Assessing Utility Function

Utility is the level of satisfaction that an individual enjoys from a consuming certain units of a commodity. There are several methods of assessing a utility function. The first step in assessing utility function is structure the decision problem. This is followed by development an appropriate measurement scale for evaluating the attributes. The decision maker then needs to acquaint with different aspect related to assessment procedure.

The second step is to identify the special characteristics relevant to the typical utility function (Li, 2014). Quantitative restrictions then need to specify for utility function of the decision maker in terms of comparison of different gambles for attributes under consideration. For assessment of final utility, a utility function is to be specified accomplishing the all the earlier mentioned characteristics.

Standard gamble is common way of assessing utility. The best outcome in a standard gamble is assigned with a utility of 1. The worst outcome in the gamble has an assigned utility of 0. The intermediate outcomes are then selected between the best and worst outcome. Decision maker then needs to make a choice between intermediary outcomes obtained for sure and gamble that involve best and worst outcomes.

The associated probability for which the decision maker is indifferent between utility obtained from intermediary outcomes and that of a gamble is determined (Kuspinar, Pickard & Mayo, 2016). The determined probability then is the utility associated to the intermediate outcomes. The process continues unless utility values of all possible economic scenarios are determined. The values of the utility are then plotted on a curve called the utility curve.

Allan Barnes decides to invest $10,000 in a government bond or share market. The interest rate for the bond is 9% whereas in the share market, when the market is good, fair or bad, the interest rates vary from 14%, 8% and 0% respectively.

1. The decision matrix showing all the possible strategies is given in the following table:

Table 1.1: Decision Matrix showing all possible strategies

Strategy	Market Condition
Good	Fair	Bad
Share Market	11400	10800	10000
Government Bond	19000	19000	19000

An optimist always hopes for the best to happen. Thus, an optimist would choose the strategy that will be giving the maximum return. It can be seen from table 1.2 that, the maximum profit can be earned from the Share Market. Thus, an optimist would choose to invest in the share market.

Table 1.2: Optimist Approach
Strategy	Market Condition	Best Profit
Good	Fair	Bad
Share Market	11400	10800	10000	11400
Government Bond	10900	10900	10900	10900

A pessimist will always imagine the worst possible scenario to happen. Thus, a pessimist would choose a strategy in which he can have the maximum profit with the minimum risk. Thus, according to a pessimist the best investment scheme would be in the Government Bond as it can earn the maximum return even when the market is bad or poor.

Table 1.3: Pessimist Approach
Strategy	Market Condition	Least Profit
Good	Fair	Bad
Share Market	11400	10800	10000	10000
Government Bond	10900	10900	10900	10900

Standard Gamble

The alternative of the Share Market is indicated by the Criterion of regret.

Table 1.4: Criterion of Regret
Strategy	Market Condition	Maximum
Good	Fair	Bad
Share Market	0	100	900	900
Government Bond	500	0	0	500

Now, assuming that the probability that the market will be good is 0.4, will be fair is 0.4 and will be bad is 0.2, the expected profits are given in the following table 1.5. From the table, it can be seen that the optimum action will be to invest in the share market. The Expected Monetary Value (EMV) hence obtained is $456.

Table 1.5: Expected Monetary Value (EMV)
Strategy	Market Condition	Expected Profit
Good	Fair	Bad
Share Market	456	432	200	456
Government Bond	436	436	218	436

The expected value given the perfect information is given by Here,  indicates the payoffs in the i^th row and j^th column and  indicates the respective probabilities in each state.

 The Expected Value of Perfect Competition (EVPI) = EV|PI – EMV = $(1110 – 456) = $654.

It has been the thought by Jim that he will be launching a new variety of men’s razor.

Return in favorable market = $100,000

Return in unfavorable market = – $60,000

The probability that the market is favorable = 0.5

Therefore, the probability that the market is unfavorable = (1 – 0.5) = 0.5.

The expected return that can be earned from the market = $((100,000 * 0.5) + (– 60,000 * 0.5)) = $20,000.

Therefore, on an average, he will incur a profit by producing the product in the market. Therefore, he should produce the product in the market.

Based on the track record of his friend, the probability that the market is favorable = (0.5 * 0.7) + (0.5 * 0.3) = 0.5 and the probability that the market is unfavorable = (0.5 * 0.2) + (0.5 * 0.8) = 0.5

The posterior probability of a good market given that his friend has provided an unfavorable market prediction = (0.3 * 0.5) = 0.15.

The expected net gain or loss from engaging his friend in the market research can be estimated as = $((100,000 * 0.5) + (– 60,000 * 0.5)) – $5,000 = $15,000.

Thus, by engaging his friend also, there is a profit from the business. Though the margin has decreased but, the probability of having a correct assumption is higher. Thus, his friend should be involved in the business.

