neither ability nor effort, so taxing income in order to reach high ability discourages effort. This formalises the equity-efficiency trade-off that real governments face, and it has become the dominant approach among tax theorists. Tax by Design, the Mirrlees Review of the UK tax system, describes optimal tax theory as balancing efficiency losses against the desire for redistribution and the need to raise revenue, with the value judgements about distribution made explicit.

The structure is familiar from health technology assessment. A cost-effectiveness decision under a fixed NHS budget constraint maximises health given a resource limit, while optimal tax theory maximises welfare given a revenue requirement and the behaviour of taxpayers.

Taxes on goods and the inverse elasticity rule

Ramsey asked how a fixed sum should be raised from taxes on commodities alone. Mankiw and colleagues state his answer as taxing goods in inverse proportion to their elasticity of demand, so that goods with inelastic demand are taxed more heavily. In Tax by Design's account the deadweight loss of a tax rises with the elasticity of demand, so the loss is smallest when higher rates fall on goods with relatively inelastic demand.

$$\frac{t_i}{p_i} = \frac{k}{\varepsilon_i}$$

where $t_i$ is the tax per unit on good $i$, $p_i$ is the price consumers pay including the tax, $\varepsilon_i$ is the absolute value of the compensated price elasticity of demand for good $i$, and $k$ is a constant set so that the taxes raise the required revenue. Tax by Design notes that this simple form holds only when goods are neither substitutes nor complements, and that the more general Ramsey rule covers those cases.

Two later results narrow the rule. Diamond and Mirrlees (1971) suggest that taxes on intermediate goods should be zero, and Atkinson and Stiglitz (1976) that final goods should be taxed uniformly when a fully nonlinear income tax is available, preferences are weakly separable between leisure and goods, and tastes for goods do not vary with ability. Mankiw and colleagues draw the lesson that a planner should not redistribute by taxing luxuries more than necessities, and they name goods with externalities as the well-known exception that justifies corrective Pigouvian taxes. Under those conditions a tax on a harmful product needs a corrective reason, since redistribution is better done through the income tax.

Income taxes, welfare weights and the top rate

Mirrlees-style models also set the shape of the income tax schedule. Diamond and Saez (2011) derive the top marginal rate that maximises revenue from the highest earners, which is optimal when society places almost no weight on their extra consumption.

$$\tau^{*} = \frac{1}{1 + a e}$$

where $\tau^{*}$ is the optimal top marginal tax rate, $a$ is the Pareto parameter that measures how thin the top tail of the income distribution is, and $e$ is the elasticity of reported income with respect to the share of a marginal unit of income that the taxpayer keeps. With $a$ = 1.5 for the United States and a mid-range elasticity of 0.25, they obtain 1 / (1 + 1.5 × 0.25) = 0.73, a top rate of about 73 per cent.

The social weights behind this are the same device that appraisal uses for distribution. If the social value of utility is logarithmic in consumption, Diamond and Saez note, weights are inversely proportional to consumption, so the weight at the USD 1,364,000 average income of the top 1 per cent in 2007 was only 3.9 per cent of the weight at the median family income of USD 52,700. HM Treasury's Green Book 2026 estimates a group's welfare weight as the national median equivalised income divided by the group's median, raised to the power 1.25, an estimate of the elasticity of marginal utility of consumption. Distributional cost-effectiveness analysis asks the same question of how much a gain to a poorer person should count.

Corrective taxes on tobacco, alcohol and sugary drinks

Tax by Design sets out the corrective case. The tax on a good whose consumption harms others should reflect the damage that a little more consumption creates, although most moderate drinkers impose no costs on others while a minority may impose very high costs. The Review adds that taxes on tobacco and alcohol rest partly on the contentious argument that consumption harms consumers themselves, an internality.

Allcott, Lockwood and Taubinsky (2019) combine the negative externality, the internality and the distributional concern in one formula.

$$t = \bar{\gamma}(1 + \sigma) + e + \frac{\mathrm{Cov}[g(z), s_{pref}(z)]}{d\bar{s}/dt}$$

where $t$ is the optimal tax per unit, $\bar{\gamma}$ is the average marginal bias (the internality) in money per unit, $\sigma$ measures how far the bias reductions caused by the tax are concentrated among people with high welfare weights, $e$ is the marginal externality per unit, $g(z)$ is the social marginal welfare weight at income $z$, $s_{pref}(z)$ is the consumption at income $z$, relative to the lowest income, that comes from preferences rather than income effects, and $d\bar{s}/dt$ is the response of average consumption to the tax, which is negative. The first two terms are the corrective motive and the last is the redistributive motive. With constant welfare weights the tax reduces to $\bar{\gamma} + e$, the core of a Pigouvian tax, and when poorer people buy more only because of their income the last term is zero.

The authors stress that externalities and internalities differ from health harms, since a consumer may rationally drink something despite the health risk because the enjoyment outweighs it. High demand elasticity makes the tax mainly corrective, and low elasticity makes its effects mainly redistributive. For the United States they estimate an optimal tax on sugar-sweetened beverages of 1 to 2.1 cents per ounce, roughly 0.8 + 1.0 × 1.2 − 0.5 = 1.5 cents once the internality is scaled up by 20 per cent and the regressive incidence is netted off, against about 0.4 cents with externalities alone. They also argue it is more efficient to tax grams of sugar than ounces of drink. The UK Soft Drinks Industry Levy moves part of the way by banding its charge by sugar content.

Worked example with an illustrative tax on sugary drinks

All numbers in this example are illustrative. A health ministry sets a per-litre tax on sugary drinks. Its analysts put the marginal externality, mainly NHS treatment costs that drinkers do not bear, at GBP 0.10 per litre and the average internality at GBP 0.15. They take $\sigma$ = 0.2 and put the redistributive term at minus GBP 0.06, because low-income households drink more partly out of preference.

