Ability, Moral Hazard, Firm Size, And Diversification

8/10/2019 Ability, Moral Hazard, Firm Size, And Diversification

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Ability, Moral Hazard, Firm Size, and DiversificationAuthor(s): Debra J. AronSource: The RAND Journal of Economics, Vol. 19, No. 1 (Spring, 1988), pp. 72-87Published by: Blackwell Publishing on behalf of The RAND CorporationStable URL: http://www.jstor.org/stable/2555398Accessed: 23/08/2009 22:00

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RAND Journal of EconomicsVol. 19, No. 1, Spring 1988

Ability, moral hazard, firm size, and diversificationDebra J. Aron*

I develop a model offirm diversification into multiple product lines that is based on theagency problem between the firm's managers and owners. The agency relationship, togetherwith a span-of-control managerial technology, determines an optimalfirm size and degreeof diversification hat are increasing in the manager's ability and therefore positively correlatedcross sectionally. I compare the benefits of merger with those achieved by using compensationcontracts based on relative performance and show that, for a particular parameterization,the relative value of merger is a nonmonotonic function of the correlation between the pro-ductivity signals of the two firms.

1. Introduction* Neoclassical microeconomic theory has generally treated the firm as being identical toa technologically determined production function. Nevertheless, it is widely recognized thatthis cost-curve approach is more appropriately applied to plants within a firm than to thedetermination of firm size or structure itself. One aspect of firm structure that has receivedespecially little attention in the economic literature is diversification.' In particular, theliterature has not succeeded in distinguishing the benefits of efficiently using capital inproduction from the benefits to common ownership of the capital. If joint use of capitalcreates efficiencies in the production of two or more goods, the joint use could in principlebe achieved by contracting over the use of the separately owned factors. This ownershipstructure does not preclude the efficient use of factors. Further, even if technological scopeeconomies create incentives to diversify, they cannot explain all diversification, becausemuch that we observe is between products that are (apparently) unrelated in productiontechnology or demand.

The purpose of this article is to analyze the incentives of firms to diversify n an economycomprising managers and capital owners whose interests do not necessarily coincide andin which information is generally not perfect. I derive the implications of the principal-agent relationship between owners and managers of firms for the optimal structure of thefirm in a competitive environment. Diversification is shown to be an optimal response to

* Northwestern University.This article is based on my Ph.D. thesis at the University of Chicago. I wish to thank my committee members,

Edward P. Lazear, Sherwin Rosen, and especially my chairman, Sanford Grossman, for many helpful discussions.I am also grateful to Alvin Klevorick, an Associate Editor, and an anonymous referee of this Journal, whosecomments greatly improved the manuscript. Financial support from the Center for the Study of the Economy andthe State at the University of Chicago is gratefully acknowledged.

'See, for example, Scherer and Ravenscraft 1984), Salter and Weinhold (1979), and Gort (1962) for empiricalevidence on the importance of diversification.

72


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the moral-hazard problem facing firms' owners. The model yields implications for firm size,degree of diversification, and choice of merger partners, as well as for the relationship betweenmanagerial compensation and firm size and the tradeoffs between merger and relative-performance evaluations.

An alternative approach to explaining the characteristics of acquisition targets focusesentirely on ownership of capital within a firm. For example, it has been suggested that firmsattempt to diversify into products whose profit streams are negatively correlated over timewith their primary product. In this way firms can avoid the secular variability associatedwith business cycles and demand or cost shocks, thereby lowering the risk facing investorsin the firms. Merger would appear to be a costly means of achieving this sort of risk reduction,however, since portfolio diversification can perfectly replicate these benefits of firm diver-sification for an investor.

A related approach views diversification as a means of reducing the risk of bankruptcy,where bankruptcy creates a deadweight cost that is not incurred if a product line within afirm goes out of business. The bankruptcy argument suggests that firms with profitabilitystreams that are negatively correlated with the acquirer's profits are the most desirabletargets. This contrasts with the empirical implications of my model, which are derived inSection 6.

The capital-ownership and bankruptcy explanations of characteristics of diversifyingfirms do not provide a model of the relationships among firm size, product line size, anddiversification. But empirical studies of these issues by Salter and Weinhold (1979) andespecially a major study by Gort (1962)2 have yielded two important regularities. First thereis a strong positive association between firm size and the number of industries in which thefirm operates. Second, the size of each product line (measured in these studies by employ-ment) increases with total firm size. These regularities do emerge as implications of themodel of firm diversification developed here.

