Consumption Choices

Action a4 is associated with an uncertain outcome that, depending on the state of the economy, may produce either a negative return or a positive return. Thus there

exists no apparent dominance relationship between action a4 and action a2, the best among the actions involving no uncertainty.

Suppose the investor believes that if the market is down in the next year, an investment in the mutual fund would lose 10 percent returns; if the market stays the same, the investment would stay the same; and if the market is up, the

investment would gain 20 percent returns. The investor has thus defined the states of nature for his/her investment decision-making problem as follows:

s1: the market is down.

s2: the market remains unchanged.

s3: the market is up.

A study of the market combined with economic expectations for the coming year

may lead the investor to attach subjective probabilities of 0.25, 0.25, and 0.50, respectively, the states of nature, s1, s2, and s3. The major question is then, how

can the investor use the foregoing information regarding investments A, B, and C, and the expected market behaviour serves as an aid in selecting the investment that

best satisfies his/her objectives? This question will be considered in the sections that follow.

outcome is known with certainty may also be included in the list of actions.

Associated with each action is a list of payoffs. If an action does not involve risk, the payoff will be the same no matter which state of nature occurs.

The payoffs associated with each possible outcome in a decision problem should be listed in a payoff table, defined as a listing, in tabular form, of the value payoffs associated with all possible actions under every state of nature in a decision problem. The payoff table is usually displayed in grid form, with the states of

nature indicated in the columns and the actions in the rows. If the actions are labeled a1, a2… an, and the states of nature labeled s1, s2, …, sk, a payoff table for

a decision problem appears as in table 14.1 below. Note that a payoff is entered in

each of the nk cells of the payoff table, one for the payoff associated with each

action under every possible state of nature.

Table 14.1: The Payoff Table State of Nature

ACTION s1 a1

a2 a3 . . . An

Example

s2 s3 …….. sk

The managing director of a large manufacturing company is considering three potential locations as sites at which to build a subsidiary plant. To decide which

95

location to select for the subsidiary plant, the managing director will determine the

degree to which each location satisfies the company‘s objectives of minimising transportation costs, minimising the effect of local taxation, and having access to

an ample pool of available semi-skilled workers. Construct a payoff table and payoff measures that effectively rank each potential location according to the degree to which each satisfies the company‘s objectives.

Solution

Let the three potential locations be sites A, B, and C. To determine a payoff measure to associate with each of the company‘s objectives under each alternative, the managing director subjectively assigns a rating on a 0 to 10 scale to measure

the degree to which each location satisfies the company‘s objectives. For each objective, a 0 rating indicates complete dissatisfaction, while a 10 rating indicates

complete dissatisfaction. The results are presented in table 14.2 below:

Table 14.2: Ratings for three alternative plant sites for a Manufacturing Company

ALTERNATIVE COMPANY OBJECTIVE Site A Site B Site C Transportation Costs 6 4 10 Taxation Costs 6 9 5 Workforce Pool 7 6 4

To combine the components of payoff, the managing director asks himself, what are the relative measures of importance of the three company objectives I have considered as components of payoff? Suppose the managing director decides that minimising transportation costs is most important and twice as important as either

96

the minimisation of local taxation or the size of workforce available. He/she thus assigns a weight of 2 to the transportation costs and weights of 1 each to taxation costs and workforce. This will give rise to the following payoff measures:

Payoff (Site A) = 6(2) + 6(1) + 7(1) = 25 Payoff (Site B) = 4(2) + 9(1) + 6(1) = 23 Payoff (Site C) = 10(2) + 5(1) + 4(1) = 29 3.3 Expected Monetary Value Decisions

A decision-making procedure, which employs both the payoff table and prior probabilities associated with the states of nature to arrive at a decision, is referred

to as the Expected Monetary Value decision procedure. Note that by prior probability, we mean probabilities representing the chances of occurrence of the identifiable states of nature in a decision problem prior to gathering any sample information. The expected monetary value decision refers to the selection of

available action based on either the expected opportunity loss or the expected profit of the action.

Decision makers are generally interested in the optimal monetary value decisions.

The optimal expected monetary value decision involves the selection of the action

associated with the minimum expected opportunity loss or the action associated with the maximum expected profit, depending on the objective of the decision maker.

The concept of expected monetary value applies mathematical expectation, where

opportunity loss or profit is the random variable and the prior probabilities represent the probability distribution associated with the random variable.

97

The expected opportunity loss is computed by:

E (Li) = Σall j LijP(sj), (i = 1, 2, …, n)

Where Lij is the opportunity loss for selecting action ai given that the state of nature, sj, occurs and P (sj) is the prior probability assigned to the state of nature,

sj.

The expected profit for each action is computed in a similar way:

E (πi) = Σall j πijP(sj)

Where πij represents profits for selecting action ai Example

By recording the daily demand for a perishable commodity over a period of time, a

retailer was able to construct the following probability distribution for the daily demand levels:

Table 14.3: Probability Distribution for the Daily Demand

sj P(sj)

1 0.5

2 0.3

3 0.2

4 or more 0.0

The opportunity loss table for this demand-inventory situation is as follows:

Table 14.4: The Opportunity Loss Table

98

State of Nature, Demand Action, Inventory s1(1) s2(2) s3(3)

a1 (1) a2 (2) a3 (3)

0 2 4

3 0 2

6 3 0

We are required to find the inventory level that minimises the expected opportunity loss.

Solution

Given the prior probabilities in the first table, the expected opportunity loss is computed as follows:

E (Li) = Σj=13LijP (sj), for each inventory level, I = 1, 2, 3.

The expected opportunity losses at each inventory level become:

E (L1) = 0(0.5) + 3(0.3) + 6(0.2) = N2.10 E (L2) = 2(0.5) + 0(0.3) + 3(0.2) = N1.60 E (L3) = 4(0.5) + 2(0.3) + 0(0.2) = N2.60

It follows that in order to minimise the expected opportunity loss, the retailer should stock 2 units of the perishable commodity. This is the optimal decision.