Kelvin Lancaster and Richard G. Lipsey, in their article “The General Theory of Second Best”, 1956, following an earlier work by James E. Meade, treated the problem of what to do when certain optimality conditions (which must be considered in order to arrive at a Paretian optimum solution in a general equilibrium system) cannot be satisfied. The main idea in this article is that, when a constraint prevents the fulfilment of one of these conditions, the other conditions are in general no longer desirable. The optimum situation in this case can be attained only by neglecting the other conditions. Indeed, this new optimum is called “second best” because a Paretian optimum cannot be attained.
This can be easily understood using the diagram depicted in the article. We start by considering a typical optimization problem, with a given production possibility frontier (PPF) considered as a boundary condition, indifference curves (green curves, in this case representing a welfare function, ω) and the optimum where the PPF is tangent to ω (point P). Since this points lies on the transformation line and an indifference curve, it defines the production and consumption optima.
When we draw a new constraint condition (red curve), it can be easily seen that point P is no longer attainable. Q could be a second best solution, since it lies both on PPF and NewCC. However, as the authors point out, the second best point would be R, inside the transformation line. This is so because an improvement on welfare can be attained by moving to point R, since it lies on a further indifference curve (ω’’), and therefore means higher welfare.
The segment MN is technically more efficient than R, but since the points on this segment cannot be attained, R is the second best solution.