Span of Control in Production Hierarchies

Transcription

1 Span of Control in Production Hierarcies Jan Eeckout and Roberto Pineiro University of Pennsylvania February, 008 Preliminary Draft Abstract Te allocation of skills in te firm is determined by te span of control of managers. Te span of control of any given manager includes te lower skilled managers and te workers tat are in te span of control of tose lower skilled managers. At eac level, skills are imperfect substitutes in te production of output and tere are decreasing returns to iring more agents wit te same skill level. In a competitive labor market wit atomless firms, we find tat:. firms ave a non-degenerate skill distribution;. larger firms ire disproportionately more skilled workers. As a result, large firms ave a skill distribution tat is more skewed and tey pay on average iger wages; 3. Te presence of a non-divisibility as in Lucas 978 will pin down te skill level of te igest skilled manager. Wen investment in skills is endogenized, we find tat te equilibrium skill distribution as a long rigt tail, even if ex ante all agents are identical. Introduction Te firm is a distribution of skills. Workers wit many different talents and skills populate a typical firm. Even te smallest firms ave occupations ranging from CEO to janitor. Most production processes involve te collaboration of agents wit different ability and skill, and typically tere is a ierarcy of decision making and execution of tasks. Wit tose ierarcies comes a distribution of skills, and as a consequence, a distribution wages. We put forward a matcing teory were firms operate in a competitive labor market and were workers optimally sort in different firms. Te teory aims to caracterize te firm as an equilibrium composition of skilled workers. Eac firm as an endogenous organigram tat maximizes output. Wit tis production tecnology, it does not only matter wo your workers are, but also wat tey do and wat te organigram of te firm is. We are grateful to numerous colleagues for valuable discussions and comments.

2 n a + bn γ Lucas n a Figure : Span of Control: n versus Lucas 978 Te starting point of our analysis is te Lucas 978 span of control model. Tere, te firm consists of a manager wo ires workers, and er productivity is limited by er span of control. Hiring more workers as decreasing returns, wic in equilibrium determines te boundaries of te firm. Because tere are complementarities between managers and workers, iger skilled managers in equilibrium ire more workers. Te firm is fully caracterized by one manager and er skill, and te equilibrium number of workers. Te key assumption is tat a firm as exactly one manager. Tis non-divisibility seems reasonable: to run a firm, we need to ire one manager wo is full time devoted to te firm. We furter build on tis model in two ways. First, wile we maintain tat a limited amount of time commitment by a manager is needed, we relax te assumption tat a firm is restricted to iring exactly one manager. Our interpretation is tat a minimum scale of managerial skill is needed, but tat we can extend beyond te minimum scale. Wen a firm ires more managers wit te same skill level, we assume tat tere are decreasing returns. We model tis by te a function n wic measures te productivity at a given skill level. In Lucas, tis is a step function valued zero if you ire less tan one manager, and valued one if you ire or more managers. We ave a smoot concave function tat reflects te decreasing returns to iring more workers. If te intercept of tat function is negative, we assume it to be bounded at zero: te firm can always opt out and not ire any manager of tat skill type at all. Tis is illustrated in Figure were n is a simple polynomial.

3 Second, we assume tat eac skilled agent faces te same tecnology witin te firm, i.e. tere is no distinction between managers and workers. A ire skilled manager as span of control over te lower skilled manager wo in turn as span of control over te next lower skilled managers. Witin tis framework we study ow te distribution of skills and earnings differs between firms. We will be able to establis weter larger firms ire more skilled workers and weter te distribution of skills inside large firms is more dispersed tan in small firms. Wile all firms ire a wide range of skills, it is possible tat large firms like GE or IBM ave a very different composition of skills tan small family firms. Te firm exists because different inputs in production are needed to produce output. And wile any particular individual or individuals of similar ability may be able to provide tose different inputs in production, typically it will be optimal to allocate differently skilled agents to different jobs even if one particular skilled agent is better at performing all oter tasks. Tis is because te firm faces a trade-off between allocating a more skilled worker wo contributes more to output at a iger wage, and a less productive agent wo commands a lower wage. Te price system, market wages for different skill levels, determines te optimal resolution of te trade-off and terefore fixes te equilibrium allocation of skills. Te optimal solution to tis trade-off fundamentally boils down to te allocation of talent according to comparative advantage, as captured by te well known metapor of te attorney and te secretary. Even if te best lawyer is also te best secretary, it is quite clear tat tis lawyer will focus on te task of being an attorney by employing a secretary instead of doing all te paperwork by imself, or by iring a secretary wo is as skilled as e is imself. Even toug e is te best secretary and te best lawyer, e can earn more by running a law firm and employing a low skilled secretary at a low wage tan by iring anoter expensive attorney to do te secretarial work. Tis implicitly involves production ierarcies as te marginal product of a lower skilled agent is affected by te skill-level of tose iger in te ierarcy. Tere are many reasons wy suc production ierarcies emerge. Tis may be due to te efficient processing of information and resolution of problems. Garicano 000 and Antràs, Garicano and Rossi-Hansberg 006 provide micro foundations for wy particular production functions may be optimal. Te ierarcies may alternatively also be due to O-ring type production tecnologies wit asymmetry as in Kremer-Maskin 996. A small mistake by one worker in te production cain can ave implications of unprecedented dimensions. One bug in te software may lead to te malfunctioning of millions of electronic devices, or te inadequate quality control for lead in paint can lead to a worldwide recall of a toy. Tere are two main implications for te equilibrium allocation of using tis tecnology:. Te firm size is endogenous and consists of a non-degenerate distribution of skills. Te imperfect substitutability of workers as inputs in production implies tat te size of te firm is 3

