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. Lindh Economics Letters 69 2000 225 –233
slowdown still remains largely a puzzle and none of the possible causes has been generally accepted to explain more than minor parts of the slowdown. Recently, the age of the capital stock has received
renewed attention as part of the explanation; see Wolff 1996. The work of Davis and Haltiwanger 1992 and their followers on gross job flows has also revived interest in putty-clay models, since the
actual gross flows observed in the economy are hard to reconcile with conventional aggregate production functions.
The point made in this paper is that the form of the capacity distribution in itself influences how technical change translates into productivity growth, and may temporarily make the relation negative.
A putty-clay model, where capital can be substituted for labor only prior to the actual investment, provides an analytical tool for the hypothesis that scarce capital to scrap may cause a productivity
slowdown. The aim here is, however, only to make a precise and simple statement of a mechanism that seems to be generally overlooked in the literature.
The basic idea is a variation on the old theme of echo effects first put forward by Karl Marx. Einarsen 1938 contains much of the early history. Salter 1960, 1965 discussed the importance of
the capacity structure extensively in a vintage context. Two contributions by Benhabib and Rustichini 1991, 1993 investigates echo effects both theoretically and empirically with models where capital
equipment is subject to fixed depreciation patterns. The model used here entails endogenous obsolescence driven by exogenous technical change in a Johansen 1972-type short-run macro
production function with a simple one variable factor version as formulated by Pomansky and Trofimov 1990. Throughout this paper, productivity refers to labor productivity since a consistent
capital aggregate independent of labor cannot be constructed in this type of model.
The essential mechanism of the model is illustrated in Fig. 1. The decrease in productivity growth is due to scarcity of old equipment, which cannot cover wage costs, and therefore only a small amount
of labor is freed to be transferred to new equipment. By raising wages more than best-practice productivity increases, more labor could be released, but that would not be the optimal thing to do,
since capital costs could not then be covered.
The paper is organized as follows. To save space, only a very simple version of the model is
1
presented. Section 2 develops the formal model of a putty-clay production structure. Section 3 treats the dynamics of the model, and Section 4 briefly discusses its empirical relevance. Section 5
summarizes the argument.
2. A putty-clay model
One homogeneous good is produced by labor, using capital equipment, heterogeneous over time but ´
homogeneous at the moment of installation. The good and numeraire can either be immediately consumed or frozen into capital equipment which is impossible to recover for consumption.
Let j denote labor input coefficients and c j the continuously differentiable density of the distribution of production capacity over input coefficients. Assume that all available labor, V, is
allocated to maximize production. Then no production unit with higher input coefficient will be used before capacity with lower input coefficients is fully utilized.
1
A more elaborate treatment investigating variations in the simplifying assumptions made here is available Lindh, 1992.
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227
Fig. 1. Labor input coefficients are measured on the vertical axis and accumulated output on the horizontal axis. Total labor is measured by the area of the bars. Each year the input requirements in best-practice technology decrease with unity i.e. productivity growth accelerates
from 0.33 to 0.5. Given that only units with the highest input coefficients are scrapped, the area of the new bar must equal that of the scrapped units. Hence average productivity growth decelerates from 0.4 to 0.21 while productivity on the scrapping margin equal to
wages accelerates from 0.2 to 0.25.
Both the input coefficient and the capacity corresponding to a given production unit are fixed. To avoid dealing with changes in the interior of the support of the distribution I assume that all new
production units will use a marginally lower input coefficient than any existing. Hence
RV,t
FV,t 5
E
cj dj, 2.1
Xt
where Xt denotes the minimal input coefficient in existing production equipment and RV,t the maximal actually used at the current time t. Furthermore,
RV,t
Vt 5
E
jcj dj. 2.2
Xt
Differentiate 2.1 and 2.2 w.r.t. V, subindexes denote partial derivatives: 1
] F 5 cRR and 1 5 RcRR
⇒ F 5
. 2.3
V V
V V
R
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. Lindh Economics Letters 69 2000 225 –233
Aggregate marginal productivity equals average productivity in the marginal production unit and, thus, the wage rate, a standard result for this type of model.
Differentiate 2.1 and 2.2 w.r.t. t. A dot over a variable denotes total time differentiation: ~
~ ~
F 1 F V 5 2 cX X 1 cRR 1 R V ,
t V
t V
~ ~
~ V 5 2 XcX X 1 RcRR 1 R V .
2.4
t V
From 2.3 it follows that all terms involving the time derivative of V will cancel out in both equations. By definition, the flow of new output capacity at any given point in time is
R X
t
~ ]
S
]
D
wt8 2 cXX 5 2 RcR ⇒
F 5 w 1 2 2.5
t
X R
by using 2.4. To simplify, V is henceforth assumed to be constant and equal to unity, and there is, therefore, no
need to distinguish the total and partial time derivatives of F and R. Note, however, that this assumption is not needed for 2.5 to hold.
The capacity flow w and the minimal input coefficient X are determined by decisions of a representative firm. The firm chooses investment, k, and labor input flow, v, in order to maximize the
present value of its future profit flow from the investment. The ex ante production function
12a a
fk,v,t 5 utk v
2.6 is a constant returns to scale Cobb–Douglas function. Let r be the expected path of the interest rate
and wv the expected path of wage increases. The firm takes w, current wages, as given and solves the problem
t 1l
z
2
E
rxdx
max
E
[ fk,v 2 wvzv]e dz 2 k subject to f wvzv for t z t 1 l,
2.7
t
k,v t
where l is the expected economic life of the investment. Expectations of strictly rising wages will then imply a finite l and also an expected period of usage that is connected.
The present value function may be convex in k and v due to the dependency of a consistent expectation of l on the capital labor ratio chosen, so a unique maximum cannot be guaranteed in
2
general. Here it is assumed that f and expectations have the properties needed for uniqueness. To simplify further, write
t 1l t 1l
z z
2
E
rxdx 2
E
rxdx
D 5
E
e dz and W 5
E
vze dz
2.8
t t
t t
and assume these expectation-dependent variables can be treated as fixed see Section 4 for a
2
Bliss 1968 gives a condition for uniqueness from which can be inferred that the set of production functions is non-empty and includes the Cobb–Douglas case for empirically reasonable parameters. For a proof in this case and a further
discussion of expectations, see Lindh 1992.
T . Lindh Economics Letters 69 2000 225 –233
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justification of this drastic assumption. The first-order conditions for maximization can then be written conveniently as
f f
1 W
a k
] ]
] ]
]] ] 1 2 a 5
and a 5 w
⇒ 5 wW.
2.9 k
D v
D 1 2 a v
The firm also has to decide whether to continue or discontinue operations on existing units. Since there is no alternative use of equipment and the capacity density is continuous, the marginal unit can
exactly cover its costs, i.e. 1
] w 5
5 Fv. 2.10
R Under these assumptions it turns out that the ratio of X to R will be constant, since 2.9 and 2.10
ensure this ratio is proportional to DW. Hence the change in output only varies with the flow of new capacity.
3. Dynamics of the model
