Question

Paul consumes only two goods, pizza (P) and hamburgers (H) and considers them to
be perfect substitutes, as shown by his utility function:
U = P + 4H
. The price of pizza
is $3 and the price of hamburgers is $6, and Paul’s monthly income is $300 (Put pizza
on the horizontal axis and hamburgers on the vertical axis).

c) 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 did not realize that she could have made him exactly as happy
by spending far less than she did. How much would she have needed to give him in
cash to make him just as well as off as with the gift certificate? Illustrate your
answer on a graph.

Answer 1

Theoretically we can solve this question by marginal analysis.

When Paul gets extra $60 for pizza. He can buy 20 pizzas (60/3=20).

His utility increases by 20.

As U=20+4H, H is constant rise in P leads to U↑ by 20.

Now to Increase his utility by 20. How many extra Hamburgers he would need?

20=P+4H. P is fixed, he would need 20/4=5 hamburgers.

5 hambugers will cost 5*$6= $30

So his grandmother could have given him $30 instead of $60 as it would have also raised the utility by the same amount.

Graphically.

We know optimal point is at the tangency of IC and BL line.

Plot the initial Budget line, BL1: 300= 4p+6h and IC1.

New kinked budget line BL2

Optimal point will be tangent to the kink.(*)

IC2 is tangent to BL2 at the kink.

IC2 can also be tangent to another BL, BL3 which is lower than BL2, 330= 4p+6h.

This point is (+).

We see the same IC, IC2 gives same utility at lower level of income also.