4 endogenous. For reasons of comparative advantage in different jobs, firms in equilibrium decide to ire workers wit different talent. Of course, quite a lot is known about te size distribution of firms for recent examples, see Luttmer 007 and Rossi-Hansberg and Wrigt 007. Te interest ere is ow firm size relates to te internal distribution of skills witin te firm.. Firms differ in teir composition of talent. Firms wit iger firm-specific total factor productivity will ire more labor wic is due to te complementarity between capital and labor inputs. More interestingly, te equilibrium distribution of skills witin different firms are not identical. We sow tat only if te elasticity of substitution between different skills is constant and tere are no indivisibilities, will te distribution of skills witin different firms be identical. We caracterize te properties of different firms for some skill distributions. Because size and firm-specific TFP are related, we can caracterize te firm s skill distribution in function of te firm size. First we analyze te general case witout any indivisibility, i.e. wit n positive everywere. We sow tat if te elasticity of substitution between different skills is constant, all firms will obtain te same distribution of skills. Firms wit iger capital stocks will be larger, but tey will not differ in te composition of skills. In contrast, if te elasticity of substitution is decreasing, te skill distribution of larger firms stocastically dominates te distribution of smaller firms. Te implication of te firstorder stocastic dominance in te skill distribution is tat tere is also stocastic dominance in te wage distribution. We analyze a competitive equilibrium in wic market wages for te same skills are te same. Larger firms terefore ire on average more skilled workers and terefore pay on average iger wages. Tis can explain a well-documented fact in te empirical labor literature, tat tere is an employer-size wage premium. Wile tis fact is typically establised after controlling for observables, it may noneteless be determined by skill eterogeneity unobserved by te econometrician. Second, we consider te case as in Lucas, were a minimum scale of output is needed. Tis is te case were is negative for some values. We find tat in equilibrium, firms wit larger capital stocks will be larger and will find it profitable to ire proportionally more ig skilled workers. Tis implies tat te skill distribution in large firms is skewed to te rigt compared to te distribution in small firms. In addition, te igest skilled manager in te large firm will be more skilled tan te CEO in a small firm. As a result, te support of skills of te small firm is included in te support of skills of te large firm. Tis is illustrated in Figure. In Section 3 below, we analyze te impact of investment in skills by ex ante identical agents and sow tat in equilibrium, tere will be an endogenous distribution of skills. Even wit no or small ex ante eterogeneity, tere can be considerable ex post inequality as tis tecnology enances eterogeneity. In equilibrium, if tere is scarcity of any one particular input, te returns to obtaining tat skill are ig. Wit increasing investment costs, te ensuing distribution of skills is decreasing in type as te 4

5 density small firm large firm skills Figure : Stocastic Dominance of Skill Distribution in Large Firms returns in term of wages must be increasing to compensate for iger investments costs. Wages can only be increasing if tere is sufficient scarcity in tat particular input. One furter application is Occupational Coice and span-of-control. A CEO cooses to optimally design te ierarcy of te firm, and se erself competes on a market were se can coose, based on equilibrium compensation scedules between a job as an employee or as a CEO. Te span-ofcontrol of te CEO determines er productivity. We extend te tecnology of production ierarcies to incorporate tis feature. Te model Population. Consider a population consisting of agents endowed wit talent x, a one-dimensional skill caracteristic. Skills are distributed according to te distribution function F x. Te measure of agents is normalized to one. Tere is a measure of capitalists, eac of wom is atomless, wo ave te property rigts to a production process k. Tis can be interpreted as firm-specific total factor productivity. Let µk denote te measure of eac type k. Production. Firms produce output y using te input k and a set of workers of different skills. Te 5

