The following post aims to explain why the diagram below is a proof without words that the Kaldor-Hicks criterion for an improvement in the economy can be non-transitive. This is in response to a request for help from a student of mine who made the request on the Facebook group Bountied Rationality, and hoped that a LessWrong post could be drawn up explaining the matter in detail.
Let me briefly explain the meaning of the terms. A Pareto improvement in an economy is a change in the state of the economy where every individual in the economy is at least as well off as before, and some individuals are strictly better off. A Kaldor improvement in an economy is a change in the state of the economy where some individuals are better off, and a re-allocation of resources would be possible so that they would be able to compensate any individuals who have been made worse off by the change, so that the net result is a Pareto improvement. The criterion of a Kaldor improvement is not anti-symmetric; it is possible for there to be two distinct states of the economy A and B such that each one is a Kaldor improvement of the other. This motivates the following. We say that a state B is a Kaldor-Hicks improvement of A if B is a Kaldor improvement of A but A is not a Kaldor improvement of B. This criterion is anti-symmetric.
In the diagram below, the x-axis and y-axis respectively represent utilities for Citizen 1 and Citizen 2. A state of affairs further to the right is preferred by Citizen 1, a state of affairs further upwards is preferred by Citizen 2. Only ordinal relations between utilities matter; we are not assuming a cardinal utility measure. The curves represent sets of combinations of utilities attainable by re-distribution from a given state of the economy. It is possible for two distinct curves to have an intersection point, that means that there are two different possible states and allocation of resources of the economy which give rise to the same pair of utilities.
The point B’ represents a Pareto improvement over the point A since it is both upwards and to the right. B is a Kaldor improvement of A since B’ is attainable from B by re-distribution and B’ is Pareto-better than A. On the other hand, no point on the curve passing through A is a Pareto-improvement over B, so A is not a Kaldor improvement over B. Thus B represents a Kaldor-Hicks improvement over A.
Using this kind of reasoning, one may confirm from the diagram that B is a Kaldor-Hicks improvement over A, C is a Kaldor-Hicks improvement over B, D is a Kaldor-Hicks improvement over C, but D is not a Kaldor-Hicks improvement over A. Thus this proves the non-transitivity of the relation in some possible circumstances.
Why the Kaldor-Hicks criterion can be non-transitive
The following post aims to explain why the diagram below is a proof without words that the Kaldor-Hicks criterion for an improvement in the economy can be non-transitive. This is in response to a request for help from a student of mine who made the request on the Facebook group Bountied Rationality, and hoped that a LessWrong post could be drawn up explaining the matter in detail.
Let me briefly explain the meaning of the terms. A Pareto improvement in an economy is a change in the state of the economy where every individual in the economy is at least as well off as before, and some individuals are strictly better off. A Kaldor improvement in an economy is a change in the state of the economy where some individuals are better off, and a re-allocation of resources would be possible so that they would be able to compensate any individuals who have been made worse off by the change, so that the net result is a Pareto improvement. The criterion of a Kaldor improvement is not anti-symmetric; it is possible for there to be two distinct states of the economy A and B such that each one is a Kaldor improvement of the other. This motivates the following. We say that a state B is a Kaldor-Hicks improvement of A if B is a Kaldor improvement of A but A is not a Kaldor improvement of B. This criterion is anti-symmetric.
In the diagram below, the x-axis and y-axis respectively represent utilities for Citizen 1 and Citizen 2. A state of affairs further to the right is preferred by Citizen 1, a state of affairs further upwards is preferred by Citizen 2. Only ordinal relations between utilities matter; we are not assuming a cardinal utility measure. The curves represent sets of combinations of utilities attainable by re-distribution from a given state of the economy. It is possible for two distinct curves to have an intersection point, that means that there are two different possible states and allocation of resources of the economy which give rise to the same pair of utilities.
The point B’ represents a Pareto improvement over the point A since it is both upwards and to the right. B is a Kaldor improvement of A since B’ is attainable from B by re-distribution and B’ is Pareto-better than A. On the other hand, no point on the curve passing through A is a Pareto-improvement over B, so A is not a Kaldor improvement over B. Thus B represents a Kaldor-Hicks improvement over A.
Using this kind of reasoning, one may confirm from the diagram that B is a Kaldor-Hicks improvement over A, C is a Kaldor-Hicks improvement over B, D is a Kaldor-Hicks improvement over C, but D is not a Kaldor-Hicks improvement over A. Thus this proves the non-transitivity of the relation in some possible circumstances.