part 3 macroeconomics of financial markets€¦ · the model consider an industry (economy)...
TRANSCRIPT
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(
4 Macroeconomics of Financial Markets
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(
5 Macroeconomics of Financial Markets
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
6 Macroeconomics of Financial Markets
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
7 Macroeconomics of Financial Markets
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(
9 Macroeconomics of Financial Markets
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(
10 Macroeconomics of Financial Markets
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
11 Macroeconomics of Financial Markets
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
12 Macroeconomics of Financial Markets
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:
13 Macroeconomics of Financial Markets
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
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
14 Macroeconomics of Financial Markets
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
,
15 Macroeconomics of Financial Markets
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:
16 Macroeconomics of Financial Markets
,)()()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
17 Macroeconomics of Financial Markets
q
0K
0q
K
18 Macroeconomics of Financial Markets
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
19 Macroeconomics of Financial Markets
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 :
Macroeconomics of Financial Markets20
,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
Macroeconomics of Financial Markets21
Inaction range
0,1 IqICq U
00 C
U
II
L qICICq
1lim1lim00
,0,1 IqICq L
Macroeconomics of Financial Markets22
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
Macroeconomics of Financial Markets23
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
Macroeconomics of Financial Markets24
)(Ic
0 I
Macroeconomics of Financial Markets25
)(qI
q Uq Lq q q
Macroeconomics of Financial Markets26
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
Macroeconomics of Financial Markets27