Question

In the academic world there is no dearth of all-knowing professors. An even more all-knowing professor now joins the fray. She claims that her research is so thorough that she can always predict the future economic outlooks without fail. Hence, instead of offering probabilities about states of nature, as the previous professor did, she proposes that she will create a white-paper on the future of the economy. If the paper predicts a negative outlook the economy is certain to be ‘depressed’. On the other hand, if she predicts a positive outlook the economy is certain to be either ‘bright’ or ‘stable’ with equal probabilities.

She says that she will conduct her research and issue the economic forecast, either positive or negative, as she finds appropriate. Given her proposal, the company is now forced to do a bit of research itself. It finds that independent of her tall claims, the past experience indicates that there is only a 96% chance that the economy will be actually positive (either bright or stable with equal probabilities) if she predicts it to be positive, but there is a 4% chance that it will actually end up being negative (i.e. depressed) even though the forecast is positive. Conversely, there is an 89% chance that the economy will be actually negative (i.e. depressed) if she issues a negative report, however an 11% chance that it will actually end up being positive (either bright or stable) even though the report is negative. Using information from the past, the research also reveals that there is a 70% chance that the professor will issue a positive report and a 30% chance the report will be negative.

· Create a new pay-off table with appropriate alternatives, states of nature, probabilities, and pay off values, and evaluate the outcome from the best course of action (5 points).

· What is the maximum you will be willing to pay the professor for her services? (5 points)

Hints: Think about the baseline return/s that you will compare to the returns with her prediction?

You will need to think back to the ‘AND’ or the ‘Multiplication’ rules and/or the ‘OR’ or the ‘Addition’ rules (Chapter 2) to calculate the probabilities in this case.

find the probability of each economy

Answer 1

There are two possibilities of economy, either it will be stable / bright or it will be depressed .In both the cases, it is quite possible that the report of professor may be positive or negative.

Case 1)- When Economy will be bright / stable :-

Subcase a) - Professor reports positive and it becomes true .

   Probability of professor predicing positive = 0.70

Probabiliy that economy will be really stable after her positvie reporting = 0.96

So, probability of bright economy in this case = 0.70*0.96 = 0.6720

Subcase b)- Professor reports negative and it becomes false

  Probabilitry of professor predicting negative = 0.30

Probability of economy being stable after her negative prediction = 0.11

Probability of economy being stable in this case = 0.30*0.11 = 0.0330

Thus total probabiltiy of economy being stable = 0.6720 + 0.0330= 0.705= 70.5%

Case 2)- When Economy will be depressed -

Subcase a) - Professor reports negative  and it becomes true .

   Probability of professor predicing negative= 0.30

Probabiliy that economy will be really depressed after her negative reporting = 0.89

So, probability of depressed economy in this case = 0.30*0.89 = 0.2670

Subcase b)- Professor reports positive  and it becomes false

  Probabilitry of professor predicting positive = 0.70

Probability of economy being depressed  after her positive prediction = 0.04

Probability of economy being depressed in this case = 0.70*0.04 = 0.0280

Thus total probabiltiy of economy being depressed = 0.2670 + 0.0280 = 0.2950= 29.50%

Henceforth, the total possibility of economy being stable is 70.5 % and its possibility of being depressed is 29.5%.