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Investment Theories
Lecture 3-4
INVESTMENT:
Previously considered how
households borrow and save to
convert fluctuating flow of income
into smoother flow of consumption.
Income then assumed exogenous.
But income must be generated
Requires productive capital.
Investment
• Concept of Investment
• The Accelerator Model
• Adjustment Cost Approach
• Q-Theory of Investment
• Optimal Capital(ınvestment) stock
INVESTMENT
• Income generation requires productive capital.
• E.g. Machines, Factories, etc
• To maintain or Increase the capital stock, some
of current output must be used to acquire
capital.
• i.e. there must be investment.
• This is inherently an intertemporal process:
Invest now→ for a higher future output.
Key Questions
a) What determines the level of investment?
b) Why does investment fluctuate so much?
• To answer (a), need to know what is
optimal capital stock at a given time.
• To answer that, how is output related to
capital stock?
Production Function
• Qt = F(Lt,Kt)
• Qt Output at time t
• Lt Labor employed at time t
• Kt Capital stock at time t
Isoquants (Combinations of L and K which
produce a particular output level)
Prod. Functions; L1>L0
MPK when L1>L0
MPK
MPK = ∂F(L,K)/ ∂K > 0
Diminishing Marginal Productivity of
capital.
0),(
2
2
K
KLF
K
MPK
Gross and Net Investment
• Distinguish between
a) Gross investment: How much invested
b) Net investment: How much capital stock changes
Different because of depreciation
Gross and Net Investment
• Capital becomes less productive over time
• Constant Depreciation Rate δ where 0< δ<1
• Capital stock changes according to
• Kt+1= Kt - δ Kt +It
• It - Gross Investment
• Kt+1-Kt = It - δ Kt
• =Jt
• Jt Net Investment
Investment Carried Out by
• Firms: Factories, Offices, Machines
• Households:Human Capital, Durable
goods
• Government:Infrastructure
• Initially think of all investment as carried
out by households
Households investmentPreviously, Could only transfer income between periods by buying/selling bonds.
Q1-C1 = S1 = B1 (1)
C2=Q2+(1+r)B1 (2)
= Q2+(1+r)(Q1-C1)
C1+ C2/1+r = Q1 + Q2/1+r (3)
Wealth
Now have more choice
Q1-C1 = S1 = B1+I1
Q1-C1-I1=B1 (4)
C2= Q2(I1)+ (1+r)B1 (5)
= Q2(I1)+(1+r)(Q1-C1-I1)
C1+(C2/1+r )=Q1-I1+Q2(I1)/(1+r) (6)
Wealth
C1+(C2/1+r )=Q1-I1+Q2(I1)/(1+r)
= Wealth2-Stage Process
• Choose I1 to maximize wealth
• Choose B1 to allocate between periods.
W= Q1-I1+ Q2(I1)/1+r
W=F(L1,K1)-I1+ F(L2,K2)/1+r
Where
K2=K1(1- δ)+I1
dK2/dI1 = 1
• To maximize Wealth
• ∂W/∂I1= -1 + 1/(1+r). ∂F/∂K2. ∂K2/∂K1
• =-1 + 1/(1+r) MPK(K2) =0
MPK(K2*)=1+r
r↑ → K2*↓
Firm’s investment
• Same outcome if decision taken by firms
which maximize present discounted value
of cash flows(V)
• Cash flows distributed to shareholders,
who are households
)(1
122211111 LwQ
rILwQV
122
1111
Vr
LwLwW
r
QIQ
1
211
Multiperiod Rule
• Two-Period analysis is special. General
rule is: MPK (Kt+1*) = r +
• Consider marginal investment, I
Produces output next period I.MPKt+1
Can then sell investment I(1-)
(could not do this in 2.period case)
rI
r
MPKIINPV t
1
)1(
1
1
Multiperiod Rule (Cont.)
rr
MPKI t
1
)1(
11 1
rI
r
MPKIINPV t
1
)1(
1
1
r
MPKrI t
1
)1()1( 1
r
rMPKI t
1
)(1
Multiperiod Rule (Cont.)
)1(
)(
10 1
r
r
r
MPKt
1)( tMPKr
r
rMPK
I
NPV t
1
)(1
r
MPK
r
r t
1)1(
)( 1
Multi period and two period
• MPK (K2*) = 1+r 2.Period
• MPK (Kt+1*) = r + Multi-Period
• Right hand side is cost of capital
• Have implicitly set Pk=1 (Pk=Price of Capital Goods)
• More general formulation:
• Pk= (See Mankiw))(
k
k
P
Pr
The link between Kt* and It*
**
1 )1( ttt IKK
)1(*
1
* ttt KKI
Invest so as to bring capital stock to desired level.
Complete and instantaneous adjustment.
Abstracts entirely from adjustment or installation
costs.
INVESTMENT: Further theory and
applications
• Have seen that investment related to interest
rates through cost of capital formula.
• Investment also thought likely to be related to:
1) Output growth
2) Stock market valuation
• We will consider each in turn.
• We will also examine how can relax assumption
of zero adjustment costs.
