Accounting Theory

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  • Topic: Decision theory, Bayes' theorem, Bayesian probability
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  • Published : May 13, 2013
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Chapter 3 – Problem 18
a) Let the probability of state 1 (the high performance state) be P(H) = 0.5

Let the probability of state 2 (the low performance state) be
P(L) = 0.5

We assume that the amount of utility or satisfaction Ajay derives from a payoff is equal to the square root of the amount of the payoff.

So, we get
Ui(a) = √x, x≥0
Where x is the amount of the payoff

The decision theory tells us that the act with the highest expected utility should be chosen.

We denote the expected utility of act a1 (AB Ltd.) by EU(a1) and the expected utility of act a2 (XY Ltd.) by EU(a2).

Thus, we get
EU(a1) = 0.5 x √1089 + 0.5 x √0 = 16.5
EU(a2) = 0.5 x √324 + 0.5 x √196 = 16

∴ Ajay should choose act a1 and invest in AB Ltd. since it has the higher utility

b) To calculate prior state probabilities, we use Bayes’ Theorem. So the posterior probability of the high performance state is
P(H│GN) = P(H)P(GN│H) / P(H)P(GN│H) + P(L)P(GN│L)
where
P(H│GN) is posterior probability of high state given good news F/S P(H) is prior probability of high state
P(GN│H) is probability F/S shows good news in high state
P(L) is prior probability of low state
P(GN│L) is probability F/S shows good news in low state
Thus, from Bayes’ Theorem, we get
P(H│GN) = (0.5 x 0.6) / [(0.5 x 0.6) + (0.5 x 0.5)] = 0.55
Then Ajay’s posterior probability of the low performance state is
P(L│GN) = 1 - 0.55 = 0.45
So the expected utility of each act based on Ajay’s prior probabilities is
EU(a1) = 0.5 x √1089 + 0.5 x √0 = 16.5
EU(a2) = 0.55 x √324 + 0.45 x √196 = 16.2

∴ Ajay should still choose a1 and invest in AB Ltd. since it has the higher expected utility.

c) Given this new information system, the posterior probability performance of high state is P(H│GN) = (0.5 x 0.8) / [(0.5 x 0.8) + (0.5 x 0.2)] = 0.8
So, the posterior probability performance of low state is
P(L│GN) = 1 – 0.8 = 0.2
Then the expected utility of each act is...
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