Download Investment

Investment
Mid-term Exam
• Tuesday, November 17th 9AM
• ?
• Semi-open Book (Bring 1 A4 size paper
with handwritten notes))
• Coverage. Lecture notes through
November 15th. this one.
Interest Rates
• Most observed interest rates are money
interest rates, i.e. how much money you
can get in the future in exchange for some
money today. 1+it
• We are more interested in goods interest
rates i.e. how many goods you can get in
the future in exchange for some goods
today. 1+rt
Real Interest Rates and Nominal
Interest Rates
• The goods interest rate is derived from the
nominal interest rate. If the price of goods is Pt
then giving $1 to the bank is equivalent to
saving 1 P goods.
t
• When you get the money back with interest
after 1 year you can buy 1  it goods.
Pt 1
Real Interest Rate
• Goods gotten per goods given up is
1  it
Pt 1
1
Pt

1  it
Pt 1
Pt
 1  rt
• Future price level is not known when the
savings decision is made. Must be forecast
by savers (and borrowers). Only ex post real
interest rate observed by economists.
Interest Rates: The Data
• Main source of international interest rate
data is IMF’s International Financial
Statistics which is only available with a
subscription fee.
• Interest rate data is typically available from
Central Banks website (List of Websites)
• US Data is most conveniently obtained
from the FRED database at the St. Louis
Federal reserve.
Real GDP: Yt
• GDP aka Nominal GDP aka Current Dollar
GDP is the weighted sum of the number of
goods produced using their current prices
as the weight.
• Real GDP aka Constant Dollar GDP aka
GDP adjusted for inflation is the weighted
sum of the number of goods produces
using the Base Year prices as yardsticks.
Price Indices: Pt
• Two most commonly used price indices
are GDP Deflator and Consumer Price
Index (CPI)
• The CPI is the price of a representative
market basket of goods relative to the
price of that same basket during a
benchmark/base year (multiplied by 100).
• The GDP deflator is the ratio of nominal
GDP to Real GDP (multiplied by 100).
Nominal GDP GDP
P  GDP Deflator 

Real GDP
Y
The Data
• The U.N. maintains annual statistical databases
of GDP and its components (investment,
consumption, etc.) for constant and current
dollar measures for a wide variety of countries
UN Main Aggregates Data Base
• Data on aggregates at higher frequencies are
typically available from national statistical
agencies Examples
– USA
– Hong Kong
– Japan
Investment is a small share of GDP
50.00%
45.00%
40.00%
35.00%
30.00%
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
China
France
Germany
Hong Kong
SAR of
China
Japan
Republic of
Korea
United
States
Investment Shares Japan
Current Dollar or Constant Dollar
0.37
0.35
0.33
0.31
0.29
0.27
0.25
19
70
19
72
19
74
19
76
19
78
19
80
19
82
19
84
19
86
19
88
19
90
19
92
19
94
19
96
19
98
20
00
20
02
20
04
0.23
Current
Constant
19
70
19
72
19
74
19
76
19
78
19
80
19
82
19
84
19
86
19
88
19
90
19
92
19
94
19
96
19
98
20
00
20
02
20
04
Relative Price of Investment goods.
Japan
1.15
1.1
1.05
1
0.95
0.9
Gross Domestic Investment
(and Components)
Table 5.2.5. Gross and Net Domestic Investment by Major Type
[Billions of dollars]
Bureau of Economic Analysis
Downloaded on 10/23/2006 At 8:46:47 AM Last Revised August 02, 2006
Line
Gross
Depreciation Net
% of GDI
1 Gross domestic investment
2454.5
1604.8
849.7 100.00%
4 Gross private domestic investment
2057.4
1352.6
704.8
83.82%
22 Change in private inventories
7 Fixed investment
2036.2
1352.6
683.6
82.96%
10
Nonresidential
1265.7
1045.6
220.1
51.57%
13
Structures
338.6
260.6
78
13.80%
16
Equipment and software
927.1
785
142.1
37.77%
19
Residential
770.4
307
463.4
31.39%
23 Gross government investment\1\
397.1
252.2
144.8
16.18%
30 Structures
248.9
127.4
121.5
10.14%
37 Equipment and software
148.1
124.8
23.3
6.03%
BEA NIPA Table 5.2.5
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
Demand for Capital
• A Cobb-Douglas firm that hires labor and
capital in rental markets maximizes profits.

