Edouard Ribes
email: edouard.augustin.ribes@gmail.com

A model of professional service firms performance with an application to US accounting firms

Keywords: Professional Services Firms, Firm performance Economic Classification [JEL]. E17, L16, L25, L84, D23

This paper models the dynamics of professional services firms performance. To do so, it first describes how those firms produce their specific outputs and how their productivity drives competitiveness. The associated equilibrium in then used to propose profit optimal organic and inorganic growth strategies, which are then discussed to account for the effect of potential future technological disruption. Applying the proposed theory to US accounting firms explains the differences in performance observed between large (i.e. more than 50 partners) and small (i.e. less than 50 partners) accounting partnerships and suggests that small (resp. large) US accounting partnerships mainly grow inorganically (resp. organically).

Introduction

Professional services are commonly defined (see [START_REF] Empson | Researching professional service firms: An introduction and overview[END_REF]) as the sum of legal, accounting, management consulting, advertising, engineering and architectural, scientific research and veterinary services1 . As per the US census, professional services have grown by 4.2% in revenue year on year (on average) between 2008 and 2018, outperforming the manufacturing sector [NAICS code 31 to 33], which year on year revenue growth has been on average of 2.1% over the same period. On the other hand, employment in professional services across the US has grown by 0.49% on average per quarter, while the manufacturing sector has shown an employment reduction of about -0.22% per quarter. At an aggregated level, those preliminary statistics stress that the manufacturing sector, on which the economic and management literature has classically been focused, and professional services obey different dynamics. Those differences have been acknowledged by the academic community in the past decade and have led to the development of a small research niche, nowadays anchored in the field of industrial organization by the seminal works of [Løwendahl, 2005] and [Maister, 2012]. If the performance and dynamics of manufacturing firms is a thoroughly discussed topic (see [START_REF] Bottazzi | Common properties and sectoral specificities in the dynamics of us manufacturing companies[END_REF] for instance), there has not been to my knowledge any adaptation made specifically for professional services firms [referred to as PSFs in the rest of this paper]. This appears as a gap, which can be easily brought to life through a few examples. For instance, manufacturing firms mostly rely on technology (and therefore capital) to produce a good, while PSFs mostly rely on labor to tailor a service. This naturally raises the question of the impact of the difference in the production engine on firms long term profitability. Similarly, if manufacturing firms rely on capital while PSFs rely on human capital, this has to entail a different growth strategy. Coming up with such examples is not difficult and this paper will start to start to bridge the associated gap. To explain the performance patterns of PSFs, this paper will build upon three main strands of the economic literature. First it builds upon the theory of firm growth. Since the seminal empirical work of Gibrat (see [Gibrat, 1931]), which was later completed by ( [Evans, 1987]), it has been thoroughly demonstrated that the growth rate of firms decreases as they become larger and that their performance becomes more predictable as they age. Additionally the larger and older firms are, the more likely they are to survive. From this empirical work has then stemmed a host of theories aimed at explaining firms' development [START_REF] Stanley | Scaling behaviour in the growth of companies[END_REF], [START_REF] Coad | Firm age and performance[END_REF], [START_REF] Haltiwanger | Who creates jobs? small versus large versus young[END_REF]...) and survival [START_REF] Anyadike-Danes | All grown up? the fate after 15 years of a quarter of a million uk firms born in 1998[END_REF], [van Stel et al., 2017], [START_REF] Cowling | Did firm age, experience, and access to finance count? sme performance after the global financial crisis[END_REF]...). The center piece of this research stream is nowadays Hopenhayn's framework ( [Hopenhayn, 1992]), which explains both growth and survival as the result of productivity shocks. However most of the growth studies focus on manufacturing and little has been done with respect to the service industry [START_REF] Nassar | The validity of gibrat's law in developed and developing countries (2008-2013): Comparison based assessment[END_REF]), even though it has been acknowledged that performance mechanism are industry dependent ( [Mansfield, 1962] )2 . This is therefore the first area of contribution for this paper. Second, this work contributes to the academic literature related to professional services firms. To date, PSFs have not attracted a lot of research because of two of their structural characteristics. First, as PSFs are mainly organized as privately held partnerships ( [Maister, 1982]), their performance is rarely publicly reported. Second, as PSFs are diversified ( [START_REF] Greenwood | Reputation, diversification, and organizational explanations of performance in professional service firms[END_REF]) and usually supply clientspecific combinations of services, their output is not as easy to measure for an outside observer as a unit of good. To overcome those difficulties and capture the performance patterns specific to this industry, the academic community has so far used PSFs' reputation as the input to predict performance (see [START_REF] Bar-Isaac | Seller reputation[END_REF], [START_REF] Bar-Isaac | Search, design, and market structure[END_REF]). Reputation is however difficult to assess consistently and the community still considers the professional services sector to lack a holistic theory (see [START_REF] Skjølsvik | Strategic management of professional service firms: Reviewing abs journals and identifying key research themes[END_REF]). This paper addresses this gap by exploring competitiveness at a service level and by constructing an "insider" picture of PSFs performance as portfolios of services3 . Finally, this work builds upon the literature on continuous optimization. It leverages Hamilton-Jacobi-Bellman [HJB] frameworks (see [Intriligator, 2002]), which are known to offer a suitable formalization of profit maximization/cost minimization problems in economy (see [START_REF] Doumic | Toward an integrated workforce planning framework using structured equations[END_REF] for an example). This paper provides an analytical solution of a PSF specific HJB model, which sits at the frontier between operational research and economy. This appears unique and innovative as there has not been previously any formalized PSFs' performance theory, let alone one which uses continuous tools. The remaining of this paper is organized as follows. Section (2) provides a descriptive analysis of the delivery of professional services. This is then leveraged in section (3) to model PSFs revenue in a competitive landscape. This model is afterwards used first to design both organic (section (4)) and inorganic (section (5)) profit-maximizing growth strategy for PSFs and second to understand the impact of technological investment on PSFs performance (section (6)). Finally section (7) discusses potential elements that can explain performance heterogeneity across firms. Note that throughout the paper, the proposed theory is illustrated by examples pertaining to US accounting firms and that additional areas of research are highlighted in the conclusion (section (8)).

