Dividend Neutrality with Transaction Costs Author(s): Gur Huberman Source: The Journal of Business, Vol. 63, No. 1, Part 2: A Conference in Honor of Merton H. Miller's Contributions to Finance and Economics (Jan., 1990), pp. S93-S106 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/2353262 . Accessed: 26/07/2011 08:21Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ucpress. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.

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Gur HubermanColumbia University and Tel Aviv University

Dividend Neutrality with Transaction Costs*

I. Introduction Millerand Modigliani(1961)arguethat in the absence of transactioncosts and other marketimperfections, corporatedividendpolicy has no effect on the welfare of individualshareholders.It is widely believed, however, that dividendpolicy does affect shareholders'welfare in the presence of transactioncosts. For instance, Brealey and Myers (1984, p. 342) illustratethis position with an example of the dividend policy of AT&T: "AT&T's regular dividends relieve many of its shareholdersof transaction costs and considerable inconvenience." The sources of transaction costs are diverse and we concentrate mostly on one, the cost of transactingwith sophisticated traders. The bidask spreadis the manifestationof the presence of sophisticated traders in the market. The spread hurtstradersby hampering risk sharing.The article's central question is whetherappropriate dividend policy affords liquidity, thereby diminishing the need to trade with sophisticatedtraders and reducingthe bid-ask spread. I construct a model of trade with endogenously determined transaction costs and study* I am gratefulto StephenRoss for helpfuldiscussionsand to the National Science Foundationand the ForderInstitute for financialsupport.(Journal of Business, 1990,vol. 63, no. 1, pt. 2)

I construct an intertemporalmodel in which investors trade shares of a firm. All tradingis done through competitivemarket makers. After the initial period and before the end of the planning is horizon, information asymmetrically distributed amongtraders, and the prices for investors who buy shares are higherthan for those who sell shares. The presence of this deviationfrom the Walrasianparadigm divinotwithstanding, dend policy does not affect the initial period's share price or shareholders'welfare. This result is robust to various extensions of the model. I also consider fixed administrative transaction costs and show that dividendpolicy is irrelevant in the presence of these transaction costs.

? 1990by The Universityof Chicago.All rightsreserved. .50 0021-9398/90/6301-0015$01 S93

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the effect of dividend policy on shareholders'welfare and on share prices. Stock markettransactionsare costly because buyers must pay the market maker's asked price and sellers receive his bid price. The marketmakers bid-ask spread arises because competitive, risk-neutral compensate themselves for tradingwith individualswho are (on average) better informedabout the future values of the risky assets. (The model is inspired by Glosten and Milgrom[1985].) I focus on dividend payment by itself, not on any resolution of uncertaintythat might be associated with the dividend. My exercise suggests that dividend policy is irrelevant to shareholders' welfare even in the presence of transactioncosts. Millerand Modigliani(1961)assume not only the absence of transaction costs but also that the firm's projects are fixed. (These projects need not be optimal.) I resort to a strongerassumption, namely that the return on the firm's marginalproject is equal to the returnof the investors' marginalprojects. The equalityof the returnsis a necessary condition for the optimalityof corporateinvestments. Dividend irrelevance in the presence of transactioncosts, although at surprising first, can be understoodintuitively.Supposethat an investor's portfolio is optimal under one dividendpolicy and then the firm announces an increase in the dividend it pays. This extra amount is withdrawnfrom the firm's marginalproject. The individualwho receives the extra dividend uses the money optimallyby investing it in his marginalproject. Thus, the net effect on the investor's welfare is zero. My model has 3 periods. In the first and third no transactioncosts only in the secare incurred.Information asymmetricallydistributed is ond period, and thereforetransactionsare costly then. Moreover,portfolio rebalancingis desired in the second period because at that time each investor learns about the relation between his third-periodidiosyncratic income and the third-periodinvestment income. I experiment with the effects of dividend payments in the second period. Presumably,a good dividendpolicy can cut down transaction costs. How do we tell a good policy from a bad one? By looking at the first-periodprice. At that time, when no transactioncosts are present, all investors will agree that the optimalpolicy is the one that maximizes the share price. I show that the first period's share price is not sensitive to the dividendpolicy. This result is robust to various generalizationsof the model. I also consider briefly a model of administrativetransactioncosts. asymmetries In it, the cost of transactingis not inducedby information but is simply a fixed cost per sale (or purchase) of stocks. Dividend policy is againirrelevantto shareholders'welfareor to sharevaluation. The model is presented in Section II. In Section III I study the effect of a change in dividend payment. In Section IV I discuss the ro-

