Bob’s utility function is given by . 0.5 0.5 U = X Y Bob earns $400. The price o
ID: 1169032 • Letter: B
Question
Bob’s utility function is given by . 0.5 0.5 U = X Y Bob earns $400. The price of X is $2 and the price of Y is $4. a. What is Bob’s optimal consumption bundle before any price change? b. Suppose the price of X increases to $4. What does Bob consume after this price change? c. Find the substitution effect, the income effect and the total effect from the price change (for good X). Based on your answer, is X a normal or an inferior good for Bob? d. Draw Bob’s demand curve for good X. Make sure all relevant points are well-labeled.
Explanation / Answer
(1)
Utility function: U = X0.5Y0.5
Budget line:
400 = 2X + 4Y, Or
200 = X + 2Y (dividing both sides by 2)
MUX = dU / dX = 0.5 x (Y / X)0.5
MUY = dU / dY = 0.5 x (X / Y)0.5
MRS (X, Y) = MUX / MUY = Y / X
In optimality, MRS = PX / PY
Or, Y / X = 2 / 4 = 1 / 2
X = 2Y
Substituting in budget line:
200 = X + 2Y = 2Y + 2Y = 4Y
Y = 50
X = 2 x 50 = 100
(2)
PX changes from 2 to 4.
MRS remains unchanged but (PX / PY) increases from (1/2) to 1 [= 4 / 4].
So, MRS = 1 implies, Y = X
Substituting in budget line:
200 = X + 2Y
200 = X + 2X = 3X
X = 200 / 3 = 66.67
Y = X = 66.67
NOTE: Out of 4 questions, the first 2 are answered.