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SLOAN SCHOOL OF MANAGEMENT

MARSHALL AND TURVEY ON PEAK LOAD OR JOINT PRODUCT PRICING P.

R.

Kleindorfer and M. A. Crew

April 1971

530-71

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 50 MEMORIAL DRIVE CAMBRIDGE, MASSACHUSETTS 02139

WISS. mST. TECH.

JUW 28

1971

0£W£r LiSMffy

MARSHALL AND TURVEY ON PEAK LOAD OR JOINT PRODUCT PRICING p.

R.

Kleindorfer and M. A. Crew

April 1971

530-71

no.S30'7l

1.

This paper is concerned with an extension of the theory of peak load or

joint product pricing.

It is prompted by a recent paper by Turvey and by Mar-

shall's early analysis of joint product pricing (Marshall 1920; Turvey 1968). Both of these authors hint at solutions to a joint product or peak load problem that have not been discussed in the major contributions on peak load pricing

(Steiner 1957, Williamson 1966).

Both Williamson and Steiner recognize the

existence of a class of peak load situations which Steiner calls the "firm peak" case.

These occur where it is not possible in the off peak period to fully util-

ize capacity even when a price equal to marginal running cost is charged.

corresponds to Marshall's "valueless straw case".

This

When the straw is worthless,

farmers concentrate on the production of a crop which has a larger proportion of ears to straw.

Corresponding to this process is a firm peak situation where a

public utility would install different kinds of facilities in order to vary its

production methods so as to reduce the costs of servicing the peak loads.

Empiri-

cal evidence of this can be noted in the electricity supply industry's production

techniques, which consist of employing plants which have different cost character-

istics according to their role in meeting demand.

This was noted by Turvey whose

paper throws some light on the problems of an electricity supply industry in meet±-ni

demand.

Although Turvey does not explicitly state what are optimal peak load

prices when an industry uses more than one kind of plant to meet its peak loads, he hints that prices equal to marginal running cost are somehow relevant.

Turvey

criticises the assumption of constant marginal running costs and constant incre-

mental capacity costs as "too simple a notion to be meaningful".

He then notes

that for an electricity system consisting of plants differing in age, location,

and type (and therefore also running costs)

curve is upward sloping.

,

the system marginal running cost

He goes on to say:

?37501

"The first consequence of this which

to marginal running costs in all off peak is relevant here is that a price equal

periods will vary between these periods."

This paper will show that the rele-

by marginal running costs vant marginal cost for pricing decisions is not given

possible to employ more than one in a joint product pricing problem when it is kind of plant to meet demand.

solution to the joint product or It is a contention of this paper that a differing costs to meet the peak load problem, where there are several plants of

Marshallian approach to joint demands, can be found, at least indirectly, in the product pricing.

pricing is While it will be argued that a kind of marginal cost

running cost mentioned relevant, it will be shown that this is not the marginal

by Turvey. analysis.

present in Marshall's An idea of the kind of marginal cost involved is

Marshall states:

"when it is possible to modify the proportions of

the whole expense of producthese (joint) products, we can ascertain what part of

tion would be saved...". (Marshall 1920, p. 390).

explains the point further. sold at a price y =

_f2(x).

4.

for all X

>_

Let b.

0.

be the constant operating cost per unit of

>_

period supplied from plant

£.

= 1,2.

i

= 1,2 be denoted by q.

2

w =

(2)

and let the capacity purchased ab initio on

Then the social welfare function, W, is given by

.

2

fx.

z

i=l

{

Jo\^y^ ^ "

per

Let the quantity supplied from plant i = 1,2

in period i = 1,2 be denoted by q

plant

x.

^y -

_ ^ ^Z%^ £=1 ^ 2

^

^a%i^ £=1 ^ ^^

-

The problem to be solved is given by

(3)

Maximize W P-i»

^2'

!

^2'

subject to

(A)

P^ = f^(x^)

(5)

q^^^

(6)

(7)

"*"

^ii

^2i

-

"

i = 1,2

^ " ^*^

^i

^U -°

P^, P^, x^, X2

^ " ^*^'

1

0, Q

^

"

-^'^

^

where

(8)

Q = (q^^. q^^, q^^. ^22'

^^l*

^^2^

The problem is solved in two stages.

