Consider a utility function ???? (X A, X B) = XA XB. Let PA = $3 and PB = $2, and income is set at m = $40. Suppose PB falls to PB′ =1. 

a. Before the price change, what was X*A and X*B, the optimal consumption bundles? Sketch the original budget line and label

b. If, after the price change, income changed so that the original optimal bundle is just as affordable. What is the new income (m’)? At (P_A, P’_B, m’), what is the new optimal bundle (X’_A, X’_B)? Sketch the budget line associated with (P_A, P’_B, m’). Label the point (X’_A, X’_B) as B. 

c. Does the substitution effect result in more X_B? How many more of fewer? After the price change, how much X_A and X_B are actually bought, (X’’_A, X’’_B)? Sketch the budget line associated with (P_A, P’_B, m). Label the point (X’’_A, X’’_B) as C. 

d. Along the vertical axis, label the income effect, the substitution effect, and the total effect on the demand for X_B. The income effect of the fall in P_B is the same as an (increase, decrease) in income of $___. (Choose “increase” or “decrease” and supply the missing value.) Does the income effect result in an increased consumption of X_B? How much more?

e. Does the substitution effect of the fall in P_B make the consumer consume more or fewer X_A? How much more of fewer? Does the income effect of the fall in the price of X_B make the consumer consume more or fewer of X_A? What is the total effect of the change in the price of X_B on the demand for X_A?