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Can we determine the optimal redistribution of wealth in an economy?

Assume that $X$ is personal net wealth. Assume, further, that utility in X is concave as depicted in the following figure.

U(Wealth)

This is a powerful argument in favor of redistribution: there exists an $X^*$ such that social welfare (the aggregation of personal utilities) is maximum. Taking away some $X$ from those at the higher end and giving this equally to all those at the lower end will be welfare improving.

I know that this problem needs to be framed in dynamic terms to make sense - because agents must have the incentive to create and keep wealth.

But it seems so simple analytically.

MY QUESTION: What am I missing?

Answer 917

What are you missing? Quite simply, the impossibility of measuring relative people’s relative utility. Just because diminishing returns to wealth exist doesn’t mean we can determine where one person’s diminishing return to retaining it intersects with another person’s need to obtain it. It’s easy to think of people that appear to have more money than they need or want and others whose needs are not the result of laziness, less easy to determine whether the marginal rich person’s bonus they worked really hard for is worth less to them than the marginal poor person’s useful extra income they might have been able to earn had they tried.

In a market system, an optimal exchange takes place when both the buyer and the seller think they will be better off given the rules of the “free” market1. We don’t have any measure of how much each party believes they have gained by (or whether they are deluded in their ex ante belief(s) ) but we know that the tradeoff has theoretically benefited both parties from their will to participate in trade at that price. Non-market redistribution implies that at least one party in the transaction believes that they will be a loser; otherwise, acting rationally, they would have been willing to agree to some form of investment loan. What economics doesn’t offer, even at the extreme end of the spectrum, is any way to quantify whether unemployable person X’s desire to eat the food they need to live is worth more than rich person Y’s desire to increase the amount in their savings account by $1.

There is a welfare economics criterion for “optimal” redistribution - the Kador-Hicks efficiency criterion - which suggests that aggregate social welfare can be improved if redistribution would create enough wealth to pay back those that had their wealth confiscated and still come out ahead. In other words if assets are taken from those using them inefficiently and given to those able to use them more efficiently you reach a social optimum: generally regarded as implying a net welfare gain happens even if those whose wealth was redistributed away from them are never repaid. The problems with this are:

(i) it’s based on the calculation of hypothetical outcomes (what the person losing the wealth would have generated based on investments or purchases they might have made versus what the people the wealth is being transferred to will generate based on things they might buy or invest in); it’s the estimation of a policymaker rather than the belief of the participants in the redistribution.

(ii) if the Kaldor-Hicks optimum could be accurately estimated it would likely indicate allowing some unproductive people to live “costs” more than leaving someone who invests in productive enterprises to keep the money. There’s a strong argument that this is ethically wrong even if is economically “optimal”.

1An market outcome in which a tax rate is announced in advance is a “free” market outcome: parties have agreed to the transaction in the knowledge that the government will take x% of income and redistribute it to someone else (Despite what passionate defenders of “free markets” might argue, confiscation or theft are quite different). Whether raising tax rates will discourage people from exchanging goods by so much that it decreases overall tax yields is another hypothetical question that ought to be considered when determining an optimal level of redistribution. Knowledge of responses to tax changes in other times/places make this easier to guesstimate than who “needs” how much redistribution.

Answer 930

As with many models, yours may not be complex enough to illustrate the full dynamics of the real system. As others have mentioned, you have assumed that agents are homogenous and have simple, smooth utility functions.

More importantly, you have not modeled the mechanism by which personal utility maximizing behavior causes global utility maximization. Although this is the central theme of introductory economics, you are suggesting a more sophisticated model may exist that accounts for differences in personal utility functions. I believe that the optimal amount and method of wealth or income redistribution depends on the mix of utility functions in the economy and the mix of externality effects of individual actions, which in turn depends on technology.

For example, redistribution may be more important than in an economy where wealthy people love polluting than in an economy where wealthy people love to protect nature.

Answer 916

First, I’m very skeptical of using a graph to depict utility versus personal worth for everyone. The amount of utility each person derives from their wealth is different from person to person. I don’t even know how one would go about measuring such a thing.

Second, when you start stealing from one group of people to give to others the incentive for people to increase their wealth diminishes. If the stealing point starts at Y then wealth producers will only produce up to Y and then stop or finds ways to hide their wealth from the criminals. Also, your hypothetical utility graph will change once you begin stealing from people.

Third, the people who receive the handouts also will have less of an incentive to increase their wealth since they can simply rely on the handout. I know of no easier way to cripple a man then to give him a handout. You can cut a man’s arm off and he will adapt and be productive. But the moment you give a man a handout he stops working hard, enhancing his skills, looking for work, etc.. When the handout stops the man is not prepared to provide for himself since he has become dependent on the handout.

Fourth, and this is the most important point, it’s immoral. Once you justify stealing some from one you can justify stealing all from any.

Thus, stealing from a group of people and giving to others will diminish the wealth of most of the people in the economy as many of the wealthy reduce their efforts (or spend considerable time and effort hiding their wealth from the criminals which diverts their energies from producing better products/services) and the recipients of the handouts also diminish their efforts to provide for themselves.

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