Question

2. Randy consumes two goods: X and Y.  Randy’s preferences over consumption bundles (X,Y) are summarized by the utility function:

u (X,Y) = X³Y.

1. Write algebraic expressions for Randy’s demand functions for goods X and Y to be

2. If P_X= 1, P_Y= 1, and m = 100, what would be Randy’s optimal consumption of goods X and Y?

3. Suppose now that the price of X rises to 3, while the price of Y and income remain unchanged.  What is Randy’s new optimal consumption bundle?

4. Calculate the equivalent variation.

Answer 1

1. The utility is given to be , and the budget line would be . The MUs would be or or and or or . The optimal combination of goods would be where or or . Putting it in the constraint, we have or or , and since , we have . These are the demand for X and Y.

2. For the given prices and income, Randy's optimal consumption of goods would be or and or .

3. For the new price of X, Randy's optimal consumption of Y would not change and remain 25, but optimal consumption of X would be or .

4. Randy's old utility level was and new utility level is . For the Marshallian demand of X be and Y be c, the indirect utility level would be as or or .

Now, for the indirect utility function be , the inverse of this function would be the expenditure function. Hence, from the indirect utility function, we have or or or . This is the expenditure function. This is the expenditure required for U* utils with prices Px and Py.

This means that for the old prices and new utility, the income required is or or or . Hence, the equivalent variation would be 100-43.86 or $56.14. This means that to reach the new utility a old price, the income must be reduced by $56.14. The graph is as below.