Question

Paul consumes only two goods, pizza (P) and hamburgers (H), and his utility function is given by U(P, H) = P + 4H. The price of pizza is $3 and the price of hamburgers is $6, and Paul's income is $300. Knowing that he likes pizza, Paul's grandmother gives him a birthday gift certificate of $60 redeemable only at Pizza Hut. Though Paul is happy to get this gift, his grandmother could have made him exactly as happy by spending far less than she did. Find the amount of cash that the grandmother could give to make Paul just as well off as with the pizza certificate.

Answer 1

To make Paul just as well off as with the pizza certificate, his grandmother could give him $60 in cash, which would allow him to buy 4 pizzas and 6 hamburgers.

Step-by-step explanation:

To determine the amount of cash that Paul's grandmother could give him to make him just as well off as with the pizza certificate, we need to find the combination of pizza and hamburgers that Paul can purchase with his income. Since the price of pizza is $3 and the price of hamburgers is $6, we can set up the following equation:

3P + 6H = 240, where P is the quantity of pizza and H is the quantity of hamburgers.

Solving this equation, we find that P = (240 - 6H) / 3.

We can substitute this value of P into Paul's utility function to find his utility:

U = (240 - 6H) / 3 + 4H = (240 - 2H + 12H) / 3 = (240 + 10H) / 3

To make Paul just as well off as with the pizza certificate, we need to find the amount of cash that gives him the same utility. Setting the utility with cash equal to the utility with the pizza certificate:

(240 + 10H) / 3 = (240 + 60) / 3

240 + 10H = 240 + 60

10H = 60

H = 6

Substituting this value of H back into the equation for P, we find that:

P = (240 - 6(6)) / 3 = 12 / 3 = 4

Therefore, Paul could be just as well off with $60 in cash, which would allow him to buy 4 pizzas and 6 hamburgers.

Answer 2

To make Paul just as well off as the pizza certificate, his grandmother could give him $0.

Step-by-step explanation:

To find the amount of cash that Paul's grandmother could give him to make him as well off as with the pizza certificate, we need to calculate the utility that he gets from the certificate first. The utility of the certificate equals the utility of the pizza he could buy with it. With the price of pizza being $3, he could buy $60/$3 = 20 pizzas. So, his utility from the certificate is 20. Now, we need to figure out how much cash would give Paul the same utility. We can use the price of hamburgers, which is $6, to determine how many hamburgers he could buy with that cash. Since his utility function is U(P, H) = P + 4H, and the utility from the pizza certificate is already accounted for, we need to calculate the utility from buying hamburgers with the cash. The desired utility level is 20, and the price of hamburgers is $6, so the number of hamburgers he could buy with the desired utility level is (20 - 20)/4 = 0. Therefore, Paul's grandmother could give him $0 to make him just as well off as the pizza certificate.