2 Delivering professional services.

Labor in PSFs is made of production and sales activities (see [Løwendahl, 2005]). Selling is a complex task as it means tailoring a credible promise to a network of prospective clients. On the other hand, production can be scripted to follow a plan and therefore can be supported by resources with less expertise. As a result, labor in professional services is allocated across two types of workers, often coined "finders" and "grinders" (see [Maister, 2012]). It takes a considerable amount of time for an individual to develop a recognized expertise in the professional services field and become a "finder" (i.e. 10 to 15 years (see [START_REF] Morris | Promotion to partner in professional service firms[END_REF])). This has led professional services firms to be mostly structured as privately owned partnerships. This form indeed allows "finders" to fully ripe the benefits of their expertise as equity partners, while providing "grinders" with a natural incentive structure to develop their expertise (see [START_REF] Greenwood | p2-form" strategic management: corporate practices in professional partnerships[END_REF] ). The concept of a professional services firm therefore emerges from a labor distribution across S partners that can both sell and produce services and E employees that can only produce services under the supervision of an expert. Within a PSF, a partner's revenue R is generated by the amount of client N (L) to which he can sell services to at a price p when supported by L employees. To increase his revenue, a partner must therefore delegate a fraction f (L) (∈ [0; 1]) of the production activities of his N (L) clients to focus on selling activities. Both partners and employees have an availability of τ units of time and it can be assumed that sales (resp. delivery) activities take T sales (resp. T prod ) units of time per client. Additionally, service delivery needs to be coordinated, which means that employees have to spend a fixed portion of time T coord with each other to articulate an output (see [Dunbar, 1993] for a discussion). Among a pool of L workers, the first one has therefore to connect to L -1 individuals, the second worker to L -2 individuals, so that overall the amount of time spent on coordination activities is given by: T coord . 1≤l≤L (L -l). This means that coordination/communication between individuals is associated to quadratic costs, which is a classic assumption in economic models (see for example [START_REF] Crawford | Strategic information transmission[END_REF] or [Dessein, 2002]) and that coordination limits the delegation of production activities (see [Williamson, 1967] for a discussion). As a result, the amount of clients a partner can manage obeys the following rules:

τ = Sales activities N (L).T sales + Production activities T prod .N (L).(1 -f (L))
(1)