Dividend Neutrality

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bustness of the results. In Section V I present and discuss the model with administrativetransactioncosts. In Section VI I provide an examplethat illustratesSection II's generalmodel. The articleconcludes with Section VII. II. The Model All investors enter the first period identical, endowed with cash and shares in a firm. A stock marketis availablein the first period. In that marketa share price is determined(but in equilibriumno trade takes place). As investors enter the second period, a fraction Q of them learn somethingabout the third-periodvalue of the shares. Moreover, each investor receives informationregardingthe relationbetween his thirdperiodidiosyncraticincome andthe fortuneof the firm.A need to trade shares arises. In the second period all shares must be sold to and boughtfrom marketmakerswho are risk neutraland competitive. The market makers do not know what the fraction Q knows. Aware that they are tradingwith investors who are better informedon average, they ask a higher price from people who buy shares than the price at which they offer to buy shares. In the thirdperiod all uncertaintyis resolved and consumptiontakes place. I assume a continuumof investors distributedin the interval(0, 1). The details of the model are given below, from the thirdperiodback to the first. Dividends are introducedin Section III and extensions are provided in Section IV. A. The ThirdPeriod All uncertaintyis resolved by the thirdperiod and consumptiontakes place then. An individualwho enters the period with n2 shares and m2 in cash consumesW3=

n2(q

+

r)

+ m2 +

Y,

(1)

where q + r is the per share income and y is the individual'sthirdperiod idiosyncraticincome. The random variablesr and y have a joint probabilitydensity function with a third parameter, a, which can be a vector. The random variable q is independentof r, y, and a. The investor's expected utility from the third-periodincome W3iSE[U(w3)Ia].

B. The Second Period 1. The uninformed investor's problem with no transaction costs. The investor enters the period with n1 shares and m1in cash. He does not consume in the second period,but he learnsthe value of the param-

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eter a. Therefore, he attempts to rebalancehis portfolio. Suppose for the moment that he can buy and sell a share for price p. The portfolio rebalancingproblem is to choose n2 and m2 to max E[U(w3)la], subject to (1) andpn2+

(2)(3)

m2 ? pn1 + mi.

Let n2(p,a,n1,m1) and m2(p,a,ni,mi) be the solution to (1)-(3).

2. The uninformedinvestor'sproblem with transactioncosts. Suppose now that the investor cannot buy and sell the sharesfor the same price; he can buy a share for b and sell a share for s. It is straightforward to modify (1)-(3) accordingly.The modifiedprogramis max E[U(w3)Ia], subject to (1) and b(n2 - n1) + m2 ' mlm2? s(n1

(4)

if n2> ni, if n 2 n2. J

(5)

- n2) + ml

Consider an investor who enters period 2 with a portfolio (n1, mi), learns the value of his a, and rebalances his portfolio optimally. His expected utility (i.e., the optimal value of the modifiedprogram[1], [4]-[5]) is denoted U*(nl,ml,b,s,a). The investor's optimalpolicy is as follows. (1) He buys n2(b,a,n1,m1) - n1 if this numberis positive. (2) He sells n1 - n2(s,a,nl,ml) if this numberis positive. (3) He does not buy or sell otherwise. (We assume that 1 and 2 cannot both take place.) LetD(b,a,n1,m1) = max[O, n2(b,a,n1,m1) - n1] (6)

andS(s,a,n1,m1) = max[O, ni - n2(s,a,n1,m1)] (7)

be the demand and supply of an investor. The aggregatedemandand supply of the uninformed D*(b) and S*(s) areD*(b) = E[D(b,a,n1,mO)] (8) (9)

andS*(s) = E[S(s,a,nl,m1)],

where the expectation is taken with respect to the parametera. 3. Informed investors. A subset of the investors are informed, in that they learn the realizationsof q and a simultaneously.Theirbehavior is also described by program (1)-(3) (or the modificationwhich allows for transactioncost), except that q is anotherconditioningvari-