6 production function is given by y kln, x were n is te vector of quantities n i and x is te vector of skills x i, and were N ] β Ln; x n i x i were is monotonically increasing and concave and β > 0. i It is important to note at tis stage tat y is a firm-level production function and tat in general it is not equal to te aggregate production function. For most of te paper, we consider a discrete distribution of types x. A continuous distribution of types is analogously represented by Ln; x ] β nxxdf k x were F k x denotes te distribution of skills in firm k. Below we derive tat tis is te continuous limit of te production tecnology wit finite skill types. Te firm s optimization problem. Markets are competitive and te atomless firms act as price takers. Given a vector of wages wx normalize te output price to firm k s problem is given by: π k; w max k n,...,n N N ] β n i x i i N n i w x i i A competitive equilibrium of te economy can be defined as follows: Definition In a competitive equilibrium in tis economy:. Firms maximize profits π k ;. workers coose te job wit te igest wage offered wx for a type x; 3. markets clear. Now te main properties, as discussed in te introduction, of tis production process are made precise.. Te marginal product of te second CEO, or te second janitor is lower. Returns are decreasing returns to iring more of te same worker since n is concave.. Te same number of more skilled workers are more productive tan low skilled workers: nx i > nx j if x i > x j. 3. Tere is a notion of scarcity. A sortage of a particular skill level can drive up te prices. A lower skilled employee x j can be more productive tan te ig skilled employee x i if se is sufficiently scarce in te firm: n i x i < n j x j provided n i n j. 4. Inputs in production can be bot complements or substitutes. 6

7 β Strc. Quasiconcave γ Substitutes and Strictly concavity Complements and Strictly concavity Once we ave set up te caracteristics of firm s problem, we must define te equilibrium Figure 3: Complements and Substitutes in our economy. Since we are assuming te approac in wic workers don t value leisure, our equilibrium is quite simple and involves only profit maximization and market clearing conditions. Later, andwe Substitutes. endogeneize decisions From te firm s on investment objective in function, education we derive and we adapt our equilibrium Complements conditions to take tese decisions in account. N ] β π kβ β n i x i n i n j x i x j n i n j i Definition 4 A Competitive equilibrium in tis economy is one in wic π Notice tat n Firms i n maximize j > 0 β >. Terefore, β determines weter x i and x j are gross complements profits π i, or substitutes. workers coose te job wit te igest wage wx, Claim Ifmarkets β >, inputs clear. are complements. If β < tey are substitutes. Let s explicitly find te equilibrium in tis economy. As presented before, firm s problem is For example, let n given by: i n γ i, ten we can summarize tis in terms of te parameter values for β R + and γ 0, ]. Te firm s problem is well-defined for β < /γ a sufficient condition for concavity is γβ <. Ten te yellow area is te " range of parameters X N β were inputs in production are NX complements, and te green π k; areaw were tey max are substitutes. a + bn γ i i# x n i w x i n,...,n N k i i Elasticity of Substitution. A key caracteristic of te firm s production function is its Elasticity of Substitution Ten, between from F.O.C.s inputs n we ave: i and n j, denoted by σ. Te elasticity of substitution is defined as " X N σ # d lnx β j/x i kβ a + bn γ i x d lnt i bγn γ RS i x i w x i, i {,...,N} i 7 4

8 were T RS dy/dx i dy/dx is te tecnical rate of substitution. Ten σ n i. n i n i Claim 3 Let n i n γ i. Ten te production function is CES σ is constant and L n; x is omogeneous of degree one. We can sow in greater generality necessary and sufficient conditions for te production function to be CES, namely tat n i a + bn γ i wit a, b constants. In te appendix, we prove tat if σ is a constant, we must ave tat n i is of te form a+bn γ i, were a and b are constants, and tat L n; x is omotetic if and only if is te form a + bn γ i. For te remainder of tis and te next subsection, we assume tat a is non-negative. CES: n i n γ i. Te CES production function can be written as: N β y k n γ i i] x. i A special case of tis CES production function is te one in Kremer 993, wic is equivalent to our model wen L N N i nγ i ] N γ wit γ 0. Finally, we can sow tat if γβ <, ten te firm s objective function as defined generally above is strictly concave. Tis Claim is proven in te Appendix.. Te equilibrium allocation We now explicitly derive te equilibrium in tis economy. F.O.C.s: Ten, rearranging, we obtain: N ] β kβ n γ i x i bγn γ i x i w x i, i {,..., N} i n i n j w xj x i w x i x j γ From te firm s problem, we obtain te Substituting back, we obtain te demand for labor quality x j as a function of wages: Market clearing satisfies: n j k kβγb xj γ γβ w x j N i n j k µ k m x j k 8 xi ] β γβ γ w x i γ

9 were mx j F x j F x j is te measure of worker type x j. Substituting for te equilibrium quantity of n j k and solving for w x j, we obtain te equilibrium wages: w x j x j m x j γ N i xi ] β γ γβ γ w x i γ k kβγb ] γ γβ µ k Now, substituting in te demand for wages, we obtain te equilibrium allocations: n j k k γβ m x j. k k γβ µ k Ten, looking at te total labor force of a firm wit capital k, we ave: n k N n j k j k γβ m k k γβ µ k were: m N j m x j. Tis expression is strictly increasing and convex in k, and te next Proposition terefore immediately follows. Proposition 4 Firms wit iger k ave a larger labor force Firms wit iger firm-specific TFP k are larger. Te productivity per worker is iger, and terefore at common economomy-wide wage rates, it is optimal for tem to ire more workers. Te question remains ow te skill distributions witin te different firms compare. Proposition 5 Wen te production function is CES, in equilibrium all firms ave te same skill distribution F k x wic is equal to te economy s skill distribution F x To see tis, look at te fraction of quality j workers in terms of te total number of workers, and we ave: n j k n k k γβ mx j k k γβ µk k γβ m k k γβ µk m x j m for every k. Terefore, te distribution of workers inside a firm is exactly te same as te one in any oter firm and mimics te distribution in te market. Firms are different in size, given different k s, but tey look alike considering te distribution of labor qualities inside te firm. Tis is a consequence of constant elasticity of substitution assumption or, more generally, omoteticity. Tis assumption imposes a lot of structure on firm s production function. 9