L.τ = Delegated production activities N (L).f (L).T prod + Coordination activities T coord . 1≤l≤L (L -l) (2)
Calling α = p.τ T sales +T prod the labor productivity of a partner and γ = T coord 2.τ the fraction of working time employees spend on coordination, the system ((1)-( 2)) yields that the revenue R a partner generates is given by: R(L) = α.(1

+ L(1 + γ) -γ.L 2 ) (3)
It has to be noted that this production system is yet subject to two constraints. First, a partner must have enough resources to service at least one client. Second a partner can not delegate more work than already available. This means that the number of employees a partner has is bounded between L and L (see lemmas (9.1) and (9.2) in the appendix for functional forms). Finally this model of production has two notable effects on firm's standard productivity indicators:

Lemma 2.1 Assuming that the partnership size remains unchanged, when partners recruit new employees, their firm's total labor productivity (ρ), defined as the revenue generated by the firm S partners divided by the amount of work realized across both the partners and employees population (i.e.

ρ(L) = α.(1 -γ.L. (1-L) 1+L
)), decreases, while the revenue per employee (

R E = ( 1 L + (1 -γ) + L.γ).α) increases if delegation is sufficient (i.e. L ≥ 1 γ ).
Example. The assumptions of this production model can be tested against real world data. The numbers of partners and employees as well as PSFs' revenue are for instance published by the Public Accounting Report on a yearly basis for the top 100 US accounting firms. This report covers about 75% of total US employment in the sector. Its 2018 edition showcases an average number of 9.1 (3.2) employees per partner and a revenue of 1.9 (0.9) M$ per partner. It shows that across all accounting firm, increasing delegation leads to a linear revenue improvement (see figure (1)) and that albeit crude, this type of model explains R 2 = 96% of the observed revenue variance in the accounting report. The estimation of the parameters of equation ( 3) via OLS shows that for every new employee, partner's revenue increases by α = 0.18M$ per year (0.007) and that across the overall panel of US accounting firms coordination does not appear to have a significant impact on partner's revenue (α.γ = -0.0003 (0.0004)).

Section (2) has therefore presented a model of service production, which appear credible against real world data. However the model does not explain how prices are set nor why there are important delegation differences within PSFs (within the 2018 Public Accounting Report the number of employees per partner L indeed ranges from 5 to 35). The proposed model will therefore be completed in section (3) to capture how professional services prices (and ultimately partners' revenue) result from a competitive equilibrium before being used alongside partners' profit maximization objectives to explain how delegation evolves as PSFs grow in sections (4) to (6).

3 Competition in professional services.

As seen in section (2), partners' revenue (i.e. R) depends in both the price of the service they offer (i.e. p) and the number of client they can manage (i.e. N ). If the previous section has shown that partners' bandwidth (i.e. N ) is a function of the number of employees (i.e. L) they delegate production activities to, prices fluctuations need to be explained. The classic mechanism used in economic theory to explain how prices are set is to consider an equilibrium where firms compete with each other based on their productivity (see ( [Hopenhayn, 1992])). In the case of PSFs, this means that the price p at which a PSF goes to market results from its delivery speed against the ones of its competitors. As PSFs are known to continuously tailor their service offerings and adjust their delivery speed as their partnerships evolve 4 (i.e. service production speed T prod (S) is a function of the partnership size) , they therefore come to clients (through requests for proposals for instance) with a price p(S) representative of their partnership structure. This price gets communicated to clients, who get a value V for the service and choose the provider who offers the most productive alternative (i.e. maximizes the client's gains for a given delivery time), a choice that can be articulated as the following maximization problem:

max S V.τ T sales + T prod (S) -α(S) (4)
This decision criteria can be used to infer the price structure of the professional services sector (see appendix for a proof).

Proposition 3.1 If production speed increases (i.e. ∂ S T prod ≥ 0) [resp. decreases] with the size of the partnership, the price of the associated services increases (∂ S p ≥ 0) [resp. decreases] and so does partners' productivity (α(S)), as professional services prices are given by:

∀S ≥ S min p(S) = V - T sales + T prod (S) T sales + T prod (S min ) .(V -p(S min )) (5)
Proposition (3.1) appears consistent with empirical evidences reported in the literature. It was indeed noted that tailored offerings (which takes longer to produce) in accounting ( [Zerni, 2012], [START_REF] Mayhew | Audit firm industry specialization as a differentiation strategy: Evidence from fees charged to firms going public[END_REF]], [Palmrose, 1986], [START_REF] Craswell | Auditor brand name reputations and industry specializations[END_REF]) and law firms ( [Hadfield, 1999]) led to higher prices. In essence, this means that the less commoditized the service, the higher its price, which is in line with the seminal ideas of [Maister, 1982].

