Journal of Public Econonucs 22 (1983) 199-214 North-Holland

COMPLEXITY IN REGULATION

Richard E QUANDT* Prmceton Unzuerszty, Prmceton, NJ 08544, USA

Received Apnl 1982, revised version received March 1983

1. Introduction

Conslderable attention has been pald m recent years to modellmg the behavior of bureaucratic orgamzatlons It has been noted that the motlvatlons of bureaucrats may differ from those of the executives of conventional firms 1 A relatively neglected question about bureaucraaes 1s how the complexity or mtncacy of the rules they admmlster 1s decided upon and who gams or loses from greater or smaller degrees of complexity

Agencies promulgate rules (or have rules promulgated for them by legslatures) which must be obeyed by mdlvlduals who wish to ‘consume’ the agency’s output Such consumption may be voluntary - an unemployed person does not have to take the steps that would lead to his being payed unemployment compensation - or they may be mandated, as m the case of the requirement to tile an Income tax return In any event, the desire to ‘consume’ the output of any agency mvolves complying with vanous rules and brings with it compliance costs A similar problem IS faced by firms m the presence of regulation Kearl (1983) has noted this and has suggested that this has caused a group of speclahst mtermedlarles to sprmg up whose activity is to sell ‘compliance services’, which they are able to do because regulations are often complex, difficult to understand, and may require heavy investment of time and effort to master

The present paper sets up some simple models for tracmg the effects of complexity Sectlon 2 presents the basic model m which the consumer 1s required to consume one umt of ‘compliance’ Sectlon 3 considers the effect of errors m compliance and section 4 deals with the case m which

*I am indebted to W&am J Baumol, Timothy Bresnahan, Therese McGmre, Robert Porter, Harvey Rosen, Oliver Wdhamson, Robert Wdhg, David Wmter and the referees for useful comments on an earher draft I am especially indebted to James Kearl, not only for helpful, cntlcal suggestions but also for suggestmg the topic Fmanclal assistance from NSF Grant No SES-8012592 IS gratefully acknowledged

‘See, for example, Nlskanen (1967), Breton and Wmtrobe (1975), Wdhamson (1964), Mlgue and B&langer (1974), and Orzechowskl(l977)

0047-2727/83/%3 00 0 1983 Elsemer Science Pubhshers B V (North-Holland)

IPE--C

200 R E Quandt, Complextty rn regulation

‘compliance’ 1s optlonal Section 5 bnefly considers some extensions and section 6 contams some conclusions

2. The basic model: The case of compulsory consumption

We consider the particular case m which there exists a ‘good’ the consumption of which, along with all other goods chosen optnnally by the consumer, provides more utility than the best he could do when he does not consume the good m question Whether such a good exists literally, irrespective of pnces, may be debatable, but as a practical matter it 1s not hard to find examples where the assumption is, to all intents and purposes, true In any event, this 1s the simplest case and we consider it first The assumption that the consumer must consume a unit of this good will be relaxed m section 4

An example 1s the case m which the government imposes on all consumers an unavoidable requirement, or rather one that may be avoided only at extreme cost, such as filing an annual income tax return Wllfully falling to file an income tax return 1s a severe offense and leads to sticlently severe penalties, so that most persons do file a tax return even if the cost of filing 1s substantial Another example 1s that of obtaining a divorce Most mdlvlduals who can no longer abide their spouses and who desire a divorce will, m fact, get one, irrespective of its cost Nor will an mdlvldual who decided to purchase a property desist from doing so because of the cost of closing the transaction

The above examples represent cases m which securing the ‘good’ is, m effect, unavoidable, but m each of these the consumer may have a choice of how to go about securmg it In the case of filing an income tax return, the consumer’s choice 1s either to fill out the tax return himself or to hire a tax accountant to do it for him In the case of divorce, the standard procedure 1s to entrust the matter to any attorney, however, mcreasmgly, ‘do-it-yourself divorces are becoming possible 2 Indlvlduals have similar choices with respect to numerous other legal procedures

