Alex and Ben live on an island. There are 15 units of X and 20 unites of Y, which are perfectly divisible. XA = 7, XB = 8, YA = 9, YB = 11. Alex does not know how many X and Y Ben has, and vice versa. Alex and Ben's respective utilities are given by uA(x,y) = x*y, uB(x,y) = sqrt(x) * y.

a) Given Alex's initial allocation, what is the maximum number of Y he would be willing to give up in exchange for incrementally more X, per unit of X

b) Is there a way to make Alex or Ben better off without making either of them worse off? If yes, explain why. If not, explain why not.

c) Alex asks Ben to give him 5 units of x. Alex offers her 7 y in exchange. (i) Would Ben have anything to gain by accepting Alex's offer? Explain why, or why not.

(ii) What is the lowest number of y that Ben would be willing to accept for her to give up x? Continuing with the same assumptions.

Ken decides to set up an auction in which he sets prices for X, px, and y, py. Ken distributes money to Alex equal to IA = x + 2y and money to Ben equal to IB = 3x + y. Ken promises to buy or sell X and Y at prices px and py, clearing all transactions at once. With the utility functions provided above, if Ken sets price equal to px and py, then Alex's total demand for X is XA*(px,py,IA) = 100 - 2px + 3py + 0.5IA and his demand for Y is YA*(px,py,IA) = 100 - px + py + 2IA, while Ben's total demand for X is XB*(px,py,IB) = 5px + 6py + 3IA and her demand for Y is YB*(px,py,IB) = - 7px + 8py + 4IA. Suppose Kens sets the price of Y to be py = 13.

(a) What is the price of x, px, that would make Alex uninterested in buying or selling either good? 

How many x and y would Ben want to buy or sell at these prices (relative to what she began with)? 

Can this be a competitive equilibrium? Explain why, or why not.

 

(b) What must the price of x be for (i) the sum total of Alex & Ben's consumption of x to equal to the total amount available (i.e. demand for x equals supply of x), and (ii) the sum total of Alex & Ben's consumption of y to equal to the total amount available (i.e demand for y equals supply of y)? What is the level of consumption of Alex & Ben?