In this article the size and structure of firms are determined not only by the productiontechnology, but also by the characteristics of the firms' managers and the incentive problemsinherent in the separation of ownership and control. As in Lucas (1978), optimal firm sizeis determined by the (exogenous) talent or managerial expertise of the manager. I assumethat this manager supplies the amount of effort to the firm that maximizes his utility, giventhe rewards and constraints he faces. To analyze this problem I adopt the methodology ofthe principal-agent iterature (Holmstrdm, 1979; Grossman and Hart, 1983). Diversification

is valuable because it mitigates the principal-agent problem by allowing the principal moreaccurately to infer the manager's behavior. For any given firm size, this leads to a tradeoffbetween increasing the diversification of the firm, thereby reducing agency costs, and in-creasing the size of each product line, thereby reducing production costs. In equilibrium,optimal firm size, product line size, and diversification are positively related.

I then compare diversification with relative-performance contracts as means of de-creasing agency costs, under the assumption that the technology of the manager's job con-strains him from applying different effort levels to different product lines. Although relative-performance evaluations are valuable, they are not perfect substitutes for diversificationwith respect to the agency benefits that each can provide. Which method dominates depends

on measurable characteristics of the observed signals on managerial input. Evaluating amanager's performance by comparing it with the performance of another firm can be valuablewhen the exogenous shocks affecting the two firms are correlated. On the other hand, eval-uations based on the productivity of different product lines that are under the common

2 Cross sectional tests on the relation between firm size and diversification were performed on 721 enterpriseswith more than 2,500 employees. Diversification was measured in several ways, including the ratio of primaryindustry output to total output for the firm, a simple count of the number of industries in which the firm is engaged,and a composite of the two. The results were qualitatively the same, regardless of the measure used.


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control of the firm's manager sharpen the measure of the manager's performance when theexogenous components of productivity across product lines are relatively uncorrelated. Ishow that the benefit to diversification vis-a-vis relative-performance contracts is not mono-tonic in the correlation between the product-specific signals, is positive over a broad rangeof correlations, and is maximized in the region of negative correlations.3

Other authors have considered the effects of managerial moral hazard on the firm(Marcus, 1982; Diamond and Verrecchia, 1982; Ireland, 1983). Work by Ramakrishnanand Thakor (1986) is closely related to the analysis in Section 6. They also analyze the valueof diversification relative to performance-comparison contracting from an agency perspective,but in a somewhat different setting. In their model, when firms merge, both incumbentmanagers maintain their positions, and after the merger, each partakes in the managementof both divisions. Consequently, they focus on issues of competition and collusion betweenthe managers that do not arise here.

In Section 2 I describe the technology of a firm, and in Section 3 I discuss the principal-agent problem and derive relevant properties of the solution. In Section 4 I analyze thefirm's optimization problem, and in Section 5 I derive comparative statics on firm size anddiversification. The implications of the model are then evaluated in light of the cross sectionalstylized facts. Section 6 analyzes the implications of the model for the firm's optimal choiceof merger partner when relative-performance comparisons are a valuable and importantcomponent of managerial incentive contracts. Section 7 contains a summary of the resultsand conclusions. The proofs of results in the text are available from the author.

2. The technology of a firm

* I consider a highly stylized model of the firm. A firm is defined to be a collection ofassets, K, organized within a technological production function and overseen by a manager.We can think of K as the total capital owned by the firm, or as a vector of inputs, includinglabor; here I treat K as capital, but this is without loss of generality. The manager I have inmind is the top decisionmaker, typically the chief executive officer, whose job is to organizethe factors under his control to increase their efficiency or total output.

Each manager in the economy has ability or managerial talent 0, which is exogenouslyendowed, and he exerts some effort, a, which he chooses. Different managers may havedifferent abilities, and ability has some distribution in the population F(O). further assume

thatthis

managerial abilityis

general:a

manageris equally productive in any industry.

Those qualities that make a manager successful involve organizational skills, business in-tuition, knowledge of economywide trends, and so forth.

A firm may produce several products. Letf(k) be the technological production func-tion associated with producing a particular output. We may think of this as the plant'sproduction function and of k as the capital per plant. The function f(k) is defined bymax {0, 1(k) c}, where 1( ) is an increasing, strictly concave function, and c is a positiveconstant (interpreted as a fixed cost). Thus, plants are characterized by U-shaped averageand increasing marginal cost curves. By assumption, any particular plant can produce onlyone product. Firms may choose to have many plants, and these plants may produce the

same product or different products.Let the number of plants producing product i be ni and the number of products thefirm produces be m. Then m describes the degree to which the firm is diversified. I assume

3 Radner and Rothschild (1975) also take a managerial approach to the determination of firm structure. Intheir model managers must allocate their time among several products or projects in which the firm is engaged.That article does not explicitly treat the incentives for diversification, because the number of projects requiringattention is exogenous. Nevertheless, the idea is appropriate to a theory of diversification and is complementary tothe one developed here.