In all these examples the mechanics of acquu-mg the good are determined by government agencies and the various branches of government Thus, the mechanics of filing income tax returns are set by statutes and by the Internal Revenue Service, the mechanics of divorce by statutes and by precedents established m courts There are thus rules that must be followed and these rules are subJect to greater or lesser complexity We shall make the fairly strong assumption that complexity possesses a cardinal measure for example, one may think of complexity as being measured by the tnne required by an average or representative person to comply with the applicable rules and

%k Sherman (1980)

R E Quandt, Complexzty zrz regulatzon 201

regulations The consumer 1s thus involved m a choice m which he must compare the utility cost of spending time on acquiring the good with the money cost of having somebody else do it for him 3 We now turn to a formal model In order to avoid awkward language, we assume henceforth that we are dealing with the filing of tax returns and thus refer to ‘purchase accounting services’ versus ‘do-it-yourself’

2 1 The consummg sector

Define x as the quantity of the composite consumption good other than accounting services, C as the measure of the complexity of the rules and regulations pertaining to tiling a tax return, p as the price pad for a purchased tax return, and M as the amount of the consumer’s income We normalize the price of the composite consumption good to be unity and define

{

0, d the consumer provides accounting services for himself, Y= 1, if the consumer purchases accounting services

By definmg x to be the consumption of a smgle composite good we are not allowing substitution by the household between time devoted to tax return preparation and time devoted to market actlvltles To the extent that tax return preparation takes more time, it will be purely at the expense of nonmarket activity time This leads to the followmg assumption about the household’s utility function

Assumption I The utility function of the representative household 1s

w, Y) = W) - (1 -YK (2 1)

where y 1s a positive parameter and where v’(x) > 0 and V”(x) < 0

The consumer maxlmlzes utlhty subject to this budget constraint

x+py=M (2 2)

Thus, if y = 0, max, U(x, 0) = V(M) - yC, and d y = 1, max, U(x, 1) = V(M -p) The consumer chooses y=O if and only if

V(M)-yC2 V(M-p) (2 3)

If all consumers have the same utility functions, the same mcomes, and face

%e SubJe-ct under d~scusmon extends beyond the topic of regulation and may be adapted to the general question of purchasing goods versus ‘do-it-yourself’

202 R E Quandt, Complexrty m regulatum

the same level of complexity, they will act alike and either all purchase accounting services or all provide them for themselves At this stage we maintain the assumptions that they have ldentlcal mcomes and face identical complexity (see section 5 for a relaxation of these assumptions), but we allow them to exhlblt different sensltlvlty to complexity by mtroducmg

Assumption 2 The parameter y IS a continuous random variable with pdf ccl(y), cdf Y(Y), and support (0, ~0)

We are thus consldermg a contmuum of households, each characterized by a particular value of y It follows lmmedlately, using (2 3), that the aggregate market demand for accounting services 1s

D(M,p, C)=Pr 1

72 V(M) - V(M -P) c I

=1-Y ( ww- wf-P) C >

(2 4)

For slmphclty we define y” = [V(M) - V(M-p)]/C and thus D(M, p, C) = 1 - Y(y”) The properties of the demand function may be summarized m

Proposition I (a) aDlaM > 0,

03) aDlap < 0, (c) aqac>o

Proof See appendix

These results are what one would mtultlvely expect An increase m personal incomes and an increase m the pnce charged by accounting firms for the preparation of a tax return have opposite effects on the demand for accounting services A cetem purlbus increase m complexity raises the utlhty- cost of ‘do-it-yourself’ and thus also increases the demand for accounting services

2 2 The service sector

We assume that accounting services are produced by firms and sold to consumers We consider here the case where accounting services are provided by perfectly competltlve firms The case m which there exists a single monopoly accounting firm 1s dealt with m section 5

R E Quandt, Complexity In regulation 203

Assumptzon 3 A firm provldmg accounting services 1s a perfect competitor m its input markets and has a contmuous production function Q =f(l, C), where 1 IS the amount of labor, with contmuous inverse 1 =h(Q, C), and with contmuous first- and second-order partial derivatives such that fi > 0, fi < 0,

fil<O, and fiz<O

Thus, the production function 1s strictly concave m the amount of labour The margmal impact of an increase m complexity 1s unfavorable on output Finally, labor’s marginal product dlmmlshes with complexity