Dividend Neutrality

S97n2(p,a,n1,ml;q)

able (in addition to a) in (2). Let

denote the solution of

the informedto (1)-(3). The behaviorof the informedtraderswho can buy for b and sell for s is similar to that outlined for the uninformed.Let U*(nl,ml,b,s,a;q) denote the optimal value of the uninformedinvestor's modifiedprogram. Similarlyto (6)-(7), we definethe demandand supplyof an informed individual:D(b,a,nj,mj;q) = max[O, n2(b,a,nj,mj;q) - nl, = max[O, n1 - n2(s,a,nj,mj;q)]. (10) (11)

andS(s,a,nj,mj;q)

The aggregatedemandand supply of the informed,conditionalon q, D*(b;q) and S*(s;q) are D*(b;q) = E[D(b,a,n1,m1;q)], and S*(s;q) = E[S(s,a,n1,mj;q)], where the expectations are over the parametera.4. Market makers and the determination of transaction costs.

(12) (13)Any

investor wishing to buy or sell shares in the second period must do so throughrisk-neutral competitivemarketmakers.These marketmakers set bid (s, in our notation) and ask (b) prices. The marketmakers are uninformed(i.e., they do not know q) and must set b > s in order to breakeven (on average)to overcome theirdisadvantagein tradingwith the informed. Furthermore,they post b and s before they know the aggregatedemandand supplyand cannotmake b and s dependentupon them. Three quantitiesare potentially relevant in determiningthe bid and ask prices. (1) The expected net income from selling shares isIS(b) = b{(l - Q)D*(b) + QE[D*(b;q)]} - E{(q + r)[(1 - Q)D*(b) + QD*(b;q)]}, (14)

where the expectation is taken over q and r. (2) The expected net income from buying shares isIB(s) = -s{(1 - Q)S*(s) + QE[S*(s;q)]} + E{(q + r)[(1 - Q)S*(s) + QS*(s;q)]}.

(15)

(3) The expected numberof shares the marketmakerholds isNS(b,s) = (1 - Q)[S*(s) + Q{E[S*(s;q)-

D*(b)] D*(b;q)]},

(16)

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where the expectation is over q. (IS standsfor income from selling, IB stands for income from buying, and NS stands for numberof shares.) Two sets of equilibriumconditions are studied. Underlyingthe first is the assumptionthat marketmakerscompete separatelyon each side expected income frombuying of the market.Therefore,in equilibrium, and that from selling shares must equal zero:IS(b) = 0, (17a)

andIB(s) = 0. (17b)

b If more than one b or s solves (17), the equilibrium is the smallest b that satisfies (17a) and the equilibriums is the largest that satisfies (17b). Another approach requires that the expected income of a market maker be zero and also that the expected inventory he holds be also zero. These amount toIS(b) + IB(s) = 0 (18a)

andNS(b,s) = 0. (18b)

(It is difficultto select among multiplesolutions here, unlike for [17]).C. The First Period

All investors enter the firstperiodidentical,endowed with the portfolio(no, mo). No investor knows the value of his a or whether he will be

amongthe informed.Every investor knows, however, that a fractionQ of the investors will be informed.A competitive stock marketis available in the first period. The share price p is derived below. Each investor will choose his portfolio (n1, ml) to maximize the condition expressed in (19):max {(1 - Q)E[U*(nl,ml,b,s,a)] + QE[U*(nl,ml,b,s,a;q)]}, (19) (20)

subject to

pn1 + ml pnO + mo.

identityof all investors)implies Marketclearing(and the first-period ni = no and hence the equilibriumfirst-periodprice p.III. Dividends

Suppose that before the second period's trade begins, the firmpays a dividend d to every shareholder.Investors know the dividend policy already in the first period. How does it affect portfolio selection and

