second-best

Theory of the Second Best

The Theory of the Second Best is a foundational result in welfare economics and public economics, first formally articulated by Richard Lipsey and Kelvin Lancaster in their landmark 1956 paper published in the Review of Economic Studies. The theory's central proposition is both startling and sobering: in a general equilibrium system where one or more Pareto optimality conditions cannot be satisfied (i.e., there exists an irremovable distortion), fulfilling all remaining Pareto conditions elsewhere in the economy need not—and generally will not—improve social welfare. The constrained optimum—the "second best"—may instead require departing from every marginal condition that would characterize a first-best allocation, even those in sectors that are themselves free of any distortion.

Formal Statement

Lipsey and Lancaster framed their argument within a constrained optimization framework. Consider an economy with $n$ variables $x_1, x_2, \ldots, x_n$ describing production, consumption, and prices across all sectors. A first-best Pareto optimum satisfies a system of first-order conditions:

Now suppose that one of these conditions—say $F_1 = 0$—cannot be satisfied due to some binding institutional, technological, or informational constraint (e.g., natural monopoly pricing, an excise tax imposed by statute, or asymmetric information). The policy problem becomes: maximize the social welfare function $W(x_1, \ldots, x_n)$ subject to the constraint that $F_1 \neq 0$ (specifically $F_1 = c$ for some nonzero constant $c$). The Theory of the Second Best proves that, except under very special separability conditions, the solution to this constrained problem will not satisfy $F_2 = 0, F_3 = 0, \ldots, F_n = 0$ either.

The intuition can be seen from the Lagrangean:

The extra constraint introduces a Lagrange multiplier $\mu$ that interacts with all $\lambda_j$ through the coupled structure of the general equilibrium system. The resulting first-order conditions for $x_1, \ldots, x_n$ are a non-trivial mixture of all the $F_j$ terms, and they cannot be solved by simply setting each $F_j$ to zero individually. The "distortion" from $F_1$ propagates throughout the entire system, "infecting" the optimality conditions of every other sector. This reveals a deep non-separability: in general equilibrium, the optimal policy for any one sector depends on the structure of distortions in all other sectors.

Intuition: A Two-Sector Economy

Consider a simple two-sector economy. Sector A is a monopoly that prices above marginal cost ($P_A > MC_A$). This distortion is irremovable—perhaps because the industry is a natural monopoly with large fixed costs that preclude marginal-cost pricing without a subsidy, and lump-sum transfers are unavailable. Sector B is perfectly competitive. A naïve policy prescription might be: "Since we cannot fix sector A, at least ensure that sector B follows marginal-cost pricing ($P_B = MC_B$)."

The Theory of the Second Best shows that this intuition can be wrong. The goods produced by sectors A and B are related—they may be substitutes, complements, or one may be an input into the other. The monopoly price in A distorts the relative price ratio $P_A/P_B$ that consumers and producers face. Setting $P_B = MC_B$ (the first-best rule for B in isolation) may actually worsen the allocation of resources between the two sectors compared to what is achievable under the constraint. The constrained optimum—the true second best—might require that $P_B$ also deviate from $MC_B$, perhaps being set above or below marginal cost depending on the cross-price elasticities of demand and the production technology linking A and B.

The key insight is that piecemeal optimization does not aggregate. An allocation that is Pareto-efficient within each sector taken separately is not necessarily Pareto-efficient for the economy as a whole when one sector is constrained. This is the heart of the "second-best" concept: since the first best is unattainable, the second best is not obtained by realizing as many first-best conditions as possible, but by solving a different optimization problem altogether.

Policy Implications: The Fragility of Piecemeal Reform

The Theory of the Second Best has profound implications for applied welfare economics and policy design. It directly challenges the rationale for piecemeal reform—the gradual removal of individual distortions one at a time—which is a common approach in trade liberalization, tax reform, and deregulation. If multiple distortions coexist, removing any single one without addressing the others may reduce rather than increase welfare.

Trade Liberalization

Consider a developing country with a protected manufacturing sector (tariff $t > 0$) and a labor market distortion (e.g., a binding minimum wage that creates urban unemployment). Standard trade theory suggests that reducing the tariff should improve welfare by moving toward free trade. However, if the labor market distortion remains, lowering the tariff may contract the manufacturing sector and release workers who cannot be absorbed into the already-rigid labor market. The Harris-Todaro model formalizes this: migration responds to expected rather than actual wages, and partial tariff reduction can exacerbate the urban-rural misallocation. The second-best tariff may well be higher than the status quo, not lower, until the labor market distortion is also addressed.

Environmental Taxation

The "double dividend" hypothesis in environmental economics argues that a Pigouvian tax on pollution can yield two benefits: (i) correcting the environmental externality, and (ii) using the revenue to reduce pre-existing distortionary taxes (e.g., labor income tax), thereby "recycling" the environmental tax revenue to improve efficiency elsewhere. But the Theory of the Second Best cautions that the interaction effects are subtle. If pollution and leisure are complements in the utility function, the Pigouvian tax may exacerbate the labor-leisure distortion created by the income tax, potentially producing a net welfare loss. Whether the double dividend exists is an empirical question that depends on the precise pattern of substitutability and complementarity across all goods—a direct consequence of the second-best logic.

