Surplus may measure waste

Journal

of Public

Economics

22 (1983) 363-374.

SURPLUS

MAY

North-Holland

MEASURE

WASTE

Yuval SHILONY* Bar-llan University, Ramat Gan, Israel

Received

April 1981, revised version

received

April 1983

When production decisions are distorted by externalities, the traditional Marshallian surplus in the input markets contains, in addition to its other components, a measure of the waste (benefit) so generated. In an extreme example of the common property type, it is shown to contain nothing but waste. That suggests another reason for caution in interpreting surpluses but also opens a new way for measuring the cost of distortions.

1. Introduction A change in input prices facing an industry produces a change in the Marshallian surplus in the input market, i.e. in the area below the industry’s demand curves for inputs. As is now well known [see Jacobsen (1979)], this change is equal to the change in the industry’s profits if the industry is monopolistic. Or, if the industry is competitive, this change is equal to the change in the industry’s prolits plus the change in consumers’ surplus in the However, an important qualification markets for the industry’s products.’ has to be added to that relationship. There must not be present in the industry any production externalities that are not taken care of by compensations and ‘bribes’ a la Coase (1960). If such distortions are present in a competitive industry, the Marshallian surplus includes some measure of the waste (or benefit, if negative) generated by the externalities, in addition to the other components mentioned. That opens a way to measure the externalities’ waste. This is shown in section 2 and in section 3, where externalities’ waste is defined and shown to be measurable. In section 4 an extreme example is chosen in which Marshallian *This work was done while I was a research fellow at CORE, Louvain-la-Neuve. I am indebted to Marcel Boyer, Giora Hanoch, Joram Mayshar and two referees for their helpful comments. ‘The consumer’s surplus may be measured ambiguously if more than one product is involved because of an inherent index-number problem expressed mathematically as path-dependence of the line integral [see Silberberg (1972)]. See also Willig (1976) for conditions under which this ambiguity may be unimportant in practice. Hausman (1981) shows how the exact consumer’s surplus can be found by deriving Hicks’ compensated demand curves from the observed market ones. 0047.-2727/83/$3.00

0

1983 Elsevier Science Publishers

B.V. (North-Holland)

364

E Shilony, Surplus may measure waste

surplus is nothing but waste. problem for another angle.

That

example

exposits

the common

property

2. Let u=(ur,..., u,) be the level of n outputs. We call this group of products an industry but it need not be an industry in any sense of the word. That way one can apply the results to externalities by one traditional industry to another, e.g. fishing and sightseeing. r =(rr,. . . , r,) is the vector of these products’ prices. Their inverse demand functions are allowed to be dependent and are denoted F’(u)=(D;~(u) ,..., D;‘(u)). The products are produced using m inputs x=(x,, . . ,x,) whose prices are p=(pr , . ,p,). Let the total number of firms in the industry be s. We denote by uif the quantity of product i produced by firm f and by xki,. the level of input k used by firm f in the production of product i, k=l,..., m, i=l,..., Iz, S=l,..., s. ui=c/.uif and xk = ci Es xkif . To avoid a still more involved notation assume that each firm produces only one product. The production possibilities and costs of firm f depend, in the case of production externalities, on the level of inputs of the other firms. That is a more general formulation than the one usually taken in the literature where externalities affect a firm through others’ outputs or just through total outputs. The present formulation can take care of the effect of the chosen input mix as well as output level. Denote by Xs the input levels of all firms other than 5 i.e. the threedimensional matrix {xkig} with the fth layer deleted: Li/(Ui/,X’)={(X,if

,..., X,ir)lf

can produce

when other firms employ The cost function

inputs

uif using (xlif,. ..,xmir) Xs}.

for firm .f is:

C,f(UiJ>X’,p)

=min

{Xp 1X E Lif(Uif, i')}.

x

Note that Shephard’s dCif

p=xki/,

Lemma

is applicable

here:

for every i, f and k,

(1)

ah

and under

perfect competition,

acif -=D;l, auif

to be assumed,

for every i, f, with uif > 0.

