Problem 1 Suppose UXY 2X 3Y Suppose Initially Px 2 Pr a Com

Problem 1 Suppose U(X,Y)- 2X + 3Y. Suppose Initially Px 2, Pr- (a) Compute the optimal bundle and Utility of the consumer from the bun Now suppose the price changes to P 4 while I, Px remain same. 1,I 10 dle. (b) Compute the new optimal bundle and Utility of the consumer from the (c) At new prices, what income level will give consumer the same level of (d) Decompose the change in demand for good X and good Y due to the price bundle. Utility as in Part(a) change into substitution effect and Income effect. Comment on the signs of the two effects (5,5,5,15 Points)

Solution

U = 2X + 3Y

(a) Budget line: 10 = 2X + Y

For a linear utility function, optimal bundle lies at one of the corner points of the linear isoquant, so either X or Y will be consumed.

When X = 0, Y = 10 and U = 2 x 0 + 3 x 10 = 30

When Y = 0, X = 10/2 = 5 and U = (2 x 5) + 0 = 10

Since utility is higher when X = 0 and Y = 10, this is the optimal bundle with associated utility = 30.

(b) New budget line: 10 = 2X + 4Y

When X = 0, Y = 10/4 = 2.5 and U = 2 x 0 + (4 x 2.5) = 10

When Y = 0, X = 10/2 = 5 and U = (2 x 5) + 0 = 10

Since utility is equal in both cases, consumer can choose either (X = 0, Y = 2.5) or (X = 5, Y = 0).

(c)

In part (a), X = 0 and Y = 10.

Total cost = 0 x 2 + 10 x 4 = 40

Required income = 40

(d)

A linear utility function means that X and Y are substitutes, therefore entire Total Effect (TE) from a price change is accounted by the Substitution Effect (SE), and Income Effect (IE) is zero.

(Case I) After price change, consumer chooses bundle (X = 0, Y = 2.5)

For good X, TE = SE = 0 - 0 = 0

For good Y, TE = SE = 2.5 - 10 = - 7.5, so substitution effect is negative.

(Case I) After price change, consumer chooses bundle (X = 5, Y = 0)

For good X, TE = SE = 5 - 0 = 5, so substitution effect is positive.

For good Y, TE = SE = 0 - 10 = - 10, so substitution effect is negative.