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Desirable Incentive Effects of Income Taxation II

Summary:
This is the second post in a series. I will discuss advantages of income taxation different from the obvious advantage that taking from people with high income hurts them less than taking from people with low income. Here again, I will assume that, in equilibrium, income tax is returned to the people who pay it as a lump sum. I do this to focus on the incentive effects of income taxation. In standard models, these effects are undesirable and amount to a deadweight loss which is second order in the tax rate. However, the standard models rely on standard assumptions which are completely implausible. They are used, because it is guessed that the policy implications don’t depend on the absurd assumptions. The policy implications always, in fact,

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This is the second post in a series. I will discuss advantages of income taxation different from the obvious advantage that taking from people with high income hurts them less than taking from people with low income. Here again, I will assume that, in equilibrium, income tax is returned to the people who pay it as a lump sum. I do this to focus on the incentive effects of income taxation.

In standard models, these effects are undesirable and amount to a deadweight loss which is second order in the tax rate. However, the standard models rely on standard assumptions which are completely implausible. They are used, because it is guessed that the policy implications don’t depend on the absurd assumptions. The policy implications always, in fact, follow from the assumptions.

In this post, for a second time, I will relax the assumption that people are 100% purely selfish and care only about their own consumption and leisure. Instead I will assume that people maximize the sum of their pleasure from consumption and leisure plus a constant far less than one times other people’s pleasure from consumption and leisure.

This is enough to eliminate the standard result that laissez faire market outcomes are Pareto efficient. As pointed out by Lester Thurow long ago (citation needed). All may benefit from redistribution from the rich to the poor. Each rich person is pleased that money is taken from other rich people and given to poor people. This can outweigh the fact that they would rather keep their money than give it to poor people. In a model with partial altruism, equality is a public good (as it leads to more efficient production of pleasure from consumption and leisure and, with partial altruism, the general welfare is a public good).

I am interested in other implications of partial altruism, so I will assume both that people belong to a few (in the model 2) identifiable types and it is possible to guess how much each person will earn and pay in income tax. This means that it is possible to eliminate the beneficial redistribution from income tax by returning to people the amount they pay as a lump sum. The key to “lump sum” is people are paid a fixed amount no matter what they do, so lump sum taxes and transfers do not affect incentives.

In the first post in this series, I consider agents who can make goods out of their labor alone and each of whom can make two goods one of which is a necessity and one of which is a luxury. For any finite population (no matter how large) partially altruistic agents balance their desire for income and their desire to help poor people a lot by producing the necessity. By reducing the first motive, an income tax can cause a lower equilibrium price for the necessity and higher welfare of the poor. This can cause higher average welfare. Finally (as noted by Thurow) everyone can prefer the outcome with higher average welfare (even if they have lower pleasure from consumption and leisure) if everyone is partially altruistic.

The model I presented in that post is odd as agents must consider the effect of their choices on market prices. I noted that this can happen for any finite population no matter how huge, so it applies also in the limit as the population goes to infinity and competition approaches perfect competition. But it is odd. The logic is an indirect economists way of saying that people want to produce a necessity, because they think it will help their customer more. It is odd, because people trade on anonymous markets and don’t know who their customers are (but can guess that poor people spend more of their income on the necessity and have a higher marginal utility of consumption).

A much more natural model has agents who interact face to face with their customers. This happens if people supply a service. Also the obvious examples are of services. For example the plastic surgeon who performs reconstructive surgery not cosmetic surgery or the lawyer who represents poor people not large corporations. Given the in person nonymous (not anonymous if I am inventing a word) provision of services, it isn’t even really necessary to consider more than one kind of service.

A Very Simple Model

The issue is the interaction of partial altruism and inequality. I assume inequality of the most nearly justified form based only on differences in ability. Agents are self employed and there is no productive capital or investment. Agents produce a service to each other. There is perfect competition and complete information. Each of these extreme assumptions eliminates a desirable effect of income taxation. The series of posts on desirable effects of income taxation will be very long.

Agents interact in pairs with one providing the service, so in a way this is a model of decentralized trade and matching with the extreme assumption that matching is instant. Importantly, complete information includes knowing the income of the customer.

I assume a possibly very small degree of altruism. Utility functions are defined in two steps. First there is pleasure from consumption of the service and leisure v(c,l) which is as usual assumed to be concave. then agent j maximizes

v(c,l) + alpha sum_i v(ci,li)

Where alpha is a positive constant which may be very small.

Agents have different ability to provide the service

y_j = (1-l)a_j

a_j is the ability of agent j. 1-l is the amount of time agent j spends working.

I will assume a takes 2 values aL <1 and aH > 1 and that half of people have a_j = aL.

With some altruism, there can be price discrimination even though there is perfect competition. Age ts may charge poor customers less. Providing the service pleases the provider in two ways – the income can be spent on consumption and the service contributes to total pleasure from consumption and leisure.

Again there is an income tax and transfers. I want to discuss only the incentive effects of the income tax and not the main benefit of taking from those who have a lower marginal utility of consumption, so I assume that the transfers are equal to the amount paid in income tax in equilibrium.

The ratio of prices charged to the able and less able depends on the income tax. Again income tax reduces the incentive to sell to the rich at a high price. Income taxation has two effects. One is an increase in leisure which implies a dead weight loss second order in the tax rate. The other is an increase in the discount given to the poor which implies a first order gain in welfare. The optimal tax rate is positive.

In this example, providing the poor with a discount is equivalent to giving them cash. The desirable incentive effect is identical to the effect of a charitable contribution deduction. The only difference is that there is administrative challenge of detecting authentic charities.

It is not strictly necessary to assume that agents can see the type of their customer – that is tell how able to produce the service the customer is. With the (frictionless) matching agents can tell that they are selling to a poor customer if the customer demands a low amount of the service, so price discrimination is a discount for low volume of sales.

Also the assumption that there is only one type of service is key. If there are two services and one is a necessity and one is a luxury, then the prices will depend on the income tax rate. The only difference is that with (frictionless) matching each seller sets prices and I don’t have to assume a finite population for people to think about their effect on the price of the necessity. Agents just set a lower price for the necessity.

Robert Waldmann
Robert J. Waldmann is a Professor of Economics at Univeristy of Rome “Tor Vergata” and received his PhD in Economics from Harvard University. Robert runs his personal blog and is an active contributor to Angrybear.

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