## On Lipsey and Lancaster and Wiens on the theory of second best

If some ideal is impossible to reach, should we get as close to the ideal as we can?

It's easy to come up with apparent counterexamples. Lipsey and Lancaster (1956) are sometimes said to have *proved* that getting as close to the ideal as we can is not the best option. Have they really?

Wiens (2020) helpfully summarizes the main result of Lipsey and Lancaster and explains how it applies outside economics. (The Lipsey and Lancaster paper is all about tariffs and taxes and Paretian conditions.)

I won't go through all the details, but here's a quick summary of the summary.

Suppose the ideal state is characterised by a list of features x_{1}…x_{n}. We assume that these features are the values of continuous variables X_{1}…X_{n}. A desirability measure V maps all possible combinations of values for these variables to a number, so that higher numbers are assigned to more desirable combinations of values. We assume that V is continuous, so that small changes in the feature values make for small changes to overall desirability.

The variables need not be independent, and we don't assume that all combinations of values are possible. For example, X_{1} might measure how many hours I spend walking in the forest today, while X_{2} measures how many hours I work. Since the day only has 24 hours, the sum of X_{1} and X_{2} is constrained to be at most 24.

Subject to some such constraint, one can compute the optimal state x_{1}…x_{n} by computing ideal "marginal exchange rates" between any two of the variables.

Now suppose we can't realize the ideal exchange rate between two of the variables – say, X_{1} and X_{2}. Loosely speaking, we can't realize the optimal ratio between X_{1} and X_{2}. We may still be able to realize the optimal ratio between other variables. Should we do that? Would it lead to the best achievable state?

Lipsey and Lancaster showed that this is so only under highly restrictive conditions. As Wiens reports, it has later been clarified that these conditions hold iff both the value function and the constraint on value combinations are separable. Without separability, all exchange rates in the best achievable state differ from the exchange rates in the best state. In that sense, the best achievable state is nothing like the ideal.

Informally, Lipsey and Lancaster's result shows that if there are a number of normatively relevant and non-separable quantities that trade off against each other, and we can't achieve the optimal ratio between two of them, then we should not realize the optimal ratio between any of them.

This is interesting and non-obvious, at least to me. But it doesn't show much about whether we should, in general, try to get close to the ideal.

To begin, Lipsey and Lancaster's result is about ratios. They don't show that if we can't achieve the optimal state x_{1}…x_{n} then the best achievable state will not involve any of the values x_{1}…x_{n}. Indeed, as Wiens shows (on pp.18-20), this hypothesis is false.

Second, Lipsey and Lancaster assume that the features that make up the possible states trade off against each other. At first I thought that this is assumed to make the cases more realistic, but I think it's actually required for the proof.

Finally, Lipsey and Lancaster assume that the relevant variables are non-separable, which suggests that they don't represent basic, intrinsic goods.

What if we're talking about basic goods? Should we get as close as we can to the ideal, if closeness is measured in terms of basic goods? Let's see.

Assuming that basic good – intrinsic bearers of value – are separable, total value has an additive representation: V(X_{1}…X_{n}) = V_{1}(X_{1}) + … + V_{n}(X_{n}). Now suppose the state x_{1}…x_{n} that maximizes this sum is unachievable. That is, we can't simultaneously maximize all of V_{1}…V_{n}. To get the best achievable state, we should still make the V_{i} as high as possible. Depending on the achievability constraint, we might have to make one of the V_{i} lower than possible in order to increase others of the V_{i}. But on balance, we should try to get V_{1}(X_{1})…V_{n}(X_{n}) as high, and thereby as close to the ideal, as we can.

It doesn't follow that we should get X_{1}…X_{n} as close to the ideal as we can. Suppose, for example, that the ideal value of X_{1} is 7, with V_{1}(7)=100. And suppose we can only achieve X_{1}-values below 6. It may well be that the maximum of V_{1} for values below 6 is at X_{1}=2. All else equal, we should then set X_{1} to 2, not to near 6.

But the following still looks plausible: If states are described in terms of intrinsically valuable features, and the subvalue functions V_{i} are all monotonic, then the best achievable state is the closest achievable state to the ideal (in terms of the intrinsically valuable features).

That the subvalue functions are monotonic means, roughly, that having more of an intrinsic good is always better than having less of the good. This looks defensible.

So perhaps there's a good sense in which the best achievable state always *is* the closest achievable state to the ideal.

*The Review of Economic Studies*24 (1): 11–32.

*Philosopher’s Imprint*20 (5).