Consider an economy with two agents, Amy and Becky, and two goods, x and y. Amy's utility function is U_A(x, y)=x_Ay_A, and Becky's utility function is U_B=\min\{4x_B, y_B\}. Amy and Becky's endowments are both \omega_A=\omega_B=(4,1). Suppose that the contract curve in this economy can be represented as y_A=mx_A. Find m.

M=3

Explanation:

1. UA(x,y) = XAYA & Ug = min{3XB,YB} WA=WB=(3,1)

Now contract curve is locus of all Pareto efficient allocations,

Where

3XB_YB

XA + XB = X = 3+3=0

→ XB = 6 - X^

YA+YB = 1 + 1 = 2

YB = 2 - YA

Thus 3(6-XA) 2 - YA =

18-3XA-2+ YA = 0

➡16-3XA+ YA = 0

➡YA 3XA - 16 =

YA = 3XA

m = 3

as

So, if

then

2. UA(XA,YA) = 6XA+ YA & UB(XB, YB)= XB + 6YB

WA WB (10,10)

Core is locus of all pareto efficient allocations that make both individuals better off, as compared to these welfare level at endowment levels,

MRSA MUX/MUY = 6, MRSB-1/6 UA-6(10) + 10 = 70 = UB

Now, min XA is such that Uª> 70

➡ 6XA + YA≥ 70 as for core, allocation is Necessarily first pareto optimal,

So, YA = 0, then

➡ 6X₁> 70

➡XA≥ 70/6

Min XA2 11.67 is the answer.

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