The inverse of the production function, I= h(Q, C), 1s defined by Q =f(h(Q, C), C) It is straightforward to verify that h1 = l/fi >O,

h,=-f,/f,>O, hl,=-flJf:>O, and hlz=fiz/f!>O In perfect competltlon the firm equates price to marginal cost, or

P=ww - VYO), cl, (2 5)

where w is the wage 4 It 1s easy to verify that (2 5) has a umque equlhbrmm solution for p

2 3 Comparatzve statzcs

Is 1s straightforward to examme how pnce IS altered when the independent variables are perturbed The substance of this 1s

Proposztzon 2 In perfect competztzon the followzng comparatzve statzcs propertzes hold

(4 ap/aM > 0, (b) ap/aw =- 0, (c) ap/ac>o

Proof See appendix

The result for perfect competltlon is what one expects Accountmg services are a normal good and an increase m M must raise the eqmhbrmm price, as must an increase m the wage of providers of the service An mcrease m complexity raises both the demand as well as the margmal cost

2 4 Welfare comparzson

Pnvate welfare 1s the sum of consumers’ and producer’s surpluses

“This assumes as a notational slmphticatlon that there IS a single firm behavmg as a perfect competitor, le equatmg pnce to margmal cost If there are n equal firms, the first argument on the nght-hand side has to be replaced by (1 - Y(u(y”))/n The analysis remains essentially the same

204 R E Quandt, Complexrty WI regulation

Producer’s surplus 1s W, = p[ 1 - Y(yO)] - wh[ 1 - Y(y’), C] Consumers who purchase the service have utlhty V(M-p) and the equivalent income 1s E= V/-‘(V(M--p))= M -p For consumers who provide it for themselves the equivalent income is E = V- ‘{ I/(M) - yC} Total consumers’ surplus 1s

w,=I1-Y(y”)l(M-p)+j&)~-l{V(M)-yC}dy 0

Total welfare 1s

w=w,+w,

=C~-Y(r”)](M-p)+~~(~)~-l{VM)-~C)d~+~U-Y(~o)l 0

- wh[ 1 - Y(yO), C] (2 6)

The effect of a change m C upon private welfare 1s obtained by differentiating W with respect to C We have

@-Cl - ‘W”N$, (2 7)

since V-‘{V(M)-y’C}=M-p, and

~=(l-Y(‘O))~-(p-whl)~(~~) “MC-p)&; -wh, 1 (2 8) Total welfare change is

X V’(M-P) 3~ Y’ ---

C ac c 1 (2 9

Consumers’ and producers’ welfare 1s affected differently by increases m complexity In perfect competition ap/aC> 0 and an increase m C unambiguously deteriorates the posltlon of consumers since aV_ ‘/X < 0 Since p= whl, producers’ welfare increases d the first term of (2 8) 1s greater than the last This 1s likely to be the case if the cost function responds only m mmor ways to mcreases m complexity thus then 1s the case noted by

R E Quandt, Complexity m regulation 205

Kearl (1983) m which the providers of services have a stake m perpetuating or Increasing the level of complexity However, the sum of consumers’ and producers’ surpluses deteriorates unambiguously with C

The level of complexity 1s set by regulators and it 1s plausible to argue that they do so on the basis of a variety of motlvatlons

On the one hand, complexity m regulation exists m order to brmg about a particular, previously agreed upon social obJectWe In the case of regulations pertammg to the tiling of income taxes there is a triple social ObJectWe to raise taxes to finance projects that are socially desirable, to stablhze the economy, and to redistribute income Too little complexity permits various of these tasks to remam (partially) unfulfilled Thus, a fully proportional tax would perhaps greatly slmphfy the tax code but would run counter to the social objective of progressivity At the opposite extreme, a set of rules more complicated than those m force at present may eliminate loopholes and thus better serve social ObJectives, but at the cost of greater complexity It IS plausible that a sufficiently great level of complexity can create such dficultles m comprehending rules and m admmlstermg them as to lmpalr the system’s ability to achieve social objectives

On the other hand, regulators have objectives of their own and complexity can serve their own utility as well 5 To achieve their more personal ends, they need to have the ‘right’ level of complexity neither too little, nor too much complexity 1s suitable for this purpose Their overall utility or welfare is thus thought to come from two sources from achieving social objectives according to their perceptions and from pursumg their own objectives Denote the welfare function of regulators m this broad sense, as perceived by regulators, by W,(C) It 1s plausible that W,(C) 1s concave m complexity

