FIG. 1. Sources of change in the auto industry. Shows plots of emission standards and gas prices against time.

may discover that the reasons for the problems in the labor and capital equations require us also to modify the materials equation, so we continue to explore other approaches in our on-going research.

The materials demand equation that we estimate for automobile model j produced at plant p in time period t has several components. In our companion paper we discuss alternative specifications for these components, but here we only provide some intuition for the simple functional form that we use.

Since we are concerned that because labor and capital may be subject to long-term adjustment processes in this industry and a static cost minimizing assumption for them might be inappropriate, we consider a production function that is conditional on an arbitrary index of labor and capital. This index, which may differ with both product characteristics, to be denoted by x, and with time, or t, will be denoted by G(L, K, x, t). Given this index, production is assumed to be a fixed coefficient times materials use.

The demand for materials, M, is then a constant coefficient times output. That coefficient, to be denoted by c(xj, εpt, β), is a function of: product characteristics (xj), a plant-specific productivity disturbance pt), and a vector of parameters to be estimated (β). In this paper, we consider only linear input-output coefficients, i.e.,

c=xjβ+εp.

[1]

Finally, we allow for a proportional time-specific productivity shock, δt. This term captures changes in underlying technology and, possibly, in the regulatory environment. (In more complicated specifications it can also capture changes in input prices that result in input substitution.) The production function is then

[2]

Then, the demand for materials that arises from the variable cost of producing product j at plant p at time t is

Mjpttc(xj, εpt, β)QJpt.

[3]

While we assume that average variable costs are constant (i.e., that the variable portion of input demand is linear in output), we do allow for increasing returns via a fixed component of cost. We denote the fixed materials requirement as µ. There may also be some fixed cost to producing more that one product at a plant. Specifically, let there be a set-up cost of ∆ for each product produced at a plant; we might think of this as a model change-over cost.c Let J(p) be the set of models produced by plant p and Jp be the number of them. Then total factor usage is given by

[4]

with Mjpt as defined in Eq. 3.

If we divide Eq. 4 through by plant output and rearrange, we obtain the equation we take to data

[5]

where is the weighted average

[6]

Except for the proportional time-dummies, δ, Eq. 5 could be estimated by ordinary least squares (under appropriate assumptions on ε).d With the proportional δ, the equation is still easy to estimate by non-linear least squares.

c  

From visits to assembly plants, we have learned that a fairly wide variety of products can be produced in a single assembly without large apparent costs. Therefore, we would not be surprised to find a small model changeover cost, particularly in materials.

d  

In the empirical work, we also experimented with linear time dummies and did not find much difference

e  

For example, firm headquarters could allocate production to plants before they learn the plant/time productivity shock ε. This assumption is particularly unconvincing if the εs are, as seems likely, serially correlated. Possible instruments for the right-hand-side variables include the unweighted average xs and interactions between product characteristics and macro-economic variables. The use of instruments becomes even more relevant once the possibility of increasing returns introduces a more direct effect of output.

f  

In particular, we do not examine the extent to which vertical integration differs among plants, and we learned from our plant visits that there are differences in the extent to which processes like stamping and wire system assembly are done in different assembly plants. Unfortunately we do not have information on the prices that guide these substitution decisions.



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