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 $$1352,$ absent any potential liability. There is a chance that division $2$ could have $$1,682$

liability for pollution damage. The potential victims have no contractual relationship with the firm. The

probability of such a loss is $0\mathrm{.}5$

$(1+),$ where s is the amount the firm spends on safety. The firm must

choose the level of s$.($Note that the derivative of $$

$(1+)$ with respect to $$s$$ is equal to $$

$(1+)2)$

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 $$1352$if no liability arises. However, if a loss of $$1,682$ occurs it cannot pay more than

the original $$1352$ 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

$1.)[2$ marks$]$

iv$)$ Is there a gain from splitting the firm up$?$ Calculate the numerical value of gain$/$loss$.[2$

marks$]($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$).)$