${}_1=Y_1-w_1{L}_1-I$

${}_2=Y_2-w_2{L}_2+(1-)K_2$

Output is determined by a Cobb$-$Douglas production function.

Show that the optimal level of investment $(I)$ of the firm is given by:

$I=\frac{Y_2}{r+}-(1-)K_1$

$[$Hint: Use $(1)$ to eliminate $K_2$ from equation $(4)$ before deriving the first$-$order conditions.$]$

How does this equation compare with equation $(12)$ in the text? From the perspective of the

present, what is the interpretation of $Y_2?$ Can you think of a way to modify equation $(6),$ such

that investment is a function of current income, $Y_1?$

Now assume that the firm has to pay taxes on its profits at a rate $.$ The total tax bill in any

period is:

;$t=1,2$

The after$-$tax cash flow in each period is now given by:

${}_1=Y_1-w_1{L}_1-I-T$

${}_2=Y_2-w_2{L}_2+$obrace$((1-)K_2{)}^{\{\begin{array}{ccc}\text{:}[& \text{V}\text{a}\text{l}\text{u}\text{e}of\text{f}\text{i}\text{r}\text{m}& ]\\ & \text{w}\text{h}\text{e}\text{n}\text{s}\text{o}\text{l}\text{d}& \end{array}}-T_2$

Derive the investment function when the firm has to pay taxes. How has the existence of taxes

affected the investment equation?

Suppose now that net investment is financed entirely by debt; that is$,$ by issuing bonds $(B).$

$B_1=I-K_1$Suppose now that net investment is financed entirely by debt; that is$,$ by issuing bonds $(B).$

$B_1=I-K_1$

In this case, the firm's debt level will always equal the value of the capital stock.

$B_t=K_t$

Using $(10)$ and $(11),$ and noting that $rB=rK,$ the cash flow in each period becomes:

${}_1=Y_1-w_1{L}_1-r{B}_1-T_1-I+B_1$

$=Y_1-w_1{L}_1-(r+)K_1-T_1$

${}_2=Y_2-w_2{L}_2+(1-)K_2-T_2-(1+r)B_2$

$=Y_2-w_2{L}_2-(r+)K_2-T_2$

Note that debt has to be repaid at the end of period $2$ and that as well the tax codes $($in Canada

as elsewhere$)$ allow firms to deduct both interest payments and depreciation $($in addition to the

wage bill$)$ from its revenues when computing taxable income.

How has the existence of debt financing affected the investment function? Does the tax on profits

affect investment? Does the tax system favour financing by debt over equity $($which is equivalent

to using retained earnings$)?$