Question

1. A consumer has the utility function U(X, Y) = (X + 2)(Y + 4).  Her income is $100, the price of X is $4, and the price of Y is $5.

1. In order to maximize utility subject to her budget constraint, how many units of X and Y will our consumer choose to purchase? Sketch a budget line – indifference curve diagram illustrating this optimum.  Label this optimum A.

2. Suppose the price of X increases to $8, while income and the price of Y remain unchanged.  How many units of X and Y will our consumer now choose to purchase? Add this second budget line and optimum to your diagram.  Label this optimum B.

3. The movement from point A to B on your diagram is the Total Effect of the price change.

In words, how would you decompose (in the Slutsky sense) this total change into a Substitution Effect and an Income Effect?

Do that.  Add the compensated budget line to your diagram and label the associated optimum C.  In terms of the points on your diagram, the substitution effect of the increase in the price of X is the movement from point ______ to _______, and the income effect is the movement from point _________ to __________.

4. Is X a normal or inferior good?  How do you know?

Is Y a normal or inferior good?  How do you know?

Might either X or Y be a Giffen good?  How do you know?

Answer 1

a) U(X,Y) = (X+2)(Y+4) = XY+4X+2Y+8

Budget constraint is P_X*X + P_Y*Y = M or, 4X+5Y = 100....(1)

Now, condition for maximization of utility is MU_X/MU_Y = P_X/P_Y

Here, MU_X = Y+4 and MU_Y = X+2

Then, (Y+4)/(X+2) = 4/5

or, 5Y+20=4X+8

or, 4X-5Y=12.....(2)

Solving (1) and (2), we get,

10Y=88

or, Y=8.8

and X=(100-5*8.8)/4 = 14

Hence, optimum level is (14,8.8)

b) If price of X increases to $8, budget line becomes 8X+5Y=100....(3)

Condition for maximization of utility is (Y+4)/(X+2) = 8/5

or, 5Y+20 = 8X+16

or, 8X-5Y= 4.....(4)

Solving (3) and (4) we get,

16X = 104

or, X = 6.5

and Y = (8*6.5 - 4)/5 = 9.6

Hence, the new bundle is (6.5,9.6).

c) We can show the entire change through a diagram.

In the above diagram , optimum bundle shifts from point A to point B. This total effect can be divided into two parts : income effect from point A to C equal to KL and substitution effect from C to B equal to JK. Income effect occurs due to change in budget of the consumer. When price of a good increases, consumers buys less of that good. Substitution effect occurs when the consumer substitutes its consumption with another good, which is relatively cheaper .

d) X is a normal good. This is because here both income and substitution effect occurs in the same direction which results in the fall in quantity demanded for good X due to its rise in price.

Here, Y is also a normal good as it is a substitute for good X.