## 4.12Consumer’s Optimization Problem

One of the topics for which calculus provides additional insight is consumer behavior. Let’s start by recalling the answer to the consumer’s problem that we derived graphically in the chapter before seeing how we can get there using calculus. A consumer maximizes her utility U(X,Y) when

You may have noticed that to solve the optimization problem in the chapter, we just gave you the consumer’s marginal utilities of goods X and Y — we didn’t actually solve for them. This is because solving for the marginal utilities requires more than just the algebra and geometry we’ve relied on so far. Calculus allows us to derive the marginal utilities directly from the consumer’s utility function. After determining the values of the marginal utilities, we can calculate the marginal rate of substitution. Calculus also provides us with another way to solve the consumer’s problem — the Lagrangian — which yields some important insights into utility maximization.

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