Investment

Chapter 14

Students Should Be Able to:

Calculate Average and Marginal product of capital.

Calculate the real and nominal rental cost of capital

Calculate the optimal capital stock as a function of the cost of capital. lculate Tobin’s q to estimate the desirability of

corporate investment. Evaluate relationship between leverage

and investment.

Terminology: Investment

We use the term investment to refer to real expenditure (public and/or private) on tangible assets.

We call the stock of tangible assets capital or physical capital.

The unit of measure of aggregate capital is dollars.

Gross Investment refers to purchases of new investment.

Net Investment is Gross Investment minus depreciation.

Components of Investment

Investment Fixed Investment

Residential Investment Business Investment

Structures Machinery & Equipment

Changes in Stocks – Inventory Investment

Gross Fixed Capital Formation:HK 2002

3%10%

54%

14%

14%

5%

Transfer Costs of Land &Building

Real Estate Developers'Margin

Machinery & Equipment

Public Construction

Private Residential

Private: Non-residential

Investment Facts

Investment expenditure is a substantial share of GDP, but not as large as consumption.

Fixed and inventory investment are closely correlated with the business cycle.

Investment is an especially volatile part of GDP.

Investment as a share of GDP: East Asia

Easterly, Rodriguez, and Schmidt-Hebbel "Public Sector Deficits and Macroeconomic Performance." (Statistical appendix) 1994 and Bruno and Easterly JME 1998.

Private Investment

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

China Hong Kong Indonesia Korea Malaysia Philippines Thailand

% o

f G

DP

Investment as a share of GDP: East Asia 1997

Easterly, Rodriguez, and Schmidt-Hebbel "Public Sector Deficits and Macroeconomic Performance." (Statistical appendix) 1994 and Bruno and Easterly JME 1998.

Public Investment

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

China Hong Kong Indonesia Korea Malaysia Philippines Thailand

% o

f G

DP

Volatility: Investment and GDPAnnual Growth Rates

Hong Kong

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

1965

1968

1971

1974

1977

1980

1983

1986

1989

1992

1995

1998

2001

% G

row

th R

ates

GDP

I

Marginal Analysis

Economists use marginal analysis to determine an optimal level of an activity.

Most activities have diminishing marginal returns. Marginal returns are the extra benefit received

from doing a bit more of the activity. Do more of the activity until that point

when marginal returns from doing a bit more of the activity start to become more than the cost of the activity.

Optimal Capital

Benefit of owning capital is that it allows us to produce more goods.

Marginal product of capital is the extra revenue from the extra goods we could produce if we had just a bit more capital.

MPK can be measured in either nominal, current price (PMPK) or real, constant price (MPK) terms.

Capital has diminishing returns. MPK is a decreasing function of the capital stock.

Productivity of Capital

The productivity or average productivity of capital is the revenue generated per dollar of capital.

APK is value of output divided by the capital stock.

Value can be measured in constant or current price terms.

Marginal productivity of capital is often thought to be roughly proportional to average productivity capital.

Value of OutputAPK

Value of Capital

K

MPK

Cost of Capital

Economists define the (time) cost of capital as the cost of holding a unit of capital for a period of time.

A firm invests in capital equipment for a period. The firm borrows money upfront to finance the

purchase. The firm produces goods and generates revenues. The firm sells the capital at the end of the period,

typically at less than the purchase price due to wear and tear.

The firm repays loan. Cost of using capital includes interest payment

plus loss on the resale of capital.

Optimal Capital Example.

A firm borrows to buy 1 capital good at interest rate 1+i.

The firm produces PMPKt+1 worth of goods and sells the capital good for .

Optimal to buy capital good as long as pay-off is greater than the cost.

Optimal Condition

Definition of Capital Cost

,K NEWtP

,K OLDtP

, , ,1[ ]

t

K NEW K OLD K NEWt t t

PMPK

iP P P

Cost of Capital

, ,1 1K OLD K NEW

t t t tP PMPK i P

Capital Cost

We can divide the capital cost into three parts.

