Journal of Economics and Business 64 (2012) 393– 398

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Journal of Economics and Business

Second-best optimality of advertising when monopoly issanctioned

Richard E. Justa, Rulon D. Popeb,∗

a University of Maryland, College Park, MD 94720, USAb Brigham Young University, Provo, UT 84602 USA

a r t i c l e i n f o

Article history:Received 7 May 2012Received in revised form 1 August 2012Accepted 8 August 2012

Keywords:AdvertisingExcessive advertisingMonopoly pricingOptimal advertisingSocial optimality

a b s t r a c t

Monopoly pricing is sanctioned by government in a variety ofcases (e.g., patent policy). We derive necessary and sufficientconditions on preferences determining when monopolists choosesocially optimal, excessive, or inadequate advertising conditionalon monopoly pricing behavior. We then derive the behavioralimplications of these conditions in an empirically tractable frame-work that is estimable with typical observable data.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

Many studies have developed conditions to determine whether advertising is socially excessive(e.g., Dixit & Norman, 1978; Nichols, 1985; see the extensive review by Bagwell, 2007). To date, how-ever, no studies have determined conditions under which monopoly advertising is socially optimal ina second-best sense for cases where monopoly pricing is sanctioned. We determine conditions underwhich advertising is socially optimal, excessive, or insufficient in such second-best cases. Through-out, we assume stable preferences such as under the Becker and Murphy (BM) assumption thatadvertising is complementary with the advertised good but without consumer-chosen advertisingquantities.

∗ Corresponding author at: Department of Economics, Brigham Young University, Provo, UT 84602, USA.Tel.: +1 801 422 3178.

E-mail addresses: rjust@umd.edu (R.E. Just), rulon pope@byu.edu (R.D. Pope).

0148-6195/$ – see front matter © 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jeconbus.2012.08.001

394 R.E. Just, R.D. Pope / Journal of Economics and Business 64 (2012) 393– 398

2. Cases where monopoly pricing is socially preferred

For many markets, governments sanction monopoly sales and pricing as an imperfect correction ofanother distortion. Examples include property rights (PRs) associated with patents, copyrights, intel-lectual property rights, and exclusive development of market goods from publicly funded research.Second-best optimality of monopoly pricing has been widely supported as practical policy undersuch PRs, and for goods causing damaging emissions or overusing resources. By comparison, first-bestoptimality conditions have proven elusive and politically impractical (Tirole, 1988; Viscusi, Vernon,& Harrington, 1995, p. 833). Conditions for optimal advertising under such second-best policies begconsideration.

3. A typical model of optimality in advertising

Following common assumptions, suppose advertising is rationed (determined by the advertisingfirm as discussed by Becker & Murphy, 1993) at no cost to the consumer and tastes are constant sothat welfare is unambiguous. Assume (i) prices other than for the advertised good are fixed so thatexpenditures on all other goods can be treated as a composite commodity z, called the numeraire;(ii) utility of the representative consumer is quasilinear in the numeraire, U(q,A) + z where q is con-sumption of the advertised good (q ≥ 0) and A is advertising expenditure (A ≥ 0); and the consumer’sbudget constraint is pq + z = m where p is price and m is income. Substituting the budget constraint,the consumer’s problem becomes maxq U(q,A) + m − pq. Demand satisfying the first-order condition,Uq − p = 0, is denoted by q = q*(p,A) or p = p*(p,A) ≡ Uq in implicit form. The monopolist has cost func-tion c(q) and profit � = pq − C(q) − A. Standard assumptions cq > 0, cqq > 0, Uq > 0, Uqq < 0, UAA <0, (Uqq − cqq)UAA > U2

qA under second-order differentiability satisfy second-order conditions below.

4. Conventional social optimality in advertising

As a standard of comparison, social optimality is defined (following BM and others) by maximizingthe sum of consumer and producer welfare,

maxq,AW = U(q, A) + m − pq + � = U(q, A) + m − c(q) − A, (1)

where, without loss of generality, the unit of measurement for advertising has unit price (also commonin the literature). First-order conditions require

Wq = Uq − cq = 0 (2)

WA = UA − 1 = 0. (3)

This equates marginal social benefits, Uq and UA, with respective marginal social costs, cq and 1. Not-ing that dW/dA = UA + (Uq − cq)qA − 1 = UA − 1 upon defining q = q(A) as the quantity that solves (2)verifies these conditions are equivalent to BM’s. While formulated consistent with BM, the advertis-ing expenditures term is employed here because separate quantity and price data on advertising arerarely available.

