Question $7$: Attention Industries has two divisions. Division $1$ makes cleaning products and the net worth of this division $($present value of cash flows$)$ is $$3,348.$ Division $2$ makes a chemical product. The net worth of division $2$ is $$1,352,$ absent any potential liability. There is a chance that disvision $2$ could have $$1,682$ liability for pollution damage. The potential victims have no contractual relationship with the firm. The probability of such a loss is $(0\mathrm{.}5)/((1+$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 derivative with respect to s to find the $"$ s $"$ that maximizes the total value. Third, plug in the value of $"$s$"$ you found into total value equation to find the maximum total value.

ii$)$ Division $1$ is riskless and has a stand$-$alone value of $$3,348.$ Since all the risk comes from division $2,$ we must consider the chosen level of safety $"$ s $"$ of this unit as a stand$-$alone entity. First, calculate the stand$-$alone value of division $2$ as an equation of $"$ s $".$ Second, take derivative with respect to s to find the $"$ s $"$ that maximizes the stand$-$alone value of division $2.$ Third, plug in the value of $"$s$"$ you found into the equation of stand$-$alone value of division $2$ to find the maximum stand$-$alone value of division $2.($Hint: the value of division $2$ is $$1,352$ if no liability arises. However, if a loss of $$1,682$ occurs it cannot pay more than the original $$1,352$ 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 $1.$ The probability of such a loss is $\frac{0\mathrm{.}5}{(1+s)},$ where s is the amount the firm spends on safety. The firm must choose the level of s $.($Note that the derivative of $\frac{a}{(1+s)}$ with respect to $"$ s $"$ is equal to $-\frac{a}{(1+s{)}^2})$

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

ii$)$ Division $1$ is riskless and has a stand$-$alone value of $$3,348.$ 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 $2.$

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 $2.($Hint: the value of division $2$ is $$2,312$ if no liability arises. However, if a loss of $$1,682$ occurs it cannot pay more than the original $$2,312$ 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$)$ Is there a gain from splitting the firm up$?$ Calculate the numerical value of gain$/$loss$.($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$).)$