Question

For each of the following situations where the price of x decreases from p¹_x to p²_x (a-b), calculate the following:

i) the optimal basket at time 1 and time 2.

ii) the income and substitution effect. What type of good does this suggest good x is?

iii) the compensating and equivalent variation. Explain in words what these mean.

(a) u(x,y) = 5xy, p¹_x = 9, p²_x = 4, p_y = 3, I = 72.

(b) u(x,y) = 7x^1/3y^1/3, p¹_x = 4, p²_x = 1, p_y = 2, I = 64

Answer 1

i) In case A, the utility function is u(x,y) = 5xy. P_1x=9, P2_x=4, P_y=3. Income=72.

We know that optimal basket of goods is when Marginal Utility of X= Marginal Utility of Y.

MU_x=d(5xy)/dx=5y.

MUy=d(5xy)/dy=5x.

Hence, At optimal basket, 5x=5y. OR x=y at optimal basket. Consider this equation 1.

We also have a price constraint. That is, the consuder cant spend more than his/her income. Here it is 72.

So, X*P_x+Y*P_Y=72. This is equation 2.

We know the prices of X and Y and we know X=Y (Equation 1). Putting values in, we get

X*9+X*3=72. OR 12*X=72. OR X=6.

Since X=Y, Y=6.

The optimal basket in time 1 is X=6 and Y=6.

In Time 2, everything remains same except P_x=4 now. So,

X*4+X*3=72. OR 7*X=72. OR X=72/7. OR X=10.29 and Y=10.29.

In Case B, utility function is u(x,y) = 7X^1/3*Y^1/3 (I hope I am getting the equation right. The questoin formatting was not proper). P_1x=4, P2_x=1, P_y=2. Income=64.

Again, optimal basket of goods is when Marginal Utility of X= Marginal Utility of Y.

MU_x=d(7X^1/3*Y^1/3 )/dx=(7/3)*.X^-2/3*Y^1/3

MUy=d(5xy)/dy=(7/3)*.X^1/3*Y^-2/3.

At optimal basket,

(7/3)*.X^-2/3*Y^1/3=(7/3)*.X^1/3*Y^-2/3

OR X=Y. This is equation 1.

Price constraint= X*P_x+Y*P_Y=64. This is equation 2.

Using these 2, we get

X*4+X*2=64 OR 6X=64 OR X=10.67. Since X=Y, Y=10.67 at time 1.

In time 2, price constraint=

X*1+X*2=64

OR 3X=64 OR X=21.33. Since X=Y, Y=21.33 in time 2.

ii) Since X=Y always, it means when you purchase X, you have to purchase Y also (or vice versa). This means both of these are perfect complement goods.

iii) Compensating Variation- This means the change in income that will let the customer consume as much as he was, before an economic change happened. For example, if the price of a good decreased (like in our case X went from 9 to 4), he would be able to consume more. A decrease in his income will return him to earlier consumption. That change is compensating variation.

Equivalent Variation- This is the change in income which would let consumer improve the utility by the same amount if the economic change had happened. For example, if prices didnt fall but income increased by an amount that would let the consumer consume the product by the amount when the prices would have fell, this would have been equivalent variation.