1. Interest cost: Net interest rate.

2. Depreciation: Defined as change in value due to aging.

3. Capital gain: Defined as change in value due to change in price of new goods.

, , ,1

,

[ ]K

K NEW K OLD K NEWt t t

P K NEWt

ck iP P P

i g P

, ,1 1

,

K NEW K OLDt t

K NEWt

P P

P

, ,1

,

KK NEW K NEW

P t tK NEWt

P Pg

P

Real Capital Cost

We can convert the optimal capital equation into real terms by dividing both sides by the price level. Define the real price of capital good as price of capital good relative to the firm’s output price.

,1

,1

,1

K

K

K

P K NEWt t

P K NEWt t

p K NEWt t t

PMPK i g P

MPK i g p

MPK r g p rck

,K NEWk tt

t

Pp

P

Example

A taxi agency can produce a certain amount of revenue with larger numbers of taxis (K = # of Taxis).

Assume earnings (revenues minus wages minus costs) per year is given by the schedule

Assume that the purchase price of a new taxi (with license) is $1,000,000. The borrowing interest cost is 4% and a taxi’s value depreciates by 8% per year. We assume that taxi’s prices increase by 2% per year.

34$200,000 K

Optimum Number of Taxis

The extra earnings generated by moving from 5 taxis to 6 taxis is less than cost of capital.

Maximum profits occurs where marginal cost equals marginal earnings.

Marginal MarginalTaxis Revenues Costs Profits Earnings Cost

1 200000 100000 100000 200000 1000002 336358.6 200000 136358.6 136358.6 1000003 455901.4 300000 155901.4 119542.8 1000004 565685.4 400000 165685.4 109784 1000005 668740.3 500000 168740.3 103054.9 1000006 766731.7 600000 166731.7 97991.42 1000007 860703.4 700000 160703.4 93971.69 1000008 951365.7 800000 151365.7 90662.28 1000009 1039230 900000 139230.5 87864.79 100000

10 1124683 1000000 124682.7 85452.17 100000

Optimal Capital: Example

Solve for Optimal Level of Capital

14

$150,000(.04 .08 .02) $1,000,000PMPK

K

14 * * 4$150,000

1.5 5.0625$100,000

K K

K

rck

K*

MPK

MPK & Optimal Capital

Q: Why does MPK slope down.

A: Diminishing returns to capital. Each additional unit of capital generates less additional revenue at a given workforce and technology level.

Q: What shifts the MPK curve.

A: Changes in productivity of capital. An increase in workforce or technology will make capital more productive and shift MPK curve out.

K

rck

K* K**

MPKMPK’

K

rck

K*

rck’

K**

MPK

Investment Volatility

The stock of capital may not be particularly volatile over the business cycle.

Capital stock is much larger than the flow of new investment in a given year, perhaps 10-15 times as large.

A 1% reduction in optimal capital stock will require a 10% reduction in investment.

Tax Rates

Corporations frequently must pay taxes on earnings. Define tax rate, .

Corporations also receive deductions for costs of capital Define deduction rates = (s1, s2, s3, ….)

Maximize after-tax profits implies that after-tax marginal product of capital = after-tax cost of capital.

Tax Wedge, tw, is defined as the extra cost of capital

beyond the interest rate.

,1 2 3(1 ) (1 ) (1 ) (1 )

KP K NEWt tPMPK s i s s g P

,K

K

P K NEWt t

p Kt t

PMPK i g tw P

MPK r g tw p

Which cost of capital?

Which interest rates should we use to calculate the cost of capital.

This depends on several things including the risk of the investment project & flexibility and duration.

If capital project is risky, we might apply a risk premium (i.e. use the interest rate on a risky bond).

