Theory of the Second Best

Theory of the Second Best in Health Economics

In welfare economics, the theory of the second best is the theorem about policy in an economy where at least one condition for Pareto efficiency is blocked and cannot be restored. Its message is that fixing the remaining conditions one market at a time carries no guarantee of a welfare gain, so the effect of any single reform depends on how that market is linked to the distorted ones. Health care is a standard case, because insurance and patents keep many prices and quantities away from the textbook optimum. The theorem shapes how health economists read marginal cost pricing rules, the pricing of medicines and the decision rules of health technology assessment.

The theorem and the efficiency conditions it unsettles

First-best welfare analysis lists the conditions that hold at a Pareto-efficient allocation, such as price equal to marginal cost in every market. Lipsey (2012), one of the theorem's two authors, states that the theory established that piecemeal satisfaction of any one of these conditions is not sufficient to increase welfare. Specific knowledge of the market in question and its relations to other markets is needed, so the welfare effect of any one policy change is context specific.

Lipsey calls the taxes, monopolies and other factors that prevent an efficient allocation "sources of divergence", or sources. A second-best situation is any situation in which the first best cannot be reached, and the second-best optimum setting of a source is the setting that maximises the objective function, given the settings of all the other sources.

Gradoz (2025) traces the result to Lipsey and Lancaster's 1956 paper in the Review of Economic Studies. When the conditions for an optimum cannot all be satisfied together, the best course is not necessarily to satisfy the feasible ones while disregarding the rest. In Lancaster's own account, quoted by Gradoz, he illustrated the result with three sectors, one competitive, one an unchangeable monopoly and one a regulated monopoly, and found that the second-best policy for the regulated sector was generally to price above marginal cost.

Why health care is a second-best setting

Culyer, writing on the welfarist economics of health care finance, describes how insurance encourages consumption beyond the point at which marginal cost equals marginal value, the ex post form of moral hazard. He then adds a second-best caveat. If hospital and other services are priced above marginal cost, the usual welfarist efficiency conditions may no longer apply, and consumers facing false price signals might need encouragement to consume more than current prices imply.

Patented medicines add a deliberate source. Danzon and Towse (2003) set out the necessary conditions for second-best efficiency in drug use and development. Price must at least equal marginal cost in each market, and the markups over marginal cost, summed across markets, must cover the joint costs of research and development, including a normal risk-adjusted return. They treat the second condition as necessary for efficient investment in research, so marginal cost pricing in every market would fail it.

These cases share one structure. A market failure or a deliberate wedge sits in one market and cannot be removed, and the question is how policy elsewhere should respond. A Pigouvian tax on alcohol set to correct an externality as if every other price were efficient need not be the best tax once the other distorted prices that drinkers face are counted.

The welfare effect of one policy lever

A two-market sketch shows the mechanism. Take welfare as the sum of consumer and producer surplus, one of the economy-wide objective functions Lipsey lists, and assume constant marginal costs, no income effects and every other good priced at marginal cost. A policy maker controls the price of good B, while a source that cannot be changed holds the price of good A above its marginal cost.

$$\frac{dW}{dp_B} = (p_A - c_A)\frac{\partial q_A}{\partial p_B} + (p_B - c_B)\frac{\partial q_B}{\partial p_B}$$

where $W$ is total surplus, $p_A$ and $p_B$ are the prices of goods A and B, $c_A$ and $c_B$ are their constant marginal costs, $q_A$ and $q_B$ are the quantities consumed, and the partial derivatives give how each quantity responds to the price of B.

The first-best rule sets $p_B = c_B$, which removes the second term. The first term remains whenever A is mispriced and the two goods are linked, so moving the price of B away from marginal cost still changes welfare. Setting the derivative to zero gives the second-best price.

$$p_B^{*} = c_B - (p_A - c_A)\frac{\partial q_A / \partial p_B}{\partial q_B / \partial p_B}$$

where $p_B^{*}$ is the second-best price of good B and the other symbols are as defined above. With substitutes, $\partial q_A / \partial p_B$ is positive and $\partial q_B / \partial p_B$ is negative, so $p_B^{*}$ lies above $c_B$. With complements the cross effect is negative and $p_B^{*}$ lies below $c_B$. The direction of the best response depends on a cross effect that the first-best rule never asks about, which is the context specificity Lipsey stresses.

