Part 3 Macroeconomics of Financial Markets€¦ · The model Consider an industry (economy)...

MacroeconomicsPart 3

Macroeconomics of Financial Markets

Lecture 9

Neoclassical investment theory

Motivation

In the previous lecture we had introduced necessary concepts of

investment theory

Now it’s time to build micro-founded dynamic investment theory to

reveal the relationship between Tobin’s q and investment

As we will argue later, this is the story of how stock market signals firms

Macroeconomics of Financial Markets2

Neoclassical investment theory

Summers L. H. (1981) “Taxation and Corporate Investment: A q-Theory Approach”. Brookings Papers on Economic Activity, 1, pp. 67-127.

Hayashi F. (1982) “Tobin’s Marginal q and Average q: A Neoclassical Interpretation”. Econometrica, 50(1), pp. 213-224.

Abel A. B. (1982) “Dynamic Effects of Permanent and Temporary Tax Policies in a q Model of Investment”. Journal of Monetary Economics, 9, pp. 353-373.

Abel A. B., Blanchard O. J. (1983) “An Intertemporal Model of Saving and Investment”. Econometrica, 51(3), pp. 675-92.

Caballero R. J. (1999) “Aggregate Investment” in Handbook of Macroeconomics ed. by J. Taylor and M. Woodford. (also NBER Working Paper No. 6264, 1997).

3 Macroeconomics of Financial Markets

The model

Consider an industry (economy) consisting of N representative

perfectly competitive firms

Capital in economy K is proportional to capital of the firm k :

tt NkK )1.9(

To simplify exposition, assume no capital depreciation. Then

ttt kkI 1)2.9(

ttt NIKK 1)3.9(


Assume that - total revenue of the firm has the property: tt kKTR ,

0

K

KMRPk

The model

The profit function is:

assume there are no other inputs and the firm relies on internal finance

tttttt ICIkKTRIk ,,)4.9(


When the firm invests I, it faces internal adjustment cost C(I)

It could be transportation cost,

assembly and maintenance costs,

training workers to operate new machinery, etc.

No investment, no cost:

Zero cost for infinitesimal investment:

Both positive and negative investments acquire adjustment cost

Adjustment cost is convex:

i.e. marginal adjustment cost increases

0)0( C

0)0( C

0)( IC

C(I)

I

Asymmetry reflects partial irreversibility of investment


Dynamic optimization

Investment should be chosen to maximize the value of the firm:

1

1

1

,

1

,)5.9(

tt

ttt

r

ICIkKTR

r

IkV

given initial capital k0 and relation between investment and capital

accumulation:

ttt kkI 1)2.9(

One can easily find FOCs by introducing and optimizing Lagrangian


0

1

0 1

,

t

tttt

tt

tttt kkIr

ICIkKTR

tt

tr

q

1

0

1

0 11

,

tt

tttt

tt

tttt

r

kkIq

r

ICIkKTR

tttt

t

ICqqICI

1,01

0,1 111 tttt qkKMRPrq

,0

11

,

11

1

1

11

1

t

t

t

tt

t

t

t r

q

r

kKMRP

r

q

k

8

Optimal investment

Consider two strategies:

#1: Sell the firm in the beginning of period t for the fare price Vt and

put money in an asset with the rate of return r to receive in the end of

period t capital income rVt

#2: Run the firm and make optimal investment It

receive profit πt in period t

face capital gain or capital loss dVt in the end of period t

The concept of opportunity cost equates revenues from two strategies

Market rewards the firm doing optimal investment by increasing its stock

price and thus its market value

ttI

t dVrVt

max)6.9(


Optimal investment

Strategy #1 does not assume any action (after the firm is sold)

Strategy #2 requires choosing I to maximize π + dV

Gain: the higher I, the higher capital and TR in all subsequent periods.

Investment increases future profits and thus determines capital gain dV

Loss: π = TR – I – C(I). Thus investment reduces current profit

Optimal choice corresponds to equality of marginal gain and loss

Remember the definition of marginal Tobin’s q: q = dV/dk

Then dV = q dk = qI

Marginal gain is q

Marginal loss is 1+C’(I)

ttI

t dVrVt

max)6.9(


tt ICq

1

Optimal investment

tt ICq 1)8.9(

Taking into account properties of C(I), we can determine optimal

investment as a function of Tobin’s q:

01,0,1)9.9( 1 ffqfqCI t

def

tt


ttttt IqdkqdV )7.9(

ttttttI

t IqICIkKTRrV ,max

ttI

t dVrVt

max)6.9(

tttttt ICIkKTRIk ,,)4.9(

The role of adjustment cost

In the absence of adjustment cost FOC (9.8) reads as q = 1

In this case we do not have a relationship between Tobin’s q and investment out of equilibrium

Thus adjustment cost is crucial for investment theory!

