Understanding Utility Function And Standard Gamble For Assessing Utility Values

Utility Function and Its Assessment

Utility function is a twice differentiable function of wealth which satisfies the principle of non-satiation and risk aversion. We can assess the utility function by mainly two methods, that is interview method and probability encoding. In interview method, a person is asked a series of questions from which his/her utility function is assessed while in probability encoding probabilities are attached to uncertain possible outcomes depending on an individual’s preference.

A standard gamble is a technique for managing outcomes by attaching probability values  to an individual’s preferences. The standard gamble is used in determining utility values by asking a series of leading questions the answer to which will help determine the utility values of the individual as they indicate an individual’s attitude towards a particular risk.

1. Decision matrix

	Good	Fair	Bad Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper
Share market	11400	10800	10000
Government bonds	10900	10900	10900

1. Optimist criterion

	Good	Fair	Bad	Maximum outcome
Share market	11400	10800	10000	11400
Government bonds	10900	10900	10900	10900

The optimist will take the option of the maximum outcome which is to trade shares under good market which its outcome is $11400

1. Pessimist criterion

	Good	Fair	Bad	Minimum outcome
Share market	11400	10800	10000	10000
Government bonds	10900	10900	10900	10900

The pessimist will take the option of maximum in the possible minimum outcome which is to trade in government bond in all seasons which results in $10900

1. Regret criterion

	Good	Fair	Bad	Minimum outcome	Maximum outcome
Share market	11400	10800	10000	10000	11400
Government bonds	10900	10900	10900	10900	10900

Regret criterion can choose to go for both the minimax and maximin take the option of trading in government bond which gives $10900.

1. Expected monetary value

	Good	Fair	Bad
Share market	11400	10800	10000
Government bonds	10900	10900	10900
Probability	0.4	0.2	0.4

The optimal action in the expected monetary value will be to trade in government bond which its outcome is $10900.

1. Expected value of perfect information

	Good	Fair	Bad
Share market	11400	10800	10000
Government bonds	10900	10900	10900
Probability	0.4	0.2	0.4

The highest expected monetary value is $10900

Expectation for maximizing profit

11100

Knowing the direction the market will go is worth $200

1. The probability of unfavorable market is 0.5

Since the expected monetary value is $20000 he can go ahead and start the producing new type of razor.

1. The friend’s probability

	Favorable	Unfavorable
Correct prediction	0.7	0.2
Incorrect prediction	0.3	0.8

The prior probabilities are 0.5 for favorable market and 0.5 for unfavorable market

The joint and marginal probability will be;

	Favorable	Unfavorable
Correct prediction	0.35	0.1	0.45
Incorrect prediction	0.15	0.4	0.55
	0.5	0.5

Posterior probability

	Favorable	Unfavorable
Correct prediction	0.78	0.73
Incorrect prediction	0.22	0.27

1. Posterior probability

Posterior probability when the friend predict unfavorable market is 0.22

1. In engaging friend; under correct prediction

Expected value of sample information (EVSI) = emv with sample information – emv without sample information.

emv with sample information =

                                = 19920

Emv without sample information

He should not engage his friend because the value of the information is less than the cost of the information.

MODEL
			Selling		Profit	Fixed
Month	RN1	Demand	Price	RN2	Margin	Costs	Profit
1	0.23297	107	$180	0.22763	20%	2000	6408
2	0.794052	147	174	0.233362	30%	2000	7956
3	0.395857	139	163	0.40283	70%	2000	11536
4	0.320458	196	174	0.240464	13%	2000	3164
5	0.245738	134	165	0.360996	90%	2000	24192
6	0.422416	153	178	0.890802	40%	2000	12408
7	0.038463	148	176	0.452313	90%	2000	19890
8	0.730945	168	178	0.737935	80%	2000	23780
9	0.503711	192	171	0.878873	40%	2000	8514
10	0.726539	149	171	0.221567	60%	2000	18795
11	0.179755	141	166	0.241994	90%	2000	15998
12	0.412426	105	163	0.142019	30%	2000	7164

DATA
Prob	Cum prob	Demand		Selling	Price	$180	$220
0.05	0	100		Monthly	Fixed cost	$2,000
0.1	0.05	120		Profit	Margin	22%	32%
0.2	0.15	140
0.3	0.35	160
0.25	0.65	180
0.1	0.9	200