Second-best pricing of medicines

Danzon and Towse describe Ramsey optimal pricing as the set of price differentials that gives the highest social welfare subject to a target profit for the producer, with prices differing across market segments in inverse relation to their demand elasticities. They call differential pricing on Ramsey principles the second best efficient way of paying for the global joint costs of pharmaceutical research and development.

Danzon, Towse and Mestre-Ferrandiz (2012) carry the argument into health technology assessment. Insurance undermines patients' price sensitivity and can push prices above second-best efficient levels. In countries with universal insurance, prices set against each payer's ICER threshold, with reimbursement limited to patients for whom the drug is cost-effective, are roughly consistent with second best static and dynamic efficiency. On that account value-based pricing against a cost-effectiveness threshold is itself a second-best device.

Cost-effectiveness decision rules in a second-best world

Claxton, Walker, Palmer and Sculpher (2010) describe cost-effectiveness analysis as supporting decision makers who try to maximise health subject to the NHS budget constraint, with the threshold expressing the shadow price of that constraint. Their simple rules, such as accepting a positive net health benefit, treat each change as marginal.

Because new technologies tend to add costs, repeated decisions eventually have non-marginal effects, displacing increasingly valuable care so that the threshold tends to fall. Claxton and colleagues call this a familiar and general problem in economics. Second best rules for a world of non-marginal changes are impossible to formulate because of their informational requirements, while applying first best rules that assume marginal changes may make matters worse. The task is to find feasible third best rules that perform better, and they read NICE appraisal policy of 2010 in this light, with a threshold value that is periodically reviewed and significant budget impact taken into account deliberatively.

Culyer makes the same point about the discipline. Economists have a well-developed set of principles for putting first-best analysis to work in a second-best world, and those principles are now a standard part of health technology assessment. The theorem therefore matters on both sides of the welfarist and extra-welfarist debate, whether the maximand is utility measured through willingness to pay or health measured in QALYs.

Third-best rules and the case for the status quo

Lipsey (2012) reports Ng's third-best argument about policy in ignorance of the second-best optimum. In a 1977 article Ng argued that, when the direction of the second-best departure is unknown and the welfare curve is concave, the expected gain from departing from the first-best rule is negative, so the first-best rule should be kept market by market.

Lipsey rejects the conclusion. Real policy makers never start from a first-best position, so the same reasoning favours keeping whatever the current setting is, which makes Ng's argument a case for the status quo. In Lipsey's illustrative model with three goods of equal weight, when good Y carries a 10 per cent tax, removing any tax on good Z of up to 9.6 per cent lowers welfare. With a 100 per cent tax on Y, removing a tax on Z of up to 61 per cent lowers welfare, even though removal establishes the first-best setting for Z.

Lipsey ends on the usefulness of welfare economics once second-best theory is accepted. Many policy decisions concern objective functions far more focused than community welfare, such as achieving a given amount of pollution at minimum cost. Maximising health from a fixed NHS budget can be read as a focused objective of this kind, which helps cost-effectiveness rules stay usable.

Worked example with a branded medicine and a substitute

All numbers are illustrative. Patients in a self-pay market buy a patented medicine A and an off-patent substitute B. Quantities are in thousands of packs per month and prices in GBP per pack. Both medicines cost GBP 10 per pack to supply at the margin, the patent holds the price of A at GBP 20, and a regulator sets the price of B.

$$q_A = 100 - 2p_A + p_B, \qquad q_B = 100 - 2p_B + p_A$$

where $q_A$ and $q_B$ are monthly quantities in thousands of packs and $p_A$ and $p_B$ are prices in GBP per pack.

1. First-best rule. With B priced at its marginal cost of GBP 10, A sells 100 − 2 × 20 + 10 = 70 thousand packs and B sells 100 − 2 × 10 + 20 = 100 thousand.

