Consider a consumer with Cobb-Douglas utility function (POINTS: 30) u(x, y) = xy and assume she has income I = $120. The prices in the economy are px = $3 and py = $4. (a) Set up the maximization problem for this consumer and obtain the demand (x , y ) (present your work). (Points: 5/30) Answer: blabla blabla 13 University of Minnesota ECON 3101 - PS3 (b) Assume the price for good y drops to py2 = $2, and obtain the new demand (x 2 , y 2 ) (present your work). (Points: 5/30) Answer: blabla blabla 14 University of Minnesota ECON 3101 - PS3 Now we will decompose the effect of the price drop into the two effects discussed in lecture, the substitution effect, which is due to the change in the price ratio, and the income effect, because a drop in price increases the agent's purchase power. To derive the substitution effect, we need a bundle (x sub , y sub) that is optimal for the new prices ($3,$2), and attains the same utility as the original bundle (x , y ). (c) What is (x sub , y sub) (feel free to leave your answer as a square root if you prefer)? (Show your calculations). (Points: 4/30) Tip: You need two equations to find (x sub , y sub) one is a tangency condition using the relevant prices, and the other one is one that guarantees its utility is the same as (x , y ). Answer: blabla blabla 15 University of Minnesota ECON 3101 - PS3 (d) Calculate the substitution effect, i.e., (Points: 4/30) substitution effect = (x sub , y sub)(x , y ) Answer: blabla blabla (e) Calculate the income the agent should have, denote it I sub, to make her consume (x sub , y sub) when prices are (px , py2). (Points: 4/30) Tip: We already discussed that to find (x sub , y sub) the tangency condition should be satisfied for the new prices. What other condition is missing for optimality? Answer: blabla blabla 16 University of Minnesota ECON 3101 - PS3 (f) Calculate the income effect, i.e., (Points: 4/30) income effect = (x 2 , y 2 )(x sub , y sub) Answer: blabla blabla (g) Draw a graph with the three bundles (the two optimal ones, and (x sub , y sub)), the indifference curves they belong to, the two budget constraints for each set of prices, ($3,$4) and ($3,$2), and the budget constraint for I sub. Remember to indicate the intercepts for each of the budget constraints. (