I have written four posts about "think at the margin, but why?" already. My exploration aims not at criticizing the important principle of marginal thinking. What I contest is that we need more natural examples to demonstrate its importance. Many examples involve sequential decisions. But they are not necessary elements in all decision problems. Giving examples like these is fine. Yet, we need examples involving non-sequential decisions.
One class of natural examples involving no sequential decisions is fixed cost. When fixed costs are substantial, marginal cost normally departs from average cost substantially. This highlights the importance of looking at the marginal cost, not average cost.
Of course, one may still say fixed cost is normally incurred prior to the marginal cost. Hence, essentially, these examples still involve sequential decisions.
Recently, I covered a topic in public finance and received questions from some students. This Q&A can be considered as an example to demonstrate why people should think at the margin. The topic is about marginal tax rate and average tax rate. A numerical example can illustrate their meanings and differences more easily.
Suppose that one's total tax burden is calculated in this way. If one earns $10,000, this income will be taxed at 5%. So, one needs to pay $500 as tax. But if one earns $20,000 in total, the tax rate for the first $10,000 is still 5% but the next $10,000 will be taxed at 10%, or $1,000. In total, one has to pay $1,500. This also means that one pays 7.5% ($1,500) of the total income ($20,000) as tax.
In this example, the tax rate imposed on the incremental income is 5% and 10% for the first and second $10,000 respectively. These tax rates are called marginal tax rates. The average tax rate is the total tax burden divided by the total income. When one's income is $10,000, the average tax rate is 5% (one pays $500 while earning $10,000). When one's income is $20,000, the average tax rate is 7.5% ($1,500 of $20,000).
In public finance, the distinction between marginal tax rate and average tax rate are important. Marginal rate is said to affect decision. The higher the rate, the less incentive one will have for working more and earning more. For example, if one knows that earning beyond $10,000 must pay tax at a much higher rate, one may choose not to earn more than $10,000, saving the work efforts.
Meanwhile, average tax rate is not about decision and incentive but about equity. When the richer has to pay a higher percentage of income as tax while the poorer pays a lower percentage of income as tax, this is said to be fair. In our example above, earning $10,000 pays 5% while earning $20,000 pays 7.5%. So, it is fair.
The above example can't show why the distinction is important as both the marginal rate and average rate are increasing with income (the former increases from 5% to 10% while the latter increases from 5% to 7.5% when income increases from $10,000 to $20,000). The following example can.
Consider a tax scheme that calculates one's tax by this formula:
Total tax = -$500+0.1*(earned income)
The interpretation of this formula is this. People will be given a fixed subsidy of $500 regardless of their earned income. So, if one has no earned income, one just has $500 for use (or a negative tax of $500). But if one starts to earn, say $1,000, the marginal tax rate is 10% on the earned income (not on the subsidy), regardless of the income levels. So, one has to pay a tax of $100 if one earns $1,000. After tax and subsidy, one can use $$1,400 (=$1,000-$100+$500) in total. One's usable income is higher than the earned income by $400 ($1,400-$1,000). So, one receives subsidy on net.
Now, if one earns $5,000, the tax is $500 while the subsidy is also $500. Then, after tax and subsidy, one can still use $5,000, the same as before.
Since the $500 subsidy will be completely offset by tax when one earns $5,000, if one earns more than $5,000, one will need to a tax on net. For example, if one earns $10,000, one has to pay a tax of $1,000. Together with the subsidy, one can use $9,500 (=$10,000-$1,000+$500). Effectively, one pays only a total tax of $500 (=$10,000-$9,500). The average tax rate is thus 5% (=500/10000).
Using the formula above, we can get the answer more quickly. The total tax for $10,000 is -$500+0.1*($10,000), or $500.
Now, if one earns $20,000, one pays $2,000 but receives $500. The after-tax-and-subsidy income is $18,500 (=$20,000-$2,000+$500). The total tax on net is $1,500 (=$20,000-$18,500). The average tax is is 7.5% (=1500/20000).
This example shows that marginal tax rate can be a constant (10%) but the average tax rate is still progressive (increasing from 5% to 7.5% when income increases from $10,000 to $20,000). We can have a fair tax system without affecting the incentive to work (working more and earning the tax rate is still 10% at the margin).
After I explained the distinction between the two rates, a student was puzzled. She thought that if the average tax rate increases from 5% to 7.5% when income increases from $10,000 to $20,000, one may still have less incentive to earn more. I said that when one earns $10,000, one will not hesitate to earn the extra $10,000 because the tax rate remains 10% for the extra $10,000. Hence, the incentive to earn more is not impeded.
The student said that one may be concerned with the reward for working like this: not working at all, one faces 0% tax; working, however, may imply 5% of one's income to be taxed away, or 7.5% of income if one works more. Hence, it seems that one's incentive to work (not at all, work or work more) will be affected by the average tax rate.
Nonetheless, the above argument involves a comparison of the non-working state with a working state (be it earning $10,000 or $20,000). If we compare only two working states (say, earning $10,000 or $20,000), the tax rate is the same at 10% and the incentive to work more won't be affected. Comparing the non-working state with the working state, however, involves an all-or-nothing choice. Either you work, or you don't work. Normally, people are not making such a choice. Normally, people will work anyway. But they will consider if it is wise to work harder for more income. They will consider if they should accept a promotion for a higher income. That's choice at the margin.
At this point, I realize that the distinction between marginal tax rate and average tax rate exactly illustrates why we should think at the margin. Sometimes, thinking at the average is exactly not right or irrelevant. In the case of tax, thinking at the average involves an all-or-nothing choice but usually we don't need to, or won't, make an all-or-nothing choice. If we usually need to make an all-or-mothing choice, thinking at the average is right. But we don’t. Choices are usually made at the margin. That's why it's important to think at the margin.
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