10 . Different Production Hierarcies We sow ere tat te equilibrium distribution of labor abilities inside a firm varies wit k according to canges in σ and te measure of labor qualities in te economy. We establis te following result, consistent wit te distributional properties discussed in Figure. Proposition 6 Let σ < 0. If te density of x is decreasing ten:. Higer k firms are larger;. Average skills and average wages are iger in larger firms tan in smaller firms; 3. Te skill and wage distribution in larger firms First-Order Stocastically dominates tose in small firms. To derive tis result, we sow ow te elasticity of substitution and te equilibrium allocation relate. In particular, te following result olds. Proposition 7 If σ is decreasing, ten iger k firms ire more of te scarce skilled workers mx < mx : m x m x σ < σ n n k > 0 k k Proof. We prove tis result ere for β. In te appendix, we provide proof for general β. Observe tat te first-order conditions, after substituting for market clearing imply k n k m x n k n k m x n Tese conditions implicitly define te equilibrium allocations n and n. Applying te implicit function teorem, we get n n k k n + k m x n and n n k k n + k m x n To simplify exposition, ere we will consider te case in wic we ave only two firms wit different managerial skills or TFP, k and k and β. We ave sown tat exactly te same results old in general, toug te derivation is somewat more involved. A sufficient condition for σ decreasing is < 0. Tis immediately follows from and te fact tat > 0, < 0. dσ dn ] n + n ] n ] 0

11 Wen k k, we obtain from using te quotient rule tat: n n k k k Te left-and side is positive provided mx mx Recall tat te elasticity of substitution is and terefore mx mx mx mx mx mx mx k mx mx > mx σ n i, n i n i mx n n m x m x k > 0 if σ < σ k k Considering te more scarce labor quality as te one wit iger levels of education or uman capital, an economy wit production ierarcies will ave larger firms iring more eavily at te top, i.e., tey will ave more skilled workers. Tis effectively means tat tey ave proportionally more managerial positions compared to smaller firms. One possible interpretation is tat of an increase in te monitoring cost. In order to manage a larger ierarcy, te demands on communication skills and span-of-control go up, leading to te iring of more skilled types. Notice tat tis is due to te relative scarcity of eac type of labor. Te result is driven by te elasticity of substitution of a given quality of labor compared to oters..3 Minimum Scale of Operation We derive under plausible conditions tat te igest skilled worker as a iger type in larger firms tan in smaller firms. Tis implies tat te distribution of iger k firms as fat tails at te top as long as te skill distribution as decreasing density. Suppose tere is some non-convexity in te production tecnology. On any given task, firms incur a fixed cost. 3 Consider te production function we used above n a + bn γ, were a < 0. A firm will ire a type x if for tat type, te equilibrium n yields positive output: n a + b n γ, were we derived n earlier as: n k k γβ m x. k k γβ µ k 3 A special case of tis tecnology is tat in Antràs, Garicano and Rossi-Hansberg 006.

12 Te firm s decision problem is terefore to coose n as long as a + b k γβ mx k k γβ µk wit capital k will terefore be indifferent between iring and not iring provided γβ k a γβ γ k γβ µ k. b mx k Kx γ > 0. A firm Te only caveat is of course tat te summation over k is for all k actively iring workers of type x. Kx denotes te set of firms actively iring type x workers. Proposition 8 Let te elasticity of substitution σ be constant, and tere is a fixed cost of employing one skill type a < 0, ten:. iger k firms ire more workers;. te support of skills ired in lower k firms is included in te support of skills of iger k firms; 3. wen te skill density is decreasing, iger k firms iger more skilled workers Example. Let skills be distributed according to te Pareto wit location and coefficient. Ten te cdf is P x x and te density is px x mx. Let te distribution of firms be uniform, µ for k 0, ]. Let n a + n /, and β. We ave: a + n if n > 0 n 0 if n 0 were a < 0. From previous calculations, we obtain: n x k k x K k dk. Define k x {k K n x k 0}. Terefore, tere exists a tresold suc tat if k < k x, { } ] max 0, a + n x k 0. Tis implies tat K k x,. Solving for k x : a + k x x kx k dk 0, and rearranging, we ave: 3 k x ax k x 3], wic defines k x. From te implicity function teorem, we ave: d k x a x k x 3] dx 3 k x + a x k ] > 0. x

