F . Verschueren Economics Letters 69 2000 289 –297
291
≠Y
t
] H 5 p
2 w 5 0 5
L t
t
≠L
t
≠Y
t
]] l 2 l
5 r l
2 p 2
dl 6
H J
t t 21
t 21 t
t
≠K
t 21
with the transversality condition
i
lim b l K 5 0
7
t t
t →
`
3. The empirical specifications
3.1. Present value model and co-integration We now give up for the moment the optimization context of Section 2 and assume that a particular
variable V and a particular profit P are linked in a present value model, so that V is generated by
t t
t `
i
V 5
O
b E P 8
f g
t t
t 1i i 50
with b and E . having the same definition as above. Campbell and Shiller 1987 have shown that
f g
t
8 can be attractively transformed into the following three equations
21
C 5 V 2 b
V 2
P 9
h j
t t
t 21 t 21
C 5 V 2 E V
10 f g
t t
t 21 t
21
S 5 V 2 1 2 b P
11
t t
t
In 9 C links contemporaneous and lagged observations of V and P . By combining 9 and 8 a
t t
t
crucial economic interpretation of this variable C is obtained in 10, since it is revealed to be the
t
expectation error from predicting V at time t 2 1. When expectations are naive, C 5 DV while in the
t t
t
perfect expectation case C 5 0. Alternatively 8 can be expressed as 11, showing how the value
t
and the profit are linked in the long run. A crucial property of the variable S emerges when it is
t
related to C . Indeed using 9, Eq. 11 can be reformulated in a dynamic way as
t 21
21 21
S 5 1 2 b DV 2 1 2 b DP 2 b1 2 b C
12
t t
t t
The latter expression indicates that 11 is the candidate co-integration equation, with S the long run
t
residual. Indeed when both P and V are I1 then from 10, C is a martingale difference which is
t t
t
I0, and which implies from 12 that the residual S is also I0, and therefore that value and profit
t 21
21
are co-integrated of order 1,1 in 11 with long-run coefficient 1 2 b
5 r 1 1 r . 1.
292 F
. Verschueren Economics Letters 69 2000 289 –297
3.2. Implication for optimal factors demand Now we study the implications of the co-integration hypothesis of 11 when this equation is
related to the theoretical model discussed in section 2. We denote V as the value associated to an
t
optimal choice of factors I , K , L and
P as the profit constructed as in 2. When these two
t t 21
t t
variables are both I1 and when they are co-integrated of order 1, 1, then with respect to results
i i
from Section 3.1 we have that P 5 P 5 PI , K
, L and V 5 max ob E P
5 ob E P
. f
g f
g
t t
t t 21
t t
t t 1i
t t 1i
In other words, co-integration implies that factors are chosen optimally at each period , since
constructed profit fits its optimal level given by the economic theory. In the neo-classical point of view with adjustment technology as described above, using 4, 5,
6 and 3, summing up from 0 to infinity, discounting and using 7, the value of the firm is checked to be V 5 1 1 r
l K . To give an observable equivalence to the multiplier
l we select the
t t
t 21 t
simplest quadratic form for the installation function defining hence ≠G ≠I
2
I a
t
] ]] G 5
13
t
2 K
t 21
with a . 0. Now using 13 and 4 we are able to rewrite the value at optimum path as
V 5 1 1 rK 1
aI 14
t t 21
t
The idea is thus to put V and
P together. When there are no adjustment costs a 5 0, 2, 11 and
t t
14 quickly leads to 1
] K
5 P 1 S
15
t 21 t
t
r So constant interest rate plays the role of long run coefficient. However when
a . 0 equations such as 15 cannot be deduced since the adjustment parameter enters the left side of the specification.
3.3. Implications for long run investment ]
In a first stage interest rate is assumed to be given and is denoted r. The sensitivity of investment to its determinants is expected to depend on the structural parameters which characterize the economy in
which firms behave. In that line when adjusting the stock of capital is costless a 5 0 we have to rely
on a parametric production technology. We chose the Cobb–Douglas functional
u 12
u
Y 5 AK L
16
t t 21
t
with u the capital–output elasticity and A a constant. Using 2, 5 and 16, when factors are chosen
optimally, profit is commonly defined as P 5 up Y 2 I , and the value–profit relation 11 can be
t t
t t
reformulated to obtain ˜I 5up Y 1 S
17
t t
t t
] ˜
with I 5 I 1rK , or using 3
t t
t 21
] K 2 1 2
d 2r K 5
up Y 1 S 18
h j
t t 21
t t
t
F . Verschueren Economics Letters 69 2000 289 –297
293
The key difference between the latter specification and the long run one originally proposed by Jorgenson 1963 is that 18 links the transformed change of capital stock to the level of output,
and not the level of both variables. In fact the latter equation may be viewed as a particular case of the former, but expressed in an attractive way to investigate co-integration assumption. When DK 5 0 the
t
] usual user cost proxy relative to price investment appears since s 5r 1
d and dividing each side of
t
18 by this constant term leads to the model of Jorgenson. Turning to the more realist case that a . 0
adjustment costs, Eqs. 2, 11 and 14 suggest that investment has the following long run representation
1
K
]] I 5
P 1 S 19
]
t t
t
1 1r a
K
] with
P 5 p Y 2 w L 2rK .
t t
t t
t t 21
Consistently with economic theory, investment is inversely related to interest rate and to the importance of installation burden. But more interestingly, the long run equation identifies structural
parameter a, so that a dynamic equation linking successive levels of investment is not required as is
1
usual in the literature . Furthermore it is important to note that when a 5 0 the latter specification
K
becomes I 5 P 1 S , implying a unit long run parameter in 19. When a parametric Cobb Douglas
t t
t
production function 16 is introduced as before, it is possible to identify both structural parameters a
and u since the above procedure gives
] u
r ]]
]] I 5
p Y 2 K
1 S 20
] ]
t t
t t 21
t
1 1r a
1 1r a
When r is to be estimated together with a a slight modification of 20 allows for identification of
these parameters since 1
r ]]
]] I 5
p Y 2 w L 2 K
1 S 21
t t
t t
t t 21
t
1 1 r a
1 1 r a
4. Data, methodology and results
