fibre content), Z),, was added to (6.4) and (6.5) to identify whether these variables affect
total fibre cost and fibre shares.^ This gives us the following static system of equations describing firms' demand behaviour for fibres:
]nC„=a, + ]n y,, + a. I n / ) + ^ a , , [ I n / p^,,)]'
In D,,, + X2 In +13 In D,,^ + (6.7) 3ht ' t^sht
where D j D2 and D j are respectively the share of pure chemical fibre products, the share of pure wool products and the share of blended products in total output.
Dynamic specification and alternative cost and share model
The translog cost function and associated cost share equations considered up to this point have been stadc in the sense that cost and share variables are taken to be a function of a set of explanatory variables observed at the same point in time, implying that the pattern of demand adjusts to changes in prices instantaneously. However, there are good reasons for believing that economic behaviour often involves a time lag during which costs and shares are adjusted to their optimal levels. For instance, a firm which finds that its current fibre shares are inconsistent with the long-run equilibrium implied by current relative fibre prices will generally spread the planned adjustment to long-run equiUbrium over a period of time. If that is the case, our cost model and share equations should incorporate this adjustment process, especially when having panel data at hand.
There are a number of ways to model the lagged adjustment behaviour of firms, including the partial adjustment model. Error Correction Model (ECM) and autoregressive model. ECM is a more generalised version of the partial adjustment model. Although empirical evidence and economic interpretation are more in favour of ECM than the restricted partial adjustment hypothesis (Maddala 1992), there is a trade- off between the use of a more desirable hypothesis and the loss of scarce observations. In
2 The inclusion of these proxy dummies can also be considered a way of capturing the so-called "fabric substitution effect" (Blau 1946), which basically relates to the indifference maps of the fmal consumers of fibre products. Ferguson and Polasek (1962) argued that fabric substitution caused, for example, by a shift in consumer preferences at a given level of prices in favour of a fabric containing less wool and more synthetic fibres, would also influence mill consumption patterns. It is therefore important to
the case of the autoregressive model, the hypothesis was rejected by a pre-test with our data. The partial adjustment model was, therefore, selected.
The partial adjustment hypothesis has often been used in applied economics to describe firms' optimal behaviour in the face of an adjustment cost. Suppose that there is a desired level of some variable that an agent would like to achieve at time t, denoted as:
+ + e , (6.8)
However, because of friction, delays, cost of adjustment and so forth, the desired level of J cannot be achieved within a single period. Starting from the previously existing level the change required to attain the desired level is but the actual change y r y , - i is only a fraction of this. Assuming that the proportion achieved is ( 1 - ^ ) , where 0 < } i < l , the partial adjustment hypothesis can be written as:
or J', + (6.9)
Small values of \ imply relatively quick adjustment, and if adjustment is complete, not partial, in a single period. Larger values of X imply that the past value of the variables concerned exerts a greater influence, and if ?i=l, nothing changes.
Applying the above partial adjustment hypothesis to (6.7) leads to the following dynamic system of cost function and share equation to be estimated:
In C,, = (1 - X){a, + In J , , ) + (1 - I n / p^,,) + (1 - a , , [In / )]' +(1 - In D,,, + (1 - X)x, In D,,, + (1 - b D,,, + X In + (1 - X) = (1 - + (1 - In (p^,, / p^,,) + (1 - In D,,,
+(1 - X)(t)2 In + (1 - In D3,, + ^ In + (1 - (6.10)
It should be pointed out that since the adjustment speed for cost and share needs to be consistent, X should be restricted so that it is equal across the two equations in (6.10). To determine whether a static or dynamic cost model is to be used, a simple t test or a log likehhood ratio test can be performed.
It is often argued that the partial adjustment hypothesis is rather ad hoc, but it can be justified in terms of a cost minimisation procedure. Ball et al. (1989) offered a more rigorous derivation of the partial adjustment model toward the translog specification, leading to basically the same result as (6.10). The authors started with a cost minimisation problem for a fixed level of a single output:^
min Q = p T p ^ + Subject to: Y, (6.11)
where C, and P^ are vectors of cost and fibre prices, F^ is a vector of fibre inputs and Y^ is a vector of output. The function g(FfF^_i) is intended to represent the cost of adjusting the level of inputs, while f(F^) represents the fu-m's production function. First-order conditions for a minimum are:
y^-f(FJ=0 (6.12) w h e r e a n d g; denote the partial derivatives of / a n d g with respect to F-. If a smooth,
increasing, strictly quasi-concave production function is assumed, then obviously the second-order sufficient conditions will always be satisfied. This ensures the existence of an optimal cost function:
C, = C,(P,y,F,F,_j) (6.13) Ball et al. (1989) select a multiplicative form of the cost function incorporating
the adjustment cost as it is consistent with the translog functional form. The partial adjustment model is written as:
Q = C/ (6.14) where G is the long-run optimum cost of production defined as C*=C*(P^, y j , X is si
partial adjustment parameter and Q is the cost of production given that no adjustment is made to the level of inputs (that is, Q = -^/^/.y)- It can be shown that this cost function