If capital project is necessarily long term, we might use a long term interest rate.

q theory & Corporate Investment

A benchmark theory of corporate investment is that investment is a function of a quantity q.

The measure of q for a firm is

The market value of a publicly listed firm without

debt is market capitalization (stock price * shares outstanding).

The market value of a publicly listed firm with debt is the market capitalization plus value of debt (i.e. the cost of owning the firm lock, stock and barrel).

Market Value of Firm

Replacement Cost of Capitalq

Calculating q: China Steel 2000

Market Capitalization = Stock Price × # of Shares

Proxy for Replacement Value of Capital – Book Value of PP&E.

Proxy for Firm Value = Market Capitalization + Book Value of Total Debt

Caveat: Intangible Assets (i.e. Technology) May Be Large for Some Firms (e.g. Acer Inc. has a 2000 q > 4).

Caveat: Book value of PP&E may underestimate replacement costs of capital as it does not adjust for inflation.

Firm Balance Sheets.Stock Price $19.5# of Shares 8,748,363,000Market Cap $170,593,078,500Book Value of PPE $118,415,993,000Total Debt $38,228,396,000Firm Value $208,821,474,500q 1.763456685

q theory

If value of firm is greater than the cost of capital (q > 1) than the value of capital inside the firm is greater than the value of capital outside the firm. If q > 1, firm should have positive net

investment. If q = 1, firm should have zero net investment. If q < 1, firm should have negative net

investment.

q as Cost of Capital Theory

We might think of q theory as similar to cost of capital theory for firms that get financing through the stock market.

Owners of equity have a claim to the profits of the firm. They might require a certain amount of profits relative to what they pay for the stock.

A firm generates a certain amount of profits per unit of capital

Profits

Market Capitalizationck

Profits

Price of CapitalMPK

ProfitsPrice of Capital

Profits

Market Capitalization

Market Capitalization

Price of Capital

PMPK MPKq

ck rck

Investment & the Stock Market

Q theory suggests that a rise in stock market prices could be thought of as a decline in the cost of raising funds through equity.

Empirically, q theory seems to do a poor job of explaining connections between the stock market and investment.

Why? Many firms change their

capital stock infrequently. Short-term fluctuations in stock market may have little effect.

Stock market bubbles may keep stock prices from reflecting a realistic assessment of value of corporate capital.

Firms may be limited in ability to raise funds in stock market.

Investment and Stock Returns

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

1965

1968

1971

1974

1977

1980

1983

1986

1989

1992

1995

1998

2001

I

Stock Return

Corporate Finance

Two kinds of Finance External Finance – Funds for investment raised

through loans or issuing securities. Internal Finance – Funds for investment raised

through retaining profits instead of paying dividends.

Benchmark M-M Theory says investment decisions and firm value should not depend on sources of financing. Requirements: No distortionary taxation Perfect financial markets with perfect

information.

Reality

Internal Funds are cheaper form of financing than external funds. Much of corporate financing is through

internal finance. Investment is more strongly affected by

cash flow than q. Cost of capital depends on collateral

value that firms can pay if they default on loans or bonds.

K

rck

K*

MPK

K

rck

K*

MPK

K**

Change in Available Internal Funds

Conclusion

Cost of capital includes interest costs plus depreciation costs plus capital losses plus tax wedge.

Capital stock that maximizes profits sets the marginal product of capital equal to the cost of capital.

Business cycle fluctuations of capital investment are due to fluctuations in productivity and cost of capital.

Investment is volatile because capital is large relative to investment in any given period. Small fluctuations in optimal capital have large effects on investment.

Conclusion pt. 2

Optimal Capital Theory implies Corporate Investment is a function of q

(market value of firm relative to the replacement value of capital).

Real Estate prices are determined by rent divided by the determinants of the cost of capital.

In reality, internal funds are the dominant source of finance for investment. External Financing interest costs may depend on the state of firms balance sheets.

Firms’ balance sheets are an additional channel of business cycle volatility.