(2)

E Shilony, Surplus may measure waste

365

The prices of inputs p are the parameters to be changed in our exercise. Under the standard assumptions (convexity in uis of the sets Lif), for each p a competitive equilibrium is attained where each firm optimizes and chooses production level and input mix. With externalities, the equilibrium is not Pareto-efficient but is a Nash solution in the decision variables of the firms which imply the externality levels. The equilibrium can, therefore, be written as a function of p: x(p), u(p), D-‘(u(p)). The total industry profits are:

n(P) = jil Differentiating

D3 ‘(U(p))Uj(p) - i] i=l

;f

/=l

C,JU,Jp),

X’(p), p).

(3)

with respect to, say, pk:

. e=li=lj=lj=lg=laXejg

aPk

sff Using (1) and (2) and rearranging

(4)

qJ,zzJ

we get:

(5) We are interested in the line integral -ILx(p)dp for some initial and terminal price vectors p and j. That is the change in the Marshallian surplus in the input markets. This line integral is unambiguous, i.e. independent of the particular path of changing prices from p to p if and only if [see Apostol m, is the kth pa&al derivative of some function (1960)] -x,‘(p), k=l,..., = F(p)-_(p). It follows that if the 0 i,. . . ,p,), and in that case -f:x(p) right-hand side of (5) is the kth partial derivative of-some function, we know that -Six(p)dp exists and also what it is. The first term is the kth partial derivative of the profit function. The second term is the kth partial derivative of the consumer surplus in the u markets, as a function of p, if it is in itself unambiguous in the same sense of path-independence. See Silberberg (1972) for the conditions; see also Willig (1976). The new term here is the last one. It turns out to be the kth partial derivative of our definition of the

I: Shilony, Surplus may measure waste

366

externalities’ next section.

waste

or deadweight

loss. This is elaborated

and shown

in the

3. Waste In order to define waste by comparing the actual situation to an optimal one we define a hypothetical minimal cost function e(~,p). It is the cost of producing u, when input prices are p, by a single owner of all firms who internalizes all the externalities, i.e. Qu,p)=min(xpIx~L(u)}, X where, as usual, vector U, i.e.

L(U)=

i

(6)

L(U) is the set of input

X(Xi~ELif(Uif,

vectors

xf),CUi/.=U if

that

produces

the output

I

The optimal situation would have attained if this sole producer were constrained to produce and sell outputs where price equals marginal cost for every commodity. The equilibrium so defined, which is a function of p, is denoted 12(p), i(p), De’(G(p)) (= t?l?(ti(p),p)/du). Comparing this hypothetical situation, which is marked by hats ‘^‘, to the actual one, leads to the following definition of nominal waste: G(P) W(P)=

i

1

D-‘(Y)~Y--G(P),P)

The first brackets show the net social surplus, consumer’s surplus, plus producer’s surplus, under optimal setting, and the second brackets show the same for the actual distorted one. Fig. 1 shows the waste for one commodity ui as the shaded area. We call (7) nominal waste because the costs both actual and hypothetical are measured in current input prices. Our interest is in measuring changes in W as the industry expands or contracts and it is modelled here to come about by a change in p. The resulting change in W can be divided into a direct price effect and a real effect. As p changes, the value of the resources wasted in the production of any given outputs changes, even if outputs do not change at all. That is the direct price effect.

367

Y Shilony, Surplus may measure waste SMC(actual

W

=

(A+B)-(A-C)

=

social

MC=c

5,jf

LTC

jf

)

B+C

Fig. 1

We are, however, interested in the real effect, the one caused by changes in input and output levels and consequently in externality levels. For a given change we try below to capture only the real part. The definition of waste in (7) takes into account the inoptimal production of a given output level as well as the inoptimal output levels. Differentiating (7) with ESpeCt to pk:

aWo aPk

4

D_ ~ttitPl) _ ae(ti(P)7 PI

i

aui

'

1ah ahIir 1 atii(P)

aC(t;(P), PI

af+(P) aPk

+cccc c i / e jsff

acij-(%f(P), kjg

aCif(uif(Ply +g

xf(P), PI dx,j,(P)

aP,

aPk

~f(P)y PI

(8) Uif,Xl

The brackets vanish for every i and f because in each regime the respective price equals the respective marginal cost. Also iYC/ap, 1~= Xk and

368

Y Shilony,

Surplus

may measure waste

so we are left with:

