INVESTMENT
Contents:
- Introduction
- A model of investment with adjustment costs (discrete and
continuous time case)
The importance of investment
Why study investment?
•
Capital accumulation is the main determinant of growth ⇒ investment demand is
important to the behavior of standards of living over the long-run.
•
investment is highly volatile ⇒ investment demand may be important to short-run
fluctuations.
The neoclassical model of investment and its shortcomings
Neoclassical approach (Jorgenson 1963): investment depends upon the interest rate
Two major failings as a description of actual behavior in this simple model of
investment:
-
A discrete change in one of the exogenous variables leads to a discrete change in
the desired capital stock. Suppose, for example, that return of capital is higher
(even slightly) in one country. Then all capital (from other countries) would flow to
this country and thus investment rate would be infinite!
-
The model does not identify any mechanism through which expectations affect
investment demand. Yet it is clear that in practice expectations about demand and
costs are central to investment decisions: firms expand (restrict) their capital stocks
when they expect their sales to be growing (falling) and the cost of capital to be low
(high).
A solution: Capital adjustment costs
The solution to the problem is provided by the presence of costs to changing the capital
stock, what we call adjustment costs. Those adjustment costs come in two forms:
1) Internal adjustment costs arise when firms face direct costs of changing their capital
stocks. (Examples: costs of installing the new capital and training workers to operate
the new machines.)
Assumption: adjustment cost increases with the size of investment (‘convex
adjustment costs’).
2) External adjustment costs arise when each firm faces a perfectly elastic supply of
capital, but where the price of capital goods relative to other goods adjusts so that firms
do not wish to invest or disinvest at infinite rates.
A model of Investment with Adjustment Costs
Assumptions:
• An industry with N identical firms.
• A representative firm’s real profits, Π, at time t are proportional to its own capital
stock k(t), and decreasing in the industry-wide capital stock K(t) ⇒
Π = π(K(t))k(t)
where π’(.) < 0
The assumption that the firm’s profits are proportional to its capital stock is
appropriate if the production function has constant returns to scale, output markets
are competitive, and the supply of all factors other than capital is perfectly elastic.
Then, if a firm doubles, for example, its capita, it employs twice as much of all
inputs and its costs and revenues are twice as high as the other’s.
The assumption that profits are declining to the aggregate capital stock is
appropriate if the demand curve (for the industry’s product) is downward-sloping.
• Adjustment costs in investment, C(.), are a convex function of the rate of change of
•
the firm’s capital stock, k . Specifically:
•
C (k ) satisfies C(0) = 0, C’(0) = 0 and C’’(.) > 0
These assumptions imply that it is costly for a firm to increase or decrease its
capital stock and that the marginal adjustment cost is increasing in the size of
investment.
• Price of capital goods is constant and equal to 1 (so, there are no external
adjustment costs).
•
• Depreciation rate equals zero i.e. δ = 0 ⇒ k (t ) = I (t )
The Firm’s Problem
Firm’s profits: Π = π(K)k – I – C(I)
Firm’s problem: maximize the present value of profits
∞
Π=
− rt
e
∫ [π (K (t ))k (t ) − I (t ) − C (I (t ))] dt
t =0
(1)
Each firm takes aggregate capital stock, K, as given and chooses investment over time
to maximize profits, Π.
Discrete form of the problem:
∞
1
[π (K t )kt − I t − C (I t )]
t
t = 0 (1 + r )
max Π = ∑
kt +1 = kt + I t
subject to
(capital accumulation)
There are infinitely many periods, so there are infinitely many constraints
⇒ Lagrangean of the firm’s maximization problem:
∞
∞
1
[π (K t )kt − I t − C (I t )] + ∑ λ t (kt + I t − kt +1 )
L=∑
t
t =0 (1 + r )
t =0
λt is the Lagrange multiplier associated with the constraint.
The firm’s problem in current-terms constraint
Profit function is written in discounted stream but this does not hold for the constraint.
We bring everything in current terms by defining:
qt ≡ (1 + r ) λ t
t
qt shows the value (to the firm) of an additional unit of capital at time t+1 in time t
currency units.
The modified Lagrangean is:
∞
L′ = ∑
t =0
1
[π (K t )kt − I t − C (I t ) + qt (kt + I t − kt +1 )]
t
(1 + r )
qt is the Lagrange multiplier in current terms.
First-order conditions
F.O.C. for investment
∂ L′
=0 ⇒
∂I
1
[− 1 − C ′(I t ) + qt ] = 0
t
(1 + r )
⇒ qt = 1 + C ′(I t )
Interpretation: the cost of acquiring one unit of capital equals the purchase price (fixed
at 1) plus the marginal adjustment cost.
⇒ The firm will invest up to the point where the cost of acquiring an extra unit of
capital equals its value.
