Editing Utility maximization problem (section)

== Utility maximisation of perfect substitutes ==
U = x + y

For a utility function with [[Substitute good|perfect substitutes]], the utility maximising bundle can be found by differentiation or simply by inspection. Suppose a consumer finds listening to [[Australia|Australian]] rock bands [[AC/DC]] and [[Tame Impala]] perfect substitutes. This means that they are happy to spend all afternoon listening to only AC/DC, or only Tame Impala, or three-quarters AC/DC and one-quarter Tame Impala, or any combination of the two bands in any amount. Therefore, the consumer's optimal choice is determined entirely by the relative prices of listening to the two artists. If attending a Tame Impala concert is cheaper than attending the AC/DC concert, the consumer chooses to attend the Tame Impala concert, and vice versa. If the two concert prices are the same, the consumer is completely indifferent and may flip a coin to decide. To see this mathematically, differentiate the utility function to find that the [[marginal rate of substitution|MRS]] is constant - this is the technical meaning of perfect substitutes. As a result of this, the solution to the consumer's constrained maximization problem will not (generally) be an interior solution, and as such one must check the utility level in the boundary cases (spend entire budget on good x, spend entire budget on good y) to see which is the solution. The special case is when the (constant) MRS equals the price ratio (for example, both goods have the same price, and same coefficients in the utility function). In this case, any combination of the two goods is a solution to the consumer problem.