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Assumption 1. Firms' owners are risk neutral, so that they maximize expected profits.5

Assumption 2. All firms are price takers, and the price of all products is the same and equalto p.6

Assumption 3. The relevant sector of the economy is, as a whole, a price taker in the capitalmarket. Call R the exogenously determined market rate of interest or opportunity cost ofcapital.

Assumption 4. The functions g( * andf(*), the amount of invested capital ki1 or all i andj, the ability level of each manager, and the form of the production function are commonknowledge.

Assumption 5. The level of output of each product qi is observable to both the principaland the agent (and to any third party enforcing the contract).

Finally, define p to be the expected payment to the manager of the firm, and let w bethe expected value of Ei or all i. Then the firm's expected profit ism m ni

Pa~h(> yj)w - R Z Z> j - p. (4)j=1 i=1 j=1

Before analyzing the firm's maximization problem, we must investigate the nature of thisterm p, the manager's compensation.

3. The moral-hazard problem

* I adopt the standard principal-agent methodology. For simplicity, I shall assume thatthe manager's (unobserved) effort can take only two values, high (a*) and low (a'), and thatfirms will always find it optimal to implement the high level of effort.7

The manager or agent is assumed to be risk averse. He has a von Neumann-Morgensternutility function U(I, a) = B(a) W(I) - G(a), where W is a real-valued, continuous, strictlyincreasing, and strictly concave function on the interval [d, oo) and W= -co for all I < d;G( ) and B( . ) are real-valued functions with B(a') > B(a*) > 0 and G(a*) > G(a') > 0.Finally, I assume that B(a')W(I) - G(a') > B(a*)W(I) - G(a*) for all I.

The principal's problem is to choose capital inputs K = (k,1, .. . , kmnm) nd the man-ager's ncentive contract I(ql ,. . . , qm K, 0) subject to incentive-compatibility and individual-

rationality constraints. This is a standard principal-agent problem with two wrinkles. First,the principal must simultaneously choose the contract and K. Second, the observed signalson effort, on which the contract is based, are functions of the chosen K as well as themanager's ability 0. If the expected cost or form of the optimal contract implementing a*,given K, depended in some complicated way on K, this problem could be quite difficult. Infact, the expected cost of the optimal contract is independent of K, and the form of thecontract depends trivially on K and 0. Given the optimal contract for any arbitrary evel ofK or 0, the optimal contract for any other level of K or 0 is a simple transformation of thatfirst contract.

This result is intuitively immediate. Observed output is a monotonic function of as,and this monotonic function is common knowledge by Assumption 4. Thus, for any level

5 The model treats shareholders or owners of the firm's capital as one principal. Any strategic interactionor delegation problem among the owners is ignored.

6 One can show that under appropriate assumptions on consumers' preferences, competitive equilibrium forthis economy is characterized by the same price for all products. See Aron ( 1985) for a discussion of the competitiveequilibrium.

7 The crucial assumption for what follows is that the optimal effort level is independent of managerial ability,invested capital, etc. Given this assumption, restricting the set of a's to two is merely a convenience.


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of capital or ability, observed output can be unravelled from equation (2) to arrive at as,which we can treat as the underlying signal on effort. One need only solve for the optimalcontract 1(z), where z af. For any levels of capital and ability, the optimal contract as afunction of output q is simply I(z) with the domain rescaled.

This establishes that the principal's contracting problem when capital is a choice variableis fundamentally identical to the standard principal-agent problem. In particular, under ourrestrictions on the utility function,8 an optimal contract exists.

The moral-hazard problem between the capital market and the manager results becausethe owners receive only an imperfect signal of the effort of managers. Intuitively, one wouldexpect that the severity (i.e., costliness) of the problem would increase with the noisiness ofthe signal. Holmstrdm (1979) and Grossman and Hart (1983) have shown this to be thecase for particular kinds of noisiness. This same intuition underlies the benefits to diver-sification in this model, as I now describe.

