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Behaving Optimally and Utility Maximization

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Consider a utility-maximizing consumer who devotes all of his weekly income, I = $720 to purchases of caviar (available at the market price, px = $10 per serving) and high-end denim jeans (available at the market price py=$120 per pair). Compute this consumer's optimal consumption of caviar (x) and jeans (y) for each of the following possible utility functions, and show your work:
a. U (x,y) = [2x +13y]^2
b. U(x,y) = Min [x,6y]
c. U(x,y) - xy^2

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Question A:
When solving utility maximization problems, we must always first take the shape of the utility function into consideration. Although the 'usual' way to solve these problems, with 'common' utility functions, is to use the Lagrangian. However, this is not always the case, as in points (a) and (b) in your questions.

Here we must solve:

max (2x + 13y)^2

subject to
10x + 120y = 720 (budget constraint)

In order to be able to use the Lagrangian, the indifference curves generated by the utility function must be strictly convex. If they are linear or concave, we can't use the Lagrangian, but the solution will be simpler: the optimal consumption will be at one of the 'extremes'; i.e., the consumer will either choose to spend his or her whole income either on x or on y.

The utility function here is:

U = (2x + 13y)^2

Let's find the indifference curves by isolating y in this equation:

sqrt(U) = 2x + 13y

sqrt(U) - 2x = 13y

y = (1/13) * (sqrt(U) - 2x)

[sqrt(U) means 'square root of U']

Clearly, this function is linear with respect to x. We know then that, at the optimal point, the consumer will either buy all of ...

Solution Summary

This solution shows steps on how to solve this consumer utility maximization problem by determining the utility function of px and py.