Dividend Neutrality

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security prices? We are especially interested in the relation between the dividend d and the first-periodshare price p. A dividendpolicy is optimalif it maximizes the initial share price p. The specificationof the dividend d is not complete until we specify the source of d, the money paid in the second period. I assume here and in the sequel that an amount equal to the dividend is subtracted from the firm's third-periodper-sharevalue, q + r. Thus, a firmthat pays d in the second period will pay only q + r - d in the third. This section's results are summarizedin theorem 1 below. They are proven in the subsequentlemmas. THEOREM Consider two dividend payment regimes. In the first, 1. no dividendis paid, and in the second a dividendd is paid. The following quantitiesremainintact across the two regimes:(1) the first-period share price, p; (2) the realized utility and expected utility of every investor in every state of nature;(3) the numberof shares held by an investor in the first and second periods. The prices at which investors can buy and sell shares, b and s, are lower by the amountof dividendd in the second regime. At this point we must alter the previous section's notations to accommodatethe dividendpayment. We changethe notationswholesale, for the sake of brevity and at the risk of ambiguity.Namely, we add another parameter, d, to the quantities defined above, for example, U*(nl,ml,b,s,a;d), D(b,a,nl,ml;d), S(s,a,nl,ml;d), and so on. Our first result shows that the dividend d has limited effect on the solution to the modifiedprogram(1), (4)-(5). Indeed, lemma 1 below shows how the dividend d alters the solution to (1), (4)-(5). LEMMA1. Consider an investor with end-of-period-Iportfolio (nl,ml). Suppose d = 0, and fix the second-period prices to be b*, and s*.

Suppose that for a given a (and q, if the investor is informed) his optimalperiod-2portfoliois (n*, m*). Finally, suppose thathis period-3 expected utility is U*. If the firm's dividendpaymentchanges from 0 to d, and if the priceschange from b* and s* to b* - d and s* - d, then the investor's

optimalperiod-2portfolio changes to (n*, m* + dn*) and his period-3 expected utility U* does not change. investorwho receives dividendd in Proof. Consideran uninformedthe second period and faces prices b* - d and s* - d. His problem is

to max E[U(w3)Ia], subject to (b* - d)(n2 - ni) + m2? ml + dn, if n2 > ni] dn 'S n+(22) M2 < (S* - d) (n - n2) + Ml + dn, if nl1 -n2, J-

(21)

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andW3

= (q +

r - d)n2

+ m2 + y

(23)

It is straightforward check that the terminalwealth W3affordedby to (22)-(23) and (n, m + nd) as the portfoliois identicalwith the terminal wealth afforded by (1) and (5), the prices b* and s* and (n, m) as the solution. Thus, the sets of terminalwealth levels are identical in both (1), (4)-(5), and (21)-(23). Thereforethe optimalityof (n*, m*) for (1), (4)-(5) with b* and s* as prices is equivalentto the optimalityof (n*, m*+ dn*) for (21)-(23) with the prices b* -

d and s* - d. That U*

remains intact follows similarly. The next lemma takes up the equilibriumperiod-2 prices, which satisfy either (17) or (18). I assume that all investors' beginning of period-2 portfolios are fixed across different dividend payment regimes, and I show that a dividend payment of d causes a cut of d in both b and s. LEMMA 2. Consider two regimes of dividend payment. In the first regime no dividendis paid in period 2, and in the second, a dividendd is paid. Suppose the end of period-I portfolio(n1,m1) of every investor is the same across the two regimes (but not necessarily across investors). If b* and s* satisfy (17) or (18) in the no-dividendregime, thenb* - d and s* - d satisfy (17) or (18), respectively, if the firm pays a

dividend d. Proof. Check that IS(b*), IB(s*) and NS(b*,s*) in the first regimeare equal to IS(b* - d), IB(s* - d), and NS(b* - d,s* - d), respec-

tively, in the second. Consider, for instance, IS, which is defined in (14). By lemma 1, D(b*) and D*(b*;q)in the first regime are equal toD(b* - d) and D*(b* - d;q), respectively, in the second. Between the

first and second regime, the income per share sold is reducedfrom b* to b* - d. This reduction is offset by the reduction in the market maker's per share inventory cost (conditionalon q), which changesfrom q + r to q + r - d. The net effect leaves IS(b* - d) equal to