We defined p to be the expected payment made to the firm's manager. Since the managerhas disutility for effort, p must be a function of the level of effort the manager is inducedto expend. Then, if effort level a* is implemented, p = p(a*). Consider a firm that producesonly one product, and let p I(a*) be the expected payment to the manager of the firm if a*is implemented. Then pl(a*) is the solution to (5):

pi(a*) = min I(z)ir(zla*)dz (5)IAz)

subject to

f U(I(z), a*)lr(z Ia*)dz > f U(I(z), a')ir(z Ia')dzand

U(I(z), a*)lr(z Ia*)dz > U0,

where U0 is the agent's reservation utility and lr(z I ) is the conditional probability densityfunction of z.

Now suppose that two projects can be undertaken simultaneously and that they haveindependent and identically distributed Ei, .e., they are both described by the distributionfunction lr(z I ). The payment scheme will be some function I(z1, Z2), where zi = afi is theoutcome of project i, given the capital input and ability. The problem facing the principalin this case is (6):

miz ff f(z1, Z2)r(z1I Ia*)r(z2 I a*)dz Idz2 (6)AI(Z ,2)

subject to

U(f(z , Z2), a*)lr(zl I a*)lr(z2 I *)dzi dz2> U(f(zI, Z2), a')ir(z1 I ')7r(z2 I a')dzIdZ2

and

fJ U(f(z1, Z2), a*)lr(zi I a*)7r(z2 Ia*)dzIdz2 > UOLet the solution to this problem be p2(a*).

Lemma 1. pi(a*) > P2(a*).9

8 Some technical integrability restrictions on the admissible set of contracts are also necessary. For a proofof existence, see Clarke and Darrough (1980). The assumption on W that bounds it below is crucial because itavoids the kind of nonexistence discussed by Mirrlees (1975).

9 Diamond and Verrecchia (1982) prove this result for a particular utility function (in the hyperbolic absolute-risk-aversion class).


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The intuition of the proof is as follows. When the principal gets two independentobservations on the agent's effort, he can reduce the risk facing the agent and still inducehim to choose a*. Since the principal must pay the agent to incur risk (because the agentrequires some minimum utility), the cost to the principal decreases when the risk facingthe agent falls. One feasible way to reduce the risk facing the agent while preserving hisincentives is to pay an amount for the observed pair of outcomes that would yield someaverage of the optimal utility payments for the two separate outcomes. That is the schemeconstructed in the proof.

Although the proposition is written for two projects, it clearly can be applied to anynumber of independent projects. Let I*(zm) be the optimal contract for zm = {Z I ,. . ., Zm }Iand let I(zm, ZmI) be the contract for the (m + 1) projects. The proof proceeds identically.Hence, we have established the following corollary.

Corollary 1. Let the cost of the optimal contract when the number of projects is m be

p(m, a*, UO).Then Ap(m, a*, Uo)/Am < 0. That is, the cost of inducing a given level ofeffort will decline as the number of product lines increases.

This result is related to results in Grossman and Hart (1985) and Holmstrdm (1979)on the value of signals. It can be interpreted as a proof that the (m + 1 th signal is indeed

informative, as defined by Holmstrom (1979).To get an idea of the magnitude of the effect, I ran some simple simulations. 10 Assuming

an additively separable log utility function for the manager, I calculated a lower bound onthe efficiency gains to diversification by using several values for the risk aversion parameter.For a benchmark case, I found that when the manager has constant relative risk aversion

equal to unity, a firm could acquire a second product line and pay the manager 29% less(in expected value) without disturbing his incentives and leave him indifferent. Under theassumptions of Section 2, this efficiency gain will be captured by the manager. This meansthat by acquiring a second product line, the manager will raise his expected utility by anamount corresponding to 29% of his expected income each year. Results for a broad rangeof risk-aversion parameters were similar.

The results in this section describe the efficiency benefits from diversification that arisebecause of the incentive problem. I have established that at any given evel of scale, the costof providing incentives for the manager is a decreasing function of the degree of diversifi-cation. Naturally, the optimal degree of diversification in a firm will also depend on the

production technology and the production-efficiency consequences of increasing scale. Ipursue these issues in the next section.