F.O.C. for capital (in period t)
∂ L′
=0 ⇒
∂ kt
1
1
(
)
[
]
π
K
+
q
−
q =0
t
t
t
t −1 t −1
(1 + r )
(1 + r )
Note: The capital stock in period t, k(t), appears in both the term for period t and the
term for period t-1.
t
(
)
1
+
r
and rearranging yields:
Multiplying the above equation by
π (K t ) = (1 + r ) qt −1 − qt
⇒ π (K t ) = r qt − ∆ qt − r ∆ qt
Ignoring r ∆ qt ≅ 0 , we have for the firm to be optimizing that the returns to capital
must equal this opportunity cost.:
marginal revenue product of capital = opportunity cost of a unit of capital
(foregone real interest minus capital gain)
Transversality condition
Transversality conditions for the finite and infinite horizon are respectively:
1
qT kT = 0
T
(1 + r )
lim
t →∞
1
qt kt = 0
t
(1 + r )
Optimization requires that at the end of the time period the firm cannot have capital
holdings (i.e. kt = 0) or the value of the firm’s capital should be equal to zero (i.e. qt =
0).
If this condition fails (infinite horizon), then, the firm is holding valuable
capital forever and so it can increase the present value of its profits by reducing
its capital stock.
The Continuous-Time Case
Firm’s problem: maximize the continuous-time objective function (1), rather than the
discrete-time objective function (2).
The current-value Hamiltonian:
•


H (k (t ), I (t )) = π (K (t ) k (t )) − I (t ) − C (I (t )) + q (t )  I (t ) − k (t )


k(t): state variable (whose value at any time is determined by past decisions)
I(t): control variable (can be freely controlled)
q(t): co-state variable (the shadow price of the state variable)
First-order conditions of the continuous-time problem
∂H
=0
∂ It
⇒
1 + C ′(I (t )) = q (t )
∂H
= r q(t ) − q (t )
∂ kt
Transversality condition:
lim e − rt q (t ) k (t ) = 0
t →∞
⇒
•
π (K (t )) = r q(t ) − q(t )
(3)
Solution for q
− r (τ − t )
− r (T − t )
(
)
(
(
)
)
q
t
=
e
π
K
τ
d
τ
+
e
q (T ) for T > t
Equation (3) implies that
∫τ =t
T
By the transversality condition the second term approaches zero as T approaches
infinity:
⇒ q (t ) =
T
∫τ
=t
e − r (τ − t )π (K (τ )) dτ
The value of a unit of capital at a given point in time equals the discounted value of its
future marginal revenue product.
Implications - Tobin’s q
q summarizes all information about the future that is relevant to a firm’s investment
decision: it shows how an additional unit of capital affects the present value of
profits. Thus the firm wants to increase (reduce) its capital stock if q is high (low)
⇒ q is a sufficient statistic.
q has an economic interpretation: an extra unit of capital increases the present value
of the firm’s profits by q, and thus raises the value of the firm by q.
⇒ q is the market value of a unit of capital.
Since the price of capital is 1, q is also the ratio of the market value of capital to its
replacement cost. This ratio is known as Tobin’s q:
(Tobin' s )
q =
market value of capital
replacemen t cost of capital
Tobin’s q and the behavior of the firm
When
q > 1 ⇒ firm invests
q < 1 ⇒ firm disinvests
This q is the average q.
But our analysis implies that what is relevant to investment is marginal q. That is:
marginal q =
market value of a marginal unit of capital
replacement cost of capital
How are marginal q and average q related?
Marginal q equals average q under two assumptions (Hayashi, 1982):
constant returns in the adjustment cost function.
constant returns in the production function.
The first assumption does not hold in our model. We implicitly assume diminishing
returns to scale in adjustment costs because we assume that adjustment costs depend
•
•
only on k , i.e. C( k ).
For example, it is more costly for a firm with 20 units of capital to add 2 more, than
it is for a firm with 10 units to add 1 more ⇒ Marginal q is less than average q.
 • 
k
C 
If the adjustment cost function was of the form  K  , then marginal q would equal
 
average q.
Model dynamics: the capital stock
The two variables we will focus on are the aggregate quantity of capital K (state
variable) and its value q (jump variable). The quantity of capital is something that the
industry inherits from the past (given), but its price can adjust freely.
Recall the first-order condition: 1 + C’(I) = q
Since C’(I) is increasing in I, this condition implies that I is increasing in q.
Since C’(0) = 0 ⇒ q = 1 for C’(0) = 0
We assumed that there are N identical firms. So, since q is the same for all firms at
equilibrium, all firms choose the same value of I:
•
K = N⋅ I
with I satisfying 1 + C’(I) = q
Putting this information together we can write:
K (t ) = f (q (t )),
f (1) = 0
f ′(⋅) 〉 0
−1
where f (q (t )) = N C ′ (q − 1) , which comes from:
C ′(I ) = q − 1 ⇒ C ′−1 C ′(I ) = C ′−1 (q − 1) ⇒ I = C ′−1 (q − 1)
When q = 1 (i.e. when market value of a marginal unit of capital equals the
replacement cost) there is no investment. Respectively K increases when q exceeds
unit and decreases when it is less than a unit.
