Attention Industries has two divisions. Division 1 makes cleaning products and the net worth of this division (present value of cash flows) is $2,250. Division 2 makes a chemical product. The net worth of division 2 is $920, absent any potential liability. There is a chance that division 2 could have $2,450 liability for pollution damage. The potential victims have no contractual relationship with the firm. The probability of such a loss is 0.5/(1+𝑠) , where s is the amount the firm spends on safety. The firm must choose the level of s. (Note that the derivative of a/(1+s) with respect to “s” is equal to −a/(1+s)^2)

i) First, calculate the total value of the firm, including both divisions at the same time, as an equation of “s”. Second, take the derivative with respect to s to find the “s” that maximizes the total value. Third, plug in the value of “s” you found into the total value equation to find the maximum total value.

ii) Division 1 is riskless and has a stand-alone value of $2,250. Since all the risk comes from division 2, we must consider the chosen level of safety “s” of this unit as a stand-alone entity. First, calculate the stand-alone value of division 2 as an equation of “s”. Second, take derivative with respect to s to find the “s” that maximizes the stand-alone value of division 2. Round the answer for s to 2 decimal places. Third, plug in the value of “s” you found into the equation of stand-alone value of division 2 to find the maximum stand-alone value of division 2. Round the value to the nearest whole number, while “s” is rounded to 2 decimal places. (Hint: the value of division 2 is $920 if no liability arises. However, if a loss of $2,450 occurs it cannot pay more than the original $920 stand-alone value due to limited liability.)

iii) What is the maximum value of the firm if we treat division 2 as a separate entity? Note that we are splitting the two divisions assuming that separated division 2 as a stand-alone firm is protected by limited liability. (Hint: Add your answer to part ii) to the net worth of division 1)

iv) What is the gain from splitting the firm up? (Hint: It is the difference between the maximum total value without split up that you found in i) and the maximum total value with split up that you found in iii))

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