The results in Lemma 1 and its corollary are robust to many of the assumptions madehere. First, the asumption of general rather than product-specific ability is merely a con-venience. I could have written ability as a vector 0 = { , . ., Om1}, where 6' denotes themanager's ability in the production of product i, and all of the proofs to this point wouldproceed identically. As in any model of diversification, product-specific ability would mitigatethe benefits of diversification, but it would not affect the incentive benefits, which are thefocus of this article. In addition, it is not necessary that ability be common knowledge. Aron(1985) analyzed the model in the context of incomplete but symmetric information about

0. The results in this article hold with 0 replaced by its estimated value.More important, it is not crucial that effort be a general input into the firm's production

function. Suppose that the technology of the firm enabled the manager to choose differenteffort levels for different products (in the spirit of, say, Radner and Rothschild (1975)).Then the manager's effort would be described by a vector a = {a, . ., am }, where ai isthe effort invested in product i, and the incentive problem would be to design a contract

10Details of the simulation procedure and results appeared n an earlier version of this article and are availablefrom the author.


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that implements effort al* n product 1, a* in product 2, etc., where in general these optimaleffort levels may not be the same. We could think of this as m contracts with contract iimplementing ai as a function of ai i for i = 1, . . ., m, while taking into account theexistence of the other contracts. When a manager holds m contracts, he could get a badoutcome (i.e., low realization of

e)in

anyone of

them, but a bad random outcome in oneproject is relatively likely to be offset by a good outcome in another project. Thus, looselyspeaking, the manager is effectively less risk averse with respect to the randomness of anyone project. This decreases the total compensation for risk required by the manager. For aformal proof that diversification improves risk sharing when effort is chosen separatelyacross products, see Ramakrishnan and Thakor (1986).

Although the assumption of general managerial input is not necessary to generate anagency incentive for diversification, the assumption of general managerial input is crucialto my comparison of merger and relative-performance contracts in Section 6. The resultsin that section do not hold if the manager may choose to apply different levels of effort

across product lines.Third, I have assumed that the output of each product line is observable within a firm.Suppose that only the total firm output were observable. Under the assumption once againthat effort is a common input in the firm, it is evident that, when firm size is held constant,increasing diversification will improve the power of inference on a. This reduces the riskthat must be borne by the manager, and therefore decreases the compensation required toprovide any given level of expected utility.

Finally, it is not necessary that the random variables Eibe independent across products.As long as the variables are not perfectly correlated, there is a benefit to diversification. Iexplore this possibility in Section 6.

4. The firm's problem

* Using the results of Section 3 and the model presented in Section 2, I can rewrite thefirm's maximization problem as:

m m ni ni m ni

max P z [a*Og(2 f(kjs)) a, f(ki,)w] - R z z kij - p(m, Uo), (7)nimki i=1 j=l s=i v=1 i=1 j=1ni,ma 1

i=1,...,m

where I suppress the a* in p(m, a*, UO).Because the plant's production function l(k) s concave, it is evident that for any product

it is optimal to invest the same amount of capital in each operating plant. Further, by theassumed symmetry of prices and technologies, it is optimal to have the same total numberof plants producing each product. Then I can simplify expression (7) to

max Pa*Oh(nmf k))w - Rnmk - p(m, UO). (8)

Notice that n and m enter expected profits perfectly symmetrically, except for the manager'scompensation function p(m, UO). ince p is decreasing in m, the firm will always prefer to

increase output by increasing m (the number of products) rather than n (the number ofplants producing each product). For any given total output, the firm would prefer to set nas small as possible and m as large as possible. Thus, the lower bound on n will be bindingat the optimum n and m, and I can set n = 1 to solve for the optimal m and k. For simplicityI shall henceforth treat m as a continuous variable, even though, by its nature, it can onlyassume integer values.

The first-order conditions for the firm's problem are:

Pa*Of k)h'(mf(k))w Rk - pm(m, Uo) = 0 (9)


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Pa*Omf (k)h'(mf(k))w - Rm = 0, (10)

where pm s the partial derivative of p(m, UO)with respect to m.We can immediately make the following observation. Taking the ratio of (9) and (10)

gives f/kf' = 1 + (pm kR), (11)

which implies that ftkf ' < 1 by Corollary 1. Thus, we have the following proposition.

Proposition 1. The optimal amount of capital per plant (now interpreted as per productline) is at a point to the left of the minimum of the average cost curve.

This result reflects the fact that there is a tradeoff at the margin between increasing theefficiency of a plant by decreasing average production costs and decreasing managerial costsby increasing diversification.

Returning to expression (8), consider the case in which there is no moral-hazard problem.In that case the manager's payment will not depend on m, and the firm is indifferent betweenusing its plants in the production of different products or having them all produce the sameproduct. Let there be any small positive cost of diversification and firms will always chooseto structure their optimal firm size by increasing n and choosing m = 1. This means thatin the model it is the moral hazard nherent to the corporate form that induces diversification.

