Utility and Utility Functions:

In the last section we talked a lot about consumers’ preferences and defined this concept as simply the rankings of consumption bundles from less desirable to more desirable. However, it would be nice if we could express these rankings in terms of some numerical index instead of being given a long (possibly infinite) list of rankings of bundles. For historical reasons, such an index is called a utility function.

Let’s assume that we ask our consumer to compare three bundles and that she expresses the following ranking. A ≥ B & B ≥ C. If we had fed these bundles into her utility function we would have gotten some numbers with the following relationship among each other:

U(A) ≥ U(B) ≥ U(C),

i.e., the utility number associated with bundle A is greater than or equal to the utility number associated with bundles B etc. Hence, the utility numbers preserve the rankings expressed by the consumer. It is important to note that the actual utility number has no meaning. This also means that we can play around with our utility function; e.g. we can add a constant to all numbers (the same constant) without changing the ranking, we can also multiply our original utility function by a constant, or take the square (or the square- root) of it, and it still is a viable utility function since it will not change the consumer’s ranking of the bundles.

Example:

Let’s assume that we have two goods (1 and 2) and three bundles: A = {3, 3} , B = {3, 2} and C = {2, 2} . If the consumer is not satiated with either good her preferences will be: A > B > C, and we want our utility function to produce numbers such that: U(A) > U(B) > U(C). Consider the following function, for example,

U(Q) = Q1 . Q2,

Where Q_i is the quantity consumed of good i. In our example we would have that,

U(A) = 3 . 3 = 9
U(B) = 3 . 2 = 6
U(C) = 2 . 2 = 4,

This does produce the required ranking. Consider now adding a constant, K = 5, so that our new utility function is, V (Q) = K + U(Q). Well, it is obvious that it would also rank the bundles in the correct way (The new utility numbers would be: V (A) = 14, V (B) = 11 and V (C) = 9.) We can also consider the following transformed functions: W(Q) = K+bU(Q), (b >0), Z(Q) = √U(Q), or Y (Q) = (U(Q))². If you apply these transformations to U(Q) you’ll find that all the new utility functions preserve the rankings of the bundles.

The point of this example is to underscore the important fact that utility numbers in themselves have no meaning, and we can transform the utility function in many ways as long as the transformation preserves the underlying rankings (a so called positive monotonic transformation). We also require some other things of our utility function, for example, if the consumer has monotonic preferences an increase in one good without a decrease in the other, should yield an increase in the utility number (since the consumer would now rank the new bundle higher than the original one). This brings us to the concept of marginal utility, and the marginal utility function.

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