The first element represents the real effect and the second the direct price effect. The real effect is the last term we have in (5). Note, incidently, that by plugging (9) into (5) we get:

(10) The waste could be traced as a residual of the surplus to the left of the input demand curves of the optimal sole producer, which are, of course, unobservable. To justify calling the first element in (9) the real one, note that in considering any cost xi pixi(p), not necessarily a wasted one, as a function of input prices, we could distinguish two effects to a change in one of the prices:

(11) The second term, xk(p), is the price effect. It is the change in cost due to the change in value of the resources already used (or wasted). The first term is the real effect, the extra resources used (or wasted) evaluated at current prices. The square brackets in (9) is the change in value of the resources already wasted, while the first term in (9) is of the same nature as the first term in (11) since

so

Alas, as the case is with the Divisia indices [see, for example, Wold (1953)], so it is in ours that the separation of a cost change into real and price factors by two line integrals may, in general, depend on the particular path of prices from p to @. Following (9) we are motivated to form the line integral:

369

Y Shilony, Surplus may measure waste

(12) where the superscript R designates it as a measure of the real change. That is our measure of the externalities’ extra real cost or waste, possibly negative when input prices change from p to p. It is the sum of all the costs that firms inflict on each other without taking them into account and possibly without being aware of them. The measure in (12) is not path-independent unless for every k,k=l,...,m:

a condition that would be sufftcient in our context but is not likely to be satisfied in general. Subject to the index-number problems concerning consumer’s surplus (CS) and real waste, WR, expressed here as the path-dependence of the line integral, we have from (5):

(13)

a modification of a known result for the case of externalities. Eq. (13) suggests another reason for caution in interpreting surpluses: they may be contaminated by externalities. On the other hand, (13) suggests a new way to measure WR as a residual:

W”(p> PI= -

j&ddp- [n(b)- @)I

- Cs(p,d.

(14)

E

WR can be given another presentation by noting that Ci & Ci, = px(p):

WRh _ PI= -

by using the definition

of rr in (3) and

fx(d dp+ CPx(ii)-p(p)1 P

D-‘(u(p))

J-,,, Assuming

I

u(r)dr+D-‘(u(p))u(~)----‘(u(p))u(p) .

that .x(p) has an inverse

system PLY) as II(~) has its inverse

D l(u),

Y Shilony,

370

we can apply integration

by parts to each of the brackets

x(P)

W”(@) = l p(x)dxX(_p)

Surplus may measure waste

to give:

u(F)

j D-‘(u) du. u(p)

(15)

The real waste in (15) is presented as the difference between the area below, not to the left of, the demand curves to the inputs and outputs of the industry. This presentation is particularly useful for the actual measurement of the real waste from observed demand functions. This WR is the new measure of externalities’ cost we would like to suggest. Possible uses of it may be to evaluate policies that affect input prices p, e.g. government production or subsidization of inputs like irrigation water (dam), the taxation of energy, etc. or to evaluate the effectiveness of public policies to curb or mitigate inter- or intraindustry externalities. Fortunately, WR and not W is advanced as the right measure to consider. To measure W would imply to observe imaginary quantities like 2, as can be seen from (9). Since WR is the central concept of this work we would like here, as a matter of conclusion before turning to an example, to justify it as the right waste measure rather than W Suppose firms in an industry inflict technological externalities on each other. That causes some resources to be wasted in comparison to an ideal situation of a sole owner of all firms who is constrained to produce where price= MC. Now a change comes to the industry, say a decline in input prices. All three supply curves in fig. 1 go down. The industry moves to a new equilibrium with a new level of wasted resources. The change in waste is composed of the change in value of the resources wasted in the old equilibrium and of the extra resources wasted in “the new equilibrium. For an infinitely small change that composition is given in (9) and we called the two parts the direct price effect and the real effect, respectively.