1
max Pt Kt ( At Lt )
 Wt Lt  Rt K t
Yt
Rt


K t Pt
Optimal K
R/P
R/P
MPK
K
K*
Cost of Capital Rises
R/P
R/P
MPK
K
K**
K*
Productivity Rises
R/P
R/P
MPK’
MPK
K*
K***
K
What is the cost of capital?
• Examine an optimizing odel of investment.
– Firm starts in period 0 with K0.
– Firms hire workers at wage rate Wt in every
period.
– Firms sell goods at price Pt
I
P
– Firms buy new investment goods at price t
– Firms use new investment goods to accumulate
capital
K  (1   ) K  I
t 1
t
t
Firm’s Intertemporal Problem
• Rather than optimizing profits in each
period, the firm managers will construct a
plan to maximizes the real net worth of the
firm.
Nominal Present Value
• Assume that the firm will last until period T.
The firm will generate a certain cash flow.
The present value (in $ terms is)

1

1
I


V0   0  
PK
(
A
L
)

W
L

P
I
t
t
t
t
t
t
t
t


1

i
t 0
t
• We can think of this as the financial value of
the firm.
Real Present Value
• The real present value is
V0  0  1

1
I
 Kt ( At Lt )  wt Lt  pt It 

P0
t 0 1  r
• Where the relative prices
Wt
wt 
Pt
I
Pt
p 
Pt
I
t
Optimization Problem
• Write the optimization problem as choosing a
stream of investment, capital, and labor to
maximize the net worth of the firm.

1
Kt ( At Lt )  wt Lt  p I
max 
t
K ,I ,L
(1  r )
t 0

I
t t
• Subject to restrictions on the path of capital
Kt 1  I t  (1   ) Kt  t
Lagrangian Multiplier
• We can solve this by assuming that there is
some large cost q to the firm of setting
tomorrows capital stock greater than the sum
of today’s capital stock.

max 
K ,I ,L
t 0
K t ( At Lt )1  wt Lt  ptI I t  qt K t 1  I t  (1   ) K t 
(1  r )t
• How large should this cost be? The minimum
necessary to insure the firm never violates
the constraint.
• The first order conditions are
I
Y
w
p
qt
(1   ) t
t
t


t
t
t
(1  r ) Lt (1  r )
(1  r ) (1  r )t

(1  r )t 1
Rt 1

Pt 1
Yt 1
qt 1
qt

(1   ) 
t 1
K t 1 (1  r )
(1  r )t
I


1



t 1 
I
I
 (1  r ) pt  pt 1 (1   )  1  r 
(1   )  ptI


1




t

1


Cost of Capital
• The rental cost of capital is the cost of
owning capital for one period.
• This cost includes:
– The foregone interest that you could have
received had you lent goods.
– The depreciation of the capital
– Any “capital loss” that occurs
Cost of Capital
• The user cost of capital is sometimes
approximated as
Rt 1
 (1  r ) ptI  ptI1 (1   )
Pt 1

 I
 1   tI1 
I
I


 1  r  
(1


)
p

p

r


 t

t 
t 1   t 1   
 1   t 1 


Dynamics of Investment
Capital Adjustment Costs
• Assume that when investment is large,
some of the investment goods are wasted.
Number of goods purchased per investment
goods installed
• Assume pI =1 for all t.
c
2
Costt  I t   ( I t )
2
Lagrangian Optimization
• Choose Investment and capital to maximize value
of the Firm

max 
K ,I ,
t 0

* 1
t t
Kt ( A L )
c
2
  I t   I t  qt  Kt 1  (1   ) Kt  I t 
2
(1  r )t
FOC
• Optimize
1 + c  I t 
qt
=
t
(1 + r) (1 + r)
t
FOC
Yt 1

