Defining isocost curves: All input combinations cost the same

82 Part II: Considering Which Side You’re On in the Decision-Making Process

You’re happiest when the marginal utility per dollar spent on each good is equal for all goods.

Choosing to Use Calculus with Consumer Choice

Dangerous curves ahead. Really! This is the section where I show you how to maximize utility by using calculus and the Lagrangian function. Calculus does make indifference curves dangerous.

Measuring indifference

Indifference curves can be described by functions. For example

shows the relationship between the quantity consumed of good x, the quantity consumed of good y, and total utility.

Constraining factors

Again, consumers face a budget constraint. For example, a consumer has a weekly budget of $400 for goods x and y. The price of good x is $10 and the price of good y is $8. The budget constraint is

where x and y are the quantities consumed of each good.

Lagrangians can make you happy

You’ll recognize this as a constrained optimization problem — the consumer is trying to maximize utility, subject to a budget constraint. This situation is ideal for a Lagrangian. (Go to Chapter 3 for more information on the Lagrangian function and how to set it up.)

The consumer wants to maximize utility, subject to the budget constraint, based upon the functions I present earlier in this section. The steps you take in order to determine the quantity of x and y that maximize utility are the following:

Chapter 5: Consumer Behavior: A Market for Anything?

1. Create a Lagrangian function. Recognize that the variable you’re trying to maximize is total utility. So, your objective function is 8x0.5y0.5 .

Second, your constraint is represented by the budget 400 – 10x – 8y = 0.

Your Lagrangian function £ is

2. Take the partial derivative of the Lagrangian with respect to x and y,

the commodities you’re consuming, and set them equal to zero. These equations ensure that total utility is being maximized.

83

3. Take the partial derivative of the Lagrangian function with respect

to λ and set it equal to zero. This partial derivative ensures that the budget constraint is satisfied.

Solving the three partial derivatives simultaneously for the variables x, y, and λ maximizes total utility, subject to the budget constraint.

Rewriting the partial derivative of £ with respect to x enables you to solve for λ.

Substituting the above equation for λ in the partial derivative of £ with respect to y yields

So

Finally, substituting 0.8y for x in the constraint (the partial derivative of £ with respect to λ) yields

Thus, you should consume 25 units of good y.