State and briefly interpret the decision rule for utility maximization.

Question

Prompt: Suppose we work at a supplement store and we observe each month a regular customer purchases 6 tubs of a carbohydrate supplement at $10 per tub and 2 tubs of a protein supplement at $20 per tub. Furthermore, after repeated small talk with this customer we learned they have the ranking of 1. (6C,3P), 2. (6C, 2P), and 3. (6C, 1P) over carbohydrate and protein tubs.

a) State and briefly interpret the decision rule for utility maximization. 

b) Based upon the purchase the regular customer makes each month, how much income can we infer they have for supplements each month? 

c) Using information from the prompt as well as our answer from part b), what combination of carbohydrate and protein tubs maximizes utility for the regular customer? 

d) Suppose the price of protein tubs this month increased from $20 per tub to $40 per tub. What happens to the regular customer’s available choices for protein tubs? Additionally, based upon this change, the customer’s income, and ranking, what would the customer’s new utility maximizing combination of carbohydrate and protein tubs be that they purchase this month? 

Answer 1

Utility maximization is achieved when the consumer buys a combination of goods (often called a bundle) where their marginal utility per dollar of each product is equal.

Specifically, it's often written as:

MU1/P1 = MU2/P2

So in this case, the consumer will be maximizing their utility when the MU of carb / $10 = MU of protein/$20 provided they have spent their entire budget.

In these types of problems its often helpful to know the budget formula, which is:

Income = P1*Q1 + P2*Q2

Here they should have an income of $100. We know the prices and quantities so you can just plug those into that budget formula and solve for income.