# Forecasting marginal costs of a multiple-output production technology

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Journal of Forecasting, Vol. 12, 421-436 (1993)

Forecasting Marginal Costs of a Multiple- output Production Technology

GEORGE M. LADY Temple University, Philadelphia, PA, U. S. A.

and

CARLISLE E. MOODY College of William and Mary, Willamsburg, VA, U.S.A.

ABSTRACT This paper presents the results of fitting a scaled translog restricted profit function to pseudo-data formed by repeated runs of a large linear programming model of domestic and international refining. The translog approximation is designed to estimate the marginal cost of producing eight petroleum products given the amounts of each product demanded and the price of crude oil. We test the model against out-of-sample data from the refinery model and historical data. The model is used in the US Department of Energys Annual Energy Outlook forecasting system.

KEY WORDS Pseudo-data Forecasting Petroleum prices Translog

The Energy Information Administration (EIA) prepares forecasts of domestic energy production, consumption, distribution, and prices which are published in the Annual Energy Outlook (AEO-EIA, 1991a). The forecasting methodology entails projecting a general equilibrium for domestic energy markets given the characteristics of the international energy markets and the domestic macroeconomy. An important component of the forecasting system is the Oil Marketing Module (OMM). This model forecasts the supply prices or refinery gate prices of domestically produced petroleum products, conditional upon the world price of crude oil and the amount of each product supplied. These prices are the marginal costs of production for each product which are then marked up to retail by the costs of distribution and marketing. In use, the OMM is iterated to convergence with product-demand models for each forecast year. Because of the amount of detail required, large linear programming models are often used to estimate the marginal costs of refining. Although the EIA has several refinery LP models available to it, they are too large and slow to be used with the other models in the AEO forecasting system.

In this paper we present the version of OMM being used by the Energy Information Administration. EIA specified that the model be small enough to incorporate into EIAs PC- based AEO forecasting system, retain a relationship to the technical detail of refinery 0277-6693/93/0SO421-16$13 .OO Received July 1991 0 1993 by John Wiley & Sons, Ltd. Revised November i992

422 Journal of Forecasting Vol. 12, Zss. No. 5

modelling, conform to the constraints implied by neoclassical production theory, and yield accurate forecasts of the refinery gate prices. Using the pioneering work of Griffin (1977a,b, 1978, 1979), we developed a model based on a scaled restricted translog profit function and its first partial derivatives. The equations are estimated on pseudo-data generated by repeated runs of a large LP refinery model known as the Oil Trade Model (OTM-DAC, 1989). The resulting model is much smaller and faster than the OTM model. The OTM model occupies 1.4 megabytes and solves in 10 minutes on a 386/25 PC. The OMM model based on pseudo- data occupies 74 kilobytes and solves in 3-4 seconds on the same computer. We test the models forecasting accuracy below against both historical data and out-of-sample pseudo- data. To our knowledge, this is the only translog approximation to a linear programming model in use as a forecasting model, although Finan and Griffin (1978) used a similar approach to estimate input-output coefficients for a macroeconometric model.

THE SCALED RESTRICTED TRANSLOG PROFIT FUNCTION

Let x be a vector of n outputs, z a vector of m inputs, and f (x, z) represent the multiple output production technology such that f (x, z) = 0 for feasible production points. For p the vector of output prices and w the vector of input prices, solutions to the linear programming representations of refining approximate solutions to the problem:

Select non-negative values for (x, z) such that, ?r = px - wz is a maximum, subject to f(x, z) = 0

Dual to the multiproduct production function f (x, z ) is the profit function, ?r = g(p, w), which gives the value of (maximum) profit found in solution to problem (1). Solutions to (1) are invariant with respect to absolute, but not relative, changes in input and output prices. As a result, the profit function is homogeneous of degree one in input and output prices. Given this, it can be shown that ax/ap; = X i and ar /aw , = - z,, respectively, for the ith output and the gth input; and further, a In r / d In p ; = p i x ; / r and a In r/a In w, = - w,z,/?r.

