The Equimarginal Principle
At this point, you may think we have exhausted all the insights we can get from the hamburger-shirt problem. We have not. The table below contains columns showing the marginal utility of shirts and the marginal utility of hamburgers. These marginal utilities are obtained from our original example, which shows the total utility of one shirt, two shirts, etc. Marginal utility is the utility of the first shirt, the second shirt, etc. Thus, the utility of the fourth hamburger is found by subtracting the utility of four hamburgers from the utility of three hamburgers. Notice that the marginal utility of each good declines as more of it is used. This is a case of diminishing returns that has the special title of "the law of diminishing marginal utility." It is based on everyday observation and introspection. After four beers, a fifth gives less pleasure than the fourth, a third hamburger gives less satisfaction than the second, etc.
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Suppose that the person is not at the optimal solution of three shirts and two hamburgers. Suppose instead that he has two shirts and three hamburgers. Can we tell from the table that he has spent his money incorrectly?
We can. Shirts and hamburgers cost the same. Suppose that each costs $1.00 and the person has $5.00 to spend. Then the last dollar spent on hamburgers gave the person only six utils, whereas the last dollar spent on shirts gave him nine utils. The dollar spent on shirts gave a much larger return, and if he could shift money from the area in which it is giving a low return to the area in which it has a high return, he will be better off. This is the basic idea of the equimarginal principle. Maximization occurs when the return on the last dollar spent is the same in all areas. In terms of a formula, a person wants
(Marginal Benefit of A)/(Price of A) = (Marginal Benefit of B)/(Price of B)
The power of this idea can be shown if we change the original problem. Suppose that the person still has $5.00 to spend, but the price of shirts doubles from $1.00 to $2.00. The old solution of three shirts and two hamburgers will no longer be affordable but will lie to the right of the budget line. To solve this new problem, two new columns must be added to our table: the marginal utility of shirts per dollar and the marginal utility of hamburgers per dollar. The table below adds them in columns MUs/(Price of Shirts) (the marginal utility of shirts divided by the price of shirts) and MUh/Price of Hamburgers).
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Price of Shirts |
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Price of Hamburgers |
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The equimarginal principle tells us to maximize utility by selecting the highest values in the columns giving marginal utility per dollar until our budget is used up. A person with only two dollars should buy two hamburgers rather than one shirt because both eight and seven are larger than five and one half. A person with $5.00, as in our example, should buy three hamburgers and one shirt. This decision does not quite equalize returns on the last dollars spent on shirts and hamburgers, but it comes as close as possible. Any other combination would give less utility and would allow for further improvement. For example, if one bought two shirts and one hamburger, the extra satisfaction from a dollar spent on shirts is only four and one half utils, whereas shifting money to hamburgers would allow one to get seven utils per dollar.
Copyright Robert Schenk