Marginal pricing

Absorption, rate-of-return, and marginal pricing have been considered here on the basis of manufacturing cost. Total cost, which is the sum of manufacturing and general costs, can also be considered as the basis. In this case the appropriate profit to consider is the net annual profit rather than the gross annual profit. 

Kremer16 then went on to make a precise estimate of the social value of an innovation or surplus under a situation of competition and under a monopoly. Thus, for example, for a price five times higher than the price fixed according to the marginal price, we detect a static distortion of 1.5, that is, the social rate of return of an innovation in a situation with marginal cost prices will be 1.5 times the return on the investment under monopoly prices. In this situation the social value of an innovation in a competitive environment would be 9.35 times the social value in a monopoly, that is, when there is no welfare loss. Kremer thus provides an estimate of welfare loss from a more thorough analytical perspective, and shows that it can be sizeable. 

Figure 4. Marginal prices for Caspian gas by transportation through different Pipelines...

It is acknowledged, however, that during certain periods of the peak hours - the super peak - a gas generator is the marginal (price-setting) unit but, due to lack of data, it is not possible to analyse the super-peak period in Germany separately. 

The demand for electricity varies by the time of day, the week and the season. The cost and price of electricity are high at times of peak power demand and low at times of low power demand. The high cost of peak power reflects the fact that the facilities needed to produce much of this power are operated for only a few hundred or a thousand hours per year. This is shown in Table 2, which lists the marginal prices of electricity in 2004 for nine US electric grids (FERC, 2004). 

The solution of the set of linear equations represented by (13) determines the shadow prices. In terms of these prices the marginal prices are given by ... 

As proposed by Fax and Mills (6), it is easier to keep track of the various derivatives if they are put into a matrix as shown in Table (1). In the general case, the shadow and marginal prices may be found from ... 

With these definitions, the shadow and marginal prices are found to be ... 

Marginal prices are more informative sional form, 0, using ... 

In each iteration with a new marginal price vector, the distributions by thermoeconomic accounting are always accessible if needed. 

The marginal prices support this conclusion. The incremental changes in decision variables, Ay, required to improve 0 result in increased capital investment. Table V shows the results of a stepwise change. 

Table V. Improving using marginal prices, example 2...

When optimizing real world problems it is often convenient to define a marginal price, 6, associated with each decision variable, y. The set of marginal prices are defined... 

The physical interpretation of these marginal prices is that they are prices associated with a differential change in the corresponding decision variables, y, and may be used to indicate the potential benefit of changing a decision variable. 

When the marginal price, k,, is negative the derivative of the objective function is negative. If we wish to minimize the objective function, this indicates that the decision variable needs to be increased in order to approach the optimum. Similarly, if the marginal price, 0 , is positive, the decision variable needs to be decreased in order to approach the optimum. 

The next step is to obtain equations for the solution of the shadow and marginal prices. This requires the evaluation of various derivatives of the constraint equation matrix. However, because not all the constraints are in algebraic form (those constraints that are functions of steam table properties) numerical derivatives must be evaluated. One other note, there are two condenser costing equations. This means that two separate derivatives must be taken and the derivative corresponding to whichever costing equation is valid for that value of condenser area, is the one that should be used. 

System Solution. The shadow prices, Xj , are evaluated using Equation 10. Table IV lists the algebraic solution for the shadow prices (made very easy by choice of diagonalized constraint equations). The marginal prices 0 are evaluated with Equation 15. Again, the choice of fixed parameters and decision variables makes it possible to solve Equation 15 explicitly for each marginal price. Table V lists these expressions. 

Table V. Marginal Price Equations for the Cogeneration System...

The solution procedure requires the designer to select a feasible set of decision variables y for the first iteration. Once this initial set of five decision variables has been chosen, the entire design (for that iteration) is fixed and the set of state variables x, and cost estimates are determined. With values assigned to all the state and decision variables, the set of shadow prices SP(i), is evaluated, and in turn, the set of marginal prices, PM(i), is determined. These marginal prices are then used to direct the iteration as described by Equation 16. 

This procedure is repeated until the marginal prices are small. One such measure of smallness for the marginal price vector is... 

Second Law quantities such as available-energy and negentropy need not be introduced explicitly into the optimization although their effects are evaluated (indirectly) by the shadow and marginal prices. Furthermore, any required Second Law based prices can be obtained by using Equations 17 and 18 or by way of a thermoeconomic accounting technique (5,15). 

The provisions of the above sections apply to profit margins, prices and discounts within the terms of other legislation. 

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