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

liability for pollution damage. The potential victims have no contractual relationship with the firm. 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. $[5$ marks$]$

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 $2.[5$ marks$]($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 $$2,3$ 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

$1.)$ marks