Kt 1
qt 1
qt

(1   ) 
t 1
t 1
t
(1  r )
(1  r )
(1  r )
Investment Equation
• We can write investment as a function of
the marginal q (i.e. the discounted benefit
of adding a unit of capital)
1
I t =  q t  1
c
qt 

Yt 1
 (1   )  qt 1
K t 1
(1  r )
Recursive Substitution
q0 
q0 
  Y1 K  (1   )  q1
q1 
1
(1  r )
  Y1 K  (1   ) 
  Y2 K  (1   )  q2
2
(1  r )
  Y1 K  (1   )  q2
1
(1  r )
1
(1  r )
q0 
q2 
  Y1 K
1
(1  r )

(1   )   
Y2
(1  r ) 2
  Y3 K  (1   )  q3
3
(1  r )
K2
(1   )  q3

(1  r )3
2
Marginal Q
• Marginal q is the discounted sum of future marginal
product of capital discounted by the real interest rate.
• The benefit of having capital inside the firm relative to
the opportunity cost of not having funds outside the firm
invested in the bank.
(1   )
q0  

MPK
j
j
j 1 (1  r )

j 1
Intuition of Marginal q
• The variable xt is the benefit of having an
extra unit of capital at the end of time t which
includes marginal product of capital.
• But any investment today will last for many
periods. The marginal benefit of investing is
the discounted sum of having capital in future
periods. Note that capital depreciates at a
rate of (1-δ) leaving (1-δ)t after t periods.
Implications of Marginal q
• Long-lasting changes in capital
productivity or interest rates have more
effect on investment.
• Investment is more sensitive to long-run
interest rates.
Steady State
• Dynamic Equation 1
1
I t =  q t  1
c
• Dynamic Equation 2
q
MPK  (1   )  q
MPK
 (r   )  q  MPK  q 
(1  r )
r 
Investment Curve
• Whenever q > 1, then invest and capital
stock increases.
• Whenever q < 1, then disinvest and capital
stock decreases.
• Whenever q = 1, leave capital stock alone.
q
ΔK > 0
1
ΔK < 0
K
Lagrangian
• Optimize
2

max 
K ,I ,L
Kt ( At Lt )1  ptI
t 0

c  It
I


K

w
L

p
Kt 1  It  (1   ) Kt 
t
t t
t I t  qt 



2  Kt

(1  r )t
FOC
wt
(1 - α)Yt
=
(1 + r)L t (1
t + r)
2

 It
ptI + ptI c 
t
(1 + r)
K t

-δ 
(1t+ r)
=
qt
t

I
 I
Yt 1
cI
 ptI1  t     ptI1c  t    t
Kt 1
2  Kt

 Kt
 Kt  qt 1 (1   )  qt
(1  r )t 1
(1  r )t 1
(1  r )t
Investment Equation
• We can write investment as a function of
the marginal q (i.e. the discounted benefit
of adding a unit of capital)
 It

p + cp  - δ 
 K t  = qt  It - δ  1 ( qt  1) 
t
I
1+r)
(
1+r)
(
Kt
p
c
t
t
I
t
I
t
It
1 qt
  (
I  1)
pt
Kt
c
Marginal q
• Recursively substitute marginal benefit of
owning
capital

I
 I 
1  Y
cI
2
t 1
 ptI1  t 1     cptI1  t 1    t 1   MBK t 1

1  r  K t 1
2  K t 1

 K t 1
 K t 1 

(1   )
(1   )
qt  MBKt 1 
qt 1 , qt 1  MBKt  2 
qt  2 ,
(1  r )
(1  r )
(1   )
qt  2  MBK t 3 
qt 3
(1  r )
2
3
 (1   ) 
 (1   ) 
(1   )
qt  MBK t 1 
MBKt  2  
MBK t  2  
MBK t 3  .....


