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
5%
3%
10%
14%
Transfer Costs of Land &
Building
Real Estate Developers'
Margin
Machinery & Equipment
14%
Public Construction
Private Residential
Private: Non-residential
54%
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
Private Investment
35.00
30.00
% of GDP
25.00
20.00
15.00
10.00
5.00
0.00
China
Hong Kong
Indonesia
Korea
Malaysia
Philippines
Thailand
Easterly, Rodriguez, and Schmidt-Hebbel "Public Sector Deficits and
Macroeconomic Performance." (Statistical appendix) 1994 and Bruno and Easterly
JME 1998.
Investment as a share of GDP:
East Asia 1997
Public Investment
20.00
18.00
16.00
% of GDP
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
China
Hong Kong
Indonesia
Korea
Malaysia
Philippines
Thailand
Easterly, Rodriguez, and Schmidt-Hebbel "Public Sector Deficits and
Macroeconomic Performance." (Statistical appendix) 1994 and Bruno and Easterly
JME 1998.
Volatility: Investment and GDP
Annual Growth Rates
Hong Kong
30.00%
25.00%
15.00%
10.00%
5.00%
0.00%
19
65
19
68
19
71
19
74
19
77
19
80
19
83
19
86
19
89
19
92
19
95
19
98
20
01
% Growth Rates
20.00%
-5.00%
-10.00%
-15.00%
-20.00%
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 of Output
APK 


Value of Capital
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.
MPK
K
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 Pt
to buy 1 capital good
at interest rate 1+i.
The firm produces
PMPKt+1 worth of
goods and sells the
capital good for Pt K ,OLD.
Optimal to buy capital
good as long as payoff is greater than the
cost.
K , NEW


Optimal Condition
Pt K1,OLD  PMPKt  1  it  Pt K , NEW
Definition of Capital
Cost
PMPK t 
iPt K , NEW  [ Pt K1,OLD  Pt K , NEW ]
 Cost of Capital
Capital Cost

1.
2.
3.
We can divide the
capital cost into three
parts.
Interest cost: Net
interest rate.
Depreciation: Defined
as change in value
due to aging.
Capital gain: Defined
as change in value
due to change in price
of new goods.
ck  iPt K , NEW  [ Pt K1,OLD  Pt K , NEW ] 
i   g
PK
Pt K , NEW
Pt K1, NEW  Pt K1,OLD

Pt K , NEW
gP
K
Pt K1, NEW  Pt K , NEW

Pt K , NEW
Real Capital Cost
PMPK  i    g
MPK  i    g
PK
t 1
PK
t 1
MPK  r    g
Pt K , NEW
ptK , NEW
pK
t 1
Pt K , NEW
p 
Pt
k
t
ptK , NEW  rckt
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
3
schedule $200,000 4 K
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.
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.
Taxis
1
2
3
4
5
6
7
8
9
10
Revenues Costs
200000 100000
336358.6 200000
455901.4 300000
565685.4 400000
668740.3 500000
766731.7 600000
860703.4 700000
951365.7 800000
1039230 900000
1124683 1000000
Profits
100000
136358.6
155901.4
165685.4
168740.3
166731.7
160703.4
151365.7
139230.5
124682.7
Marginal Marginal
Earnings Cost
200000 100000
136358.6 100000
119542.8 100000
109784 100000
103054.9 100000
97991.42 100000
93971.69 100000
90662.28 100000
87864.79 100000
85452.17 100000
Optimal Capital: Example

Solve for Optimal Level of Capital
PMPK 
$150, 000
1
4
 (.04  .08  .02) $1, 000, 000
K
$150, 000 14 *
 K  K *  1.54  5.0625
$100, 000
MPK
rck
K
K*
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.
MPK
MPK’
rck
K
K*
K**
MPK
rck’
rck
K
K**
K*
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.
PK
(1   ) PMPKt  (1  s1 )i  (1  s2 )  (1  s3 ) g
Pt K , NEW

Tax Wedge, tw, is defined as the extra cost of capital
beyond the interest rate.
PMPK t  i    g
MPK t  r    g
PK
pK
 tw Pt K , NEW
 tw ptK
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
Market Value of Firm
q
Replacement Cost of Capital
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).
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
# of Shares
Market Cap
Book Value of PPE
Total Debt
Firm Value
q
$19.5
8,748,363,000
$170,593,078,500
$118,415,993,000
$38,228,396,000
$208,821,474,500
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
Profits
ck 
Market Capitalization
they pay for the stock.
A firm generates a certain amount of
Profits
MPK 
Price of Capital
profits per unit of capital
q
Market Capitalization

Price of Capital
Profits
Price of Capital
Profits
Market Capitalization

PMPK MPK

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
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
65 9 68 9 71 9 74 9 77 9 80 9 83 9 86 9 89 9 92 9 95 9 98 0 01
19
1
1
1
1
1
1
1
1
1
1
1
2
-20.00%
-40.00%
-60.00%
-80.00%
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.
MPK
rck
K
K*
Change in Available Internal Funds
MPK
rck
K
K*
K**
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.