13 Claim 9 x as k x. Proof. Assume tat tere is a x R suc tat k x. But ten, from we must ave: 3 k x ax }{{} k x 3 0 }{{} wic is a contradiction. Ten, we cannot ave k x for x finite. Since d kx dx > 0, k x 0,, we must ave k x as x. Claim 0 k > 0, i.e., some firms sut down in equilibrium. Proof. From, we ave: 3 k a k 3] Now, observe tat te LHS of tis equality is strictly increasing in k, wile te RHS is strictly decreasing. But if k 0, we ave LHS < RHS, so we must ave tat k > 0. Te fact tat k is increasing in x of course also implies tat te larger firms k ave iger cut-off types for teir igest skilled employee. Te maximum quality of x tat a given k firm ire: x k 3k a k 3 and is increasing in k. Te lowest firm tat as positive profits in tis market x 3k 0.5 k 3 k 0.5 Finally, we also verifty tat te demand in te rigt tail is in fact decreasing as x increases: { } 3k d dn x k x kx 3 ] 3k {x k x 3] } 3 k x d kx dx x dx dx x 4 k x 3] Substituting k x and rearranging, we ave: dn x k dx xk x 4 + a x k ] x k x 3] < 0 So, te demand is strictly decreasing in x, for a given k and a cut off rule is optimal. 3

14 For tis example, we now explicitly ave te measure of skills witin a firm nx k 3k x k x 3] were k x solves. Normalizing tis measure to sum up to one, we obtain te firm s distribution of skills. Larger firms ire more workers of all skill types, but from simple comparison of te normalized densities, we see tat te low k firms ire proportionally more low skilled workers. Te ig k firm s skill distribution is terefore eavy in te tail and skewed to te rigt. 3 Applications 3. Investment in skills: Endogenous eterogeneity Wit production ierarcies, ex ante identical agents ave incentives to take on different levels of investment. Because all skill levels are needed in production, it cannot be an equilibrium were all agents coose to invest te same amount and obtain te same level of investment. We now sow tis for te constant elasticity case wit n i n γ i. Let cx i be te cost associated wit obtaining skill level x j. We first find an expression for wages. From our previous calculations, we obtain: w x j x j m x j γ N i m x i γ x iβ k kβγb γβ µ k] γβ. x Notice tat w x j depends on te ratio j, were m x mx j γ j is te aggregate supply of skill j. Ten,te worker s problem is given by: max {w x c x,..., w x N c x N } {x,...,x N } since in equilibrium all skills must be offered 4 and workers are ex ante symmetric, we must ave: were v is a constant. w x c x... w x N c x N v Ten w x i is identical to te cost function up to a constant. Using tis, we can calculate m x i, wic is te supply of skill i. Observe tat: v + c x j x j m x j γ N i m x i γ x iβ k ] γβ kβγ γβ µ k 4 Wit n i n γ i offered in in equilibrium. te Inada conditions old and te marginal productivity at ni 0 is infinity. Hence all skills will be 4

15 and tat: v + c x j v + c x i Ten, from te expression for v + c x j, we ave: v + c x j x j m x j γ m x j γ N i m xi m x j Substituting te above expression and rearranging: Ten: x j m x j v + c x j m x j x j γ v + c x j γ N γ i N γ i x i v + c x i ] γ x i v + c x i ] γ x j mx j γ x i. mx i γ ] γ x i β k k β γβ γ β γβ γ kβγb γβ µ k] γβ. kβγb ] γβ µ k k kβγb ] γβ µ k v + c x j c x j x j v + c x j If tere is no fixed cost, m x j < 0, for every x j. Te density of skill types is downward sloping. Te iger te skill level, te iger te cost of obtaining tose skills. As a result, wages must be iger for iger skill level to compensate for te cost. For tat to be te case, tere must be fewer people in equilibrium wo invest to obtain ig skill levels. 5 Observe tat te properties of te distribution ere are derived in te context of a competitive market witout any externalities Occupational Coice and Span-of-Control Models of occupational coice ave increasingly received attention as a way of explaining te aggregate outcomes by means of micro-founded allocation problems. Lucas 978 uses a matcing problem were differently skilled agents decide to become eiter workers or entrepreneurs. Te span-of-control of te manager determines te size of te firm, wic in equilibrium generates an equilibrium distribution of firms. In our context, we can extend Lucas framework to allow for CEO s to run te firm, rater tan capitalists. Te production tecnology ten becomes y xln, x instead of y kln, x. Tere will 5 If tere is a fixed cost, ten tere is te possibility tat m x j is positive for small x js, and tere can be a skewed unimodal distribution wit a long upper tail. Te intuition comes from te compensation for te fixed cost. Since tese skills are not tat valuable per se small x s, but tere is tis fixed cost tat workers ave to pay, we need to increase wages by decreasing m x, so we ave tis initial increase in m x as we increase x and ten we start decreasing. 6 For a framework wit spillovers from tecnology adoption and te ensuing endogenous eterogeneity of ex ante identical agents, see for example Eeckout and Jovanovic 00. 5