The first-best solution to the problem of maxlmlzmg overall welfare 1s to maximize W(C) + (welfare from achieving social ObJectives) + (regulators’ private welfare), where W(C) = W,(C) + W,(C) 1s the private welfare derived by mdlvlduals and firms from production and consumption Since we have only regulators’ perception for the second term of the maximand, a second- best solution is to maximize W(C) + W,(C) Regulators may, however, use different weights m the maxlmand and attempt to maxnnlze &W(C)+ (1 -c1)W3(C) (05 a5 1) (If they are very myopic they may even use a=O) The first-order condltlon for a maxlmum 1s aW,(C) + (1 -LX)&(C) = 0 Differentiating totally and evaluating at the second-best solution (which 1s obtained when c1= 1 - LY) yields dC/da = - 2 W’( C)/[crW”(C) + (1 - a) W;(C)] which 1s negative This leads to

Proposztlon 3 Zf regulators maxtmlze a W(C) + (1 -a) W,( C), and weight thezr

‘See, for example, Mlgue and BBlanger (1974) or especially Downs (1965) on the comphcated set of goals that bureaucrats hope to achieve

206 R E Quandt, Complextty m regulation

own welfare W, more heavzly than przvate welfare W, the level of complexzty zn perfect competztzon exceeds the soczally optzmum level

3. Errors in compliance

Pursuing the analogy with fihng an mcome tax return, one must recognize that the consumer who ‘does-it-himself may commit an error m attempting to comply with regulations The error may remam undetected, m which case nothing further happens However, d the error 1s detected, the consumer must pay a penalty and may also mcur a direct utility cost due to the tlme- loss and embarrassment he may suffer 6

In the present section we explore the consequences of errors To simplify the analysis, we assume that errors are always detected We retam the defimtlon of y from section 2 and define

0, if the consumer commits no error, z=

1, if the consumer does commit an error

It 1s also assumed that accounting firms do not commit errors, the consumer can thus always guard hlmself agamst errors by purchasmg the service The utlhty function of the representative household can then be wntten as

~(~,Y,z)=~(x)-(1-Y)Y,C-(1-Y)zY,, (3 1)

where yZ measures the utility cost of commlttmg an error, and the budget constraint is

(3 2)

where pz 1s the price of accounting services and p2 the penalty assessed if an error 1s committed The remaining features of the model are expressed m

Assumptzon 4 (a) If an error 1s committed, a fixed penalty pz 1s paid (b) Consumers commit errors with a fixed probability 4 which depends

only on the regulators’ level of momtormg and not on the level of complexity

(c) yZ 1s identical for all consumers (d) As m assumption 2, y1 has pdf $(yJ 7

6A somewhat slmllar problem arises if the consumer commits fraud by mtentlonally falsifying his income tax return See Alhngham and Sandmo (1972)

‘These assumptions represent considerable snnphticatlon In reality, for example, people with higher incomes usually face more complex tax return problems and are also audited more frequently by the Internal Revenue Service

R E Quandt, Complextty m regulation 207

If the consumer does not purchase, his expected utlhty 1s E(U) = (1-q)[V(M)-ylC]+q[V(M-p2)-~IC-~z] If he purchases, his utlhty 1s V(M-p,) Followmg section 2, the aggregate demand IS

DW, Pl, Pz, c, 4) = 1 - w4, (3 3)

where Y? = CU -cdW@ +~VM--P~)- I/(~-PA-wJ/C

The development from this point on 1s similar to that of section 2, the relevant proposltlons are stated but the proofs are omitted

Proposltlon 4

(a) If p2Sp1, then aDlaM >O, aDlap, < 0, aDlap, >O, aD/aq> 0, and aD/X > 0 when y,, > 0 and the partzal derzvatzves are zero If y,, 50,