IS(b*). So far we have consideredthe effect of changingdividendpolicy on the behaviorin period 2. We have establishedthe equivalencebetween period 2's prices, portfolios, and expected utility levels, assumingthat period-I behavior is unaffectedby a change in dividendpolicy. In the next lemma I justify this assumption. LEMMA 3. Consider two regimes of dividend payment. In the first regimeno dividendis paid in period 2, and in the second a dividendd is paid. Then, the end-of-period-Iportfolio(n1, m1) of every investor and period 1 share price p are the same across the two regimes. Proof. Comparethe indirectutilities U*(nl,ml,b*,s*,a) and U*(nl, mi,b*,s*,a;q) in the firstregimewith U*(nl,ml,b* - d,s* - d, a;d) andU*(ni,mi,b* - d,s* - d,a;q;d) in the second. By lemma 1, they are

Dividend Neutrality

S1o0

equal (across the regimes)for all values of the portfolio(n1,m1). Therefore, given a period-I share price p, every investor's optimalperiod-I portfoliodoes not dependon period2's dividendpolicy. Consequently, period-I's share price, p, is independentof the dividendpolicy. IV. Robustness Dividend irrelevance in the presence of transactioncosts holds under more general assumptionsthan those made in the precedingsection. I entertain a few generalizationsand point out why they do not affect dividend irrelevance. 1. It is often mentioned that a generous dividend policy attracts investors who need liquidity. These people presumablyprefer to receive a dividend ratherthan sell shares to finance their consumption. This view is not supported by the model; if we incorporatesecondperiod consumptioninto the model, dividendirrelevanceis still valid. The reason is that lemma I remainsvalid, and the other results follow from it when we apply backwardinduction. Second-periodconsumptioncan be incorporatedinto the model by allowing the utility function U to depend not only on third-period wealth, W3, also on second-periodconsumption,say c2, and adding but that consumptionc2 to the left-handsides of the second-periodbudget constraint(5). 2. The extension of the model to more than 3 periods is straightforward but entails cumbersome notation. The results are similarif the model contains K + 2 periods such that the firmliquidatesin the last period and a competitive market with symmetric informationtakes place in the firstperiod. The shareprice in the firstperiodis unaffected by the dividend policy duringthe intermediateK periods, althoughin these periods, investors may have to buy shares for higherprices than those at which they can sell. 3. It has been assumedthat the dividendis known alreadyin the first period. What if it were known to the investors only in the second period and in the first period every investor merely has a subjective probabilitydistributionover the possible second-perioddividendpayouts? Lemmas I and 2 above are left intactunderthis new assumption. The statement of lemma 3 is also correct, but in the proof one has to replace U* by its (subjective) expected value over the possible dividend payments. 4. Suppose the firm's liquidationvalue is the sum of three random variables, r, q, and z, and suppose the realizationof z is public knowledge at the beginning of period 2. Even if the dividend payment deprice or pends on z, d = d(z), it will not affect the share's first-period investors' welfare. To see this, note that once z is known, the prices b* and s* depend on z. Subject to this modification,lemmas 1, 2, and 3 remainthe same as before.

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5. Individualsare assumed homogeneous as they enter (and leave) period 1; subsequently, either a person knows q (the informationregardingthe ultimatevalue of the shares) or not. The three lemmas are robust to heterogeneity in initial endowments. Also, the simple information structure in period 2 could be generalized to allow different individuals to possess different signals regardingthe firm's ultimate value (ratherthan either know q or not). 6. Preferences are assumed identical in period 1. Again, this is not crucial; allow the investors to possess, in period 1, informationabout period 2 realizationof their a's and you do not vitiate any of the three lemmas. 7. We assume that investors find out that they are informed (i.e., know q) only in period 2. Whatif some of the informedinvestors knew that they were informed(but they did not find out the realizationof q) already in period 1? Again, none of the three lemmas is vitiated. 8. We assume that the rate of returnon the marginal project is zero, that is, that one dollarin period 1 becomes one dollarin period2, which becomes one dollar in period 3. The results do not depend on this assumption or on the assumption that the marginalproject is safe. What is importantis that the rate of return on the marginalproject availableto the individualis equal to the rate of returnon the marginal projectavailableto the firm.(Contrastthis with Huberman[1984],who assumes otherwise and concludes that externalfinancingis important.) V. Administrative TransactionCosts The cost of transactingis also the time cost of the contact with the market(and the marketmaker)and the clerks en route to it. It is also the cost of the paperworkand the utilization of the communication network. I bundle all these as administrative costs and model them as having a fixed size c per sale (or purchase)of stocks. I arguebelow that dividendpolicy is irrelevantto shareholders'welfare (or to share prices) in the presence of administrativetransaction costs. The argument is similar to that in the case of informationinduced transactioncosts. ThereforeI describe it brieflyand stop with lemma 4 below, which is the analogueof lemma 1, the crucial step in proving dividend irrelevancein the previous sections. To study the impact of administrativetransactioncosts I resort to the 3-period model used in the previous sections but modify it by disposingof asymmetryof information(and the bid-askspread).Thus, I set q = 0. The analysis of the model is similarto the earlieranalysis, namely, I first study equilibriumin the absence of second-perioddividend payment and then introduce dividend distributionand show that every in equilibrium the new regimeis equivalentto an equilibrium the noin dividend regime and vice versa.

