Question

Question 1 Jon Snow consumes pizza and burgers. His utility function is u(P, B) = PB where P is the number of pizzas and B is the number of burgers. Jon Snow has $30 to spend, and he plans to spend it all on pizza and burgers. The price of one pizza is $10 and the price of one burger is $3.

(a) Find and label Jon Snow’s initial optimal bundle on a graph where pizza is on the x-axis and burgers are on the y-axis. Label this point A.

(b) Suppose that the local burger restaurant is running a promotion selling burgers at $2 a burger. Solve for Jon Snow’s new optimal bundle. Label this bundle B on your graph from part (a).

(c) Find the substitution effect, income effect, and the total effect for both goods following the decrease in the price of burgers from $3 to $2. Label the intermediate bundle used to find the substitution and income effects point C on your graph from part (a).

(d) Are pizza and burgers normal or inferior goods? Explain your answer.

Answer 1

Utility is maximized when MUP/MUB = PP/PB

MUP = U/P = B

MUB = U/B = P

MUP/MUB = B/P

(a) Initial budget line: 30 = 10P + 3B

B/P = 10/3

10P = 3B

Substituting in budget line,

30 = 10P + 10P = 20P

P = 1.5

Again,

30 = 3B + 3B = 6B

B = 5

From budget line,

When P = 0, B = 30/3 = 10 (Vertical intercept) and when B = 0, P = 30/10 = 3 (Horizontal intercept).

In following graph, EF is the initial budget line and utility is maximized at point A where indifference curve IC0 is tangent to EF with optimal bundle being (P0, B0) = (1.5, 5).

(b) New budget line: 30 = 10P + 2B, or 15 = 5P + B

B/P = 10/2 = 5

B = 5P

Substituting in budget line,

15 = 5P + 5P = 10P

P = 1.5

B = 5 x 1.5 = 7.5

From new budget line,

When P = 0, B = 15 (Vertical intercept) and when B = 0, P = 15/5 = 3 (Horizontal intercept).

In above, GF is the new budget line and utility is maximized at point B where new indifference curve IC1 is tangent to GF with optimal bundle being (P0, B1) = (1.5, 7.5).

(c) Utility at initial prices = 1.5 x 5 = 7.5

Total effect (TE) for B = 7.5 - 5 = 2.5

Total effect (TE) for P = 1.5 - 1.5 = 0

To find substitution effect (SE), we keep utility unchanged and substitute B = 5P (i.e. P = B/5) in utility function:

B x (B/5) = 7.5

B² = 7.5 x 5 = 37.5

B = 6.12

P = 6.12/5 = 1.22

Therefore,

SE for B = 6.12 - 5 = 1.12

SE for P = 1.22 - 1.5 = -0.28

Income effect (IE) = TE - SE

Therefore,

IE for B = 2.5 - 1.12 = 1.38

IE for P = 0 - (-0.28) = 0.28

In above graph, to find substitution effect, a line is drawn parallel to new budget line GF which is tangent to IC0 at point C with decomposition bundle being (P2, B2) = (1.22, 6.12). TE for Pizza is zero and TE for Burger is movement from B0 to B1. SE (movement from point A to point C) for Pizza is (P2 - P0) and SE for Burger is (B2 - B0). IE (Movement from point C to point B) for Pizza is (P0 - P2) and IE for Burger is (B1 - B2).

(d)

For Pizza, IE > SE So Pizza is a Giffen (inferior) good.

For Burgers, SE > 0 and IE > 0, and demand increases due to fall in price. So Burger is normal good.