Suppose a consumer purchases only two goods food and clothing. Her preferences between two goods can be represented by the following utility function

\(U(x,y)=x^{\alpha}y^{1-\alpha}\)

where 0 < div Py is clothing and Px food of price market current The dollars. I income monthly given exogenously Her consumed. amount the denotes y consumed, x 1,>

a) Set up the consumer's utility maximization problem.

b) Using the Lagrangian Multiplier method, solve the consumer's utility maximization problem to derive the consumer's demand curves for both food and clothing as a function of prices and income.

c) Is clothing a normal good? Explain

d) What can be said about the cross-price elasticity of food with respect to the price of clothing

Now assume that, a = 1/3, Px=$5, Py=$1, and I= $300

e) Calculate her optimal amount of food and clothing consumption.

f) Calculate the total price effect, substitution effect, and income effect on her food consumption which would result from a decrease in the price of food from $5 to $4. Illustrate all of these effects on a clearly labeled diagram.

g) Calculate the compensating variation of the reduction in the price of food from $5 to $4. Illustrate this compensating variation on a clearly labeled diagram.

h) Calculate the equivalent variation of the reduction in the price of food from $5 to $4. Illustrate the equivalent variation on a clearly labeled diagram.