Question

Consider an exchange economy with two goods (1 and 2) and two consumers (Anna and Bob). Anna’s utility is uA = xA1xA2, while Bob’s utility is given by uB = xB1 + xB2. Initial endowments are (3, 0) for Anna, and (0, 2) for Bob. (a) Find all Pareto efficient allocations. (b) Find the competitive equilibrium. (c) Draw a clearly labeled Edgeworth box. (d) Are the preferences of Anna and Bob any different? Does it make sense to treat them differently? Briefly justify your answer.

Answer 1

A) at Pareto efficient allocations

MRSA = MRSB

MRS = MU1/MU2

So at PO:

xA2/xA1 = 1

So, xA2 = xA1 : set of all Pareto efficient allocations

b) at CE : xA1 + xB1 = total X1 = 3

as for A, Cobb Douglas utility function

So, xA1 = M/2P1, xA2 = M/2P2

MA : endowment income = 3P1

MB = 2P2

As at CE : MRSA = MRSB = P1/P2

So , P1/P2 = 1

If P2 = 1, P1 = 1

So, at CE , xA1 = 3/2, xA2 = 3/2

So, xB1 = 3-3/2 = 3/2

xB2 = 2-3/2 = 1/2

So CE allocation

(xA1, xA2) = (1.5, 1.5) & (xB1, xB2) = (.5,.5)

c) box dimensions : 3×2

E : initial endowment point

d) yes, two have different preferences

Anna has Cobb Douglas preferences.

while , Bob has perfect substitutes preferences

yes, two are treated differently