(b) lfpz >pl and y. SO, the partial derlvatlves are zero, (c) If pz >pl and y. >O, then aDlap, -c 0, aDlap, >O, aD/aq>O, and

aDfiX > 0, in addltlon, aDjaM > 0 If q 1s small but aDlaM < 0 If q 1s large

These outcomes are analogous to those of the previous section There are new (and expected) results an increase m the penalty or m the probability of commlttmg an error increases the aggregate demand for accounting service Furthermore, the existence of a nonzero probability of making an error may cause everybody to buy the service, m that case parameter variations do not affect demand on the margm Finally, it 1s possible for an increase m M actually to reduce the amount demanded The mtultlve reason for this result is the following Consider the margmal consumer who 1s Just indifferent between purchasing the service or doing it himself For such a consumer it must be true that E(U) = V(M-pi), or

(1 -q)CV(M)-y,Cl+qCI/(M-p,)-y,C-y,l= VM-PI)

A small change m M changes the left-hand side by [( 1 - q)V’(M) + qV (M-p,)] dM and the right-hand side by V’(M-p,) dM If q IS large (and the critical size also depends on p1 and p2), the former change will (by concavity) exceed the latter and the marginal consumer becomes a ‘self-doer’

Whether the penalty is greater or smaller than p1 depends on cu-cumstances and will vary from case to case The penalties associated with innocent income tax errors are frequently small In the income tax case It 1s also plausible to argue that q IS fairly small It has been estnnated, for example, that 7 percent of tax returns contam computational errors [Leapman (1981)] That suggests that on the whole aD/aM >O ~111 be the rule

The perfectly competltlve equlhbrmm is defined as m (2 5) which leads to

JPE D

Proposition 5 If yy >O, 8p1/Sp2 >O, dp,/iq >O, ip,/iC>O, and ap,law>o Furthermore, under the condztzons In whzch aDfdM >O, we also have ap,/aM>o

These results are smular to those of proposltlon 2 and require no further comment The (private) welfare function 1s also analogous and 1s

-yiC-ml dy, t-d1 - W31 -wW - ‘f’u(y’3, Cl (3 4)

The previously noted conflict of interests between consumers and producers 1s again evident, this time not only with respect to C but with respect to q and p2 as well (see appendix) The producers have an unambiguous interest m higher penalties and a hgher degree of ‘error-proneness’ Of course, this latter 1s not directly set by regulators (although m a more general framework it will tend to depend on the level of complexity), but the situation 1s open to the alternative interpretation m which self-doers always commit errors but are detected only occasionally Thus, producers have a stake m more rigorous momtormg by regulators Finally, assuming that regulators’ welfare function 1s concave m pz, q, and C all three parameters will tend to be set at socially excessive levels

4 A generalization: The case of noncompulsory consumption

In the previous sections we assumed throughout that the consumer must either buy or provide for himself the service m question or suffer a catastrophic dlmmutlon of utility In the present section this assumption 1s relaxed Purchasing a house, or mcorporatmg a small family business, or suing m a small claims court, or getting a passport, or employmg a domestic servant are all activities that the consumer may or may not consume, and if he does, he may cope with the applicable regulations by either ‘doing it himself or by hlrmg somebody for the purpose

We retam the defimtlon of y as the indicator of whether or not the consumer provides the service for himself We also define

0,

{

If the consumer does not consume the service at all,

‘= 1, d the consumer consumes the service

The utility function 1s now defined by

Assumption 5 The representative consumer’s utlhty function

U(x, y, z) = V(x) + pz - y( 1 - y)Cz (4 1)

R E Quandt, Complexzty zn regulatzon 209

The budget constraint is, as before, x + yp= M Given the same assumptions about V(x) as before, there are four possible outcomes

Action ut111ty

Buy, consume Buy, not consume Not buy, consume Not buy, not consume

UM-p)+p V(M-P) ~(‘(M)+P-YC V(M)

The form of the utility function expresses the restriction that if the consumer does not consume the service, he will m fact not provide the (unnecessary) service for himself Moreover, since V(M - p) < V(M -p) + p he will never buy the service when he does not consume it We now introduce

Assumptzon 6 The parameters p and y are contmuous and m$ependent random variables with support (0, co) and pdfs denoted by gl(p) and g,(y) and cdfs by G,(p) and G,(y), respectively