The efficiency gains achieved by diversification cannot be replicated by manipulatingshareholders' portfolios or by complicated contracting over the use of capital. In Section 6I show that relative-performance ontracting is also not a perfect substitute for diversificationin alleviating the agency problem in firms. I first turn to the comparative statics of thediversification model.

5. The comparative statics

* One would intuitively expect that optimal firm size would be increasing in managerialability. To perform the comparative statics, however, one must be careful to account forthe effect of ability on U0, which is determined endogenously. I assume that the market formanagers is perfectly competitive and that the supply of managers of any ability level isfixed, i.e., the supply of managers is perfectly inelastic at any 0. Together with Assumption3, this implies that managers earn all the rents they generate. In equilibrium, then, U0(O) sdetermined by the equation:

Pa*Oh(m*f k*))w - Rm*k* - p(m*, UO(O)) 0 (12)

for all 0 such that managers of ability 0 are active in the market, and where m*, k* are theoptimal values of m and k, given 0. Differentiating (12) with respect to 0 and applying theenvelope theorem give

Pa*h(m*f (k*))w/puo = dUoldO. (13)

One can show that for all U(I, a) that are either multiplicatively or additively separable,pu0(a, m, UO)> 0. Hence, dUoldO s positive.

Now consider the effect of ability 0 on firm size and diversification. Examination of

the first-order conditions indicates that the sign of the effect of 0 will depend on the signsof the partials Pmu0 and Pmm, hat is, the effects of increased diversification and increasedability on the (moral-hazard-reducing) marginal benefit of diversification. Consider pmmfirst. One expects the marginal benefit of diversification to be decreasing in diversification,that is, Pmm > 0 (since pm < 0). The precise effect depends on the form of the optimalcontract, but using the proof of Lemma 1, one can show that Pmm > 0 on average. Forsimplicity, in this section I shall assume that utility is exponential, in which case pmu0 0(see Aron ( 1985) for the proof), but see footnote 11 for a discussion of the greater generalityof the results that follow.


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Under these conditions we can unambiguously sign the following:1'

dm/d6 > O (14)

d(mf(k))/d > 0 (15)

df(k)/d6 > 0 (16)

dm/dP > 0 (17)

dk/dP>0. (18)

Thus, we have the following proposition.

Proposition 2. Let U(I, a) = B(a)[-exp(-I/6)], 6 > 0. Then expected firm size (defined asexpected revenue or value of capital), product line size (defined as Pa*0g(mf(k))f(k)w),and diversification are monotonically increasing in ability.

Corollary 2. Expected firm size, product line size, and diversification are positively correlatedwith each other, cross sectionally.12

If we specify some reservation utility outside the managerial labor force for potentialmanagers, equation (12) determines a lower bound on the ability of managers who willactually run firms. For example, if the reservation utility in nonmanagerial alternatives is0, the lower bound 0 is given by

Pa*Oh(m(q)f(k(O)))w m(O)k(O)R = p(m(O), 0). (19)

This lower bound always exists by Proposition 2 and Corollary 1.

The results in Proposition 2 and Corollary 2 correspond precisely to the stylized factspresented in the Introduction. I am aware of no other model of diversification that generatesimplications for the cross sectional distribution of firm size and degree of diversification.The alternative models of diversification I discussed have no explicit implications for theoptimal degree of diversification short of expansion into every product in the market.

Proposition 3. If U(I, a) = B(a)[-exp(-I/6)], 6 > 0, as in Proposition 2, then the slope ofthe optimal incentive contract with respect to output is independent of ability and thereforeis independent of firm size.

This result relies on exponential utility. More generally, one can show that for the class

of hyperbolic absolute-risk-aversion utility functions described in footnote 11, the slope isnondecreasing in firm size.

Inequalities (17) and (18) do not rely on the form of the utility function. Inequalities (14) and (15) holdunambiguously when the W(I) are hyperbolic absolute-risk-aversion utility functions, where

U(I, a) = B(a)[-exp(-I/6)]

U(I, a) In (I + 6) - G(a)

U(I, a) = B(a)(1/y - 1)(yI + 6)(y - l)/,y, oy> 0.

In each case W(I) is a hyperbolic absolute-risk-aversion utility function with risk tolerance yI + 6, and the utility

function has been constrained to be either additively or multiplicatively separable. Recall that a utility functionW(x) belongs to such a class if it exhibits linear absolute risk tolerance (the reciprocal of absolute risk aversion) inx. That is, for some numbers a and 3, - W'(x)/ W (x) = a + Ox.

