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3.2.2.2(a). Static logit model

Considine and Mount (1984) introduced more elaborate linear logit models for a system of cost share equations, which are designed to satisfy three theoretical properties of the input-demand function derived from the producer‟s cost minimization assumption. The properties are (i) non-negativity of inputs, (ii) zero-degree homogeneous function in prices, and (iii) negative own-price along with symmetric cross-price effects. The non-negativity is automatically ensured by adopting exponential functions for cost shares. The other properties are guaranteed by imposing additional conditions. Firstly, a static model will be examined in the following section. A static linear logit model of cost share is given by

∑

where is the share of input cost in total cost, and are the price and quantity of the

input respectively, is the total cost, and is a function of all input prices, , and the level of output, , which is as follows

∑

where , , , and are unknown parameters; are random-error terms. The share elasticities with respect to prices and output under non-restriction are derived as

∑

∑

Using Shephard‟s Lemma and the share elasticities equation (3.12) and (3.13), the price elasticity for the input and the cross-price elasticities are derived as

The linear logit model of cost share satisfies the second property, zero-degree homogenous function in prices which implies that the sum of the N price elasticities should be zero. Given the equations (3.14) and (3.15), this property is summarized in the following equation which means that the sum of the share elasticities should be zero;

∑

∑

∑

Symmetry of the cross-price effects in the third property implies that

To satisfy these above equations, two constraints are imposed in the model. The first one is given by

∑

where is an arbitrary constant. Another constraint imposed is as follows:

The price coefficients are redefined as

where is defined as the predicted shares for each observation to deal with an endogeneity problem in the stage of estimation. See the estimation part below for further information.

Then, using the redefined equation (3.20) and imposing constraints (3.18) and (3.19), the linear logit function (3.11) can be rewritten as:

( ) ( ) ( )

where

( ∑

∑

)

Adding an error term, into each and using logarithms for the share equation, (3.10) can be written as a linearized form for estimation:

( )

∑ ( ) * ∑ ∑

+ ( )

∑ ( )

where

∑

³⁰

Three identifying restrictions, , are required to identify the remaining coefficients for estimating the system of equations. These normalizing constraints do not affect the estimates of the elasticities. Finally, using the symmetry condition ( and homogeneity condition ( ∑ the cross-price elasticities are derived as

and the own-price elasticities are as follows:

As for estimation methods, either the iterative Zellner‟s seemingly unrelated regression estimation or the full information maximum likelihood estimation can be implemented.

30 The terms are simply the predicted logarithmic share ratios which are obtained via estimating the equation (3.24).

3.2.3.2 (b) A dynamic linear logit model

The above static linear logit model assumes that demand converges to long-run equilibrium immediately. However, the factor demand would be not flexible due to various constraints such as fixed factors for production, technological elements, and producers‟

rational expectations meaning that they respond to expected prices. Therefore, demands for input factors respond to price changes gradually. For those reasons, Considine and Mount (1984) applied Tradeway‟s (1971) optimal path to adjustment process in the linear logit model, and called it the dynamic logit model. The optimal path equation is given by

̇ [ ]

where is the stationary equilibrium, and is an matrix with elements that are functions of and the discount rate, r.

Substituting , where the asterisk denotes the equilibrium level of the corresponding variable, into a logarithmic approximation of equation (3.27) gives a dynamic form of the linear logit model. Therefore, the quantity change can be written as

∑ ( )

where is an element of the matrix in the equation (3.27), and is the element of in the equation (3.27). Given the equation (3.28), the quantity ratio can be derived as

(

) ∑( )( )

Imposing conditions that for and for all in the equation (3.29) yields the following cost share ratio:

(

) (

) (

) (

) (

) Using a linear logit model corresponding to the equation (3.10) and (3.11), the equilibrium share ratio can be defined as:

(

) ∑( )

where , , and are the parameters in the equilibrium state. Substituting the equation (3.31) and the equation (3.30) yields the following equations

(

)

∑( )

(

) (

)

The equations (3.32) are for estimation purposes, which corresponds to the following linear logit model for the share

∑

with

∑

The dynamic model is distinguished by introducing a time subscript and the lagged quantity with partial adjustment coefficient, . The short-run price elasticities are same as in the static model and the long-run price elasticities can be computed as

where is the short-rum elasticity given in the equations (3.25) and (3.26). If the partial coefficient, is greater than 0, it guarantees the Le Chatelier principle that the short-run price elasticities can never be greater than the long-run price elasticities.

As for estimation, a two-step iterative Zellner estimation is employed to estimate a system of implicit share equations. Note that the iterative Zellner estimation method provides the parameters which are invariant to the selection of the base input. Besides, the two-step iterative procedure handles an endogeneity problem of the cost shares variables. In the first step, using actual data for the endogenous variables on the right side of the equations (3.23), estimate parameters in the equations (3.23) and compute predicted shares given by the equation (3.24). In the second step, these initial predicted shares from the first stage are used to re-estimate parameters in the equations and generate the predicted shares. Then, the predicted shares instead of the actual data are re-entered into the endogenous variables in the first step. This iteration is repeated until the convergence criteria for the parameter changes between two steps, which is set at less than 0.1%, is reached.

Overall, the logit model outperforms the translog model when the two models‟

estimation results are compared. According to Urga and Walters (2003)‟s work, the static

translog model has parameters that violate concavity conditions of the cost function.

Furthermore, the residual of the model is serially correlated and has non-normal errors.

Another study, Kim (2006), employing the translog model for analysing Korean industrial sectors‟ energy demand, observed that the translog model often yields non-negative price elasticities of energy demand. On the other hand, the logit model has superiorities; the model satisfies the non-negativity of the input share, and shows reasonable parameters for own- and cross-price elasticities. In the case of the dynamic logit model, the model guarantees the Le Chatelier principle by giving short-run elasticities that are less than their long-run counterparts. For those reasons, we adopt the dynamic logit model among inter-fuel substitution models for the energy demand block.