(1  r )
 (1  r ) 
 (1  r ) 
Intuition of Marginal q
• The variable xt is the benefit of having an extra unit
of capital at the end of time t which includes
marginal product of capital less capital adjustment
costs plus the degree to which having more capital
reduces future capital adjustment costs.
• But any investment today will last for many periods.
The marginal benefit of investing is the discounted
sum of having capital in future periods. Note that
capital depreciates at a rate of (1-δ) leaving (1-δ)t
after t periods.
Implications of Marginal q
• Long-lasting changes in capital
productivity or interest rates have more
effect on investment.
• Investment is more sensitive to long-run
interest rates.
Tobin’s q
• Difficult to measure q for an individual firm.
• Researchers typically use Tobin’s q as a
proxy for marginal q:
tq 
Market Value of Equity+Book Value of Debt
Replacement Cost of Capital
Equivalency
• Multiply the first order condition by
qt K t 1  MBK t 1 K t 1 
(1   )
qt 1 K t 1
(1  r )
2

 
1 
c I  I t 1
I  I t 1
MBK t 1 K t 1 
   K t 1  cpt 1 
   I t 1 
 Yt 1  pt 1 
1 r 
2
K
K
 t 1

 t 1
 

Dt 1
2
q I



Pt 1 t 1 t 1
1 
c I  I t 1

I

    wt Lt   pt 1  qt 1  I t 1  
Yt 1  pt 1 
1 r 
2
K
1 r
 t 1



q [(1   ) K t 1  I t 1 ]
(1   )
1 Dt 1
qt K t 1  MBK t 1 K t 1 
qt 1 K t 1 
 t 1
Pt 1
(1  r )
1 r
1 r
qt K t 1 
q K
1 Dt 1
 t 1 t  2
Pt 1
1 r
1 r
– Recursive substitution
qt 1 Kt  2
qt  2 Kt 3
1
1
qt Kt 1 
 t 1 
, qt 1K t  2 
t 2 

1 r
1 r
1 r
1 r
2
3
1
 1 
 1 
qt Kt 1 
 t 1  
 t 2  
  t 3  ....
1 r
 1 r 
 1 r 
Vt
qt
Vt
qt Kt 1  Vt  qt 
 I  I
Kt 1
pt
pt Kt 1
Tobin’s q
• A firm’s q is measured as the financial value
of a firm divided by the replacement cost of
capital.
• When a firm is valued by financial markets at
levels greater than the sale value of its
capital, then capital is worth more inside the
firm than out and the firm should add capital.
• When financial value of a firm exceeds value
of its capital, capital is worth more inside the
firm than outside so the firm should move
more capital inside the company.
Data
• Book Value of Debt can be obtained from a
firms balance sheet
• Market Value of Stock is the product of the
stock price and the number of shares
• Market value of physical capital is defined
recursively. Define pKt+1 as the end of period
replacement cost of capital.
Pt I
pKt 1  (1   ) I pKt  I t
Pt 1
Empirical Results
• Tobin’s q has only weak effects on
investment.
– Reason: Fluctuations in the stock market are
not taken by firms as a measure of future
value of capital.
• After controlling for q, cash flow still affects
the level of investment.
Tobin’s q
• Difficult to measure q for an individual firm.
• Researchers typically use Tobin’s q as a
proxy for marginal q:
tq 
Market Value of Equity+Book Value of Debt
Replacement Cost of Capital
Tobin’s q
• A firm’s q is measured as the financial value
of a firm divided by the replacement cost of
capital.
• When a firm is valued by financial markets at
levels greater than the sale value of its
capital, then capital is worth more inside the
firm than out and the firm should add capital.
• When financial value of a firm exceeds value
of its capital, capital is worth more inside the
firm than outside so the firm should move
more capital inside the company.
Data
• Book Value of Debt can be obtained from a
firms balance sheet
• Market Value of Stock is the product of the
stock price and the number of shares
• Market value of physical capital is defined
recursively. Define pKt+1 as the end of period
replacement cost of capital.
Pt I
pKt 1  (1   ) I pKt  I t
Pt 1
Empirical Results
• Tobin’s q has only weak effects on
investment.
– Reason: Fluctuations in the stock market are
not taken by firms as a measure of future
value of capital.
• After controlling for q, cash flow still affects
the level of investment.
Assymmetric Information
• There are a set of entrepreneurs with risky
investment project.
– Each project requires 1 unit of good to
implement.
– Each entrepreneur has 0 < W < 1 units of
goods.
– Each entrepreneur must borrows (1-W) from a
risk neutral financial system which also
accepts deposits at a rate 1+r.
Project Risk
• Investment projects will have a random payoff with
a uniform distribution over the range [0, 2γ]
• Expected value of the outcome is γ.
• Each entrepreneur has its own level of γ - Riskier
projects have higher expected returns.
1
2
γ
2γ
Symmetric Information
• To make them willing to provide loans, banks
demand an expected return of 1+r.
• If all information is public and free, banks can lend
(1-W) and collect an equity payment of
(1  r ) W