16 be a wage for all types bot as a worker and as a CEO, and te equilibrium allocation is determined by te occupational coice of eac type, driven by te maximum over bot wages. Wit a distribution of skills, CEOs of different skills will manage teams, eac potentially wit a different composition and distribution. Tis relates to te findings in Gabaix and Landier 008 wo analyze te matcing problem of CEOs to firms wit different capital stocks. managing te composition and distribution of te work force. Here we interpret te task of te CEO as We derive te equilibrium condition tat determines te occupational coice decision for te continuous type distribution te derivation of te continuous type formulation as te limit of te discrete type case is at te end of tis section. If te distance between two sequential qualities is, we ave N + x x. Ten, m x i F x i F x i F x i F x i. Wages and profits, after taking te limit for 0, satisfy: w x i γx α i ] E x γ γ f x dx f x i γ and γ π x, w γ x x γ γxx α γ i i dxi. E c w x i ten, substituting w x i, we obtain: Ten te condition to become a manager is: γ x γ π x, w ] γ x i f x i ] γ dx i. E x γ E j f x j dx c j γ x γ ] γ x i f x i ] γ dx i E x γ E j f x j dx c j ] γx α E x γ γ f x dx f x γ Rearranging, we ave: x α+αγ γ f x γ γ γ ] E x γ f x dx E x c i f x i ] γ. dx i We can also go furter and introduce span-of-control at eac skill level. More generally terefore, we can use our set up and specify as n i ; n i, x instead of n i : te returns to eac occupation do not only depend on te number of people of te same skill, but on te number of people of oter skill levels. As a result, tere is an occupational coice decision for eac job, and eac occupation, driven by te local span-of-control of tat occupation. As in Lucas 979, tis partitions te set of skills into different distributions. Tis generalized production function tat combines te optimal allocation of skills wit an occupational coice decision is really getting to te eart of production ierarcies. At all levels witin 6

17 te firm, managers can be interpreted as aving span-of-control of different degrees over workers wit different skill levels. One of te main objectives of tis paper is to furter elaborate te links between production ierarcies and span-of-control and occupational coice. 7

18 4 Appendix Claim If σ is a constant, we must ave tat n i is of te form a + bn γ i, were a and b are constants. Proof. Since σ is a constant, we ave tat: n i + σn i n i 0 is a omogeneous second order linear differential equation. Considering n i g n i we reduce it to a first order ODE. Solving it, we obtain: n i n 0 e n i n 0 σy dy were n 0 is te initial condition. Taking te integral on bot sides, we obtain: Ten, notice tat: Substituting back, we ave: Substituting back again, we ave: Solving te integral, we obtain: Ten, rearranging, we ave: Terefore: n i n 0 n 0 ni n 0 e z n 0 σy dy dz z n 0 σy dy n0 σ z y dy σ ln y n 0 z σ ln n 0 z e z n 0 σy dy e ln n 0 z ] σ n i n 0 n 0 ni n 0 n0 σ z n i n 0 n 0 n σ 0 ni n0 σ dz z n 0 z σ dz n i n 0 n 0 n σ σ 0 σ z σ σ n i n 0 σ σ n 0 n 0 + n i n 0 ] σ σ n 0 n σ 0 n σ σ i n i a + bn γ i 8

19 were: a : n 0 σ σ n 0 n 0 b : σ σ n 0 n σ 0 γ : σ σ. Claim L n; x is omotetic if and only if is te form a + bn γ i. Proof. We know tat, by definition, L n; x is omotetic if for any i, j {,..., N} and for any t > 0, we ave tat: But ten, we sould ave: rearranging: Ln;x n i Ln;x n j Since tis must always be satisfied, we must ave: were c is a constant. But ten, we must ave: Ltn;x n i Ltn;x n j n i n j tn i tn j tn j n j tn i n i tn i n i c tn i c n i since te function f β t β, wit t > 0, is continuous and as image on 0,, by mean value teorem we ave tat tere is a γ 0, suc tat t γ c. Terefore, we ave: tn i t γ n i Terefore, is a omogeneous function of degree γ. Since is a univarite function, it is easy to see tat it must be of te form dn γ i, were bd is a constant Note tat n i n i n γ dn γ, were d. But ten, we ave: n i i n i dn i i dn γ i dn i d γ nγ i + a Define b d γ, so we ave: n i a + bn γ i. 9

