Question

13.30 The CEO of a chemicals firm must decide whether to develop a new process that has been suggested by the research division. His decision tree is shown in Figure 13.18 There are two sources of uncertainty. The production cost is viewed as a continuous ran dom variable, uniformly distributed between $1.75 and $2.25, and the size of the market Figure 13.18 Decision tree for Problem 13.30 Cost of arket Size (X) Net Profit S(4 - C)X- $20,000 Develop Do Not Develop Net Profi $0 564 CHAPTER 13 RISK ATTITUDES (units sold) for the product is normally distributed with mean 10,300 units and standard deviation 2200 units The firm's CEO is slightly risk-averse. His utility function is given by U(Z) 1 e-Z/20,000 where Zis the net profit Should the CEO develop the new process? Answer this question by running a computer simulation, using 200 trials. Should the decision maker be concerned about the fact that if he develops the new process, the utility could be less than or greater than the utility for $0? On what basis should he make his decision?

Answer 1

We will need to use Excel to solve this Question. Please follow the instructions

Open Excel

In Column A, generate a random number for Size of market (X) using the following formula:

=NORM.INV(RAND(),10300,2200)

Here, mean = 10300 and standard deviation = 2200

This formula generates random numbers which follow normal distribution with given mean and standard deviation.

Copy this formula from cell A2 to A201

In column B, generate random number for cost of production (C) using the following formula:

=1.75 + 0.5*RAND()

Copy this formula from cell B2 to B201

In column C, calculate net profit (Z) using the following formula:

=((4-C)*X)-20000

For C and X use the corresponding numbers from column B and A respectively.

For example, Cell C2 = =((4-B2)*A2)-20000

In column D, calculate utility function as follows:

=1-exp(-Z/20000)

Thus, Cell D2 = =1-(EXP(-C2/20000))

In cell E2 calculate average of D2 to D201

This will give you an average of utility function for 200 simulations.

You can refresh the excel to generate more number of simulations.

Results of simulation 1, Average U(z):

-0.00146

Results of simulation 2, Average U(z):

0.017731

Results of simulation 3, Average U(z):

0.020276

Results of simulation 4, Average U(z):

-0.00367

Results of simulation 5, Average U(z):

0.009476

Now, Utility of Do not develop step with Net profit = 0 is 1-exp(-0/20000) = 0

if E2 > 0, Develop (as the CEO is risk averse)

If E2 < 0, Do not develop

However, with our simulations each time there is a new Utility function which may or may not be greater than zero. However, every time the number of times U(Z) < 0 is less than 100 even if the average U(z) is negative. Thus, the CEO should develop the process. And the average U(Z) should not be the decision criteria but the number of times U(z) > 0 should be the decision making criteria.