12 It should be noted that exponential (or even hyperbolic absolute-risk-aversion) utility is not necessary for(14)-(16) to hold. We only require that Pmuo ot be excessively large. The intuition is as follows. The total amountof capital devoted to a firm increases with the manager's ability. Some of the increased size would normally takethe form of increasing each plant size, and some would take the form of increasing the number of plants. At higherlevels of ability, however, the manager would earn a higher expected salary. If risk aversion decreases very quicklyas expected income rises (i.e., P.uo is sufficiently large), the incentive to diversify (add plants) falls. In this case thedecreased incentive to diversify may dominate the size effect of increased ability, with larger firms' being lessdiversified.


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01 A reinterpretation f the problem. We have, to this point, treated the contracting problemas one in which firms maximize profits subject to the constraints that managers receive agiven level of expected utility and that the incentive-compatibility constraints are satisfied.This is the way the problem is generally formulated in the literature. The problem could

be described equivalently, however, as the dual one in which managers maximize utilitysubject to the incentive-compatibility constraints and a zero-expected-profit constraint onthe firm. Indeed, in the model presented here it is the managers who benefit when a firmdiversifies, because they receive all of the rents they generate, and shareholders always earnzero profit.

It is perhaps more intuitive and realistic to think of the manager, rather than theowners, as actually making the decisions whether to diversify, how much to diversify, andhow large the firm should be. But this is perfectly consistent with the model, and by solvingthe dual rather than the primary problem for an optimal contract, it is explicitly the managerwho maximizes his utility by choosing among different firm structures. Thus, one can in-terpret the model as one in which managers decide to diversify the firm because of the gainsthat will accrue to them personally. This yields a social gain because it optimally shifts tothe shareholders the risk that the managers face.

6. Relative performance comparisons and themerger/contracting tradeoff

* We have thus far analyzed a firm's optimal contracting problem when the only infor-mative signals on a manager's effort are generated by the firm itself. In general, a numberof signals in the economy may be informative with respect to the manager's input. Forexample, in industries in which competition is not perfect, it might be difficult by lookingat the firm alone to distinguish between a poor performance by a firm's manager and anexogenous decline in demand. In such a case observing the performance of other firms inthe industry would improve the ability to evaluate the manager. If all firms did poorly, itis likely that demand decreased, and it would be inefficient to punish the firm's managerfor the full decline. An efficient contract would adjust for the amount of decline (or im-provement) attributable to market performance, to the extent this could be ascertained. Weterm such contracts relative-performance contracts.

Holmstr6m (1979, 1982) has analyzed the value of relative-performance comparisons.Related work in which incentives are created via competition among agents is the work ontournaments (Lazear and Rosen, 1981; Green and Stokey, 1983). I have explicitly eliminatedthe benefit to such contracting by assuming that the random component of productivity isindependent across firms. In this case there is no benefit to writing the compensation contractfor firm i's manager as a function of the productivity of firm j, since firm j's output has noinformative value for firm i (Holmstr6m, 1982). I now consider the case in which thecorrelation between ci and ?j can vary between -1 and 1, and examine the role of relative-performance contracts vis-a-vis diversification.

The questions I address are: Would the desirability of a firm as an acquisition targetvary with the correlation between the target's product and the acquirer's own product? Howcan the correlation be exploited in optimal incentive contracts using relative-performancecomparisons? And finally, can one describe a tradeoff between relative-performance com-parisons and acquisition?

0 The information structure. For simplicity, assume that firm i produces only one product,qi. Recall that the owners of firm i can effectively observe aci and that the problem is toimplement a. Similarly, owners of firm i can observe ajej of any other firm j in the economy,where a1 is the effort level of firm j's manager. Indeed, they can observe cj itself since the


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owners of firm i know aj. This follows because, ex post, all parties know what effort levelsare chosen by all managers, since managers are responding to publicly observed incentives. 13

Mergers are differentiated from relative-performance contracting by the signal that canbe used in each case to create incentives for managers. If any firm i producing qj were to

acquire product j, output qj would be a function of manager i's effort, a. Thus, his com-pensation would be a function of ac as well as aci. If product j were not acquired, thenmanager i's compensation would be a function of ej and aci. Which pair of signals is moreinformative about a depends on the correlation between ci and yJ.

In what follows I wish to isolate the informational issues in the tradeoff between con-tracting and acquisition. Thus, I shall treat product j as a pure signal-producing activity. 4This allows me to disregard the issues of optimal firm size treated in previous sections.Further, I analyze the tradeoff from a partial equilibrium perspective and ignore the optimalcontracting problem of the second firm.15 Finally, to derive definite results I shall considera parameterization of the random variables.