MODEL
			Selling		Profit	Fixed
Month	RN1	Demand	Price	RN2	Margin	Costs	Profit
1	0.23297	192	$187	0.22763	20%	2000	4700
2	0.158694	103	218	0.479955	30%	2000	8259
3	0.179437	126	186	0.672223	70%	2000	17550
4	0.285063	187	191	0.671108	13%	2000	2978
5	0.482977	187	197	0.800577	90%	2000	37080
6	0.636752	132	199	0.133095	40%	2000	12408
7	0.395724	168	203	0.651491	90%	2000	19890
8	0.19685	116	214	0.334298	80%	2000	23780
9	0.226749	113	188	0.495291	40%	2000	8514
10	0.592614	150	185	0.459049	60%	2000	18795
11	0.223638	123	199	0.774456	90%	2000	15998
12	0.15323	142	210	0.031765	30%	2000	7164

Report of the changes observed

It can be seen than the changes cause the increase of profit from $159805 to $177116. This will bring about the benefit to the company as it obvious the profit margin also increased. The drastic effect it can cause is that with time the increase in price of the tires can lower the demand leading to low sales which will eventually lower the profit. This can force the profit margin to reduce. Some players can come in the market to take the opportunity to sell tires at relatively lower price which can bring stiff competition lowering the profits, demand even further. Therefore, increasing the varieties of tires customers can choose from can make them understand the increase in prices setting in a continuous trend which can be realized by customers in later days. This will ensure continuous demand keeping our increase in profit margin. (Working Group on Radiative Corrections and Monte Carlo Generators for Low Energies, 2010)

1. Regression between Overhead cost against machine hours

The cost equation I from Overhead cost and Machine hours is

From the above table we see the p-value is less than 0.05 which shows that in this case the variable Machine Hours is statistically significant and can be used for future projections

The regression analysis between Overhead cost and batches

Regression Statistics
Multiple R	0.999144163
R Square	0.998289059
Adjusted R Square	0.998098954
Standard Error	6299.920723
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2.08417E+11	2.08417E+11	5251.262	9.17576E-14
Residual	9	357201010.1	39689001.12
Total	10	2.08775E+11
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	119.1919192	2317.767593	0.051425311	0.96011	-5123.962644	5362.34648	-5123.96	5362.346482
X Variable 1	267.345679	3.689277512	72.46559201	9.18E-14	258.9999535	275.691405	259	275.6914046

The cost equation I from Overhead cost and Batches is

From the above table we see the p-value is less than 0.05 which shows that in this case the variable Batches is statistically significant and can be used for future projections

The regression analysis between Overhead cost both Machine hours and Batches

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.999169
R Square	0.998338
Adjusted R Square	0.997923
Standard Error	6585.241
Observations	11
ANOVA
	df	SS	MS	F	Significance F
Regression	2	2.08428E+11	1.04214E+11	2403.15558	7.62471E-12
Residual	8	346923227.2	43365403.4
Total	10	2.08775E+11
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	66.4435	2425.159896	0.027397575	0.97881375	-5525.985247	5658.872	-5525.99	5658.872249
X Variable 1	1.018835	2.092790071	0.486830774	0.63943573	-3.807147947	5.844817	-3.80715	5.844817169
X Variable 2	253.6505	28.39446871	8.93309405	1.9576E-05	188.1726972	319.1282	188.1727	319.1282217

The cost equation from Overhead cost and both Machine hours and Batches is

From the above table the p-value of Batches is less than 0.05 which makes it statistically significant to the equation and can be used for future projections while the p-value of Machine Hours is more than 0.05 making it statistically insignificant hence can be dropped for future projection.

1. I will use the simple regression of Overhead cost and Batches. Batches proves to be more statistically significance than Machine Hours because when they are regressed with Overhead cost the p-value of Batches is very smaller than 0.05.

References

Colleen M. Norris, W. A. (2006). Ordinal regression model and the linear regression model were superior to the logistic regression models. 9.

Working Group on Radiative Corrections and Monte Carlo Generators for Low Energies, S. A. (2010). Quest for precision in hadronic cross sections at low energy:. Monte Carlo tools vs. experimental data, 102.