Firm’s investment decision

• Arbitrage Equation: equates the returns from two different types of investment.  For example, putting money in a bank, and investing in new capital.

• Capital gain (loss): the growth rate of the asset price, ∆pk /pk , where ∆pk  = pk,t+1−pk,t .

• User cost of capital (uc): the total cost to the firm of using one more unit of capital.

• Desired Capital

R + d¯−  

1 − τ       .

– The right-hand side is the uc. The firm should invest in capital until the value of the extra output that capital produces falls to equal the user cost.

– The uc depends on the interest rate R , the depreciation rate d¯, the (expected) capital gain , and the tax rate τ .

– The arbitrage argument:  cost of funds, R , equals the return from investing in capital, (1 − τ )MPK − d¯+  .

•  Investment Rate

It             gK  + d¯

Yt            3 · uc  .

Sketch of the proof:

– Recall that Kt+1  = It  + (1 − d¯)Kt , so ∆Kt+1  = Kt+1  − Kt   = It  − d¯Kt , hence gK  =  =  − d¯ =Y(I)t(t)     − d¯.

– Recall that with Cobb-Douglas production function Y  = K1/3L2/3 , MPK  = K −2/3L2/3  =   . Since MPK = uc,  = 3 · uc.

– Plug it into the first equation, and we get gK  =Y(I)t(t)   · 3 · uc − d¯, which leads to the conclusion.

Question 1

An economy begins in a steady state with an investment rate of 20 percent, a corporate tax rate of 25 percent, a real interest rate of 2 percent, a depreciation rate of 7 percent, and a price of capital that falls at an annual rate of 2 percent.

1. What is the user cost of capital?

R + d¯−      0.02 + 0.07 − ( −0.02)

1 − τ                        1 − 0.25

2.  Suppose the real interest rate rises from 2 percent to 4 percent.  By how much does

the user cost of capital rise?

∆uc =  = 0.0267 .

3. How would your answer have differed if the corporate tax rate had been zero? Explain the effect that taxes have on the user cost of capital (and hence the investment rate). If τ = 0, ∆uc = ∆R = 0.02.  Given the tax rate of 25%, a 2% increase in the interest rate causes a 2% increase in the after-tax MPK, so the pre-tax MPK must increase by 0.0267, which is due to the tax wedge between the pre-tax and after-tax MPK. Also, the increase in user costs lowers the investment rate .

Question 2

Suppose the user cost of capital in an economy with no corporate income tax is 10 percent.

1. What is the user cost if the corporate tax rate rises to 20 percent? uc =  = 0.125.

2. Suppose an economy’s steady-state investment rate I/Y is 30 percent when the corpo- rate tax rate is zero. What happens to this investment rate if the corporate tax rate rises to 20 percent?

If τ = 0, uc = 0.10, I/Y = 0.30, gK   = 0 (in steady-state),  we get 0.30 = ,  so d¯= 0.09.

If τ = 0.20, uc = 0.125, so I/Y =  = 0.24.

The Stock Market

• Arbitrage equation: return in bank account = return from the stock

R · ps  = dividend + ∆ps    ⇐⇒ R = dividend + ∆ps

– Dividend: a payment by a firm to its shareholders.

– The percentage return in bank account = the dividend return (percent) + the capital gain (percent).

dividend

R − ∆ps

ps

– Assume the capital-gain term is a constant. When dividend grows at a constant rate, g, the capital-gain term is also g .

– The stock price equals the current dividend divided by R − g .

•  Price-earnings ratio: equal to the dividend-earnings ratio divided by R − g .

•  Informationally efficient market: financial prices fully and correctly reflect all available information. Impossible to make economic profits by trading based on that info.

V         stock market value

pK K         value of capital   .

– When q > 1, the value of the firm is greater than its capital, and the firm should invest in more capital. When q < 1, we would expect the firm to be “disinvesting.”

– q often exceeds  1:  firms also own their brand names  (often called “goodwill” capital) and the ideas they have created.

Question 3

Use the equation for the stock price to answer the following questions.

1. Your stockbroker calls to tell you she has a great deal on a stock.  She gives you the following information: the real interest rate is 4 percent, the capital gain on this stock is 3.5 percent, and it pays $2.50 in dividends. How much should you pay for this stock?

dividend           $2.50     

R − ∆ps                 0.04 − 0.035

ps

2.  The next day, your stockbroker calls back and tells you about another stock for sale for $75 (i.e., the market price). She does not know the dividend growth, but she does know the dividend payment, which is $0.35 per share, and the real interest rate, which is 4 percent. How much annual dividend growth should you expect at that price?      Using the same formula, we get $75 = , so g = 0.035.

3. A hot stock tip comes from a friend.  The price of the stock is $50, the capital gain is 2.5 percent, and you know the real interest rate from your previous two discussions with your broker. How much of a dividend payment will you want?

Using the same formula, we get $50 = , so dividend = $0.75.