Special case. Recent empirical evidences can be used to propose an analytical form for professional services prices and partners' revenue. On one hand, labor productivity has been shown to be distributed across firms according to a power law (see [START_REF] Aoyama | Labour productivity superstatistics[END_REF]). This means that the number of partnerships that have a productivity α is ∝ α 1 φ . On the other hand, it is also known that firms distribution follow a zipf law with respect to their size (see [Axtell, 2001]) The number of partnerships with S partners can then be assumed to be proportional S -1 . Those two findings therefore implies that partners' productivity follows a power law in their firm's partnership size (i.e. α(S) ∝ S -φ ). As service prices and partners' productivity are linked (see appendix), professional services prices and partners' revenue can be analytically expressed as:

p(S) = p(S max ).V p(S max ) + (V -p(S max )).e -φ.(Smax-S) ) R(S, L) = α 0 .S -φ .(1 + (1 + γ).L -γ.L 2 ) (6)
Example. The functional form (6) can be tested against the 2018 US Public Accounting report data set, which was used in section (2). To do so, a simple two step approach was used. First, for a given productivity elasticity (i.e. φ), the sensitivity of a partner's revenue to its leverage model (α 0 ,γ) was estimated. The procedure was then iterated over a range of value for φ (i.e. φ ∈] -2; 2[ with a step of 0.01 and the regression that yielded the highest explanatory power (i.e. R 2 ) was kept. This was done on the entire data-set (i.e. 100 firms) as well as on the subset of partnerships that had respectively less than 50 partners (i.e. 54 small partnerships with less than 500 employees) and more than 50 partners (i.e. 46 large partnerships (very large firms) with more than 500 employees). data-set (R 2 grows by 1 to 2%). This also highlights significant differences across small and large accounting partnerships. First, in small partnerships, partners' productivity appears to decrease as the partnership expands (φ ≥ 0), while it increases in large partnerships (see figure ( 2)). Second, adding an employee in a small partnership generates twice as much additional revenue than in a large one. Third, coordination does appear to be negligible in small partnerships while it becomes a significant topic in large ones5 . Sections ( 2) and ( 3) have shown that partners revenue (R) is dependent in both the amount of employees that ensure the service production (L) and the partnership size (S). Those two variables are yet not independent: partners indeed seek to maximize the firm profitability (which drives their earnings) and have to adjust their production structure as the partnership evolves (i.e. L is a function of S).

Results (summarized in the

The associated performance optimal trajectory will now be discussed throughout sections (4) to (6).

4 PSFs organic growth.

As seen in sections ( 2) and (3), partners' revenue (R) changes with both the size of the partnership (S) and the number of employees (L) they leverage. As partners are compensated based on their profit Π, while their employees perceive a fixed wage c (so that Π(S, L) = R(S, L) -c.L), partners have to adjust their leverage so as to maximize their earnings as their partnership is scaling up (i.e. L is a function of S). This evolution is made of two components. On one hand, partnerships grow organically as employees absorb enough knowledge over time to become able to sell services and be promoted to partners. On the other hand, partnerships can grow inorganically as their constituents search and recruit new peers externally. Organic growth has traditionally been considered as the main strategic option for PSFs. The associated promotion opportunities indeed act both as a long term incentive for employees (see [START_REF] Lazear | Rank-order tournaments as optimum labor contracts[END_REF] for a discussion on tournaments incentives) and as a way for the firm to increase its revenue (and possibly its employment) by identifying individuals with potential (see [Maister, 2012], [Maister, 1982]). Meanwhile, inorganic growth remains uncommon (see [START_REF] Morris | Promotion to partner in professional service firms[END_REF]), because it is expensive and risky (see [Farhadi, 2007] for a discussion on accounting firms). As such, this section will explain how partners can optimize their earnings in a PSFs that grows organically while section (5) will detail the changes entailed by the use of inorganic growth mediums. PSFs organic growth has been thoroughly discussed in the literature. The most prominent school of thoughts on the subject (see [START_REF] Galanter | Tournament of lawyers: The transformation of the big law firm[END_REF] for a review pertaining to law firms) stresses that it obeys a tournament structure, where a fixed portion θ of employees make it to partner [i.e. dS t = θ.E t .dt = θ.S t .L(S t ).dt]. For partners to maximize their earnings over a period T , they must therefore find a delegation strategy (L(.)) that solves the following problem:

Π = max L(.) T t=0 [R(L(S t ), S t ) -c.L(S t )].dt ↔ ∂ t Π + max L(.) (θ.S.L.∂ S Π + R(S, L) -c.L) = 0 (7)
This delegation strategy depends in the service delivery structure of the firm and notably changes if individual coordination (γ) is important as seen in the following proposition (proof can be found in the appendix)6 .

Proposition 4.1 When individual coordination is not negligible (i.e. γ = 0), partners' optimal delegation strategy (L(.)) depends in their productivity (α):

L(S t ) = 1 + γ.L(S 0 ) 2 γ . α(S 0 ) α(S) - 1 γ (8)
If partners' productivity decreases as the partnership expands (∂ S α ≤ 0), the firm's partnership size increases organically in an "exponential" fashion (i.e. has a increasing growth speed over time). Otherwise, the partnership grows but exhibits a decreasing organic growth rate.

If individual coordination can be neglected (i.e. γ = 0), the optimal delegation strategy is independent of the service price and depends only in the firm delivery speed (T prod (S t )):

L(S t ) = L max (S t ) = ( T prod (S t ) T sales ) (9)
The partnership organic growth speed then increases only if the service production activities slow down as the partnership expands (i.e. ∂ S T prod ≤ 0)

There are two notable features of the model developed through sections (2) to (4). First increasing PSFs' human capital (i.e. expanding the partnership) does not necessarily translate in growth. This highlights PSFs specificity in the industrial organization landscape as most of the literature considers firm growth as synonym of employment. Second the model shows that to achieve organic growth means speeding up service delivery (i.e. to standardize/ commoditize), whether or not delegation exhibits decreasing return to scale (for instance because of coordination). This is interesting because it shows that setting up a PSF as a partnership may not always be a suitable answer as the human capital accumulation does not necessarily entails the desired returns. Although this discussion is not in scope of this paper, this could constitute an interesting avenue for further research.

Example. The proposed model can be used with the 2018 US Public Accounting report to infer some properties on US accounting firms, assuming of course organic growth as the main medium of partnership expansion. First, employees promotion chances can be estimated as the data-set records both the growth rate of each firm's partnership ( ∂tS S ∈ [-17.5%; +50%] with an average of 7.4% (11.5%)) and their leverage. This can be done directly via OLS (as ∂tS S = θ.L) and results in θ = 0.52% (0.14%), which is on par with the literature7 . Second, the data-set can be used to infer how partnerships production speed is evolving. In the case of small [pres. large] partnership (i.e. less [reps. more] than 50 partners), leverage is negatively [resp. positively] correlated with the partnership size. Proposition (4.1) therefore entails that delivery speed increases [resp. decreases] as small [resp. large] accounting partnerships. This however appears counter factual in the case of large partnerships. As the market is subject to firms consolidation (see US census statistics) to gain in competitiveness, one would expect that large partnerships keep on improving their service delivery model while expanding.

In summary, the performance model developed until section (4) is the reflect of the current knowledge available in the literature and provides a performance optimal development strategy for partnerships. However, its core assumption is that partnerships grow organically which, as shown in the previous example, may lead to counter factual results. The next section will therefore expand the model to cover partnerships dynamics in a more holistic fashion and explain the observed revenue and employment patterns.

5 Profit-optimal PSFs growth strategies.

Although organic growth has been the main focus of the management and economic literature pertaining to PSFs performance, it is a slow process8 . To speed up their development and maximize partners earnings over a given time horizon, PSFs have the opportunity to compete in the labor market to source/attract partners from their competitors. The expertise of each partner is indeed general rather than firm specific. As such, on top of their organic growth patterns, PSFs bring in H ≥ 0 new partners and also loose T ≥ 0 partners to other firms. Their partnerships' evolution dynamics thus become:

dS = ( Organic growth θ.S.L(S) + External hire(s) H(S) - Competition pressure T (S) ).dt (10)
This search for new partners represents a significant investment, which is usually shared equally by all the current partners (see [START_REF] Groysberg | The risky business of hiring stars[END_REF] for instance). It is therefore subject to an optimization as the associated costs are empirically known to be super linear (see [START_REF] Christensen | On-the-job search and the wage distribution[END_REF]) (i.e. the unit cost of bringing a new partner in increases with the total of amount of partners externally hired). In the literature, those costs are commonly assumed to be quadratic (see [Burdett, 1978] for instance) (i.e. c H .H 2 ). To maximize their earnings, partners have therefore to find a strategy (L(.),H(.)) which solves an enhanced version of the problem (7) defined in section (4):

