Question

Consider an unincorporated firm with a two period (1 and 2) time horizon. At the beginning of period 1, the firm has a predetermined capital stock, K1 . During

period 1, gross investment expenditure, I, financed out of retained earnings, are incurred with the purpose of both maintaining and increasing the capital stock in period 2. In each of the two periods, the capital stock depreciates at a rate δ, so at the beginning of period 2, the firm's capital stock will be given by:

K2 =I+(1−δ)K1 (1) Investment is undertaken with the purpose of maximizing the present value V of the

net cash flows over the two periods, defined as Π1 and Π2 :
V=Π1 + Π2 (2)

1+r (1+r)2

At the end of period 2, the firm is liquidated, its remaining assets sold at their replacement value (1−δ)K2 . For simplicity, we assume that there are no adjustment costs associated with changing the firm's capital stock and labour’s input is ignored.1 If Yt is the firm's output in period t, the cash flows from the firm to the owners during the two periods will then be:

Π1 = Y1 − I (3)

Π2 =Y2 +(1−δ)K2 (4)

Equation (4) includes the revenue from the sale of the firm's remaining capital stock at the end of period 2. Output in each period is given by a Cobb-Douglas production function:

Yt =AtKtα ,t=1,2 (5) The term A is productivity.

a) To set the problem up, sub equations (3) and (4) into (2). Then sub in the production function (equation (5)) for period 2. Finally, use equation (1) to eliminate K2 where it appears, which will allow you to highlight gross investment, I. Now you are ready. Start by taking the 1st derivate of the resulting

1 Labour costs would in any event drop out of the final equation.2

equation with respect to I. Next, based on the expression you found, show that the optimal level of investment has the following form:

Hint: For period 2 profits, first evaluate the marginal product of capital (MPK2)

as it relates to Y2 and K2 and then use equation (1).
b) Suppose now that the firm has to pay taxes on its overall profits. Note that based

on Canadian tax law the firm can deduct all costs, which in this case is assumed to be depreciation.

T1 =τ(Y1 −δK1) (7)T2 =τ(Y2 −δK2) (8)

Equations (3) and (4) now become:
Π1 = Y1 − I − T1 (3’)

Π2 =Y2 +(1−δ)K2 −T2 (4’)

Derive a new investment equation based on the existence of the tax on profits. Discuss how the existence of the profit tax affects investment. Would an increase in the tax rate lower investment?

c) With taxes in place, debt financing is introduced. In particular, assume that depreciation, δK, is financed by retained earnings, while net investment, I − δ K

is financed entirely by debt or bonds, B . In this case, the firm's stock of debt Btwill always be equal to its capital stock, that is:

Bt =Kt, t=1,2 (9) From the above, the firm’s revenue from new borrowing during period 1, ∆ Bt ,

will be equal to its net investment during that period:
∆B 1= I −δK1 (10)

Using (9) and (10) and noting that the firm’s interest payments on its debt will be rBt = rKt , the net cash flow in each period will be:

I=αY2 −(1−δ)K (6)

1

r+δ
What interpretation would you give to Y2 in equation (6)?

3

Π1 = Y1 − rB1 − I − T1 + ∆ B1=Y1 −(r+δ)K1 −T1

Π2 =Y2 +(1−δ)K2 −T2 −(1+r)B2=Y2 −(r+δ)−T2

(11)

(12)

The above assumes that the firm must pay off all of its debts with interest at the end of the period. Allowing for interest and depreciation deductibility, the tax bill faced by the firm in each period will be:

T=τY−(r+δ)K ,0<τ<1,t=1,2 (13)t ⎡⎣ t t ⎤⎦

Based on the above, derive the firm’s investment function. Does the profit tax affect investment? Is the tax system neutral towards the choice of financing? Be sure and explain your results.

d) Go back to the original model (equations (1) through (5)) and now assume that there are adjustment costs arising with the introduction of new capital. In particular assume that the relationship is linear:

c(I)=(1+c)I, 0<c<1 (14)

Derive an investment equation that shows the role of installation costs. What aspect of the equation is affected by this change in the model? [Note: To simplify your work, assume that the cross product c × r ≈ 0 and can safely be ignored; in point of fact, it would be small.]

Answer 1

Consider an unincorporated firm with a two period (1 and 2) time horizon. At the beginning of period 1, the firm has a predetermined capital stock, K1 . During

period 1, gross investment expenditure, I, financed out of retained earnings, are incurred with the purpose of both maintaining and increasing the capital stock in period 2. In each of the two periods, the capital stock depreciates at a rate δ, so at the beginning of period 2, the firm's capital stock will be given by:

K2 =I+(1−δ)K1 (1) Investment is undertaken with the purpose of maximizing the present value V of the

net cash flows over the two periods, defined as Π1 and Π2 :
V=Π1 + Π2 (2)

1+r (1+r)2

At the end of period 2, the firm is liquidated, its remaining assets sold at their replacement value (1−δ)K2 . For simplicity, we assume that there are no adjustment costs associated with changing the firm's capital stock and labour’s input is ignored.1 If Yt is the firm's output in period t, the cash flows from the firm to the owners during the two periods will then be:

Π1 = Y1 − I (3)

Π2 =Y2 +(1−δ)K2 (4)

Equation (4) includes the revenue from the sale of the firm's remaining capital