Regulatory Reform in Finance

In financial regulation, deposit insurance creates moral hazard by insulating depositors from bank risk, encouraging excessive risk-taking. A well-intentioned regulator might respond by tightening capital adequacy requirements—a seemingly prudent first-best fix. However, if deposit insurance (the irremovable distortion) persists, stricter capital requirements may induce banks to shift their portfolios into even riskier assets to maintain target returns, potentially increasing systemic fragility. The second-best regulatory package must coordinate capital requirements, resolution mechanisms, and market discipline simultaneously rather than optimizing each instrument in isolation.

The Separability Exception

Lipsey and Lancaster did identify conditions under which piecemeal reform does remain valid—namely, when the irremovable distortion is separable from the rest of the economy. Formally, if the social welfare function (or the underlying utility and production functions) can be written such that the distorted variables are additively or multiplicatively separable from the others, then the remaining Pareto conditions retain their validity in the second-best solution.

Suppose the welfare function takes the form:

where the distortion only affects variables $\{x_1, \ldots, x_k\}$ and $g$ and $h$ have zero cross-partial derivatives. In this case, optimizing over $\{x_{k+1}, \ldots, x_n\}$ is independent of the distortion's precise magnitude, and the remaining first-best conditions hold. However, Lipsey and Lancaster emphasized that such separability is extremely restrictive in practice. Nearly all goods are substitutes or complements to some degree; production technologies exhibit complex input-output linkages; and information asymmetries create feedback loops that violate separability. The burden of proof, they argued, should rest on those who claim that piecemeal reform is welfare-improving.

Subsequent Developments and Criticisms

The Theory of the Second Best has spawned extensive theoretical refinements and remains a cornerstone of modern public economics.

Optimal Taxation

James Mirrlees and Peter Diamond, building on earlier work by James Meade and Frank Ramsey, explicitly incorporated second-best reasoning into the theory of optimal taxation. The Ramsey rule for commodity taxation—tax goods inversely with their elasticity of demand—is itself a second-best result: it tells us not to seek "no distortion" (which would require lump-sum taxes) but to optimally design distortions to minimize the excess burden given an irreducible revenue requirement. Similarly, Mirrlees' nonlinear income tax model treats informational constraints (the government cannot observe individual productivity) as the irremovable distortion, and derives the optimal tax schedule that balances equity and efficiency under those constraints. The resulting marginal tax rates can be positive, negative, or zero depending on the skill distribution—a classic second-best pattern.

The General Theory of Distortions

Subsequent work generalized the Lipsey-Lancaster framework into what is sometimes called the "general theory of distortions." Jagdish Bhagwati and T. N. Srinivasan showed that the welfare ranking of alternative policies in the presence of multiple distortions cannot be determined by counting the number of distortions removed—a count that has no theoretical relationship to welfare. Arnold Harberger's cost-benefit analysis and the Hicksian compensating variation framework provided practical tools for evaluating piecemeal reforms using shadow prices that reflect second-best interactions, but these methods require detailed knowledge of cross-price elasticities that is often unavailable.

Criticisms and Limitations

Two main lines of criticism have been leveled against the Theory of the Second Best. First, it is fundamentally a negative theory: it tells us that simple policy rules are invalid without providing clear alternative rules that are robust across contexts. The direction and magnitude of optimal deviations from first-best conditions depend on parameters (elasticities, factor shares, distributional weights) that are difficult to estimate with precision. This has led some economists to dismiss it as "anything goes" nihilism, though its proponents counter that the theory correctly diagnoses the complexity of real economies rather than evading it.

Second, the theory abstracts from transaction costs and political economy. Implementing a fully optimized second-best policy requires detailed information about preferences and technology that governments rarely possess. Furthermore, actual reform processes are constrained by political feasibility, administrative capacity, and legal frameworks. In this context, Herbert Simon's concept of satisficing—pursuing reforms that are "good enough" given informational and cognitive limits—may be more practical than attempting to compute and implement the theoretically optimal second-best. Proponents of piecemeal reform, such as Jan Tinbergen, argued that sequential policy adjustments with feedback mechanisms can converge toward a constrained optimum even without full information.

Conclusion

Despite its limitations, the Theory of the Second Best stands as one of the most important conceptual contributions of 20th-century welfare economics. It provides a rigorous and general caution against the fallacy of composition in policy design: what is optimal for a part is not necessarily optimal for the whole. Its legacy is visible across virtually every field of applied economics—from trade policy and environmental regulation to tax design and financial stability. The theory does not paralyze policymaking but demands that any reform be evaluated with a systemic, general-equilibrium perspective, recognizing that in an already-distorted world, pursuing first-best rules in isolation may be the surest path to the worst of all outcomes.