Π = max L(.),H(.) T t=0 [(p.N )(S t ) -c.L(S t ) -c H . H(S t ) 2 S t ].dt (11)
As seen in section (4), the overall growth strategy of a partnership is dependent in the nature of its service delivery model and the partners earnings maximization leads to the following proposition (please refer to the appendix for a proof).

Proposition 5.1 When service delivery is not subject to frictions due to coordination (i.e. T coord = 0), partners maximize their earnings by delegating all of their production activities to employees (i.e. L = L max (S) = (

T prod
T sales )(S)) and seeking peers externally at a rate H(.) given by 9 :

H(S t ) = T (S t ) -θ.L max .S t + (T (S t ) -θ.L max .S t ) 2 - S t c H .((α -c).(L max + 1) -C)) (12)
In a competitive landscape, the external search therefore secures the partnership growth (i.e.∂ t S ≥ 0) in the absence of labor coordination (e.g. the growth of small US accounting partnerships according to the previous examples). However, when coordination occurs, the optimal partnership growth strategy entailed by problem (eq. ( 11)) becomes more complex.

Proposition 5.2 When the delivery of professional services requires significant coordination (i.e. γ = 0), if the optimal external search speed (H(.)) of a partnership grows with partner productivity and leverage, it becomes asymptotically negligible as the partnership expands (i.e. H(S)/S → 0):

H = α(S).(2.γ.L -(1 + γ)) + c 2.c H .θ (13)
The delegation strategy of partners however grows more complex and differs from the partnership organic patterns as delegation and promotion potentially becomes a way to mitigate the partner market competitive pressure (i.e. c -θ.c H .2.T (S)) 10 :

L(S t ) = -(c -θ.c H .2.T (S t )) ± (c -θ.c H .2.T (S t )) 2 -4.(θ 2 .c H .S t + α(S t ).γ).M (S t ) θ 2 .c H .S t + α(S t ).γ (14)
Looking back at section (3), the case of US accounting partnerships can be used to illustrate the proposed theory on PSFs profitable growth.

Example. In small US accounting partnerships (i.e. firms with less than 50 partners) coordination is not significant. Therefore, partners productivity (α(S)) 11 and delegation strategy (L max (S)) 12 evolution with the partnership size can be directly estimated from the data-set. Assuming that the cost of an employee in the US is c = 0.06k$ per year (as per the US census), that partners turnover rate (i.e. T (S)/S) is proportional to their productivity (see [Jovanovic, 1979]) and worth 10% on average, that search costs (c H ) for new partners are worth 1M$ ( [START_REF] Groysberg | The risky business of hiring stars[END_REF]), the proposed theory predicts that the growth rate of US accounting firms is stable (see figure ( 3)). This is aligned with the data set (i.e. there was no correlations between small partnership size and their growth rates) and comes with two main observations. First, external hiring appears to be responsible for most of the growth of those small US accounting partnerships. Second, as those partnerships expand, partners turnover increases and so does external hiring as a compensation mechanism. ) and C = -α(S0).(γ.L(S0) 2 + 1 -

T (S 0 ) S 0 .θ .(2.γ.L(S0) -(1 + γ))) -(α(S 0 ).(2.γ.L(S 0 )-(1+γ))+c) 2 4.c H .θ 2 .S 0
11 Running the regression α = a + b.S yields that partner's productivity is worth a = 0.19k$ (0.012) per year and decreases by b = -480 $ per year (380) for every new partner that is added to the firm structure.

12 The leverage sensitivity to the partnership size can be estimated with the regression Lmax = a + b.S. An OLS estimation on the US accounting data-set yields that a = 22.86 (1.91) and for every new partner the leverage decreases by b = -0.34 employee (0.060).