5. Advertising when monopoly pricing is socially optimal

We first represent monopoly pricing as a second-best social optimum. Where (q) is a standardmeasure of external or user cost, or an indirect measure of future benefits from innovation incen-tives, social welfare maximization without advertising is represented by maxqW = U(q) − (q) + m −c(q), which has first-order condition Wq = Uq − q − cq = 0. This social optimum is attained by amonopolist solving maxq� = p∗q − c(q) if and only if q = −Uqqq because the monopolist’s first-ordercondition is �q = p∗

qq + p∗ − cq = 0 or, equivalently, �q = Uqqq + Uq − cq = 0.1 Accordingly, the case

1 The external cost (q) need not be additive. If the social welfare problem is maxq W = w(U(q) + m − c(q), q), then thefirst-order condition, Wq = w1(Uq − cq) + w2 = 0, is equivalent to the monopolist’s first-order condition if w2/w1 = −Uqqq.

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where monopoly pricing is optimal is represented by assuming q = −Uqqq or, equivalently, by solvingthis differential equation, (q) = U(q) − Uqq. While this monopoly pricing condition may only approx-imate reality, we investigate advertising optimality assuming it is exact (patent policy is optimal).

Adding advertising, the social welfare maximization problem becomes maxq,AW = U(q, A) − (q) + m − c(q) − A where (q) represents an external or user cost ( q > 0, qq > 0). The first-orderconditions are

Wq = Uq − q − cq = Uq + Uqqq − cq = 0 (4)

WA = UA − 1 = 0. (5)

Eqs. (4) and (5) equate marginal social benefits, Uq and UA, with respective marginal social costs,cq − Uqqq and 1, as in (2) and (3) where marginal social cost now includes marginal external or usercost.2

If no solution satisfying (5) exists, then Kuhn–Tucker conditions for the more general problemconstraining A ≥ 0 implies A = 0 if WA = UA − 1 < 0, so the social optimum would involve monopolypricing with no advertising. Equivalently, because UA is monotonic (UAA < 0), no advertising is optimalif UA(q∗(0), 0) < 1 where q*(A) represents the solution of (4). Thus, conditional on monopoly pricingas a second-best policy, positive advertising is socially preferred if its marginal benefit (UA(q∗(0), 0))for consumers exceeds its cost (1).

6. Social optimality of advertising conditional on monopoly pricing

If the industry is monopolistic, then first-order conditions in price-dependent form for the monop-olist are

�q = p∗ + p∗qq − cq = Uq + Uqqq − cq = 0 (6)

�A = p∗Aq − 1 = UqAq − 1 = 0 (7)

(assuming �qq = Uqqqq + 2Uqq − cqq < 0 and �AA = UqAAq < 0). Eqs. (6) and (7) equate the monopolist’smarginal private benefits from production and advertising, Uq + Uqqq and UqAq, to the correspondingmarginal private costs, cq and 1, respectively. Condition (7) requires UqA > 0. If no such solutionexists, then Kuhn–Tucker theory with constraint A ≥ 0 implies that a monopolist chooses not toadvertise.

Proposition 1. Conditional on monopoly pricing, a monopolist will choose the socially optimal levelof advertising for all levels of q iff preferences are linearly additive in advertising with respect to theadvertised good, i.e., U(q, A) = f (q) + g(A) · q.

Proof. Conditions (6) and (7) are equivalent to the social optimality conditions in (4) and (5) if andonly if UA = UqAq, which is a differential equation solved by U(q, A) = f (q) + g(A) · q.

Proposition 1 imposes no conditions on preferences if advertising is not present. The additionalterm, g(A)·q, is general with respect to A but implies that incremental benefits of advertising areproportional to consumption. Standard conditions on preferences imply fq > 0, fqq < 0, and gAA < 0. Amonopolist chooses to advertise only if gA > 0. If gA < 0, then advertising is not socially preferred becauseUqA < 0, which implies UA < 0.