The accelerator model
• Dates back to 1917.
• Idea is that there is a stable relationship
between optimal capital stock and output:
• Kt* = h Qt (1)
• h positive constant
The accelerator model; a particular
form of production function
• Consider a particular form of production
function, Cobb-Douglas:
)2(1 LKQ
The accelerator model, cont.
11LK
K
QMPK
In this case:
K
LK 1
= =
K
Q
Since MPK = r +
K = Qr
(4) i.e h =
r
(3)
The accelerator model, cont.Constant only if r and constant.
If r and vary, (1) only true if L-shaped is quants or
“fixed coefficients”.
If (1) true:
tttt hQhQKK 1
*
1
= tt QQh 1
or
Jt = hQt (5)
Where
Qt = Qt+1Qt
“Accelerator Model”
The accelerator model, cont.
Predictably,
empirical support for
model is limited.
Subsequently
modified to
incorporate partial
adjustment
mechanism
Partial adjustment mechanism
1
*
1 tttt KKgKK
0<g<1
If g=1, back to accelerator
With g<1, flexible accelerator (exercise: show how (6)
derived from optimal behavior with adjustment and other
costs)
Rewriting (6) and using (1)
1
*)1( ttt KggKK
=
1)1( tt KgghQ (7)
(6)
Partial adjustment mechanism, cont.Rewriting (6) and using (1)
1
*)1( ttt KggKK = 1)1( tt KgghQ
Same holds for preceding periods
211 )1( ttt KgghQK (8)
...2 tK Substituting into (7)
...)1()1( 2
2
1 tttt ghQgghQgghQK
To get in terms of investment, use lagged version of (9), giving,
....)()1()()1( 32
2
2111 tttttttt QQggQQggQQghKK (10)
(7)
(9)
Partial adjustment mechanism
• Current investment related not just to current output change, but lagged change too.
• Current investment includes continuing adjustment to past changes.
• Flexible accelerator fits data better than simple accelerator.
• Costs of adjustment seem to be important. However, investment forward looking-Models don’t capture this.
• Cannot observe a firm’s expectations…But for quoted companies can observe something like market expectations
q Theory• q or Tobin’s q defined as
• q=(stock market value) / (replacement cost of capital)
• If q>1 capital is worth more installed in firm than it actually costs.
• q>1 firm could potentially issue new shares, which more than pay for new capital.
• Formal theory developed by Fumio Hayashi, Lawrence Summers and Andrew Abel in early 1980s.
• Sachs and Larrain present as if alternative to adjustment cost approach- But adjustment costs central.
• Based on intertemporal optimization- will consider Two-Period Model (similar to Burda And Wyplostz)
Optimum investment level
• Firm chooses investment and capital stock to
maximize:
(11)
• Price of output and capital goods equal to 1.
222211111121 ),()1(
1)(),(),( LwKLF
rICILwKLFKIV
Installation cost of ınvestment
C(I1)
I1
0)0()0(
0)(
0)(
1
1
CC
IC
IC
So the larger the
investment, the higher
are adjustments or
installation costs
(Marginal and
Average)
Optimum investment
Constraint is
112 )1( IKK
Choose I1 and K2 to maximize (11) subject to (12)
(12)
Lagrangian Function
)13()1(),(1
1)(),(),,( 211222211111121
KIKqLwKLF
rICILwKLFqKIZ
q is shadow price on the constraint, (see the seminar exercise)
Optimum investment
• Differentiate (13):
0)1(
1
22
q
K
F
rK
Z
OR
r
KMPKq
1
)(*
2 (14)
R.H.S is present values of MPK
n.b Pk=1
LHS is marginal q
Optimum investment
0)(1*
1
1
qIC
I
Z
OR
)(1*
1ICq (15)
If optimal to invest, q>1
If optimal not to invest, q=1
Optimum investmentCan combine (14) and (15) to eliminate q
)(1)1()(*
1
*
2 ICrKMPK (16)
Similar to cost of
capital for formula-but
invest less because of
adjustment cost
Optimum investment
Since observe q not MPK, can concentrate on (15), or its inverse
)1(*
1 qDI )51(
D( )= 1'C ( )I1
Comments
• Some empirical support
• Summers:10% rise in stock market → 0.9% rise in I/K
• Rather weak link
• Preceding theory requires firms to be able to borrow freely to finance investment.
• May not be able to do this: face credit rationing
• Credit rationing may be rational response of lenders faced with differential risks of repayment.
• Loan designed for firm with average risk of default attractive to high-risk firms.
• Problem of adverse selection
• → don’t just rely on price (r) to allocate loans, but restrict quantity too.
• In fact- see Bond/Jenkinson-firms don’t borrow much to finance investment
• Especially in U.S. and U.K. rely on retained earnings.
• Implication for examining investment: include measures of current cash flow.
(See Schiantarelli)
• Possible policy implication: Reform financing of industrial investment: less emphasis on stock markets, more on banks
• Ongoing debate (see Bond/Jenkinson)
The end
• Questions
• Discussion