20 Claim 3 γβ < is a sufficient condition for strictly concavity of firm s objective function, wenever a 0. Proof. Notice tat: π n i N ] β kβ β a + bn γ i x i b γ n γ i x i + i N ] β kβ a + bn γ i x i bγ γ n γ i x i. i Rearranging: N ] β π n kβ a + bn γ i x i bγn γ i i i i x i { N ]} β bn γ i γx i + γ a + bn γ i x i Ten, π < 0 if we ave: n N ] β N ]} kβ a + bn γ i x i bγn γ x {β bn γ γx + γ a + bn γ i x i < 0 Wic implies: Rearranging, we ave: i N ] β bn γ γx + γ a + bn γ i x i < 0 i ] N N γβ bn γ x + γ a x i + b n γ i x i < 0 From our assumption tat > 0, we must ave b > 0. However, initially we don t ave any assumptions on a. If we consider a 0, we notice tat a sufficient condition would be γβ < I m already assuming by concavity of tat γ <. To get Inada conditions, we necessarily ave a 0. If a < 0, ten we wouldn t ave strict concavity olding for all n. Let s now consider te second principal minor. Ten, our condition is given by: N ] β 3 k β a + bn γ i x i b γ n γ n γ γβ b n γ x x γ x + n γ x + γ a N i x i + b ] N i3 nγ i x > 0 i i Again, for te case in wic a 0, γβ < is a sufficient condition, since γ <. Let s now consider te tird principal minor. Ten, our condition is given by: N ] 3β 4 k 3 β 3 a + bn γ i x i b 3 γ 3 n γ n γ n γ 3 x x x 3 γ i i i γβ b n γ x + n γ x + n γ 3 x 3 + γ a N i x i + b ] N i4 nγ i x < 0 i i 0

21 Ten, again, for te case in wic a 0, γβ < is a sufficient condition. We also can see te pattern for tese conditions, meaning tat γβ < is a sufficient condition for any N and a 0. Terefore, γβ < is a sufficient condition for strict concavity of te objective function wenever a 0. Proof of Proposition 7 for general β. Proof. Equilibrium conditions. Two firms, two skills; Endogenous Variables: n, n, n, n, w, w. k β n x + n ] β x n x w k β n x + n ] β x n x w k β n x + n ] β x n x w 3 k β n x + n ] β x n x w 4 n + n m x 5 n + n m x 6 For te general case, wen β, we can reduce te system to: k n x + n ] β x n k m x n k n x + n ] β x n k x + m x n x x m x n + m x n x β β m x n 0 F m x n 0 F Te main problem is tat tis is a non-linear non-separable system. From F F, we ave: n n m x n m x n Ten, let s prepare ourselves for te IFT: D k F F F k k F F k k were: F n x + n ] β x n k F m x n x + m x n ] β x m x n k F n x + n ] β x n k F m x n x + m x n ] β x m x n k

22 And, were: F n F n D n F F n F n k {β n x + n ] β x n ] x + n x + n ] β x n } β m x n x + m x n ] β x m x n ] x k m x n x + m x n ] β x m x n k β n x + n ] β x n n x + k β m x n x + m x n ] β x m x n m x n x F n F n F n k β n x + n ] β x n n x + k β m x n x + m x n ] β x m x n m x n x F n k {β n x + n ] β x n ] x + n x + n ] β x n } β m x n x + m x n ] β x m x n ] x k m x n x + m x n ] β x m x n Ten, we ave: So: det D n F F n F n F n F n F n F n k {β n x + n ] β x n ] x + n x + n ] β x n } β m x n x + m x n ] β x m x n ] x k m x n x + m x n ] β x m x n k {β n x + n ] β x n ] x + n x + n ] β x n } β m x n x + m x n ] β x m x n ] x k m x n x + m x n ] β x m x n

23 Rearranging: F n F n k n x + n ] { β x β n ] x + n x + n ] x n } k m x n x + m x n ] β x β m x n ] x m x n x + m x n ] x m x n k n x + n ] { β x β n ] x + n x + n ] x n } k m x n x + m x n ] β x β m x n ] x m x n x + m x n ] x m x n and F n F n k β n x + n ] β x n n x + k β m x n x + m x n ] β x m x n m x n x k β n x + n ] β x n n x + k β m x n x + m x n ] β x m x n m x n x Now, consider te symmetric equilibrium in wic k k, n mx and n mx. Ten, we ave: F n F n k k k ] m x m x β ] β mx x k x + x ] + mx x + mx x mx ] β mx x ] + mx x + mx x mx and F n F n k β m x x + m x ] β ] m x x m x x x 3

24 Ten, det D n F becomes: m det D n F 4k x m x x + ] β mx mx x + mx ] + x + x mx mx mx x ] β 4 mx mx ] x ] If β <, tis is necessarily different tan zero. Oterwise, tis could be zero but te set of parameters in wic tis occurs as mean zero. Ten: Ten: Substituting, we ave: n n k k n n k k D n F n n k k n n k k det D n F det D n F F n F n F n F n Dn F D k F F n F n F n F n F F k k F F k k Ten: Ten: n F k det D n F n F F k n F k F n F k k {β n x + n ] β x n ] x + n x + n ] β x n } β m x n x + m x n ] β x m x n ] x k m x n x + m x n ] β x m x n n x + n ] β x n at k k k and symmetric equilibrium, we ave: F n F k k k k ] ] m x m x β 3 β k x + x mx x + ] x + x mx mx mx m x 4

