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Asset Economy with Expected Utility

创建日期：2024-05-27 16:10:06

所属分类：经济Economics

标签： Economy

Asset Economy with Expected Utility

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Asset Economy with Expected Utility

Overview
• I will be covering Q1 and Q2, please do Q3,4 in your own time
Question 1
Suppose one agent is risk averse and one agent is risk neutral.
(a) Draw the contract curve.
Recall the contract curve is all the feasible trade outcome that are Pareto efficient.
How can we arrive at this graph so quickly ? Well, suppose that that is was efficient
for the risk averse agent to carry some risk (i.e not on the 45 degree line). You can
then increase the total utility by giving that risk to the risk neutral individual who is
not affected by it. It must then be true that the risk averse agent carries no risk on
the contract curve.
If the edgeworth box was a square then the risk neutral individual would also carry no
risk. That is because there is no aggregate risk, a square edgeworth box means that
the total resources are the same in both states.
We have Wilson’s Mutuality Principle:
In complete markets all idiosyncratic risk (i.e risk purely from the allocation of endow-
ments) is insured in the competitive equilibrium
(b) How risk averse is the representative consumer?
The representative consumer behaves like the consumers on average in that they as a
single person economy reach the same outcome.
In our economy risk neutral agents don’t care about risk and the risk averse agents
don’t take any risk. The representative consumer is then risk neutral.
1
(c) Suppose now both are risk averse with CARA utility, what would you expect the
contract curve to look like?
Wilson tells us that the marginal share of risk is proportional to the risk tolerance.
Here both have constant risk tolerance and so the contract curve is somewhere between
the two dashed lines and parallel to them. If agent one on the left was less risk tolerant
the line would be closer to the left, while if the agent on the right was less risk tolerant
the line would be closer to the right.
Note that if there was no aggregate risk, the the two dashed lines would be the same
line as neither has to carry any risk.
Question 2
Suppose we have utilities v1(y) and v2(y), endowment vectors w
1 and w2 and arrow prices
α.
We assume there is no aggregate uncertainty. The total endowments must be the same for
both states, let that be w
w := w11 + w
2
1 = w
1
2 + w
2
2
(a) For this part assume they agree on the state probabilities (π, 1− π)
Claim: π
1−π =
α1
α2
We take π as given so we need to find the arrow prices to prove the claim.
The agent maximises expected utility subject to her BC, that is
max πv1(y
1
1) + (1− π)v1(y12)
s.t α1y
1
1 + α2y
1
2 = α1w
1
1 + α
1
2w
1
2
First order conditions give us
π
1− π
v′1(y
1
1)
v′1(y
1
2)
=
α1
α2
2
An identical condition for agent 2 would also hold.
If one of the agents is risk neutral then v′ is constant and so we have proven our claim.

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MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 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