 Payoff
• This will pay an expected value of (1+r)(1-W) for
banks.
• The expected payoff for entrepreneurs is
γ -(1+r)(1-W).
• Entrepreneurs will invest if γ -(1+r)(1-W) > (1+r)W
Asymmetric Information
• Assume that entrepreneurs naturally
observe their own level of output. B
• Banks must pay a cost of c in order to
monitor output. This makes equity contract
expensive to implement.
• Risky debt is the efficient contract
– Banks receive a predetermined payment D.
– If entrepreneurs do not pay D, the banks pay
monitoring cost and collect entire output level.
Expected return of banks
• If D < 2 γ , the probability that an entrepreneur
will be able to fully repay the loan is
the probability of default is D
2
1
2
γ
D
2γ
2  D
2
and
Payoff if there is a default
• What is the average payoff of payoffs of
project which are low enough that the
entrepreneur must default? D
2
• The entrepreneur will only default if their
payoff is less than D.
• Given a uniform distribution, half of
projects with payoffs less than D, will have
payoffs less than D
2
Expected return of banks
• If D > 2 γ , the entrepreneur will always default. The
payoff to the banker would be γ-c.
• If D < 2 γ, the payoff would be
2  D
D D
2  c
1 2
D
(  c) 
D
D
2
2 2
2
4
• The payoff is a quadratic function of D with a maximum
at
MAX
D
 2  c
If the lender asks for debt repayments beyond that point,
they will be paying monitoring costs so frequently that
they will lose money
Returns as a function of Debt
RMAX
2γ-c
2γ
D
Expected payoff to banker
• Bankers demand an expected payoff at
least equal to (1+r)(1-W).
• They will never lend if RMAX < (1+r)(1-W).
• If RMAX > (1+r)(1-W) then in a competive
market, the debt levels will be
2  c
1 2
D
D  (1  r )(1  W )
2
4
D  2  c 
 2  c 
2
 4 (1  r )(1  W )
Payoff to the entrepreneur
• The entrepreneurs will have to pay returns
to bankers to compensate them for their
expected monitoring costs. The expected
value of monitoring costs is
2  c 
D
A
c
2
 2  c 
2
 4 (1  r )(1  W )
2
c
• The expected return to the entrepreneur is
γ-(1+r)(1-W) – A. They will invest if
γ-(1+r)(1-W) – A > (1+r)W.
Investment Decision
• We can write this function as
  (1  r )(1  W )  A  (1  r )W
• Hurdle for investment is higher when W <
1.
• The greater is W, the less will be the
average monitoring costs.
Midterm Exam 2
• Tuesday, November 16 9-10:50
– Lecture Theater E
• Coverage: Frictional, Unemployment,
Consumption, Investment,
• Bring Calculator, Writing Instruments, 1 A4
sized paper with written notes.