The market demand 1s the analogue of (2 4) and 1s given by

D=Pr{V(M-p)+p> V(M), V(M-p)+p> V(M)+p-yC}

= Cl - G,(p”)l Cl - G(Y~)I,

where p” = V(M)- V(M -p) and y”=po/C It is straightforward to verify that proposition 1 holds m the present case as well Equating pnce to MC and requu-mg market eqmhbnum yields

wM1- G,(P’))(~ - ‘%(rO)), Cl = P,

from which proposltlon 2 can easily be verified Private welfare 1s again the sum of the consumer surplus’ equivalent

income and producer surplus In the present case, however, consumer surplus 1s of three types Letting C* denote ‘consume’ and B* denote ‘buy’, the three types are those of (C*, B*), (C*, -B*) and (- C*, -B*) We have

w= wc*p+ w,. _lz* + w_c., ._p + WFIRMS, (4 3)

where WC.Be, W,. _g. and Wet. _Be are the income eqmvalent surpluses given

210 R E Quandt, Complexity m regulation

W c*B*= JpI J?Wf-_p) +dg,(dgz(y) dpdy

= Cl - W”)l $ J,‘- ‘1 VW-i.4 + p)gl(d dp,

WC, _ B’ = ll v-‘{1/‘(M)+p-yC)g,(p)gz(r)dpdy,

W_c* ,-B*=TI v-‘{V(M)}gl(p)gz(y)dydp,

and where

W,IRMS=PC~ -G,b")lCl -W")l--W -GI(P~))(~ -Gz(Y~)), Cl

Analog&sly with previous results It can be shown that dW/X IS unambiguously negative Hence, m this case, too, the level of complexity IS set at a socially excessive level if the regulatory agency itself has a welfare function concave in C

5. Extensions

The basic model can be extended expected and some contrasting results

5 1 A monopoly servxe sector

m several directions to yield some

Since at price p and complexity level C the firm can sell D(M, p, C) units, the monopoly firm will wish to choose p so as to maximize

n =pD(M p, C) - wwwf, P, Cl, C) (5 1)

This requires

aD ~‘o+(P-whl)7p=0 ap and

~=2~+(p-whl)$vhll g <o ( > 2

(5 4

(5 3)

R E Quandt, Complexity m regulation 211

Eq (5 2) 1s eqmvalent to the usual monopoly eqmhbrmm condltlon E =p/ (p-MC), where E 1s the elasticity of demand This case differs from the competitive case in several respects

(1) The solution to (5 2) does not necessarily correspond to a profit maximum It wtll do so if +‘(~“)V’(M-p)/t+h(yo)Ccr, where r 1s the consumer’s absolute risk aversion I= - V/V Ceterzs parzbus, this 1s assured by sufiiclently large values of r and C by a small value of I/’ It 1s also assured if 1+9(r) 1s umform

(2) The comparative statics results of the competitive case carry through partially In the uniform-$(y) case, ap/aM and ap/aw are posltlve as before but iYp/X is of ambiguous sign To sign ap/X, further restnctlons are needed on V”(M-p) If V”(M -p) IS numerically small, ceterzs parzbus, ap/X will be positive If, for example, V(X) =x”, a < 1, ap/aC>O will tend to be the case If p IS small relative to M, 1 e if the price of accounting services 1s a small fraction of income

(3) The welfare comparison IS ambiguous This 1s partly because ap/aC IS ambiguous m sign and partly because the second term m (2 8) does not vanish If ap/XcO, consumers might benefit and producers suffer from further increases of complexity Unlike the perfectly competltlve case, the actual result will depend, among others, on the magnitude of Y”

Although the pure monopoly case IS an extreme scenario, it 1s an important benchmark because more reahstlc cases intermediate between perfect competltlon and monopoly are likely to share the amblgultles inherent m the latter

5 2 Varzatlons m mcome and complexrty

Assume that complexity depends on income M as well as on a pohcy parameter 0 C = C(M, 13), with C1 > 0 and C2 > 0 Thus, we assume that richer consumers face more complexity and that the pohcy parameter measures ‘mtrmslc’ complexity The analogue of eq (2 3) still governs the choice of y = 0 At the margm of indifference between y = 0 or 1,

V(lM) - VW-p) = yC(M, O),

which lmphcltly defines a value M’=p(p,~, 0) with the obvious properties aMoldy co, aM”/ap ~0 and aM”/&I< 0, which follows from the concavity of V Given a pdf u(m) descnbmg the mcome dlstnbutlon, demand then 1s