Dividend Neutrality

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Suppose no dividendis paid in the second period. Denote the share price in the second period by P, and define the functionQX fO if x = 0, C(x)=c otherwise. (4

(24)

The optimization problem faced by an investor in the second period is max E[U(w3)Ia], subject toW3=

(25)(26)

n2r

+

m2

+

Y,

andP(n2 - nl) + m2 - ml+ C(n2 - nl) < 0. (27)

From (25)-(27) derive the individual'sdemandfunction, aggregateit over the investors (the a's) and find the second-periodmarket-clearing price P. Given the equilibrium price P, compute an investor's indirect utility as a function of his end-of-period-iportfolio(n1, m1). Investors choose theirend-of-period-Iportfoliosin orderto maximizethose indirect utilities, and hence period l's market-clearing share price p. Suppose that the firmdistributesa dividendd per shareat the beginning of period 2, and that the money is subtractedfrom the liquidating dividendof period 3. Then the optimizationproblemfaced by an investor in period 2 is max E[U(w3)Ia], subject toW3=

(28)Y, (29)

n2(r

-

d)

+

M2

+

and P(n2 - n) + m2 - ml - nld + C(n2 - nl)

.

(30)

Recall that in the case of information-induced transaction costs lemma 1 was the crucial step in provingthat shareholders'welfareand period-i share price were not sensitive to dividend policy. The analogue of lemma I is lemma 4. The proofs are also similar. LEMMA 4. Consider an investor with end-of-period-iportfolio (n1,m1). Suppose d = 0, and fix the second period price to be P*. Suppose

thatfor a given conditioningvariablea, his optimalperiod-2portfoliois(n *, m?*).

If the firm's dividend policy changes from 0 to d, and if the pricechanges from P* to P* - d, then the investor's optimal period-2 port-

folio changes to (n*, m* + dn*). Proof. First, note that the terminalwealth, W3, affordedby (n2,M2) under (26) and that afforded by (n2, m2 + dn2)under (29) are equal.

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Second, note that if P = P* in (27) and P = P* - d in (30), then (n2, satisfies (27) if and only if (n2, m2 + dn2) satisfies (30). Thus, m*) optimalityof (nW, underthe no-dividendregimeis equivalentto the optimalityof (n*, m* + dn*) in the second regime.M2)

VI.

An Example with Exponential Utility and Normal Payoffs

I return to the information-inducedtransaction costs to illustrate how trade can come about, and I performcomparativestatics regarding the buy-and-sellprices b and s. Suppose the utility function U is exponential:U(w) = - e-W. (31)

Suppose also that the randomvariables(q, r, y) are trivariatenormal with covariance matrix V. The diagonalelements of V are vqq, Vrr, and vyy.The off-diagonalelements are zero, except the covarianceof r and y, which is denotedby vry. This covarianceitself is a randomvariablein period 1 but becomes known to every investor in period 2. Thus, v,y is the parametera in this example. to It is straightforward solve (1)-(3) both for the informed(those who know q) and the uninformedinvestors. The solutions aren2(p,Vry,n,mlm) =

[q + E(r)

-

v

- P]IVrr,

(32a)

for the informed, andn2(p9,Vry,ni9,m) =

[E(q + r) -

Vry -

P/I[Vrr + Vqq],

(32b)

for the uninformed. When (1)-(3) is replacedby (1), (4)-(5), we can definecutoff covariances for the uninformedv,.*(b) and v *(s). An uninformedinvestor whose v,y > v *(s) will sell shares, and an investor whose v,y < v *(b) will buy shares. Similarly, we define the cutoff covariances for the informedinvestors, v,.*(b;q)and v,.*(s;q). The cutoff covariancessatisfyn2[b,v *(b),ni,mn] - ni = 0 (33a)

andn2(s,v,*(s),n1,m1) - ni = 0, (33b)

for the uninformed;they satisfyn2[b,v,.*(b;q),n1,m1] =

00,

(33c)

andn21s,v,*(s;q),ni,m1]-

n=

(33d)

for the informed.

