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4.14 Utility Maximization

The ratio of the marginal utilities is one piece of the puzzle — the preferences side. To finish the consumer’s optimization, we need to relate consumer preferences to the prices of the goods and the consumer’s income. Let’s start by looking at a consumer whose utility function is the standard Cobb –Douglas functional form, *U*(*X*,*Y*) = *X*^{α}*Y*^{1 – α}, where 0 < *α* < 1, and whose income is *I* = *P*_{X}X + *P*_{Y}Y. This consumer’s utility-maximization problem is written formally as

This is known as a *constrained optimization problem,* in which *U*(*X*,*Y*) is the objective function and *I* = *P*_{X}X + *P*_{Y}Y is the constraint. In other words, how much utility the consumer can get is *constrained* by how much income she has to spend. In the chapter, we solved the same constrained optimization problem graphically with the objective of reaching the highest indifference curve while staying on the budget constraint.

If this were an unconstrained maximization problem, finding the optimal combination of variables would be fairly straightforward: Take the partial derivatives of the objective function with respect to each of the variables, set them equal to zero, and solve for the variables. But the presence of the budget constraint complicates the solution of the optimization problem (although without the constraint there wouldn’t even be a finite solution to the utility-maximization problem because in an unconstrained world, the consumer consumes infinite amounts of each good).

There are two approaches to solving the consumer’s utility-maximization problem using calculus. The first relies on what we already demonstrated in this chapter: At the optimum, the marginal rate of substitution equals the ratio of the two goods’ prices. First, take the partial derivatives of the utility function with respect to each of the goods to derive the marginal utilities:

Next, use the relationship between the marginal utilities and the marginal rate of substitution to solve for *MRS*_{XY} and simplify the expression:

Find *Y* as a function of *X* by setting *MRS*_{XY} equal to the ratio of the prices:

Now that we have the optimal relationship between *Y* and *X*, substitute the expression for *Y* into the budget constraint to solve for the optimal consumption bundle:

You can see that the resulting optimal bundle
is dependent on all

three pieces of the consumer’s problem: the consumer’s relative preferences (*α*, 1 – *α*), the consumer’s income *I*, and the goods’ prices (*P*_{X}, *P*_{Y}).