2. Second-best price. The cross effect on A is 1 and the own effect on B is −2, so the second-best price of B is GBP 15, since 10 − 10 × (1 / −2) = 15. At that price A sells 100 − 2 × 20 + 15 = 75 thousand packs and B sells 100 − 2 × 15 + 20 = 90 thousand.

3. Welfare change. Each extra pack of A adds its GBP 10 margin, so 5,000 extra packs add 10 × 5 = GBP 50,000 a month. The margin on B climbs steadily up to GBP 5, an average of GBP 2.50 over a fall of 10,000 packs, so B loses 2.5 × 10 = GBP 25,000 a month. The net gain over the first-best rule is 50 − 25 = GBP 25,000 a month.

4. Overshooting. With B priced at GBP 20, each medicine sells 80 thousand packs. The gain on A is 10 × 10 = GBP 100,000 and the loss on B is 5 × 20 = GBP 100,000, with an average margin of GBP 5 over a fall of 20,000 packs, so welfare is back at its first-best-rule level.

5. Complements. If the cross effect were −1, the second-best price would fall below marginal cost to GBP 5, since 10 − 10 × (−1 / −2) = 5.

Pricing B at marginal cost draws patients away from A, which the patent markup already leaves under-used relative to its cost. The second-best price limits that switching, but the gain depends on knowing the cross effect, since the wrong sign or a large overshoot loses it. The surplus measure weights every GBP equally, so an analysis counting health gains or the access of poorer patients could reach a different verdict.

Boundaries with neighbouring ideas

Pareto efficiency and the fundamental welfare theorems describe the first best, the benchmark that the theory of the second best assumes cannot be reached, and Lipsey (2012) contrasts their generality with the context specificity of second-best theory. Partial moves towards allocative efficiency can therefore lower welfare.

Market failure names the sources, while the theory of the second best concerns policy once some sources are fixed. A shadow price in cost-benefit analysis is one practical response, valuing resources at their social opportunity cost when market prices are distorted. Competitive equilibrium and the utility possibility frontier describe the first best, while a technological externality is one kind of source of divergence that the theorem can take as given.

Limitations and common misreadings

A common misreading treats the theorem as proof that intervention is pointless or that any policy is as good as another. Lipsey's conclusion is narrower. There is no general a priori presumption for adopting any specific policy, including the status quo, so welfare gains have to be shown from context-specific evidence on how the markets involved are linked.

A second error reads "second best" as a polite word for a compromise. In the theory it is a technical term for the best attainable setting given the sources that remain, and that setting can lie further from the first-best rule than current policy does.

Sources

• Claxton K, Walker S, Palmer S, Sculpher M. 2010. Appropriate perspectives for health care decisions. CHE Research Paper 54. York: Centre for Health Economics, University of York. https://www.york.ac.uk/media/che/documents/papers/researchpapers/rp54_appropriate_perspectives_for_health_care_decisions.pdf
• Culyer AJ. 2012. The Humble Economist: Tony Culyer on Health, Health Care and Social Decision Making. Cookson R, Claxton K (eds). York: University of York and Office of Health Economics. https://www.york.ac.uk/media/che/documents/books/The%20Humble%20Economist%20corrected%20PDF%20version.pdf
• Danzon PM, Towse A. 2003. Differential pricing for pharmaceuticals: reconciling access, R&D and patents. International Journal of Health Care Finance and Economics 3(3): 183-205. https://doi.org/10.1023/a:1025384819575
• Danzon PM, Towse AK, Mestre-Ferrandiz J. 2012. Value-based differential pricing: efficient prices for drugs in a global context. NBER Working Paper 18593. https://www.nber.org/papers/w18593
• Gradoz J. 2025. A note on the origins of The General Theory of Second Best: the accounts of Kelvin John Lancaster, Richard George Lipsey and Harry Gordon Johnson. CHOPE Working Paper 2025-01. Duke University. https://hope.econ.duke.edu/sites/hope.econ.duke.edu/files/documents/A%20Note%20on%20the.pdf
• Lipsey RG. 2012. A critique of Ng's third-best theory. Department of Economics Working Paper 12-02. Simon Fraser University. https://summit.sfu.ca/_flysystem/fedora/sfu_migrate/10928/dp12-02.pdf