You may follow the methods shown in the mp4 on Decision Analysis for a way to do part (b) of this question if you wish.
(a) Discuss how a utility function can be assessed. What is a standard gamble and how is it used in determining utility values?
(b) Alan Barnes invests primarily in the share market. Recently he has become concerned about the share market as a good investment. During the next year he must decide whether to invest $10,000 in the share market or in a government bond at an interest rate of 9%.
Alan expects the share market to be good, fair or bad, giving a return of 14%, 8% or 0% respectively on his money.
1.Develop a decision matrix showing the two possible strategies, the three states of the share market and the monetary gains or losses under the six possible actionstate scenarios.
Answer the following questions. Each answer must be supported with appropriate calculations and/or a table of figures, and you must state for questions 2 to 5 which alternative would be selected.
 Which alternative would an optimist choose?
 Which alternative would a pessimist choose?
 Which alternative is indicated by the criterion of regret?
 Assuming probability of a good market = 0.4, a fair market = 0.4 and a bad market = 0.2, using expected monetary values what is the optimum action?
 What is the expected value of perfect information?
Assessing the Utility Function
An individual usually enjoys the consumption of certain units of a commodity. This provides them with satisfaction which is known as the utility of the commodity. The utility of the commodity can thus be expressed in the form of a function. The function is designed by describing the decision problem. There are several characteristics associated with the utility function which is important to be identified as well. Various quantitative restrictions can be used to specify the function of utility which will be of help in making decisions by comparison of different strategies or strategies (Garriga, 2014).
One of the common ways with which utility can be assessed is the standard gamble. When the utility of a commodity is 1, that is the best possible outcome of the gamble. Similarly, when the utility of a commodity is 0, that is the worst possible gamble. In between the best and the worst cases, there is also a case in which intermediary outcomes are available. From these intermediary cases, the best and the worst outcomes are considered by the decision makers in order to gamble. Probabilities are associated with the intermediary outcomes and gambles, with the help of which the decision maker makes the decision (Hutchins et al., 2015).
 Investment Amount = $10,000
The amount is to be invested in Share Market or Government Bond.
Government Bond interest rate = 9%
Share Market interest rate is 14%, 8% and 0% when the market is good, fair and bad respectively.
 Considering the condition stated above, the decision matrix that can be obtained is given in table 1.
Table 1: Decision Matrix with all possible Strategies
Strategy 
Market Condition 

Good 
Fair 
Bad 

Share Market 
11400 
10800 
10000 
Government Bond 
19000 
19000 
19000 
 The optimist approach in choosing the best strategy is to invest in the share market. The results are shown in table 2.
Table 2: Optimist Approach in Selecting Best Strategy 

Strategy 
Market Condition 
Best Profit 

Good 
Fair 
Bad 

Share Market 
11400 
10800 
10000 
11400 
Government Bond 
10900 
10900 
10900 
10900 
 The approach of the pessimist in selecting the best strategy is to select the investment in the Government Bonds. The results are shown in table 3.
Table 3: Pessimist Approach in Selecting the Best Strategy 

Strategy 
Market Condition 
Least Profit 

Good 
Fair 
Bad 

Share Market 
11400 
10800 
10000 
10000 
Government Bond 
10900 
10900 
10900 
10900 
 Table 4 shows the criterion of regret.
Table 4: Criterion of Regret in Selecting Strategies 

Strategy 
Market Condition 
Maximum 

Good 
Fair 
Bad 

Share Market 
0 
100 
900 
900 
Government Bond 
500 
0 
0 
500 
 Probability that the market is good = 0.4
Probability that the market is fair = 0.4
Probability that the market is bad = 0.2
Thus, the expected monetary value is given by the following table 5. It can be seen from the table that the expected monetary value is high in case of investing in the share market.
Table 5: Expected Monetary Value (EMV) from different Strategies 

