A consumer has the following utility function:U(x,y)=xy

1. Why does this utility function represent the same preferences as the Cobb-Douglas utility function U(x,y)=x ^a +y ^b witha =b =0,5?
2. Write down the consumer'sutility maximization problem (UMP) and derive the consumer's Marshallian demand functions.
3. Assume that initially the consumer faces pricesp1 =9, p2 =1 and. The consumer's income is given byI = 72 . What is the utility level in optimum?
4. Suppose then that the price of good x decreases toP1` = 4. Compute the change in demand of good x following the price change.
5. Decompose the total effect into the substitution effect and the income effect. Hint: Derive the Hicksian demand function to calculate the substitution effect. Notice, that the utility level should be given by the answer to question 3 above.
6. Compute the Marshallian own-price elasticity of demand for good x using the actual percentage changes of demand and price whenp1 changes top1 .