An oversimplified model for optimal distribution of wealth

Question

Consider the following, overly simplified, model for determining an optimal wealth distribution for society:

Let $X$ be a random variable, which will model the distribution of wealth in a society. The goal is to maximize the expected value of a utility function of $X$, for example $\mathsf E(\ln(X))$, under the side constraint that $\mathsf E(X)=c$ for some constant $c>0$ (i.e. under the side constraint that the total wealth of the society is fixed).

The problem with this model is that, by Jensen's inequality, we have $$\mathsf E(\ln(X))\le\ln(\mathsf E(X))$$ with equality achieved when $X=\mathsf E(X)$, so that we immediately end up with communism (everyone having the same wealth being judged as the optimal solution).

We thus enrich this model with an additional term that penalizes wealth redistribution. Assume for instance that, in an idealized free market society, wealth would be distributed like a Paréto distribution, i.e. we would have $$\mathsf P(X_{\text{idealized}}\le t)=1-\left(\frac{w_{\text{min}}}t\right)^k$$ for some parameters $w_{\text{min}}>0$ and $k>0$ and all $t\ge w_{\text{min}}$, as well as $\mathsf P(X_{\text{idealized}}\ge t)=0$ for all $t<w_{\text{min}}$.

The cost of wealth distribution shall now be the Wasserstein-$1$ metric $$W_1(X, X_{\text{idealized}})=\int_{\mathbb R} \lvert\mathsf P(X\le t)-\mathsf P(X_{\text{idealized}}\le t)\rvert\,\mathrm dt.$$

My question: Is there any literature on the optimization problem

\begin{align*}\text{maximize }&\mathsf E(\ln(X))-W_1(X,X_{\text{idealized}}) \\ \text{given }&\mathsf E(X)=c, X\ge 0\end{align*}

over all possible distributions for $X$ ?

Answer 1

Here is an ultra-simplified model of the sort you seem to be looking for.

Everyone shares the same utility function $$\log(C)-A\log(1-L)$$ where $A$ is a constant, $C$ is consumption and $L$ is the fraction of the day spent working.

A person who works produces $TL$ units of output, where $T$ is specific to the individual. The log of $T$ is normally distributed, with standard deviation $\sigma$.

The planner is ominiscient and omnipotent in the sense that he knows the value of $T$ for each individual, can mandate the number of hours that each individual works, and can mandate any amount of income redistribution. The redistribution process is costless.

Per your observation, it's clear that total output should be divided equally because of Jensen's inequality. But it will still be the case that different people work different amounts. We can ask, for example, what fraction of the population ought to be unemployed.

Consider a worker with talent $T$ who works $L$ hours. Let $K$ be the output of all other workers. Then total output is $C=K+TL$, divided among all workers (of whom there are, say, $N$), so consumption per worker is $(K+TL)/N$. Total utility is then $$N \log((K+TL)/N)+ALog(1-L)+G$$ where $G$ is the disutility of working for all the other workers.

This expression is maximized at $L=max(1-AC/T,0)$, which is the amount a worker should work as a function of $T$.

The average worker therefore produces the amount $$\int_{AC}^\infty (T-AC)f(t)dt$$ where $f$ is the density function for the lognormal distribution with standard deviation $\sigma$.

On the other hand, the average worker consumes $C$ and so must produce $C$. This gives $$\int_{AC}^\infty(T-AC)f(T)dT=C$$ which can be solved explicitly for $C$.

Any worker with talent less than $AC$ is unemployed. Therefore optimal unemployment is $$\int_0^{2C}f(t)dt$$ which can be calculated explicitly because we've previously calculated the value of $C$.

Real world estimates suggest that reasonable values for $A$ and $\sigma$ are about $A=2$ and $\sigma=.4$. I believe you'll find that if you plug those values into the above, you'll find that optimal unemployment is about 23% of the population.

Now: Notice that in this model, everyone consumes equally, but those with more talent are commanded to work more. This means that it pays to conceal your talent, and in particular to act as if your talent falls below $AC$ so you won't have to work at all. We've made the extreme assumption that you can't get away with this because the planner is omniscient. So let's go to the other extreme and assume the planner has absolutely no knowledge of anyone's talent. (The truth surely lies somewhere properly between those extremes.)

Now all the planner can do is announce a pair of functions that determine, as a function of talent, how many hours you must work and how much you get to consume. Workers must be induced to reveal their talents honestly, which means that in equilibrium (i.e. after everyone optimizes), utility must be constant across the population. (Otherwise, if your utility were greater than mine, I would claim to have your talent so that I could earn your utility.) This imposes a constraint on the properties of the pair of functions. The planner chooses the functions to maximize total utility subject to that constraint. I believe that with $A=2$ and $\sigma=.4$, you'll find that optimal unemployment is somewhere under 1%. There is still significant redistribution, but much less than in the first version of the model.

These are the sorts of toy models that I teach my undergraduates when they are first encountering these ideas. There is a vast literature of more sophisticated models, some of which has led to Nobel prizes. You might Google for the work of Jim Mirrlees on optimal income taxation.