It is first noted that the minimum operating

5.

and capacity costs for supplying specified output quantities, x- and x„ 1

and 2 are given by the solution to the following linear program.

(9)

Minimize Y = b^ q^^ + h^ q^^ + b^

subject to (5),

Let Q

(x^

,

Q

+ ^2 ^22

^1 ^1

"*"

^2 ^2

x„) denote an optimal solution to (9) for specified x^ ,

= (x2, x^ - X2, X2, 0, X2, x^ - X2) for

be used; and if 3, - B~

^

b„ - b,

,

)

_Z

_

+ b- so that >

0, P.

^t

f rom

that y-o = 0*

But q2

Furthermore, q„^ = x^ - x„ = b„

= f.(x.) =

+ g^-

X,

>

>

implies

Since (12) and (14) imply, with

the optimal price, P^

,

is given by

P^ = b2 + 62*

(20)

To obtain P„ we note that since q,, = q,2 = x^ X. = y,. 1 li

+ b^ for 1

i = 1,2. '

Therefore,

>

0, it follows by

(16)

that

^1 + ^2 " ^11

(21)

Now

= x. >

q

"^

"^

^12

^^1

implies by (15) that y

X^ + X^ = 2b^ + e^.

+

= 3

y

so that (21) yields

Using P^ = X^, i = 1,2, and (20) it follows that

P2 = 2b^ + ^1 - (^2 + 32)

(22)

It may easily be verified from (1)

Thus, if x^ > X2

X

-

5*

that b^

0, i = 1,2,


_

f

'.




"^

.

(25), and (27) that

(22),

^1 " ^^2 ^ ^2^

x„ > 0.

It can now be shown that cases i and ii are

Assume that there exist x^

(29)

Then since

1

3

,

x„, and x such that x^

i

^o

^^"^

+ 62 = ^l^^l^

2b^ +

^x-

(^2

+

2^

" ^2^''2^

it follows from (29) and (30) that x

>

\.

>

= &i +

\x^2

and ^2 ^ ^1 b^^

"*"

~ ^

^'^^ 1^21

^12 ^^^^

"^12

~ ^* ^

°'

Since in the firm peak

- u.-.

and from (17), ^12 ~ ^' ^^ follows that

^2.

"^

^1

"*"

^1 ^^^ ^^°^

that X- = b-.

Weil (1968) has an interesting analysis of this problem which makes basically the same point, while examining the beef and hides joint product pricing problem. his example there is no complication of running costs.

correspond to the

2b,.

+

B-

of this analysis.

In

He simply has 10 units to

However, his result is just the same.

His price for beef is the cost of the cow as given by the demand curves, less the

15.

value of the hides - in terms of his example

X^

= 10 - X„.

S.

C. Littlechild

has made a similar point on the relevance of the Marshallian analysis to the

problem of production over time, Littlechild (1970).

The criteria for performance are to be found in two British Government Publications (1961, 1967).

Profit maximization was at the heart of Marshall's approach.

For example, it is

implied by the last sentence of the mathematical note XIX.

Q

This point is made clearly by Weil (1968)

ginal revenues.

In footnote

5

,

who describes his multipliers as mar-

this point was ignored, because it makes no dif-

ference to the fundamental principles involved.

Irrespective of whether profits

or W are maximized in Weil's problem the two dual variables sum to the same amount.

The only difference is that in Weil's case they are marginal revenues while in the case described in footnote 5 they are prices (which are equal to marginal costs as

illustrated in the analysis).

Profit maximizing simply requires that demands be

incorporated explicitly into the analysis through the marginal revenue functions instead of the demand functions. results. X.

- X ),

This applies to both the firm and shifting peak

For the firm peak case, where x^ the results are that

^1=^2+

the shifting peak case, with x^ = x_ = x

>

x^ >

and Q = (x-, x^ - x., x- 0, x-,

^^ ^'^^ ^2 ^ ^^1 >

"^

^1 ~ ^^2

"*"

^2^'

^°^

and Q = (x, 0, x, 0, x, 0), ^i + ^9 ~

2b^ + B^.

9

See Officer (1966).

Actually constant returns to scale prevail and so the result

applies with the usual qualification about constant returns and the "indeterminacy of the purest competition", Samuelson (1947).

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