1. Externality only. The tax is GBP 0.10 per litre.

2. Internality with equal welfare weights. The tax becomes 0.15 + 0.10 = GBP 0.25 per litre.

3. Progressivity of correction. Lower-income drinkers have larger biases, so 0.15 × 1.2 + 0.10 = 0.28, or GBP 0.28 per litre.

4. Redistributive motive. The tax falls to 0.28 − 0.06 = 0.22, or GBP 0.22 per litre, about 0.22 × 0.33 = 0.0726, or 7p, on a 330ml can.

5. Less responsive demand. If demand were half as responsive and the other terms stayed fixed, the redistributive term would double, since it divides by the consumption response, and the tax would fall to 0.28 − 0.12 = 0.16, or GBP 0.16 per litre.

The result sits well above the externality-only tax, so most of the case rests on the internality, the hardest number to measure. A ministry that counted only health gains would push the tax higher, because nothing in its objective records the enjoyment the tax removes.

Welfare maximisation and health maximisation

Health taxes show the welfarist and extra-welfarist split. Allcott and colleagues argue that maximising health cannot be the right social objective, since the way to maximise health would be to ban sugary and fatty foods, and their framework instead trades health-related externalities and internalities against consumer surplus, producer surplus and government revenue. That is the welfarist position of welfare economics, with outcomes judged by utility.

Brouwer, Culyer, van Exel and Rutten (2008) note that some writers read extra-welfarism as replacing utility with health, while Culyer cautions against discarding either. For a health tax the maximand changes the answer, because a QALY-maximising analyst records no value for the pleasure lost and so tends to favour a higher tax than a welfarist analyst using the same evidence.

Optimal taxation in cost-benefit analysis and appraisal

Optimal tax theory explains why public money can cost more than its face value. The European Commission's 2014 guide to cost-benefit analysis notes that a euro in the public budget may be worth more than in private hands because taxation distorts, so under non-optimal taxes a marginal cost of public funds other than one should adjust public flows, with one as the default where no national guidelines exist.

HM Treasury takes a different line. Green Book 2026 paragraph 6.44 says taxes create distortions but that practitioners should not generally include the costs of raising public funds, because most proposals are funded from pre-determined departmental budgets. A health programme appraised within a fixed NHS budget is in the same position, since its opportunity cost is the care displaced, not the tax raised.

Boundaries with neighbouring ideas

A Pigouvian tax is one instrument inside optimal tax theory, set to offset an external cost, while optimal taxation designs the whole system. Deadweight loss is the efficiency cost the theory keeps low, and Pareto efficiency is its benchmark. Diamond and Saez call any top rate above the revenue-maximising rate second-best Pareto inefficient, since cutting it would raise revenue and leave top earners better off. Progressive taxation describes the shape of a schedule, whereas optimal taxation asks which shape is best.

Limitations and common misreadings

Mankiw and colleagues show that optimal income tax schedules turn on the shape of the ability distribution and on disputed labour supply elasticities, and that the degree of concern for inequality is a normative question that data cannot answer. Tax by Design notes that the theory cannot readily handle many concepts of fairness, and that taxing fatty foods is harder than taxing alcohol or tobacco because moderate consumption of most foods is beneficial and a harmful category is hard to define.

A common misreading equates the optimal tax with the revenue-maximising tax, which holds only for the top income rate when society gives almost no weight to top earners' consumption. Another treats the goal of a health tax as the lowest possible consumption, when the optimal tax offsets only the harm that consumers do not take into account.

Sources

• Allcott H, Lockwood B, Taubinsky D. 2019. Regressive sin taxes, with an application to the optimal soda tax. Quarterly Journal of Economics 134(3): 1557-1626. NBER Working Paper 25841. https://www.nber.org/papers/w25841
• Allcott H, Lockwood B, Taubinsky D. 2019. Should we tax sugar-sweetened beverages? An overview of theory and evidence. Journal of Economic Perspectives 33(3): 202-227. NBER Working Paper 25842. https://www.nber.org/papers/w25842
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. 2008. Welfarism vs. extra-welfarism. Journal of Health Economics 27(2): 325-338. https://eprints.whiterose.ac.uk/id/eprint/3789/
• Diamond P, Saez E. 2011. The case for a progressive tax: from basic research to policy recommendations. Journal of Economic Perspectives 25(4): 165-190. CESifo Working Paper 3548. https://eml.berkeley.edu/~saez/diamond-saezJEP11opttax.pdf
• European Commission, Directorate-General for Regional and Urban Policy. 2014. Guide to Cost-Benefit Analysis of Investment Projects. https://ec.europa.eu/regional_policy/sources/studies/cba_guide.pdf
• HM Revenue and Customs. 2026. Check if your drink is liable for the Soft Drinks Industry Levy. Last updated 25 March 2026. https://www.gov.uk/guidance/check-if-your-drink-is-liable-for-the-soft-drinks-industry-levy
• HM Treasury. 2026. The Green Book (2026). https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• Mankiw NG, Weinzierl M, Yagan D. 2009. Optimal taxation in theory and practice. Journal of Economic Perspectives 23(4): 147-174. NBER Working Paper 15071. https://www.nber.org/papers/w15071
• Mirrlees J, Adam S, Besley T, Blundell R, Bond S, Chote R, Gammie M, Johnson P, Myles G, Poterba J. 2011. Tax by Design: The Mirrlees Review. Oxford: Oxford University Press for the Institute for Fiscal Studies. https://ifs.org.uk/sites/default/files/output_url_files/taxbydesign.pdf