Assignment 10 Economics 31 Fall 1999

Introduction to Decision Theory

Reading: See handouts, refer to other texts (Wonnacott & Wonnacott,
Introductory Statistics for Business and Economics, Smith, Statistical
Reasoning, Mansfield, Statistics for Business and Economics).

1. The owner of a sporting goods store says that he is indifferent between
the certainty of receiving 7500 and a gamble where there is a .5 chance of
receiving 5000 and a .5 chance of receiving 10000. Also he says that he is
indifferent between the certainty of receiving 10,000 and a gamble where
there is a .5 chance of receiving 7500 and a .5 chance of receiving 12,500.

A) draw four points on the utility function of the owner of the store.
B)Does the owner of the store seem to be a risk averter, risk lover, or
risk nuetral. Explain (courtesy of Mansfield 17.3).

2. A publisher intends to sign a contract for an accounting text with one
of three authors, Smith, Brown, or Jones. If the text turns out to be very
successful, profits (excluding any extraordinary accounting costs) will be
$250,000; if the book is only moderately successful, these profits will be
$80,000. In the event that the text fails, a loss of $80,000 will result.
The probabilities given in the first table have been conjectured for these
states of nature for the three books.

Name	Very Successful	Moderately Successful	Failure
Smith	.2	.6	.2
Brown	.1	.8	.1
Jones	.3	.2	.5

The publisher also has the option of mounting, at a cost of $30,000, an
extraordinary advertising campaign for the book once it is published. It is
estimated that if this were to be done, the probabilities for the three
states of nature would be as shown in the following table:

Name	Very Successful	Moderately Successful	Failure
Smith	.4	.4	.2
Brown	.3	.6	.1
Jones	.5	.2	.3

 

a. Draw the publisher's decision tree.

b. According to the expected monetary value criterion, which author should
be signed, and should the extraordinary advertising campaign be mounted?

c. Following the calculations in part b., the publisher has signed a
contract with the chosen author. At this point, it is discovered that a
clerical error has been made in the marketing department and that in fact
the actual cost of the advertising campaign is $40,000. According to the
EMV criterion, should the publisher offer to pay the chosen author to
withdraw from the contract, and if so, what is the highest sum he should
offer.

3. A Steel firm is interested in merging with an oil firm. If the oil
firm's current drilling activities are a success, the steel firm will gain
20 million from the merger. If the oil firm's current driling activities
are not a success, the steel firm will lose 10 million from the merger. The
steel firm believes there is a .4 probability that the oil firm's current
drilling activities will be a success (courtesy of Mansfield 17.9.

A) What is the expected opportunity loss if the steel firm carries out the merger?

B)what is the expected opportunity loss if the steel firm does not carry
out the merger?

4. The IRS has studied your last year's income tax return and has sent you
a bill for $500 in back taxes. You now have the choice of paying the bill
or disputing the audit. If you dispute it, you must pay an accountant $50
to prepare your case. After preliminary talks with your accountant your
chances of winning the dispute are 0.05.

a. Should you dispute the case based in monetary expectations?

b. Assume that large losses of money are disastrous to you as a struggling
student.This is reflected in your utility function, which indicates that U (-$50)
is -4 utils, U (-$500) is -425 utils, and U(-$550) is -440 utils. Based on
expected utility, what is your best course of action?

5. You have been offered $2 or a raffle ticket with a 1 percent chance of
winning 100. Your utility function is U(x)=x + .02 x^2 where x is your
monetary gain. Which choice maximizes your expected return? Which maximizes
your expected utility? Explain any difference (Smith 19.15).

6. Substitute Teachers. Substitute teachers for a school can be hired on a
contract or noncontract basis. Contract substitutes receive $30 each school
day for the whole year simply for being available to teach, and also
receive an additional $150 on any day they actually teach. When the number
of contract substitutes is inadequate on any school day, noncontract
substitutes must be called in. Noncontract substitutes receive $220 for
each day they teach but nothing when they do not teach. The school
administrator must now decide how many substitutes should be given
contracts for the coming school year. Records show that the following
probability distribution describes the number of substitutes required on
any school day:

si	0	1	2
P(si)	.4	.5	.1

Construct the payoff table for the administrator's decision problem,
showing the acts, outcome states, and payoffs for any given school day.
Display the costs as negative payoffs.

7. Jill's wealth consists entirely of stock in company NEW, which may be
taken over by company GLOM.If so, her stock will be worth 360,000., if not
it will be worth 160,000. She considers the two possible outcomes to be
equally likely. Her utility function is U=W^.5where W is her
wealth.

A)What is the expected value of her wealth?

B)What is the expectedvalue of her utility?

C)What is the minimum amount for which she would be
willing to sell her stock? Why is this minimum amount larger or smaller
than the expected value of her stock? (Smith 19.17)

8.Good, Bad, and Ugly are to fight a three-cornered pistol duel. All know
that Good's probability of hitting a target is .3, Ugly's chance is .5 and
Bad never misses. They are to fire at their choice of target in
succession-Good, then Bad, then Ugly-until only one is left unhit. What
should Good's strategy be? (Smith 19.50)

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