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