Strategy 
Market Condition 
Expected Profit 

Good 
Fair 
Bad 

Share Market 
456 
432 
200 
456 
Government Bond 
436 
436 
218 
436 
 Expected value given perfect information (
Thus, Expected Value of Perfect Information (EVPI) = EVPI – EMV = $(1110 – 456) = $654.
Returns from Market:
Favorable = $100,000
Unfavorable = –$60,000.
Probability of favorable market = 0.5
Probability of unfavorable market = 0.5
 Expected Return = $((100,000 * 0.5) – (60,000 * 0.5)) = $20,000.
Thus, it can be seen that the business will earn a profit. Thus, Jim can start the business of producing men’s razor.
 Probability of favorable market depending on the track record of his friend = (0.5 * 0.7) + (0.5 * 0.3) = 0.5. Therefore, the probability of unfavorable market = (1 – 0.5) = 0.5.
 The required posterior probability = (0.3 * 0.5) = 0.15.
 Expected net gain / loss = $((100,000 * 0.5) + (– 60,000 * 0.5)) – $5,000 = $15,000.
Engagement of friend has decreased the expected profit margin but there is still profit from the business. Engaging the friend is thus a better choice as his engagement will provide the market conditions more accurately.
 The formulas used in the construction of the simulation is shown with the help of the following excel sheet given in figure 1. shows the results of the simulation from the formulas listed in figure 1.
 Over the 12month period, Tully tyres have a monthly average profit of $21,584.16.
 By increasing the selling price from $200 to $220 and increasing the profit margin from 22% to 32%, the sales is unaffected, and the average profit has been obtained as $28,811.12. The revised simulation results are given in figure 3.
According to the initial information provided, it has been observed that the average profit is around $21,000. But with the increase in the selling price by $40 and increasing the profit margin by 2%, it is known that there will be no effect in the sales of the product. From the increased credentials, the results have shown that there is an increase in the average profit, which is around $28,000 now. Thus, the increased rates must be used for the business.
Thanks and Regards
Name and Signature
 The table 6 shows the necessary calculations to estimate the overhead cost using high low method.
Table 6
Machine Hours (x) 
Overhead Cost (y) 

Highest Activity 
3,800 
$48,000 
Lowest Activity 
1,800 
$46,000 
Marginal Cost = $(48,000 – 46,000) / (3800 – 1800) = $1
Fixed Cost = $(48,000 – 1 * 3800) = $(46,000 – 1 * 1800) = $44,200
Equation to estimate overhead cost:
Overhead Cost (y)= 44,200 + Machine Hours (x)
When, machine hours = 3000, overhead cost = $(44,200 + 3,000) = $47,200
 Table 7 gives the regression results to estimate the overhead cost with the help of machine hours.
Table 7: Overhead Cost against Machine Hours 



Regression Statistics 


Multiple R 
0.104 

R Square 
0.011 

Adjusted R Square 
0.113 

Standard Error 
15447.614 

Observations 
10 

ANOVA 



df 
SS 
MS 
F 
Significance F 

Regression 
1 
2.1E+07 
2.1E+07 
0.088 
0.774 

Residual 
8 
1.9E+09 
2.4E+08 

Total 
9 
1.9E+09 


Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
59198.785 
21473.783 
2.757 
0.025 
9680.152 
108717.417 
MH 
2.304 
7.774 
0.296 
0.774 
20.230 
15.621 
The model developed is insignificant as the significance F is more than the level of significance (0.05). Thus, Machine hours is not a good predictor of overhead cost.
Table 8 gives the regression results to estimate the overhead cost with the help of batches.
Table 8: Overhead Costs against Batches 

