Other
Microeconomics FHS Problem Set 6 Sketch Solutions
- Course
- Institution
Sketch solutions for select questions in Problem Set 6: Risk & Expected Utility; Insurance
[Show more]
Seller
Follow
ib45pointer
Reviews received
Content preview
UNIVERSITY OF OXFORD Last updated: September 16, 2013Department of Economics
Undergraduate - Microeconomics
Tutorial/Class 6: Risk & Expected Utility; Insurance
Sketch solutions
FOR USE BY TUTORS ONLY
DO NOT CIRCULATE, ESPECIALLY TO STUDENTS
, Problems and Multipart Questions
1. Your Utility Function
There is no ‘correct’ answer to this first question – your answer will depend on your
own attitude to risk.
Suppose you are o↵ered a gamble (£x, £y; ⇡) – that is, with probability ⇡ you get £x
and with probability 1 ⇡ you get £y. What would be your certainty equivalent (CE)
for the gamble if x = 0, y = 1000, and ⇡ = 12 ? Call this amount c. Now write down
your CEs for each of the gambles (0, £c; 12 ) and (£c, £1000; 12 ). Use this information
to plot the shape of your utility function.
(You need to think carefully about your CEs – what would you choose if you were
really o↵ered these gambles. See Kreps 3.6 for discussion of this kind of exercise.)
In this context, their utility function u(·) is cardinal, i.e. the only
transformations that ensure that we still have an EU representation of preferences
over lotteries is v(·) = ↵+ u(·), for any ↵ and any > 0. With w denoting initial
wealth, deciding where to plot (0, u(w)) fixes the location parameter ↵, and deciding
where to plot (1000, u(w+1000)) fixes the scale parameter ; the other points are
determined only by the three CEs they are asked to come up with -- see (b) below.
Although there is no ‘correct’ answer to this question, there are some incorrect
ones.
(a) I would be surprised if u(w) > u(w+1000), and in general if u were plotted
as not increasing.
(b) Whatever the value of c, u(w + c) should be plotted half way between u(w)
and u(w + 1000) -- this is from the definition of CE.
Similarly the first of the other two CEs should be plotted half way between u(w)
and u(w + c), and the second half way between u(w + c) and u(w + 1000).
Apart from that, pretty much anything goes ...
2. Certainty Equivalent; Risk Premium; Risk Spreading and Pooling
(a) Bill’s initial wealth is 22. He can invest it all in a risky project that has a 50:50
chance of succeeding. If he invests and the project succeeds, his final wealth will
be 40, but if it fails, his final wealth will be 10. Bill’s preferences over monetary
outcomes, y, are summarised by the utility function u(y) = ln y. For him, what is
the certainty equivalent of this project? What is the risk premium? Why should
he reject the opportunity to invest in the project?
Expected Utility = 12 ln 10 + 12 ln 40 = 1
2 ln 400 = ln 20, i.e. u(20), therefore the
certainty equivalent is 20.
1
Expected outcome = 2 10 + 12 40 = 25, therefore the risk premium is 5,
i.e. 25 20.
Bill’s initial wealth of 22 is greater than 20, the certainty equivalent of
the project, so he should reject the opportunity to invest.
1
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller ib45pointer. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
82191 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now