Linear additivity in advertising is a useful reference point where the firm’s marginal benefit coin-cides with society’s marginal benefit. It imposes weak complementarity (Mäler, 1974), UA

∣∣q=0

= 0, a

common assumption for problems with external or user costs, whereby advertising affects utility

2 The term Uqqq is the conventional monopoly rent. Although a benefit to the monopolist, we call it a social cost in keepingwith our second-best conditional terminology.

396 R.E. Just, R.D. Pope / Journal of Economics and Business 64 (2012) 393– 398

only if the advertised good is consumed.3 This seems a mild and sensible assumption for mostadvertised goods such as pharmaceuticals, although linear additivity in advertising is a specialcase.

Proposition 1 is perhaps surprising and certainly at odds with many papers that suggest monopolyadvertising is unambiguously excessive. This result supports the Bayh–Dole Act adopted by theU.S. Congress in 1980, which gave a major boost to privatization of research in the public sec-tor. Previously, universities were not allowed to retain and sell patents produced by publiclyfunded biological and information science research. Without title, advances were made freely avail-able but were not picked up for private market development without exclusive opportunities toappropriate costs of market development. The Bayh–Dole Act established a uniform standard ofintellectual property rights for all federally funded research, which permitted patents developedat public expense to be sold to private firms for monopolistic market development. With subse-quent expansion to plants, animals, and micro-organisms, the Bayh–Dole Act has been a majorforce behind university patenting, licensing, and technology transfer, with subsequent monopolis-tic market development under patents (Henderson, Jaffe, & Trajtenberg, 1998; Jaffe, 2000; Massing,2000).

Proposition 1 is illuminated by decomposing social welfare into consumer, producer, and externalcomponents, W = U + � − , where U = U(q, A) + m − pq. Because p = Uq under consumer optimiza-tion, the marginal effect of advertising is UA = UA − UqAq for consumers and �A = UqAq − 1 for themonopolist. The first term in both cases is a marginal benefit while the second is a marginal cost.The marginal cost for consumers offsets the marginal benefit for producers yielding the advertisingcondition of social optimality. The condition of Proposition 1 equates the marginal social benefit, UAto the monopolist’s marginal private benefit, UqAq.

Proposition 2. Conditional on monopoly pricing, a monopolist will choose a socially excessive, opti-mal, or insufficient level of advertising as the marginal utility of advertising with respect to productionis elastic, unitary, or inelastic (ε > 1, ε = 1 or ε < 1) respectively.

Proof. Comparing (5) and (7), if UA < (=)(>) UqAq then monopoly advertising at �A = UqAq − 1 = 0implies WA = UA − 1 < (=)(>) 0, meaning that advertising is excessive (optimal) (insufficient) for bothconsumers and society. Proposition 1 follows where the elasticity of the marginal utility of advertisingwith respect to production is ε = (∂UA/∂q)(q/UA) = UqAq/UA.

Proposition 2 illuminates an empirical weakness of the BM framework, which is based on aprice of advertising (with consumer-determined quantity). Hence, the marginal utility of advertisingis observable in its price, which trivializes the question about optimality of advertising. Realisti-cally, the price of advertising is rarely observable.4 Our framework enables empirical feasibilityby assuming advertising is rationed (determined by the monopolist) and observable only as totalexpenditure.

Weak complementarity gives welfare significance to areas between compensated demands asadvertising changes (Just, Hueth, & Schmitz, 2004). Quasilinearity of utility in the numeraire—asimposed for conceptual convenience to equate compensated and ordinary demands—gives ordinaryconsumer surplus exact welfare significance as a measure of advertising benefits, but this assumptionis easily relaxed.

Consider a change in advertising and price from (A0,p0) to (A1,p1) where the indirect utility functionis V(pi, Ai, m) ≡ U(qi, Ai) + m − pqi and qi ≡ q∗(pi, Ai), i = 0,1. Quasilinearity of utility in the numeraire

3 This rules out advertising that is intrinsically entertaining.4 Even if advertising is not rationed, its price is usually unobservable in media such as newspapers, magazines, sporting events

and television services.