Regression Statistics 


Multiple R 
0.91 

R Square 
0.83 

Adjusted R Square 
0.81 

Standard Error 
6379.22 

Observations 
10 

ANOVA 


df 
SS 
MS 
F 
Significance F 

Regression 
1 
1.06E+09 
1.06E+09 
39.427 
0.000 

Residual 
8 
3.26E+08 
4.07E+07 

Total 
9 
1.93E+09 


Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
6555.556 
7666.868 
0.855 
0.417 
11124.274 
24235.385 
Batches 
234.568 
37.357 
6.279 
0.000 
148.422 
320.714 
The model developed is significant as the significance F is less than the level of significance (0.05). Thus, batches is a good predictor of overhead cost. Moreover, batches can predict 83% accurately the variations in the overhead cost.
Overhead Cost = 6555.556 + 234.568 * Batches
Table 9 gives the regression results to estimate the overhead cost with the help of batches and machine hours.
Table 9: Overhead Costs against Machine hours and Batches 



Regression Statistics 


Multiple R 
0.913 

R Square 
0.833 

Adjusted R Square 
0.785 

Standard Error 
6783.922 

Observations 
10 

ANOVA 



df 
SS 
MS 
F 
Significance F 

Regression 
2 
1.6E+09 
8.0E+08 
17.468 
0.002 

Residual 
7 
3.2E+08 
4.6E+07 

Total 
9 
1.9E+09 


Coefficients 
Standard Error 
t Stat 
Pvalue 
Lower 95% 
Upper 95% 
Intercept 
9205.658 
12704.918 
0.725 
0.492 
20836.700 
39248.016 
MH 
0.931 
3.422 
0.272 
0.793 
9.022 
7.161 
Batches 
233.827 
39.820 
5.872 
0.001 
139.667 
327.987 
The model developed is significant as the significance F is less than the level of significance (0.05). Moreover, the model can predict 83% accurately the variations in the overhead cost.
Overhead Cost = 9205.658 + 233.827 * Batches – 0.931 * MH
 Model developed in table 8 and 9 has shown the same value of R Square but the adjusted R Square is higher in the model in table 8. Thus, the better will be considered as the model which involves only batches to predict overhead cost.
 Overhead Cost = 6555.556 + 234.568 * 150 = $41,740.74
 Contribution margin for Product A = $(12 – 8) = $4
Contribution margin for Product B = $(15 – 10) = $5
 For Product B, break even = (5000 / 5) = 1000 units
 For Product A, break even = (5000 / 4) = 1250 units
 Ratio of manufacture of A and B is 3:1.
 Profit of $3,500 will be earned if 1333 units of product A and 667 units of product B are sold. The necessary calculations are shown in table 10.
Table 10
Product 
A 
B 
Total 
Sales Unit 
3 
1 
4 
Sales price 
$36 
$15 
$51 
Variable cost 
$24 
$10 
$34 
Total fixed costs 
$5,000 

Total Contribution 
$12 
$5 
$17 
Weighted average Contribution 
$4 

Targeted Profit (Before Tax) 
$3,500 

Targeted Sales Volume 
1333 
667 
2000 
 Profit of $8,400 will be earned after tax, if 855 units of product A and 428 units of product B are sold. The necessary calculations are shown in table 11.
Table 11
Product 
A 
B 
Total 
Sales Unit 
3 
1 
4 
Sales price 
$72 
$15 
$87 
Variable cost 
$24 
$10 
$34 
Total fixed costs 
$5,000 

Total Contribution 
$48 
$5 
$53 
Weighted average Contribution 
$13 

Targeted Profit (After Tax) 
$8,400 

Tax Rate 
30% 

Targeted Profit (Before Tax) 
12000 

Targeted Sales Volume 
855 
428 
1283 
References
Garriga, E. (2014). Beyond stakeholder utility function: Stakeholder capability in the value creation process. Journal of Business Ethics, 120(4), 489507.
Hutchins, R., Viera, A. J., Sheridan, S. L., & Pignone, M. P. (2015). Quantifying the utility of taking pills for cardiovascular prevention. Circulation: Cardiovascular Quality and Outcomes, 8(2), 155163.
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