Assume preferences can be represented by the following utility function:

u(x₁, x₂) = x₁ x₂²

a. Is the utility function monotonic? Justify.

b. Set up the consumerís utility maximization problem for prices p₁, p₂ and income m (the general case)

c. Solve the problem. You will obtain demand functions x₁* (p₁, p₂, m) , x₂* (p₁, p₂, m) in terms of the parameters (p₁, p₂, m) : Obtain price elasticity of demand for good one. Obtain income elasticity of demand for good 2.

d. Assume that, originally, the consumer faces:

prices p₁ = 2, p₂ = 5 and income m = 180:

Then, the price of good 1 increases to p₁, = 3. Obtain the income and substitution effiects for good 1 with Slutsky compensation (that is, compensating the individual so that it can still buy the old bundle at the new prices).

e. Find the amount of compensation needed for Hicks compensation (that is, compensating the individual so that he is indiffierent to his old bundle). To do this plug the old bundle into the utility function to obtain the level of utility you want to acheive. Then plug the demand functions into the utility function. Then replace prices with new prices and equate the two utilities. By now you should have a function of income equal to a number. Solve for the appropriate income level. That is the compensation needed to make the individual indiffierent to the old bundle. The amount of compensation needed should be lower than with Slutsky compensation, but because the price change is very small, there should be barely any diffierence between the two.

f. Graph your results in (e) by plotting the old and new indiffierence curves, the old, compensated and new budget sets and the old, compensated and new choices (quantities demanded).