Adjustment cost is widely used in modern dynamic macroeconomics

in modelling price rigidities

labor demand

currency bands, and so on


tt ICq 1)8.9(

tt qfI )9.9(

Marginal Tobin’s q

Tobin’s q is the shadow price of investment (additional capital)

So we have standard microeconomic interpretation of the FOC: price of capital should be equal to its marginal cost

Remember the definition from Lecture 8:


tt ICq 1)8.9(

1 1

,)10.9(

tt

t

tt

r

kKMRP

dk

dVq

Thus, optimal investment is chosen with respect to future productivity of capital

1 1

,)9.9(

tttt

r

kKMRPfqfI

Marginal Tobin’s q

r

q

kKMRP

q

qq

t

tt

t

tt 111 ,

)11.9(

The revenue from additional capital, MRP, plus capital gain, qt+1 - qt,

should be equal to the opportunity cost, rqt


r

q

r

kKMRP

r

kKMRP

rr

kKMRP

r

kKMRP

r

kKMRP

r

kKMRPq

ttt

tt

tt

tt

tt

ttt

11

,

1

,

1

1

1

,

1

,

1

,

1

,

111

21

11

2

11

1

Average and marginal Tobin’s q

Average Tobin’s q :

dk

IkdVq

,

k

IkVq

,


Firm chooses optimal investment on the basis of marginal Tobin’s q

average Tobin’s q is easy to compute,

marginal Tobin’s q is a theoretical variable

they are the same if the value of the firm is linear-homogenous with respect to

capital

Marginal Tobin’s q :

Dynamic system

ttttt rqkKMRPqq 111 ,)11.9(

Write in continuous time for convenience:


,)()()13.9( tqftK

,)14.9( tKMRPtrqtq k

ttt NIKK 1)3.9(

tt qfI)9.9(

ttt qNfKK 1)12.9(

Steady state and vector field

rKMRPqtq

qtK

k

**

*

,0)(

,1,0)(

,1)(,0)(

,1)(,0)(

tqtK

tqtK

rKMRPtqtq

rKMRPtqtq

k

k

**

**

)(,0)(

,)(,0)(

,)()()13.9( tqNftK

,)14.9( tKMRPtrqtq k


q

0K

0q

K


Equilibrium

The system “capital stock–capital price” has a saddle-type steady state

There is only one trajectory – saddle path – that leads to equilibrium

We can rule out all unstable trajectories as they mean capital over-accumulation or price (of capital) bubbles

Formally, they violate transversality condition (didn’t introduce for humanity)

We will discuss No-bubbles condition in the next lecture

Economy should start on the saddle-path. This is possible because

Capital is a sluggish variable, i.e. K0 is given

Investment and Tobin’s q are jump variables

Forward-looking firms choose investment which corresponds to the Tobin’s q on the saddle path

That means that stock market determines the shadow price of capital that rules out asset-price bubble


Extension: Uncertainty and reluctance to invest

In the baseline neoclassical model equilibrium Tobin’s q is 1

If then investment is a continuous increasing function of q :


,1 ICq

00 C,01 Iq

,01 Iq

01 Iq

Uncertainty about future profits does not change this relationship:

r

q

kKMRPE

q

qqE

t

ttt

t

ttt 111 ,

)'11.9(

1 1

,)'10.9(

ttt

t

ttt

r

kKMRPE

dk

dVEq

Does it mean that uncertainty does not alter investment decisions?

)(qI

0 1 q


Inaction range

0,1 IqICq U

00 C

U

II

L qICICq

1lim1lim00

,0,1 IqICq L


Suppose that in general case and assume asymmetry in marginal costs for positive and negative investment

This creates inaction range: small deviations from q = 1 do not induce firm to invest As marginal cost of infinitesimal investment are nonzero, marginal gain

can be smaller than marginal cost

When q fluctuates within the inaction range [qU,qL], the firm postpones positive or negative investment

This explains empirically observable infrequent investment

)(qI

Uq 1 Lq q


Lumpy Investment

Another observation is that investments are lumpy

These are micro-level observations. Macroeconomists did not pay attention until 1990s as aggregate investment does not demonstrate infrequent and lumpy character.

But micro-level frictions are potentially important for macro dynamics!

Lumpy investment will appear in the neoclassical model if we introduce fixed cost

Baseline model assumes only variable costs, which is not general

So, in general, adjustment costs are complex (non-convex)

Facing fixed adjustment cost the firm has to invest big money for the gain in the value of the firm to overcome this fixed cost

As the new investment theory shows, in this case there are some ranges of Tobin’s q when there is no theoretical relationship between investment and q


)(Ic

0 I


)(qI

q Uq Lq q q


Uncertainty and reluctance to invest

High enough marginal adjustment costs of negative investment

can make investment completely irreversible

Higher uncertainty (together with non-convex adjustment costs)

widens inaction range that makes the firm’s investment even

more infrequent

Thus, higher uncertainty decreases aggregate investment

Note that introducing uncertainty into the baseline model with only

convex adjustment cost does not reveal any channel of how uncertainty

affects investment


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Part 3 Macroeconomics of Financial Markets€¦ · The model Consider an industry (economy)...

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