Dividend Neutrality

S105

Equations(33a)-(33d) can be solved explicitly to v *(b)= [E(q + r) -

b]

-

(Vrr + Vqq)nl, (Vrr + Vqq)fli,

(34a)(34b)

v *(s) = [E(q + r) - s]

v, *(b;q) = [q + E(r)v,*,(s;q) = [q + E(r)

b] s]

-

Vrrfl, Vrrfl1.

(34c) (34d)

Note that if b > s, then v,*(b) < v,*(s) and v,*(b;q)< v,*(s;q).Investors whose personal incomes (y) are highly correlatedwith the investment income (q + r) sell shares, whereas investors whose v, 's are low buy shares. How do the buy-and-sellprices, b and s, vary with the variance of the informed'ssignal Vqqand with the probability beinginformed,Q? of The higher Vqq, the better the informationof the informed.The higher Q, the more menacing it is to the market maker. I consider the equilibrium condition (17a)-(17b), and let b* and s* be the prices satisfying (17a)-(17b). The dependence of b* and s* on Vqqand Q is summarizedin proposition 1. PROPOSITION Under the equilibrium 1. (17a)-(17b), the buy price b* increases with Vqqand Q; the sell price s* decreases with quantities. Proof. Considerfirstthe dependenceof b* on Vqq. Equations(17a)(17b)imply that the revenue function IS(b) is increasingat b*. (Otherwise, competitionwould drive the equilibrium price b* lower.) It is buy evident from (6), (8), and (32b) that the aggregatedemandof the uninformedD*(b) decreases as Vqqincreases. Completelydifferentiate (17a) with respect to Vqqand apply the above observationsto conclude thatb* increases withVqq.

Under (17b), s* decreases with Vqq. The argumentis similarto the one above. The revenue function IB(s) is decreasingat s*. The uninformed's supply, S*(s), decreases as Vqqincreases. Next, study the dependence of b* and s* on Q. The derivative of IS(b) with respect to Q is the expected revenue per informedinvestor minus the expected revenue per uninformedinvestor. The expected revenue per informed investor is negative, and that per uninformed investor is positive. (Recallthat their Q-weightedaverge is zero at b*.) Therefore, IS(b) decreases with Q. Since IS(b) also increases with respect to b at b*, the equilibriumb* must increase with Q. A similar argumentshows that the equilibrium decreases with Q. s* VII. Conclusion The originalintent of this work was to show that dividendpolicy affects investors' welfare and securityprices if transacting costly. For is the transactioncosts I consider, I reach the opposite conclusion.

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Journal of Business

Millerand Modigliani(1961) argue that dividendsdo not matterbecause the investor can buy (if the dividendpaymentis too high for his taste) or sell (in the opposite case) shares to suit his needs. This argument is suspicious in the presence of transactioncosts. Although transactioncosts are present in my model, dividendpayments have no effect on shareholders'welfare. A dollarpaid today is a dollarmissing tomorrow.This forward-looking observationimplies an adjustmentin investors' optimal portfolios in reaction to changes in dividend policy. These adjustmentsneutralizethe change in dividend payment, notwithstanding transactioncosts. theReferencesBrealey, Richard,and Myers, Stewart. 1984. Principles of Corporate Finance. 2d ed. New York: McGrawHill. Glosten, LawrenceR., and Milgrom,Paul R. 1985.Bid, ask and transaction prices in a specialist marketwith heterogeneouslyinformedtraders.Journal of Financial Economics 14:71-100. Huberman,Gur. 1984.Externalfinancing liquidity.Journal of Finance 39:895-908. and Miller,MertonH., andModigliani, Franco. 1961.Dividendpolicy, growthandthe valuation of shares. Journal of Business 34 (October):411-33.