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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
In this section the relation between capacity structure and productivity growth is derived. With V fixed the proportional rate of change in labor productivity is identical to the proportional rate of
change in aggregate production. So the percentage change in productivity growth is by definition 2
2 2
ˆ ~
F 5F 2F 3.1
where one hat indicates the logarithmic differential operator w.r.t. t, and two hats the repeated application of that operator. The growth rate of output is positive by the assumptions made, so the
sign of the first term will be sufficient, although not necessary, to tell whether productivity growth slows down or not. Define the elasticity of the capacity density at input coefficient j as
c9j j ]]]
ej 8 .
3.2 cj
The first term on the RHS of 3.1 can be related to wage changes. By 2.5 we have 2
2 X
~
S D
ˆ ]
ˆ F 5 w 1 1 2
5 w,
R since X R is constant. From the definition of capacity flow in 2.5 it follows that
2 c9RR
ˆ ˆ
ˆ ~
ˆ ]]]
w 5 R 1 R 1 R 2X.
cR 2
2 ˆ
ˆ ˆ
ˆ ˆ
~ ˆ
By 2.10, R 5 2 w and noting that R 5R 1 R and, since X R is constant, R 5X. Finally, we arrive at 2
2 ~
ˆ ˆ
F 5 2 1 1 eRw 1w. 3.3
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. Lindh Economics Letters 69 2000 225 –233
3
Wage changes can be shown to be proportional to changes in u, the technique factor, and so from 3.3 it follows that productivity will decelerate for certain values of eR even if wages, or
equivalently technical change, accelerate. 2
2 ˆˆ
ˆ
Proposition 1. u ,F and eR 2 1 implies F 0. Proof. By inspection of 3.1 and 3.3.
h Thus, even as technical change accelerates, productivity may decelerate. A short run schedule of
supply, that is concave at the margin, would be even more sufficient, since
Proposition 2. The price schedule of supply is concave at the margin if and only if eR 0. Proof. Let C j be production as a function of the highest input coefficient used. Then
C 9j 5 cj and C 0j 5 c9j . Invert C j to obtain the price schedule of supply jC in terms of labor units. It follows that
ej 1
C 0 ]
]] ]]]
j 9C 5 and j 0C 5 2
5 2 ,
3 2
C 9 C 9
jcj proving this assertion for all open sets where C 9 j ± 0. Due to our assumption of a connected
support of c it therefore holds for X j R. h
4
Note that in Fig. 1 the bars depict a reversed supply schedule, which when smoothed will be concave at the marginal unit.
The elasticity eR is positive whenever cR is increasing, e.g. if past investment has been insufficient to raise capacity flow at the same rate as input coefficients decreased. In steady state with
positive growth, eR 5 2 2 everywhere Pomansky and Trofimov, 1990. It can be shown that the steady-state structure is actually a boundary case, since if eR is only
slightly greater than 22 and the capacity distribution otherwise has the steady-state elasticity, except for a small neighborhood of R, there could be deceleration even for a constant rate of technical
change. Generalizations of the assumptions of course modify these conclusions, but the essential thrust of
5
the argument is preserved. The effects of some generalizations are summarized below. An elasticity of substitution in the ex ante production function below unity would tend to slow
down productivity growth. Intuitively the labor restriction is restricting growth more seriously when labor is harder to substitute for.
A decreasing ratio WD will make a slowdown likely when eR . 2 1, because the difference in
3
From the first expression in the first-order conditions 2.9 and the assumption of fixed expectations it follows that ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
w 2 k 5 0 and by direct logarithmic differentiation w 5 1 2 ak 1 av 1 u. From the last expression w 5 k 2 v, so it is clear ˆ
ˆ that aw 5 u.
4
Accumulating supply from the least efficient to most efficient units instead of the other way around.
5
Proofs are in Lindh 1992 and will be made available on request.
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231
growth rates in W and D is then negative and the relative gap in input coefficients will decrease. This would be consistent with increasing scrapping rates and a relatively abundant supply of labor at the
same time as quasi-rents diminish. This could be the case in the later stages of a slowdown due to scarcity of scrapping and would hence tend to prolong the time to recovery.
Probably, growth in labor supply will accelerate labor productivity growth, but there are also mechanisms working in the other direction so the contribution cannot in general be determined.
4. Empirical relevance
