Question

1. Suppose Mike’s utility function is u(x,y)=2lnx +lny.

a. Derive the demand functions.

b. Is y a Normal good?

c. Is x an ordinary good?

d. Assume the price of x is initially 1 dollar, and the price of y is also 1 dollar. Given that income is 9, if the price of x doubles to 2 dollars, decompose the change in consumption of x into substitution effect and income effect. Illustrate your answer with a graph.

e. Now, given the price increase in part b, compute the Compensating variation (CV) and the Equivalent Variation (EV).

Answer 1

(a) The marginal utilities would be as or and or . The optimal bundles would be where or or or or . This would be the utility maximizing combination of goods. Hence, putting this in the budget constraint , we have or or or and since , we have or . These are the required Marshallian demand functions.

(b) We have , meaning that as the income increases, so does the demand for y. Hence, y is indeed a normal good.

(c) We have , meaning that as the price increases, the demand decreases. Hence, x is an ordinary good.

(d) The utility function is . Putting the utility maximizing combination or in the utility function, we have or or or or or is the Hicksian demand for x.

Initially, the demand for x is or or , and demand for y is or or . In this case, the agent is having a utility of or or , and hence .

After the price change, the demand for x is or , and demand for y does not change. Hence, the utility would be or or .

The hicksian demand at previous utility and changed price would be or . The total change in demand is 3 - 6 = -3. The substitution effect would be 4.7622 - 6 = -1.2378. The rest will be the income effect, ie 3 - 4.7622 = -1.7622.

(e) The Hicksian demand for y can be found by putting in the utility function as or or or or .

The expenditure function would be or or .

To remain in the same utility level at new prices, the expenditure/income required is or or or or dollars. Hence, the compensating variation would be the difference between actual income and this income, which is $14.29 - $9 or $5.29.

To be in the new utility at new prices, the expenditure required is or or dollars. The equivalent variation would be the difference between this price and the actual price, which is $9 - $5.67 or $3.33.