Utility-Maximizing Rule

Suppose that a consumer’s preferences can be represented by the utility function *U* = *f*(*X*, *Y*) where *X* and *Y* represent the amounts of goods *X* and *Y* consumed. Our consumer’s problem is to maximize utility subject to the constraint that total expenditures on the two goods equals income: *I* = *P _{X}X* +

To solve this problem, we form the function *V* = *f*(*X*, *Y*) + l
(*I* - *P _{X}X* -

¶
*V*/¶
*X* = ¶
f/¶
*X* - l
*P _{X}* = 0

¶
*V*/¶
*Y* = ¶
f/¶
*Y* - l
*P _{Y}* = 0

¶
*V*/¶
l
= *I* - *P _{X}X* -

These three conditions represent three equations in the three unknowns—*X*, *Y*, and l
. We can use the first two equations to eliminate l
: From the first, we see that l
= and the second equation can also be solved for l
to obtain l
= . Since both of these expressions equal l
, they must equal each other: = . The terms in the numerators are the marginal utilities of goods *X* and *Y*, respectively, so this is the condition that we require. Maximum utility is achieved when the marginal utility per dollar of each good is the same: . Of course, the final equation implies that the consumer’s budget must be exhausted as well. (We assume that the second-order conditions for a maximum are fulfilled.)

Alternatively, the first two equations could be solved as follows. Add l
*P _{X}* to both sides of the first equation and l