San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

The True Price of Manufactured Energy

Discussion of energy sources which are alternatives to fossil fuels typically find that the cheapest alternatives are higher in costs than energy from fossil fuels. That leads to the anticipation that in the future with higher prices for petroleum one or more alternative sources will become economically viable. There are two major flaws in this anticipation.

First the price of petroleum is not established by its cost of production. Its price is established by supply and demand and the supply is limited by natural factors. If a new energy source were discovered that was cheaper than the energy from petroleum the price of petroleum would immediately drop to a level just below that of the new energy source. Thus the new energy source would not displace petroleum, it would only lower its price.

Second, the cost of energy from an alternative source is based upon upon energy having the relatively low price based upon fossil fuels. If fossil fuels are not available the price of the energy involved in the manufacture of wind turbines or photovoltaic cells would be higher. The amount by which it is higher depends upon how energy intensive is the manufacture of the new energy. For example, the production of ethanol requires a major energy input for its distillation from the fermented carbohydrates. Biodiesel requires numerous energy inputs for the growing and processing of the oil-containing agricultural crops.

The material below addresses the problem of the dependence of the cost of energy on the price of energy and the consequences of a shift from fossil fuels to manufactured energy.

Analysis

Let the set of commodities be separated into two classes. In the first class are those commodities whose price is determined by the cost of production. For these commodities their supply adjusts to the level of demand at a price equal to the cost of production. Let the number of these commodities be denoted as n. The column vector of these prices is denoted as P.

The other class of commodities are those whose prices are exogenously determined, either because they are set administratively or because their quantity is determined exogenously. The number of these commodities is denoted as m.

The scheme of input-output (interindustry) analysis in which the input requirements for the production of a commodity is given by a column vector is used.

Let A be the n×n matrix of input coefficients for inputs of the manufactured commodities required for the manufactured commodities (shown in the above diagram colored pink). Let B be the m×n matrix of input coefficients of the exogenously supplied commodities for the manufactured commodities (shown in the above diagram colored light blue).

Let P stand for the prices of the commodities whose prices are equal to the cost of production and R stand for the column vector of the prices of the commodities whose prices are exogenously determined. The price of energy pE is the first component of R.

Prices as being equal to the costs of production can then be represented as:

P = ATP + BTR

where AT and BT are the transposes of the matrices A and B. The dimensions of BT are n×m.

The above equations have the solution

P = (I − AT)-1BTR

where I is the n×n identity matrix.

It is notable that the prices which are set by the cost of production are determined by the prices of those commodities whose prices are exogenously determined.

The Case of Manufactured Energy

Let B1 be the top row of the matrix B; i.e., the energy inputs required per unit production of the manufactured commodities. When energy becomes a commodity manufactured some process then row B1 gets switched from being the top row of B to being appended to the bottom of matrix A.

A new column has to be created for the process of manufacturing energy. Let C be the column vector of the commodity input requirements for the manufactured energy other than energy itself. The input of energy required per unit production of energy is represented as b. The inputs of the commodities whose prices are exogenously determined are represented by the column vector D.

The prices of the manufactured commodities then satisfy the equations

| P    |     | AT     B1T| | P |      | B'T |
               =                              +              R'
| pE |     | CT         b | | pE|       |DT |

where B' is B with the first row, the one concerning energy, deleted. D is the additional column added representing the input requirements of manufactured energy from the commodities whose prices are exogenously determined. Thus B' has the dimensions (m-1)×n and B'T is n×(m-1). R' is just R with the first component deleted. The solution to this system of equations is then

   | P |          |         | AT   B1T | |-1
                         =     | I  −                   | | | B'T | R'
            | pE |        |          | CT    b   | |   | DT |

This display shows how energy price and other commodity prices are affected by the various influences including the energy input per unit of manufactured energy output.

The matrix whose inverse is needed for the solution for commodity prices can be represented as

| (I−AT)     −B1T |
|    −CT       1-b   |

The inverse can be represented as

| M1,1     M1 |
| M2        m |

where M1,1 is an n×n matrix, M1 an n×1 matrix, M2 a 1×n matrix and m a simple real number, a 1×1 matrix.

These elements of the inverse matrix have to satisfy the following conditions.

(I−AT)M1,1 − B2TM2 = I
−CTM1,1 + (1-b)M2 = O
(I−AT)M1 − B1Tm = O
−CTM1 + (1-b)m = 1

where O represents a matrix of zeroes.

From the second relation it is found that

M2 = (1/(1-b))CTM1,1

For convenience let the expression 1/(1-b) be denoted as k. When the above relation is substituted into the first relation the result is

[(I-AT) + kB1CT]M1,1 = I
and thus
M1,1 = [(I-AT) + kB1CT]-1
which can also be expressed as
M1,1 = [I + k(I-AT)-1B1CT]-1(I-AT)-1

Since M1,1 is known then M2=kCTM1,1 is also known.

From the fourth relation above it follows that

m = k(1 + CTM1) = k + kCTM1

The substitution of this expression for m into the third relation above the result is

(I-AT)M1 − kB1TCTM1 − kB1T = O
or, equivalently
[(I-AT) − kB1TCT]M1 = kB1T
and thus
M1 = k[(I-AT) − kB1TCT]-1B1T

With M1 determined then m is determined through the relation m=k(1+CTM1).

The True Price of Manufactured Energy

The equation determining the price of energy pE can be separated from the rest of the solution. It is

pE = M2B'TR' + mDTR'

The price of manufactured energy when its manufacture does not rely the relatively cheap energy from fossil fuel can be quite different from the value computed when such cheap energy is available. Furthermore the manufactured energy which is the cheapest when fossil fuel energy is available may not be the cheapest when it is the only energy source. The true price of manufactured energy for a particular alternative is the price that would prevail if that alternative energy source is the only one available.

(To be continued.)


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