O Parameterizing the signals. Assume that has a log-normal distribution. For purposesof information extraction, observing as is equivalent to observing log(ae) since one cantransform one to the other without loss of information. It will be more convenient to considerobserving the transformed signal log(ae) = log(a) + log(e) since log(e) has a normal distri-bution.

Consider the problem faced by firm 1. Let the signal firm 1 observes on its own outputbe X1, where X1 =o a + q1j, a log(a), qij log(c), qij N(O, cry). Consider a second firm,with output X2 = a2 + '12, where a2 is the log of a2, and a2 equals the effort of firm 2's

manager. Under the assumption that firm 1 knows that firm 2's contract with its managerimplements a2, firm 1 can observe q2. Thus, without merger firm 1's signal on the effort ofthe manager of firm 1 is (X1, pq2)- We assume that q1iand 2 are bivariate normal with meansequal to zero, correlation X, and variances a2 and a2, respectively.

This simple structure generates immediate implications about acquisition choices. Sup-pose that, for some reason, the product of firm 2 had a perfect negative correlation withthe product of firm 1. With perfect negative correlation between its products, a diversifiedfirm could achieve the first best. This is the case that, at first blush, one might expect to bethe most attractive for merger. Our analysis indicates, however, that the value of acquisitionrelative to relative-performance contracting would be zero. The firm would be indifferent

between the two: In either case the firm could write a riskless contract implementing a. Thepair of signals in the merger case, {X1, X2}, allows perfect inference of a, as does the pairof signals {X1, 7q2} in the relative-performance contracting case. This gives us the followingproposition.

Proposition 4. When the correlation of m, q2i s-1, the firm is indifferent between acquisitionand relative-performance contracting with respect to solving the agency problem.

Thus, with zero correlation merger is valuable but relative-performance contractinghas no value, while at perfect negative correlation the two are equally valuable. One would

expect that if 11and 2 were identical, the value of merger would be zero, since the additionalsignal would carry no additional information. On the other hand, relative-performancecontracting would achieve first-best since one could perfectly infer a (and therefore a) from

13 It is not necessary that the contracts themselves be observed. What a will optimally be implemented forany firm is common knowledge.

I am grateful to an Associate Editor for suggesting this simplification.'5 Aron (1985) analyzes the problem within a general equilibrium framework as a matching problem. The

qualitative results are virtually the same.


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This is quite intuitive, because for X E (-1, 1) this is equivalent to

2 cov(71l, 12) < var(1 1). (25)

Further, when agency costs are approximately linear in the variance of the signal, the valueof merger over relative-performance contracting is monotonic in Vij.17 The function Vij isplotted in Figure 1 for various values of b.

Two results emerge for our parameterization. First, there is a tradeoff between con-tracting based on relative performance and merger, and the tradeoff is not monotonic inthe correlation between the firms' signals. In particular, the relative value of merger isincreasing as correlation increases from -1 and decreasing in the neighborhood of X = 0.Second, in the region of zero correlation, the relative benefits of merger are a decreasingfunction of the variance of the acquired firm's signal (when holding the variance of theacquiring firm's signal and the covariance between the two signals constant).

By assuming product j to be merely a signal-producing activity, I have avoided theissue of collusion between managers. When the compensation of manager i is contingent

FIGURE 1

THE VALUE OF A MERGER AS A FUNCTION OF SIGNAL CORRELATION

iV

2

4

-1 ~~~~~~~~~~~~0

17 Vijwas derived under the hypothetical condition that only two firms exist. Nevertheless, the derivation isperfectly general. Suppose we want to determine whether firms i and j, in a population of N + 2 firms, shouldmerge. In either case both firms will condition their contracts on the signals generated by the N other firms in theeconomy. Let nN = { k, k = 1, . . ., N, k # i, j }. Without merger the two firms, i and j, will have contracts basedon the signals {Xi, tj, 1N} and {X, li, IN}, respectively. If the firms merge, the contract will be written on the signal{Xi, X;, my}. Let -y be a sufficient statistic for {Xi, 1N}. Then we can write the nonmerger signals as {yi, qj},{^yj, qi}, and the merger signal as {^yi, yj}. This is equivalent to the case of a two-firm economy in which firm i'ssignal is -yj rather than Xi and similarly for j.


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Ability, Moral Hazard, Firm Size, And Diversification

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