There have been a few posts I’ve read lately involving people scaling down their future work expectations based on planned Obama tax increases. Starting with this hypothetical example, you can quickly go from Jonah Goldberg’s desire not to be one of the suckers having to foot the deficit spending orgy to working fewer hours, or, like in Cafe Hayek’s comment of the week, not working at all (or working abroad). Here is a quick and very simple model to show the potentially counter-productive disincentives involved in marginal tax increases.

Suppose we have a world with one marginal tax rate (i.e., a flat income tax) of 0%. In order to raise revenue to pay for spending, the federal government institutes two marginal tax rates: 0% up to $75,000 and 80% after that point. We will have individuals making various amounts of money, but the important cases to recognize are those in which the tax change can potentially make a difference, so we can look at three cases: just shy of $75K, just over $75K, and moderately over $75K (with those who make well more than $75K having the same effects as those moderately over).

I’ll work out the second case, as the mathematical process works the same way for all three instances. We first should calculate a utility function for the individual. I decided to go with Y = -26X^2 + 2950X, where X is the number of hours worked per week and Y is the gross yearly income. The individual makes $29.50 an hour and the weight on marginal disutility has been set in this example to give reasonable numbers of hours per week. For our individual making $29.50 an hour—with the somewhat unrealistic assumptions of no extra overtime payments, no need to worry about a minimum number of hours to work (to cover personal costs), and a flexible number of hours to work each week—he would work 56.73 hours per week and earn $83,676.75 over the course of a year. You can get this by taking the derivative of Y with respect to X and setting Y equal to zero (which allows for profit-maximizing) and solving for X. X will be 56.73, and you can calculate earnings as Wages * Hours worked * Weeks worked. Wages will be $29.5, hours worked 56.73 (which we solved for), and weeks worked 50 (which is an assumption based on the average American’s time worked). This value of X means that once an individual works 56.73 hours, the income derived from the next unit of time will be less than the disutility of working that extra moment, so the individual will choose not to do this.

But now suppose that we have that 80% tax. This changes the utility function a bit, as instead of making $29.50 an hour for any dollar amount over $75,000, the individual is only making $5.90 an hour. We want to find out what the individual’s incentives look like at the $75,000 mark, so to do that, solve the equation 75000 = $29.5 * X * 50 weeks (where 50 weeks is an estimate of the number of weeks worked a year) and you get 50.85. Up until 50.85 hours per week, the utility function will look the same, but after that point, it will look like Y(2) = -26X^2 + 590X. At 50.85 hours per week, it does not make sense for the individual to work any more hours, as there will be a net disutility of work—in fact, if 80% were taxed on all dollars earned, the individual would only work 11.35 hours per week. So in this case, the individual would stop working at 50.85 hours per week and not pay any income tax.

However, let us assume that people decide that an income tax is necessary for paying for desired government services, so a 10% rate is levied on dollars up to $50K and a 30% marginal rate on anything over $50K. First, we’ll do the simple case for comparison: a 10% rate across the board. In this case, the worker, instead of receiving $29.50 an hour, will get $26.55 an hour, so we have to update our utility function to be Y = -26X^2 + 2655X. After solving this, you can find that the individual will work 51.06 hours per week and earn $67,779.08 (net), paying $7531 in income taxes. Note that we can already see an distortionary income effect from the tax: the individual is working 5 fewer hours per week as opposed to the profit-maximizing case and thereby creates approximately $10,000 less than in the first instance.

Now, let us look at the case with the extra marginal rate in place. In this case, the new rate kicks in at $45,000 net, which the individual reaches at approximately 33.9 hours per week (estimating 50 weeks worked a year, it would be $45000 = $26.55 * X * 50 weeks). After that point, the new equation is Y(2) = -26X^2 + 2065X (where the 2065 is $29.50 * .7). In this case, the individual’s profit maximization process will have him work 39.71 hours total: 33.9 hours a week, which will build up the old rate, and 5.81 hours a week at the higher rate. The total net income will be ($26.55 * 33.9 * 50 ) + ($20.65 * 5.81 * 50) = $51,001.08. The government will take $5000 in taxes from the first set and $2570.92 from the second set (because $20.65 * 5.81 * 50 = $5998.83. $5998.83 / .7 = $8569.75, which is the gross income. Taxed income is $8569.75 * .3). Total tax receipts are $7570.92. In this particular case, the government will get approximately $40 in additional receipts, whereas if the person had not cut down on hours in this second scenario, they would have taken roughly $5062 more. So even though tax receipts go up, a rather large marginal tax rate increase (tripling it at $50K/year) ends up with a minimal increase in revenue because the individual shifts away to non-taxed activities (leisure). I should also note that if the marginal tax rate were, say, 35%, the governmental revenue would actually be less for that individual than at the 10% flat rate.

The conclusion here is that saying “taxes change the incentives of workers” really does mean something. The people who earn the most in any given year also tend to be the people who can afford to cut down on their work. A couple making $400,000 a year can cut down to $199,000 by working fewer hours if punitive taxes after earning $200K are too high. Their lifestyle will change in an undesirable fashion (given that they presumably had the opportunity to earn $199K a year while working fewer hours), but many will do it because the alternative options would bring less utility and because it’s not like $199,000 a year’s going to leave you in bread lines. Other productive individuals may drop out of the workplace entirely and live off of annuities or shift their earnings into non-taxed benefits (or benefits taxed at lower rates). The net effect will be a drain on the economy, by giving incentives to productive individuals not to produce (not to mention on small businesses which happen not to be corporations and are taxed using income tax rates).