2. Consider that Abebe consumes two goods, cloth (Good X) and pizza (Good Y).
The price of
cloth and pizza are 2 Birr and 4 Birr respectively. Assume also that Abebe has a total income of
100 Birr. He faces a preference represented by U (X, Y) = X1/3Y2/3
A. Write the budget line equation and plot it on the graph.
B. Compute the optimum level of consumption of cloth (X) and pizza (Y).
C. Suppose that Abebe’s income increases from 100 Birr to 150 Birr, then determine the new
optimum consumption bundle.
D. Suppose that the price of cloth (Good X) raises to 3 Birr, then determine the new optimum
consumption bundle.
A,
The general equation of budget line is:
Y = M/Py – (Px / Py)X
Where:
● X is the quantity of cloth.
● Px is the price of good X.
● Py is the quantity of good Y.
● Y is the number of pizza.
● M is the buyer’s income.
In the problem, the values are:
Px = 2, Py = 4, M = 100
Substitute the Values:
Y = 100/4 – (2/4)X = 50 – 0.5X
Equation:
Y = 25 – 0.5X
Graph
Abebe's Budget Line for Cloth and Pizza
30
25
Quantity of Pizza (Y)
20
15
10
0
0 10 20 30 40 50 60
Quantity of Cloth(X)
B,
Given:
U(X,Y) = X1/3Y2/3 (Utility Function)
2X + 4Y = 100 (Budget Constraint)
Find MRS and Price Ratio:
MUx / MUy = Px / Py
MUx = 1/3(X-2/3Y2/3)
MUy = 2/3(X1/3Y-1/3)
MRS = MUx / MUy = Y / 2X
Simplify:
Y/2X = 1/2
Y=X
Plug into Budget Line:
Substitute Y = X into: 2X + 4Y = 100
2X + 4X = 100
6X = 100
X = 100/6 = 16.6
Since Y=X, Y= 16.6
X = 16.6 , Y = 16.6
The optimum level of consumption for Abebe is to buy 16.6 clothes and 16.6 pizzas.
