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Homework answers / question archive / Ashly and Betty consume X, and Y
Ashly and Betty consume X, and Y. Ashly’s utility function is ?? = ?? 0.6?? 0.4 annd Betty's is: ?? = ?? 0.4?? 0.6 Their initial endowments are ?? = 10,?? = 20,?? = 20,?? = 10.
a) Compute the set of Pareto optimal allocations (i.e. the “contract curve").
b) Using your answer from part a, suppose you are on the contract curve and ?? = 5. Find the values of ??,??, ??.
c) Draw the Edgeworth box describing this economy. Include the initial endowments, the contract curve, and the equilibrium allocation. (Note: Start the box with Ashly)
d) Assuming that the price of Y is equal to 1, compute the competitive equilibrium of this economy (i.e. the price of X at which demand for X equals supply of X and demand for Y equals supply of Y).
Given information:
a) Compute the set of Pareto optimal allocations (i.e. the “contract curve").
Ashly has a relative preference for good-X and Betty has a relative preference for good-Y. Hence, we might expect that the Pareto-efficient allocations in this model would have the property that Ashly would consume relatively more X and Betty would consume relatively more Y. To find these allocations explicitly, we need to find a way of dividing the available goods in such a way that the utility of Ashly is maximized for any pre-assigned utility for Betty. Setting up the Lagrangian expression for this problem, we have:
L( ?? , ??) = ??(??,??) + λ [??(30 - ??, 30 - YA) - U'B]
Substituting for the explicit utility functions assumed here yields
L( ?? , ??) = ??0.6?? 0.4 + λ [(30 - ??)0.4(30 - YA)0.6 - U'B]
and the First Order Conditions for a maximum are
∂L/∂?? = 0.6(??/??)0.4 - λ* (0.4)*[(30 - YA)/(30 - ??)]0.6 = 0
∂L/∂Y? = 0.4(X?/Y?)0.6 - λ* (0.6)*[(30 - XA)/(30 - Y?)]0.4 = 0
Moving the terms in λ to the right and dividing the top equation by the bottom gives
(0.6/0.4)*(??/??) = (0.4/0.6)*[(30 - YA)/(30 - ??)]
or
?? / (30 - ??) = [(9/4)*?? ]/ ( 30 - YA)
This equation allows us to identify all the Pareto optimal allocations in this exchange economy i.e., the Contract Curve.
b) Using your answer from part a, suppose you are on the contract curve and ?? = 5. Find the values of ??,??, ??.
Using the result from part-a
?? / (30 - ??) = [(9/4)*?? ]/ ( 30 - YA)
Given- ?? = 5
The above equation would become
5 / 25 =[(2.25)* ?? ]/ ( 30 - YA) so ?? = 2.448, YB= 27.552 , XB= 25
c) Draw the Edgeworth box describing this economy. Include the initial endowments, the contract curve, and the equilibrium allocation. (Note: Start the box with Ashly)
Initial endowment is given by w.
d) Assuming that the price of Y is equal to 1, compute the competitive equilibrium of this economy (i.e. the price of X at which demand for X equals supply of X and demand for Y equals supply of Y).
For competitive equilibrium, we need
MRSXY for Ashly = MRSXY for Betty = PX/PY
1. MRSXY for Ashly = (∂UA/∂XA) / (∂UA/∂YA) = (3/2)*??/??
2. MRSXY for Betty = (∂UB/∂XB) / (∂UB/∂YB) = (2/3)*?B/?B
Using, ?? = 2.448, YB= 27.552 , XB= 25, XA = 5
MRSXY for Ashly = (3/2)*(2.448/5) = 0.7344
MRSXY for Betty = (2/3)*(27.552/25) = 0.7344
Hence, MRS are indeed equal, implying a price ration of 0.7344. Given that Price of Y=1, Therefore Price of X=0.7433 units
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