### A Tale of Two Welfare Functions, or Mankiw vs Mirrlees

7/16/2013 03:04:00 PM
Tweetable This is a rather late response to Greg Mankiw's provocative essay on the philosophy of redistribution, called "Defending the One Percent." In his essay, Mankiw concedes the need for government policy to ensure that the distribution of resources is "efficient," but opposes any other kind of redistribution, pushing instead of a philosophical principle of "just desserts" whereby people are allowed to keep all the wealth they "earn." This is in contrast to the Mirrlees approach to public policy, which maintains that aggregate well-being (aka "social welfare") should be maximized, subject to all the feasibility constraints.

Mankiw rails against the "utilitarian" framework of Mirrleesian policy design, but the truth is that both Mirrleesian and "just desserts" approaches rely on the utilitarian philosophy (Mankiw's notion of "efficiency" is a utilitarian idea). Here's the Mirrlees approach: Each individual has preferences over labor and consumption, in that they dislike the former and enjoy the latter. So if $C_i$ is how much individual $i$ consumes and $L_i$ is how much he works, then we can summarize the net amount of value he gets out of that combination of work and consumption by the utility function $u_i\left(C_i,L_i\right)$. Now, we can just sum utility functions across individuals to get the social welfare function $$W=\sum_{i=1}^I\theta_i u_i\left(C_i,L_i\right).$$ We then choose a tax policy that maximizes $W$ subject to all the various constraints. Notice that there is a set of coefficients in the social welfare function: $\left\{\theta_i\right\}_{i=1}^I$, these are the individual-specific weights that are proportional to how much we care about the corresponding individual. For example, as Mankiw is fond of pointing out, we typically set $\theta_i=0$ for individuals who live in other countries. But contrary to Mankiw's claim, this is not a logical inconsistency on the part of practitioners of Mirrleesian policy analysis--we are quite clear and consistent with how we define the weights. Generally, we set $\theta_i=1$ for everyone in the US, and $0$ for everyone else. Yes, that is a philosophical choice, but there is no logical inconsistency about that choice.

Mankiw's "just desserts" philosophy is almost identical. He too believes that the government should enact policies to maximize the social welfare function $W$. Afterall, Mankiw is a member of the Pigou club, which advocates using activist tax policy to eliminate inefficiency, which in turn is defined as any point on $W$ other than it's maximum. But where "just desserts" differs from Mirrlees is in the specification of the set of weights $\left\{\theta_i\right\}_{i=1}^I$. You see, where as Mirrleesians maintain consistent, explicit values for each $\theta_i$, Mankiw does not. As long as $W$ is maximized for some set of weights, never mind how extreme or unfair those weights are, the government should do nothing. In this respect, it is Mankiw that is engaged in a logical inconsistency, implicitly arguing for policy based on one set of weights $\left\{\theta_i\right\}_{i=1}^I$ some of the time, but other sets of weights $\left\{\theta_i^{'} \right\}_{i=1}^I$ at other times, never making it clear which weights he is assuming at any given point of time.

In reality this is all a smoke screen. Mankiw values the rich and high-born more than he values the measly grimy working class. So in his policy analysis he gives himself and others like him very high $\theta_i$'s, while giving very low weights to the rest of us, but he doesn't want us to know that's what he thinks, so he scribbled a Chewbacca defense inventing the nonsensical notion of "just desserts" to try to throw us off the track.