4. An extreme example Let us see an example of the common property type, in which input price goes down and An= ACS=O, so the generated surplus is all waste. The industry is a competitive one of fishing firms in a lake. The industry uses one, possibly composite input x and the production function of every firm f is uf = (12 -2x)x/. The externality is effected through the total input x, and not the input of the others, Xf, assuming that the difference is of no consequence. The firm’s cost function is:

(16)

Y. Shilony,

Surplus may measure waste

when p is the price of x. In the aggregate function of the fishing effort:

we have the catch

u(x)= 12x-2x2,

371

of fish u as a

(17)

that has the shape shown in fig. 2. This is the classical intraindustry externality. Firms crowd each other but do not take account of it. Under that modelling, firms have constant marginal cost. That marginal cost, in a competitive equilibrium, is equal to price. That makes the firms’ output and therefore the firms’ number indeterminate. To avoid that, one could assume that each firm has a critical capacity after which costs increase because, say, control becomes intractable.

U’

16 16

3

2

4

Fig. 2

From

(17) one can get the inverse: x(u) = 3 * $z&,

and also total cost TC when

(18) p

is the price of x:

TC(u,p)=px(u)=p(3+d9-) and average

(see fig. 3);

(19)

(see fig. 4).

(20)

cost:

AC(u,P)= If r is the price of fish, the market

equilibrium

is attained

when all firms

312

Y Shilony,

Surplus may measure waste

3p__-_--_--_-_-_-_

Fig. 3

AC

_______-__----_-!-

I t ’

‘,!A

16 18

Fig. 4

8r -

4r

-

TX

2

4

6 Fig. 5

Y Shilony, Surplus may measure waste

equate

their marginal

I=----’

or, writing

cost to r, i.e.

P(U) u

373

(21)

’

it as

r4x) P= -=

12r-2rx,

X

(21’)

one gets the industry’s inverse demand for the input x (see fig. 5). We shall now consider a change of p that would move the industry from point A to point B in fig. 2. Output and price remain the same and therefore the demand curve can be picked arbitrarily. Since r does not play any role, assume r = 1. Alternatively, one could assume a horizontal demand curve at r= 1 and although u would go up, the same result holds and the surplus is all waste. We start from an equilibrium point p = 8, x = 2, u = 16 and move to another where p =4, x =4, u = 16. This decrease of p generates in the market for x (assuming for simplicity that only this fishing industry uses the input x) an increase of 12 in surplus - area D+E in fig. 5. On the other hand, An=0 because profits are zero before as well as after the change, which only caused an influx of new firms to the industry. Also, ACS=O, because the price and quantity of the product did not change (or the demand curve is horizontal). -J:x(p)dp=12 IS . therefore the extra waste generated in the industry. That can be explained by noting that, because in our case r and u are constant and so also must be the product x.p [see (20)], so D+ E= E + F in fig. 5. As the price p goes down from 8, more and more additional units of x are employed in the industry up to a total of two extra which, we know, are simply wasted. If we evaluate each of them by its moving or current price, we get 12. In our case of a linear demand for x it amounts to evaluating the two units of x at their average price over the change which is 6. Formally, from (9) we have:

From

(16): aC, _ -_

ax

2~9 (12-2x)2

Y Shilony, Surplus may nmsure

374

waste

so from (17) and (21):

From

(21)

axlap=

-3, SO:

WR(8,4)=j2x(p)(-_t)=!

8

8

-x(p)dp=

$6:p)dp=12.

8

Note in passing that exactly the same displacement of equilibrium from A to B in fig. 2 could be the result of an increase in the price of fish from 1 to 2 when p= 8. That would move the industry along its supply curve (average cost) from A’ to B’ in fig. 4 and generate there an addition to producers’ surplus. That addition we could separate into its components along the same lines, but we already know that it comprises nothing but waste. Producers’ supply is dealt with in a separate paper.

References Apostol, T.M., 1960, Mathematical analysis: A modern approach to advanced calculus (AddisonWesley, Reading, Mass.). Coase, R.H., 1960, The problem of social cost, Journal of Law and Economics, l-44. Hausman, J.A., 1981, Exact consumer’s surplus and deadweight loss, American Economic Review 71, 662-676. Jacobsen, S.E., 1979, On the equivalence of input and output market Marshallian surplus measures, American Economic Review 69,423428. Silberberg, E., 1972, Duality and the many consumer’s surplus, American Economic Review 62, 9422952. Willig, R.D., 1976, Consumer’s surplus without apology, American Economic Review 66, 589597. Wold, H., 1953, Demand analysis (John Wiley).