In An Exchange Economy With Two Consumers And Two Goods Consumer A Has Utility F

In an exchange economy with two consumers and two goods, consumer A has utility function U(xA, yA) = (xA)^2yA, consumer B has utility function U(xB, yB) = xByB.

Let (¯xA, ¯yA) represent the endowment allocation of consumer A and (¯xB, ¯yB) represent the endowment allocation of consumer B.

The total endowment of each good is 10 units. That is, ¯xA + ¯xB = 10 and ¯yA + ¯yB = 10. Set y as a numeraire.

a) Derive the contract curve in terms of xA and yA . As ¯xA increases (without changing the total endowment of x), how does the contract curve change?

b) Given any price of x, for each consumer, derive the demand for good x as a function of the consumer’s endowment and the price of x.

That is, express xA as a function of ¯xA, ¯yA and Px , and xB as a function of ¯xB, ¯yB and Px.

c) Let Px* represent the equilibrium price of x. Express Px* as a function of ¯xA and ¯yA . (Hint: use the market clearing condition and the demand functions in the previous part.) As ¯xA increases, does the equilibrium price of x increase, decrease, or stay the same?