Welfare Analysis Just Do It! (Part 2: Doing the Impossible)

In a previous post I discussed some of the properties of consumer preferences that were required to rationalize a demand function. This came down to properties of what is known as the Slutsky substitution matrix which was required to be symmetric and negative semi-definite. These properties satisfy the strong axiom (SA) of revealed preference. As stated in the widely adopted graduate micro text by Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green (MWG) chances of the SA "being satisfied by a real economy are essentially zero."

In a follow up post I discussed the idea of a 'normative representative consumer.' In order to have a normative representative consumer, we have to assume a social welfare function, and assume it is maximized by an optimal distribution of wealth according to some specified wealth distribution rule.

Making more 'impossible' assumptions didn't seem to help. And in fact according to Arrow's Impossibility Theorem, they really were practically impossible. So....when it comes to policy analysis (like for instance policies related to climate change) how do economists include social welfare in a cost benefit analysis?

There was a really great discussion about this in a Macro Musings podcast with James Broughel hosted by David Beckworth.

James Broughel: "And the welfare measure that they use is a social welfare function that they derive from the Ramsey neoclassical growth model, which is a famous economic growth model. So they take a welfare function from that model, they say this is society's preferences or this is the social planner's preferences or something along those lines. And then their goal is to maximize that....Well, the most obvious problem with this approach is that it relies on this social welfare function, which is supposed to describe the aggregated preferences of everyone in society. And aggregating the time preferences of everyone in society is really just a special case of aggregating the preferences in general, which runs into this issue of Arrow's Impossibility Theorem."

?Arrow's theorem* requires that in order for any social welfare function to represent society's preferences (which are an aggregation of individual preferences) it must obey six axioms:

1) It must rank all social states

2) It must obey transitivity (see my previous post about symmetry of the Slutsky substitution matrix)

3) The ranking must be positively related to individual preferences

4) New social states should not affect the ranking of original social states - also referred to as independence of irrelevant alternatives

5) The ranking should not be based on customs overriding individual preferences

6) Rankings are not made by a dictator

Arrow's theorem states that there is no social welfare function that can aggregate preferences or a social decision rule that can satisfy all six axioms. Like I mentioned in my previous posts, it seems like based on 'the math' and the theory, welfare analysis for applied policy work isn't feasible. Maybe we should just limit ourselves to positive analysis (focusing on efficiency). So how do economists approach normative welfare related policy questions?

James Broughel: "they just say, well, that's society's preferences. And this has become a convention in economics, it's done all over the place."

David Beckworth: "Because it's tractable, right? It's easy to do. The math is easy."

James Broughel: "Yeah, you can do the math. But, there really isn't any basis for it. I think that they would, the advocates of this approach would acknowledge that. They would say, our approach is normative, but hey, lots of economists agree on it."

So the tongue in cheek answer is how do you do welfare analysis despite all of the challenges I have discussed? You make some impossible assumptions and 'just do it' because the math is easy....sort of. You have to accept there are a number of problems that require these kinds of simplifying assumptions to motivate more critical thinking about the alternatives we face in a policy and decision making environment, as imperfect as that may be.

Most of the podcast was actually about two major schools of thought regarding the appropriate discount rate for doing cost benefit analysis for policies with long term impacts (again like climate change). Even if we are able to achieve scientific consensus on the impacts of climate change, the actual policy solutions have to be evaluated in terms of the costs today vs. the benefits of mitigating future climate events. That requires a discount rate, which as David and James discuss, there is no solid consensus on what is appropriate. That merits a future post!

*Microeconomic theory:basic principles and extensions. 8th Edition. Walter Nicholson (2002)