5. Sheila and Bruce are having a party. Sheila has brought with her 18 cans of b
ID: 1111774 • Letter: 5
Question
5. Sheila and Bruce are having a party. Sheila has brought with her 18 cans of beer (x) and 12 bags of nuts (y). Bruce has brought 10 cans of beer and 20 bags of nuts. Sheila and Bruce have preferences that can be represented by the following utility functions: U)-2nny and UB(XB.y") = 1.5 Iran, InyB a) b) c) d) e) Given their preferences is the endowment point a Pareto efficient allocation of nuts and beer? Given their marginal rates of substitution at the endowment point who values beer more highly? What will be the pattern of trade associated with every mutually beneficial trade? Assume that the terms of trade are 2 bags of nuts for 1 can of beer. Find one trade that is mutually beneficial (you must verify that it is mutually beneficial). Assume that the terms of trade are 2 bags of nuts for 1 can of beer. Find one trade that is NOT mutually beneficial (you must verify that it is NOT mutually beneficial) In an Edgeworth box diagram illustrate the endowment point and a pair of indifference curves through the endowment. Illustrate the two trades that you found in (c) and (d).Explanation / Answer
a)Pareto optimal allocation of nuts and beer is that allocation where it will not be possible to make one person (Sheila or Bruce) better off without making any other person worse off.
The slope of an indifference curve dy/dx is just the ratio of the marginal utilities of goods 'X' (beer) & 'Y'(nuts) (or: dy/dx = MUx / MUy = MRSxy)
Thus the condition for Pareto optimality may be properly defined as that point where:
MRSS = MRSB .
We have, US (xS, yS) = 2lnxS + lnyS
Thus, MRSS = 2yS/xS (Where, x=18 and, y=12)
= 2*12S/18S = 4/3
We have, UB (xB, yB) = 1.5lnxB + lnyB
Thus, MRSB = 1.5yB/xB (Where, x=10 and, y=20)
= 1.5*20B/10B = 3
Since, MRSS MRSB thus, this is not pareto optimum allocation
b) Sheila will give up four units of nuts to obtain three additional units of beer, so for one beer she is willing to give up 3/4th units of nuts.
However, Bruce is willing to give up three units of nuts to obtain one additional unit of beer. Hence, Bruce value beer more. Hence as per the trade pattern, Bruce will trade nuts for beer.
c)
The terms of trade are 2 bags of nuts for 1 can of beer.
So we are given initially that, Sheila has 18 cans of beer and 12 bags of nuts. Since, Bruce values beer more and would give up nuts for beer hence he trades for receiving beer cans from Sheila and giving away bags of nuts in return. Now, Sheila gives 1 can of beer for 2 bags of nuts. So out of 18 cans of beer, she gives away 1 can and in return receives 2 bags of nuts. Thus, now Sheila has :
17 cans of Beer and,
14 bags of nuts.
We were also given that Bruce initially had 10 cans of beer and 20 bags of nuts. After trading with Sheila, he receives 1 can of beer and gives away 2 bags of nuts. Thus, now Bruce has:
11 cans of Beer and,
18 bags of nuts.
Now to check whether this trade is mutually beneficial or not, we check for utility functions before and after trade using the given utility functions:
US (xS, yS) = 2lnxS + lnyS
UB (xB, yB) = 1.5lnxB + lnyB
Before Trade:
US (xS, yS) = 2ln18 + ln12
= (2*2.89)+2.48
=8.26
UB (xB, yB) = 1.5ln10 + ln20
= (1.5*2.3) + 2.9
= 6.35
After Trade:
US (xS, yS) = 2ln17 + ln14
= (2*2.83)+2.64
= 8.3
UB (xB, yB) = 1.5ln11 + ln18
= (1.5*2.398) + 2.89
= 6.49
Since, the utilities are more after trade thus, there are mutually beneficial gains from trade in this case.
d)
The terms of trade are 2 bags of nuts for 1 can of beer.
A mutually non beneficial trade would be one wherein Bruce gives 1 can of beer for 2 bags of nuts.
So we are given initially that, Bruce had 10 cans of beer and 20 bags of nuts. So out of 10 cans of beer, he gives away 1 can of beer and in return receives 2 bags of nuts. Thus, now Bruce has:
9 cans of Beer and,
22 bags of nuts.
We were also given that Sheila has 18 cans of beer and 12 bags of nuts. After trading with Bruce, she receives 1 can of beer and gives away 2 bags of nuts.
Thus, now Sheila has :
19 cans of Beer and,
10 bags of nuts.
Now to check whether this trade is mutually beneficial or not, we check for utility functions before and after trade using the given utility functions:
US (xS, yS) = 2lnxS + lnyS
UB (xB, yB) = 1.5lnxB + lnyB
Before Trade:
US (xS, yS) = 2ln18 + ln12
= (2*2.89)+2.48
=8.26
UB (xB, yB) = 1.5ln10 + ln20
= (1.5*2.3) + 2.9
= 6.35
After Trade:
US (xS, yS) = 2ln19 + ln10
= (2*2.94)+2.3
= 8.18
UB (xB, yB) = 1.5ln9 + ln22
= (1.5*2.197) + 3.09
= 6.2
Since, the utilities are less after trade thus, there are no mutually beneficial gains from trade in this case.
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