WP, 8, Y) = s Y(M) - WM -P) I YC(M, 0)

u(M)~M=~~u(M)~M=~-T(M~),

which again follows from the concavity of V It follows immediately that

212 R E Quad, Complexzty m regulation

aDlap ~0, aDlay >O and aD/i% > 0, as expected The perfectly competltlve equlhbnum 1s gven by the analogue of (2 5) and it 1s easy to verify the comparative statics results, ap/atl>O and ap/ay >O An increase m the basic level of complexrty or m mdlvlduals’ aversion to complexity increases the market price for compliance services Finally, using for consumers the income-equivalent surplus, aggregate private welfare is

W=I$M-p)u(M)dM+j’ V-l{V(M)-$(M,O)}u(M)dM 0

+P[l - r(M')l- wh[l- qM"), C(M, O)],

from which a W/a0 CO IS immediate The substance of the previous analysis 1s not altered by assuming income to be variable and by letting C depend on M

6. Conclusions

Several cases were examined m some detail In the first, consumers must consume a unit of ‘compliance good’ and their choice 1s whether to provide comphance services for themselves or whether to purchase them from a firm In the second case, they may commit errors if they provide the service themselves In the third case consumers also have the option of not consummg the ‘compliance good’ at all Some extensions considered the cases m which (a) compliance services are provided by a monopoly firm, and (b) incomes and the complexity faced by mdlvlduals vary

Several questions remam for future mvestlgatlon One of these 1s how overall welfare depends on the frequency with which rules are changed Regulatory pohcy may require adaptation to changed cn-cumstances, yet It 1s unlikely that It would be optimal to have even the most minute change m circumstances immediately reflected m rule changes Another question 1s how the analysis may be modified If the complymg agents are firms themselves, if the level of complexity faced depends on the firm’s monopoly power (eg more monopoly power subjects the firm to more regulations), the complexity faced by the firm may be endogenously determined

Among the cases examined, the first three establish the opposmg interests of consumers and of the firms that produce compliance services, as was conjectured by Kearl (1983) The net welfare effect of complexity 1s unambiguous m perfect competltlon and the level of complexity and the penalty for erroneous compliance will be set by regulators at socially excessive levels The basic conclusions are confirmed m the extensions except m the case of monopoly m which the results hold only under stronger assumptions

R E Quandt, Complexzty m regulatron 213

The result that complexzty will tend to be set at soczally excesszve levels 1s analogous to market failure m the presence of public goods A latter 1s a good with the property that mdlvzduals cannot be excluded from zts consumptzon Complexzty, on the other hand, 1s a bad with the property that its consumptzon cannot be avoided Effzczency would require that the marginal benefit of complexzty be equated with the sum of the margmal personal utlhty costs In the absence of a mechanism that reveals these marginal utzhty costs, regulators may have little choice but to behave hke a firm that generates unfavorable externalities and produces socially excessive levels of output ’

Appendix

Proof of proposztzon 1 By straightforward dzfferentzatzon, aD/aM = -t,Q’)[V’(M)- V’(M-p)]/C>O by concavity We further have aD/ap= - $(r”)V’(M -p)/C>O and aD/aC= $(y”)yo/C>O

Proof of proposztion 2 Defining L= wh,[l - Y(y’), C] -p from (2 5) yields the total differential LM dM + L, dp + L, dC + L, dw =O, where L, = -wh,,ll/(y”)[V’(M)-V’(M-p)]/C>O by concavzty,

L,= -l-wh,,$(y”)V’(M-p)/C<O,

and L, = h, > 0 The concluszon follows

The derwatzves of (3 4) Using the functional dependence of p1 on the parameters pZ, 4, and C, we obtain

E=f Il/(rJ av-1C(1--4)I/(M)+qT/(M-P,)-y,C-q,y,l dy

aPz 1,

g=p *trl) av’C(l-q)~(M)+ql/(M-P,)-y,C-qyzl dy,_wh, ac 7

which are all negative

*The questIon of social costs of regulation has been exammed from several diferent pomts of view See, for example, Posner (1975)

214 R E Quandt, Complexity In regulation

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