The linear programming representation of refining technology is homogeneous of degree one in all inputs since, for a given solution, if all constraints were changed by a given proportionate amount the solution values for outputs and variable inputs would change by the same proportionate amount. As a result, all model solutions must be constrained by bounds on facilities. In the short run such bounds are due to the assumption that existing capital is fixed. For the long-run solutions, facilities can be added, but only if it is economic to do so. Thus, for any long-run solution, capital is fixed at the margin in that no facilities can be added if product prices are too low. Since some factors are fixed, g ( p , w) is termed the restricted profit function and the technology represented by f (x, z) exhibits (almost) homogeneity of degree k < 1. In particular, if f(x, z) = 0, then f (Xkx, Xz) = 0. For the multiproduct production function almost homogeneous of degree 1 and k, it can be shown that

Revenue = px = (1/(1 - k ) ) r As a result, the restricted profit function is homogeneous of degree l / ( l - k ) in p and homogeneous of degree - k/(l - k) in w.

and Cost = wz = (k/(l - k ) ) x

The derivations in this and the next section are based upon Lau (1978).

G. M. Lady and C. E. Moody Forecasting Marginal Costs 423

APPROXIMATING THE SCALED RESTRICTED PROFIT FUNCTION

Griffin (1977a) first approximated the profit function from pseudo-data generated by a linear programming refining model using the translog functional form. However, this earlier effort differs in several important particulars from the problem being considered here. Griffin formulated the problem as one of revenue maximization constrained by fixed total costs and the multiproduct technology. Given solutions to this problem, Griffin scaled output prices such that profits equalled zero. Under these circumstances, the profit function corresponding to the multiproduct production function identifies the price possibility frontier for which profits are constant and equal to zero. The profit function was then submitted to a linear transformation by adding (constant) total costs. The first partial derivatives of the log of the transformed profit function to the logs of output and input prices are, respectively, the revenue share of each output and the negative cost share of each input. These share equations as approximated by the translog functional form are the basis for the coefficient estimation from the pseudo- data. Significantly, Griffin did not estimate the transformed profit function itself.

In the OMM model we must forecast the marginal costs in both absolute and relative terms. This made it impossible to scale output prices such that profits are zero. However, the share equations remain desirable and convenient as a basis for the estimation. As a result, we selected the restricted profit function scaled down to its (1 - k)th root as the function to be approximated by the translog functional form. For v = T ' - ~ , it can be shown that a In v/d In pi = pixi/Revenue and a In v / a In wg = - wgzg/Revenue.

The OMM model is also required to forecast the amount of petroleum products imported into the USA. Therefore, we must also make product imports endogenous to the model used to forecast product marginal costs. We did this by treating product imports as 'inputs' to the enterprise of satisfying domestic product demand. In solution, the refinery gate price and the (landed) import price are equal for each product category. Therefore, in the approximation of the restricted profit function with imports treated as inputs, the prices of the seven products that are imported appear in two roles, first as an output price and then as an input price corresponding to imports of the product. Because the activity of importing petroleum products is part of the process of satisfying domestic demand, but not part of the technology of refining crude oil, we assume that imports are separable from outputs in the multiproduct production function. As a result, import prices do not appear as arguments in the output share equations and output prices do not appear as arguments in the import share equations. The prices of all other inputs to refining except crude oil were held constant for all the OTM runs made to generate the pseudo-data. Accordingly, the restricted profit function to be estimated has sixteen arguments: eight supply prices (i.e. marginal costs) for output product categories (gasoline (GSH), LPG (LPG), jet fuel (JET), petrochemicals (PCH), distillate (DST), low- sulphur residual fuel (LSR), high-sulphur residual fuel (HSR), and other products (OTH)), seven prices for product imports as inputs, and the refinery's acquisition price for crude oil. For w , the price of crude oil, the translog approximation of the scaled restricted profit function is given as

15 l 8 8 ln(7r'-k)=ao+ C a i l n p i + - C C c ; j l n p i l n p j + C d i ,16 lnp i ln w i = l 2 i = l j = 1 i = 9

424 Journal of Forecasting Vol. 12, Zss. No. 5

The equations to be estimated are the first partial derivatives of equation (2). There are eight output share equations:

8

j= 1 (1 - k) a In ~ / a In pi = piXi/R = ai + C C i j In pj + di,16 In W , i = 1,2, ..., 8 (3)

where R is revenue, and seven input share equations,

There is a sixteenth equation for crude oil input to refining: 8

j=1 (1 - k) a In r/a In w = - w z l 6 / ~ = 616 + C dl6j In Ph-8 + d16.16 In w . ( 5 )

The characteristics of the restricted profit function constrain the values of the coefficients. The constraints are:

Symmetry:

c. . IJ - - Cji, e8h = ehg, dk.16 = dl6.k. T l - k is homogeneous of degree one in output prices:

T l - k is homogeneous in input prices: I5 15 16

g=9 h=9 k = 9 C ezh + C (dk.i6+ d i a , k ) = O .

The sum of output shares (= I ) is invariant to a change in any one price: 8 8 C C i j = 0 for each J , C di.16 = 0. i = l i=l

Also, the degree of homogeneity is 16

g = 9 k = - C bg.

The coefficients of equations (3)-(5) were estimated from the pseudo-data subject to the constraints (6)-(9). As discussed in the next section, equation (2) was not included in the estimation. Thus, the coefficient uo of equation (2) was not estimated directly. Instead, as discussed in the fifth section, this coefficient was set when calibrating the forecasting equations to a base case.

GENERATION OF PSEUDO-DATA

OTM is a highly detailed linear programming model (with some non-linear features) that simulates the domestic refining sector and world markets for petroleum products for ten discrete geographic regions. As such it contains endogenous supply functions for crude oil and petroleum product imports. The pseudo-data are to be used to estimate the coefficients in

G. M. Lady and C. E. Moody Forecasting Marginal Costs 425

equations (3)-(5). Accordingly, we specified the OTM model for variations in each output price holding all other prices constant. Two data sets were created representing a short-run and long-run version of OTM. The short-run version fixed capacities and other operating features of refineries at the most recently available historical values (1988). In the long-run version of the model we removed all capacity constraints and allowed OTM to add capacity using the models own optimization criteria. For each of the short- and long-run data sets we considered three scenarios corresponding to alternative assumptions about the world oil price which spanned the range of crude oil price variation typically assumed for the annual AEO forecasts.

For each world oil price scenario, we designed a base or reference case. To do this, we ran OTM such that it satisfied US product demand quantities set equal to 1988 actual levels. To get a fully consistent set of prices and quantities several additional model runs were necessary. The point of the additional runs was to adjust the estimates of prices and quantities such that if one of them (e.g. prices) were assumed as model inputs, then the other (e.g. quantities) would be found in the models solution. Given a consistent base case, we made scoping runs by varying each price above and below its base case price. The purpose of these runs was to find corner solutions for each price; that is, the smallest price variation, both above and below, from the base case price (all else equal to the base case values) such that further price variation would produce no significant changes in domestic product output. The corner solution prices were the maximum and minimum prices used for pseudo-data generation. Sixteen price-variation runs were conducted for each product price by dividing the range of variation into 15 equal segments. We changed each price individually with all other prices set at base case values. This design provided a base case and 128 additional price-variation cases for each world oil price scenario. As a result, the short- and long-run pseudo-data sets each consist of the results of 387 OTM runs.

MODEL ESTIMATION

The model to be estimated consists of the eight output share equations (3), the seven input import share equations (4), and the crude oil input share equation (3, to which we append a vector of independently and identically normally distributed random errors e = [Ei, cg, &16]. The estimation is done subject to the constraints (6)-(9).

The use of pseudo-data allows us to avoid many problems that are common in non- experimental data. In generating the pseudo-data, each model run is independent so that there is no autocorrelation among the data. Further, the model is the same from run to run so that there is no heteroskedasticity. We generated the pseudo-data by varying relative prices to assure sufficient sampling variability. We chose the prices to be orthogonal to avoid collinearity. Finally, the exogenous variables are truly exogenous.

The estimation technique is iterated Zellner Efficient Least Squares (IZELS), also known as Iterated Seemingly Unrelated Regressions (ITSUR), which minimizes the trace of the residual covariance matrix. The error terms are correlated across equations because the revenue shares sum to one. To restore non-singularity to the residual covariance matrix, one equation must be deleted. Here, the share of other products (S,) equation is rem...

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