•
q=1⇒ K =0
•
q>1⇒ K >0
•
q<1⇒ K <0
(4)
Graphical analysis of capital stock dynamics
q
•
K >0
•
K =0
1
•
K <0
K
Model dynamics: shadow price of capital q
•
Rewriting the second first-order condition as an equation for q yields:
•
•
q (t ) = r q (t ) − π (K (t ) ) ⇒
∂ q (t )
>0
∂ K (t )
(5)
Π (K (t )(t ))
This expression implies: q(t ) = 0 ⇔ q (t ) =
r
•
•
Since Π’(K) < 0, the set of points satisfying q(t ) = 0 is downward-sloping in (K, q).
•
Also, q (t ) is increasing in K:
•
•
q (t ) > 0 to the right of q(t ) = 0
•
•
q (t ) < 0 to the left of q(t ) = 0
Graphical analysis of q dynamics
q
•
q>0
•
q <0
•
q=0
K
The phase diagram
The phase diagram combines the information from the previous two figures by
showing how K and q must behave to satisfy (4) and (5) at every point in time given
the initial value of K. According to this combined picture there is a saddle path leading
to equilibrium. Outside equilibrium for a given level of capital stock K there is a
unique level of q that produces a stable path. Specifically, there is a unique level of q
such that K and q converge to the point where they are stable (point E in the diagram).
q
E
K = 0
q=1
q = 0
K
K
Assume, for example, that capital is equal to K1. Only the value q1 > 1 will bring the
economy back to equilibrium. For this value it is profitable to invest and capital
accumulates. But as capital accumulates, and because of the presence of adjustment
costs, it becomes for the firm less and less profitable to invest. Thus, adjustment cost is
a mechanism that stops investment of being infinite and we return to equilibrium
following the unique saddle path.
q
q1 〉 1
E
K = 0
q =1
q = 0
K1
K
K
Let’s take an example of a point in the diagram that is not equilibrium, like point A.
•
Since q > 1, firms invest and increase their capital stocks, thus K > 0 .
Since K increases, profits decrease and this can only be sustained if q increases more,
•
i.e. q > 0 . Thus, both K and q move up and to the right in the diagram.
Similarly, if q < 1, firms eventually move into the region where both K and q are
falling and remain there.
q
A
E
K = 0
q =1
q = 0
K
K
Summary of equilibrium
The long-run equilibrium, point E, is characterized by q = 1, which implies that
•
K = 0 , and q = 0 .
The fact that q = 1 means that the market value of a marginal unit of capital is equal
to the replacement cost, thus the firms have no incentive to increase or decrease
their capital stocks.
From equation (3), for q = 0 when q = 1, the marginal revenue product of capital
must equal the foregone interest rate r.
The Effects of Output Movements
An increase in aggregate output raises the demand for industry’s product, and thus
raises profits for a given capital stock.
Thus the natural way to model an increase in aggregate output is an upward shift of the
π(K) function.
We consider two kinds of shifts:
- a permanent shift (like an improvement in technology)
- a transitory shift (like good weather conditions)
Permanent shift in output
Assume that the industry is initially in long-run equilibrium (point E in the following
figure) and that there is an unanticipated, permanent upward shift of the π (K ) function.
The horizontal line remains unaffected (q = 1) but the q = 0 locus shifts up.
q
q′
E′
E
q=1
q ′ = 0
q = 0
K
K = 0
K
K
Implications of a permanent shift in output
The value of installed capital jumps immediately to the point on the new saddle path
for the given capital stock. (As firms expect rise in their future profits, value of K
capital increases in q’ ). Since q’ > 1, it is profitable for the firms to invest and capital
accumulates. But as capital accumulates profits will decrease because of the
adjustment costs and it becomes less and less profitable to invest. Due to this
procedure K and q move down that (along the new saddle) path to the new long-run
equilibrium point, E’.
⇒ The economy ends up with higher capital stock K but again q equals 1. Thus:
A permanent increase in output leads to a temporary increase in investment,
which leaves the economy with a higher capital stock.
Transitory shift in output
Now consider an increase in output that is known to be temporary:
• The economy is in the long-run equilibrium.
• There is an unexpected upward shift of the profit function.
• When this happens, it is known that the function will return to its initial position at
some later time T.
The key insight needed to find the effects of this change is that there cannot be an
anticipated jump in q.
We know that at time T, both K and q must be on the saddle path leading back to the
initial long-run equilibrium. At the time of the shock the marginal value of capital q
rises but not that much as in the permanent shift case.
⇒ q jumps from point E to point A in the following figure.
q
q′
A
E′
E
q =1
B
K = 0
K 〈 0
q′ = 0
q = 0
K
K
K
q and K then move gradually to point B, arriving there at time T. Finally, the economy
then moves up the old saddle path to E.
Implications of a transitory shift in output
q rises less than it does if the increase in output is permanent since q determines
investment, investment responds less.
the path of K and q crosses the K = 0 line before it reaches the old saddle path, that
is before time T ⇒ the capital stock begins to decline before output returns to
normal.
The comparison of permanent and temporary output movements shows that investment
is higher when output is expected to be higher in the future than when it is not. Thus:
Expectations of high output in the future can raise current investment.