A sample of the simulation run in excel is given in figures 3.1 and 3.2. Here 3.1 represents the simulation results and figure 3.2 shows the formulas that has been used for the simulation.

 average monthly profit of Ajax Tyres over the 12-month period is $21,452.36.

The results of the revised simulation when selling price increases by $40, from $200 to $220 and by increasing the profit margin from 22% to 32% is given To

Quantitative Restrictions

The Manager,

According to the information provided, the range of the selling price of Tully Tyres has increased by $40. Thus the increased range has become between $200 to $220.This increase did not affect the sales of the product. On the other hand, by increasing the selling price, the profit margin has increased from 22% to 32%. It has been observed from the results of both the simulations that average profit has been more when the selling price has increased.

Thanks and Regards,

Name and Signature

Table 4.1: High Low Method to Estimate Overhead Cost

	Machine Hours (x)	Overhead Cost (y)
Highest Activity	3,800	$48,000
Lowest Activity	1,800	$46,000

The marginal cost per unit can be estimated with the help of the following formula: Therefore, the total fixed cost is given by Thus, the equation that can be used to estimate the overhead cost is given by 

Therefore, the estimated overhead cost when 3000 machine hours were used is $(44,200 + 3,000) = $47,200

The regression results for estimating the overhead costs against batches is given by the following table 4.2.

Table 4.2: Regression Results for Estimating 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	P-value	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

  From table 4.2, it can be seen clearly that the overall significance value is less than 0.05, the specified level of significance at which the test is conducted. Thus, the model developed is significant. From the value of the R Square (0.83), it can be said that 83 percent of the variations in the overhead costs can be explained by the number of batches.

The estimated regression equation is given by

From the estimated regression equation, it can be said that with each unit increase in the number of batches, the overhead cost increases by $234.568.

The regression results for estimating the overhead costs against machine hours is given by the following table 4.3.

Table 4.3: Regression Results for Estimating Overhead Costs 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	P-value	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

From table 4.3, it can be seen clearly that the overall significance value (0.774) is greater than 0.05, the specified level of significance at which the test is conducted. Thus, the model developed is insignificant. From the value of the R Square (0.011), it can be said that 1.1 percent of the variations in the overhead costs can be explained by the number of batches.

The estimated regression equation is given by From the estimated regression equation, it can be said that with each unit increase in the machine hours, the overhead cost decreases by $2.304.  

The regression results for estimating the overhead costs against machine hours is given by the following table 4.4.

Table 4.4: Regression Results for Estimating Overhead Costs against Machine hours and Number of 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	P-value	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

From table 4.4, it can be seen clearly that the overall significance value (0.002) is less than 0.05, the specified level of significance at which the test is conducted. Thus, the model developed is significant. From the value of the R Square (0.833), it can be said that 83.3 percent of the variations in the overhead costs can be explained by the number of batches and overhead cost. Moreover, it can be seen from the model that the variable Machine hours is insignificant but the variable number of batches is significant.  

The estimated regression equation is given byFrom the estimated regression equation, it can be said that with each unit increase in the machine hours, the overhead cost decreases by $0.931 and with each unit increase in batches, the overhead cost increases by $233.827.

From the regression models developed, it can be seen clearly that for the regression model 1 and 3, the value of r Square are the same, but the value of the adjusted R Square is higher for the first model. Thus, it can be said that Batches is a better estimator of overhead cost than the overall model.The estimated overhead cost in a month when 150 batches of the product has been produced is:

For product A, the sales price per unit is $12 and the variable cost per unit is $8. Thus, the contribution margin is $ (12 – 8) = $4.

Similarly, for product B, the sales price per unit is $15 and the variable cost per unit is $10. Thus, the contribution margin is $ (15 – 10) = $5.If the manufacturer specializes only in Product B, then the manufacturer needs to sell (5000 / 5) units = 1000 units.If the manufacturer specializes only in Product A, then the manufacturer needs to sell (5000 / 4) units = 1250 units.

If the manufacturer decides to manufacture both the products A and B, in the ratio of 3:1 respectively,

• In order to earn a profit of $3,500 before tax for the month, the number of products of A that has to be sold is 1333 and the number of products of B that has to be sold is 667. The necessary calculations are shown in table 5.1.

Table 5.1

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

In order to earn a profit of $8,400 after tax for the month, the number of products of A that has to be sold is 855 and the number of products of B that has to be sold is 428. The necessary calculations are shown in table 5.2.

Table 5.2

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

Kuspinar, A., Pickard, S., & Mayo, N. E. (2016). Developing a valuation function for the preference-based multiple sclerosis index: comparison of standard gamble and rating scale. PloS one, 11(4), e0151905.

Li, W. (2014). Risk assessment of power systems: models, methods, and applications. John Wiley & Sons.