R.E. Just, R.D. Pope / Journal of Economics and Business 64 (2012) 393– 398 397

p1

p1 f

a

e

b

UqqA = 0

UqqA < 0

UqqA > 0

q

p(q, A1) = D1

p0

p1

p(q, A0) = D0

0 q0

d

c

'

*

Fig. 1. Potential effects of advertising on demand.

implies that compensating variation, equivalent variation, and the change in consumer surplus areidentical,5

CV = EV = �CS = U(q1, A1) − U(q0, A0) + p1q1 − p0q0

= V(p1, A1) − V(p0, A0)

= V(p1, A1) − V(p1, A1) − V(p0, A0) + V(p0, A0)

=∫ p1

p1

Vp(p, A1) dp −∫ p0

p0

Vp(p, A0) dp

=∫ p1

p1

q∗(p, A1) dp −∫ p0

p0

q∗(p, A0) dp.

(8)

Defining a choke price pj as the minimum price such that q∗(pj, Aj) = 0, j = 1, 2, the third equality of(8) follows from weak complementarity (regardless of linear additivity of preferences in advertising)because V(p1, A1) = V(p0, A0) = m when q = 0. The latter equality of (8) follows from the envelopetheorem whereby Vp(p, A) = (Uq − p)q∗

p − q∗ = −q∗.

Proposition 3. Conditional on monopoly pricing and weak complementarity, monopoly advertisingwill be socially excessive, optimal, or insufficient as UqqA > 0, UqqA = 0, or UqqA < 0, respectively, forall relevant q and A.

Proof. From the proof of Proposition 2, advertising is socially excessive (insufficient) as the marginalprivate benefit of advertising (UqAq = pAq) is greater (less) than the marginal social benefit (UA) Dif-ferentiating the difference, (d/dq)(UqAq − UA) = UqqAq where UqAq − UA = 0 at q = 0 because weakcomplementarity implies UA = 0 at q = 0. Hence, for any positive q, UqAq − UA > (=)(<) 0 as UqqA >(=)(<) 0.

5 Empirical implementation with differences among these three measures is easily accommodated for the case where quasi-linearity is relaxed. See Just et al. (2004).

398 R.E. Just, R.D. Pope / Journal of Economics and Business 64 (2012) 393– 398

Using the fundamental theorem of calculus, Proposition 3 presents a companion result in the largeanalogous to Proposition 2 by using weak complementarity. The strength of Proposition 3 is that itanalyzes an intuitive geometric characteristic of demand curves. That is, the proof of Proposition 3suggests that demand curves can be usefully differentiated depending on whether demand curves fanclockwise (UqqA < 0) or counter-clockwise (UqqA > 0) as they are increased by advertising. Suppose inFig. 1 that advertising shifts the demand curve upward in vertically parallel fashion so that the slopeof demand in price–quantity space at A1 (the curve labeled UqqA = 0) is the same as at A0 where A1 > A0.With quantity fixed at q0, the marginal consumer cost of advertising, UqAq yields the incremental wel-fare cost, area p1abp0. The marginal consumer benefit, UA, yields the incremental welfare benefit, areacabd. These areas are equal when UqqA = 0 for all q, which yields the reference condition of Proposition1.

In contrast, if advertising causes the demand curve to fan counterclockwise as it moves up, UqqA > 0then the incremental consumer benefit, area cfbd, is less than the incremental consumer cost, areap′

1fbp0, so private advertising is excessive, UA < UqAq With clockwise fanning, UqqA < 0 the incrementalconsumer benefit, area cebd, exceeds the incremental consumer cost, p∗

1ebp0, so private advertisingis sub-optimal. Consumers pay less and the monopolist gains less revenue than the additional benefitgained by consumers.

If the mathematical form of Proposition 1 is generalized to U(A, q) = f (q) + g(A)h(q), then demandsfan counterclockwise (clockwise) as advertising increases if hq > 0 (hq < 0). Crossing of demand curvesis ruled out by UqA > 0, as required for positive advertising.

7. Concluding comments

The cases of Proposition 3 generate an empirically tractable framework to test for both sociallyexcessive and inefficient advertising in cases where monopoly pricing is sanctioned by second-bestpolicies. Alternatively, a Box–Cox transformation on q can be used to test for linear additivity underweak complementarity.

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