25 and again, at k k k, we ave: k β F n F k k β n x + n ] β x n n x + k β m x n x + m x n ] β x m x n m x n x n x + n ] β x n m x x + F n F k k k k m x x ] β 3 m x m x Putting everyting togeter at k k k, we ave: F n F k F k k k n F k k k k ] ] m x m x β 3 β k x + x mx x + ] mx x + mx x mx ] m x m x β 3 m k β x + x x m x ] m x m x β m k x + x x m x Terefore: n k k k k { k mx x + ] β mx x det D n F mx x } mx m x x > 0 Now, let s calculate n k. Ten, we ave: n F k k k k det D n F n F F k n F k 5

26 Ten, let s substitute tis step by step: F n F k k n x + n ] { β x β n ] x + n x + n ] x n } k m x n x + m x n x ] β β m x n ] x m x n x + m x n ] x m x n n x + n ] β x n at k k k, we ave: F n F k k k k ] m x m x β 3 k x + x + and Ten, at k k k, we ave: k β mx F n F k β x + ] x mx ] x n mx k β n x + n ] β x n n x + k β m x n x + m x n ] β x m x n m x n x n x + n ] β x n m x x + Ten, we ave: F n F k ] m x m x β 3 k x + x k β k m x m x x + x + F n F k k k k m x k k k + m x m x x ] β 3 m x F n mx x ] β 3 m x x ] β m x 6 mx ] m x x m x F k k k k ] β mx x ] m x x + n mx x mx ] m x x m x

27 Ten: Ten: n k n n k k k Terefore, we ave: Rearranging, we ave: { k k mx x + mx n n k x + det D n F m x m x ] β mx x n det D n F k n n k n n mx mx ] β x n n k > 0 if k k m x mx m x mx > m x mx > mx mx mx + mx } mx > 0 mx mx mx m x m x m x wic is exactly te same condition we obtained before for te case in wic β. mx Example in wic σ n < 0 Case witout Inada Conditions: arctan n : but note tat n as n 0, and n arctan n n + n > 0 Ten: solving it: n n + n < 0 σ n n n. +n σ n σ + n n +n n 7

28 and σ n 3 < 0 but note tat: 3n n + n 3 notice tat tis derivative is negative until n 3 and ten becomes positive. Since we want a bounded function, unless lim n n is not defined, we must ave lim n n 0, and ten we need tis long tail. If < 0 tis would be impossible. Example wit Inada Conditions X Ci-Square wit one degree of freedom. were: Pdf of Ci-square X k : n Γ k k Γ x k e x. k Remark: we are considering te distribution as. Ten: 0 t k e t dt We can sow tat: Γ π. Ten, te pdf of X is: Finally: n π n e n. n π n 3 e n + n σ + n. Notice tat: n as n 0 and < 0. Again, we ave > 0. Looking at n d n dk Anyway, I actually can sow a general proof of simplest case in wic β. n d n dk Proposition 4 If β and < 0, we ave tat < 0 if we assume < 0 at least for te n d n dk < 0. Proof. First of all, remember tat, simplifying te system of equilibrium conditions, we end up wit te following two conditions: 8

29 k n k m x n k n k m x n Ten, rearranging eq., we ave: k k m x n n Now, consider tat we increase m x. Since LHS is constant and < 0, we must increase n to increase te numerator and decrease te denominator. Terefore, an increase in m x increases bot m x n and n. Now, using te IFT, we ave: n n k k n + k m x n Similarly: n n k k n + k m x n Considering tat m x > m x, using a similar argument as te one we used above, we ave n > n and m x n > m x n an increase in m increases n but less tan proportionally. Ten, using te fact tat < 0 and < 0, we ave tat n < n and k n + k m x n ] > k n + k m x n ] since n is more negative tan n and so for so on. Terefore, n k < n k. But ten: n k n n k n < 0 and n d n dk n k n n k n n < 0. Now, I tink we can ave a broader solution. First of all, notice tat: n n k k n + k m x n dividing above and below by n, we ave: n k k n n k 9 mx n n

30 Since n k k m x n, we ave: n k { k n n mx n mx n A similar argument can be made for n k. Ten, for n k n n k n < 0, we ave: { } < n n mx n n n mx n } { n n mx n n n mx n Substituting te elasticity of substitution, we ave: n n n m x n n m x n < n n n m x n n m x n σ n n + m x n σ m x n < σ n n + m x n σ. m x n } Derivation of te continuous case. We need to be careful about wic assumptions we impose on n x for writing down te continuous case. If we rewrite te model wit s, we are using a partition/refinement argument, wic delivers a Riemann integral 7. Based on tis, we must ave a piecewise continuous n x. Consider a partition P and an associated set of points X in wic X i I i, were I i is an interval in te partition P. Ten, S P, X, f] is defined by: A function f is integrable if and only if: S P, X, f] N i lim S P, X, f] P 0 n X i X i I i. x x n x xdx for any P, X. We can sow tat any piecewise continuous function satisfies integrability. Te continuous case can derived from taking te appropriate limit for 0 ] β Ln, x nxdx A special case wit CES: ten becomes in te continuous case: N ] β Ln, x n γ i xα i i ] β Ln, x n γ i xα i dn i. 7 A function is Riemann integrable if it is continuous almost everywere, i.e., it is discontinuous in at most a zero measure set. 30

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