2. Utility

The utility maximisation problem, given the assumptions we’ve made, asks you to choose the bundle that gives the highest utility for a given expenditure, or to determine the minimum amount required to be spent to reach a given utility.

These are two different ways to reach utility maximisation:

1. Marshallian (uncompensated) demand – Max U(q₁,q₂) subject to Y = p₁q₁ + p₂q₂.  In other words, with a given income Y, what combination of goods (q₁,q₂) will maximise utility U?  If income and/or prices change, the bundle will also change.
2. Hicksian (compensated) demand

The Lagrange method can be used to solve Marshallian demand functions.

.  The solution to the utility maximisation problem are Marshallian (uncompensated) demand functions q₁(p₁,p₂,Y) and q₂(p₁,p₂,Y).

Note that preferences are assumed to be strictly monotonic (which is the technical name for more is better):

• if bundle A contains at least as much of every commodity as the bundle B, then A is at least as good as B
• if bundle A contains strictly more of every commodity than B, then A is strictly better than B.

Due to strict monotonicity of preferences the optimal bundle will be on the budget constraint (rather than strictly inside the budget set) – all income will be spent. We also assume that the entire of a consumer’s available income is spent – none is saved.  All bundles on a budget line represent complete expenditure of all income.