In large US accounting partnerships (i.e. more than 50 partners), the dynamics are yet different. As labor needs to be coordinated when firms are large (see section (3)), the proposition (5.2) stresses that in large US accounting partnerships, external partner sourcing rapidly decreases to becomes negligible, a property that is empirically supported by the data-set (see figure ( 4)).

In summary, this section has shown that, in a competitive environment, if external search can be used to improve profitability, it only ensures the growth of nimble partnerships (for instance small US accounting ones) and not the growth of less agile ones. This duality in PSFs has already been empirically observed, although, to my knowledge, never explained. For instance, [START_REF] Galanter | Why the big get bigger: The promotion-to-partner tournament and the growth of large law firms[END_REF] reported a difference in growth modes in law firms as they grew bigger and broader evidences in the field of industrial organization (see [Evans, 1987]) have also stressed that firm growth is dependent in its maturity. The proposed theory however does not explain how service delivery speed (T prod ) improves as the partnerships expands. Although the topic has been usually approached in the management literature under the lens of process improvements (see [Løwendahl, 2005]), recent findings ( [START_REF] Frey | The future of employment: How susceptible are jobs to computerisation?[END_REF]) suggest that technology has the potential to replace labor at scale. Those type of investments could impact the proposed performance paradigm, a topic which will be the focus of the following section.

Technology and PSFs performance: a discussion

The sections (2) to ( 5) have demonstrated how PSFs can maximize their profitability in a market where competition is productivity-driven. This was notably shown to be synonym of growth when service delivery does not require coordination. Although the proposed framework closely aligns with the current literature related to PSFs ( [Maister, 2012]), recent findings (see [START_REF] Arntz | The risk of automation for jobs in oecd countries[END_REF]) have stressed that in upcoming years, 15% to 25% of PSFs labor could be replaced by technology. This could change the competitive equilibrium of the sector in the future as partners can invest an amount I in technology to speed up delivery13 . Assuming that technology becomes the main lever to pull to lower production time (i.e. T prod (I) with ∂ I T prod (I) ≤ 0) means that partners' productivity (α) and maximum leverage (L max ) become a function of technology. For partners to maximize their earnings, they must therefore find an optimal strategy around the delegation of their production tasks (L(.)), their peers external search (H(.)) and their accumulated technology investments (I(.)). Assuming that technology investments are shared equally within the partnership, the work done in section (3) to (5) entails that partners' optimal strategy is defined as a solution of:

∂ t Π + max L(.),I(.),H(.) ((θ.S.L + H -T )∂ S Π + α(I). (1 + L.(1 + γ) -γ.L 2 ) 1 + ( T prod T sales )(I) -c.L - (c H .H 2 + ∂ t I) S ) = 0 (15)
As in previous sections, several cases appear depending in the need for coordination while delivering services (note that proves can be found in the appendix.) Proposition 6.1 If labor coordination can be neglected (i.e. T coord = 0) and technology is an available option to become more competitive, partners investments are driven by the following optimal program (with C and p constants):

∂ S I = 2. c H .S.( V.τ T sales -p + C) + c H .(c + p).( (c + p) 4.c H .θ 2 - T 2.θ ) ( 16
)
They then delegate as much work as possible (i.e. L = L max (I)) and their optimal external search effort (H) becomes a function of their accumulated technology investments (H

= c+p 2.c H .θ + ∂ S I 2.c H ).
Technology therefore entails two main changes to the strategies of nimble partnerships (i.e. partnerships where labor does not require coordination). First, when technology is available, nimble PSFs profit maximization is no longer synonym of partnership growth. Second, technology leads to reduction of organic growth opportunities and an increased reliance on external searches. When employees coordination (T coord = 0) can no longer be neglected, the effect of technology on PSFs performance depends in the magnitude of the disruption.

Lemma 6.2 If technology represents a minor disruption (i.e. if there is a level of investment in technology I * which minimizes the production time (i.e. ∂ I T prod (I * ) = 0) and if the maximum leverage post investment (L max (I * )) is still superior to the current leverage (L(S 0 ))), then the optimal investment strategy for partners is to perform a single up-front investment (i.e. ∀S ≥ S 0 , I(S) = I * ) and the optimal delegation and external hiring strategies (L(.) and H(.)) still follow the patterns described in section (4).

Proposition 6.3 However if technology is disruptive (i.e leads to a drastic reduction in the delegation needs), partners profit maximization entails maximizing delegation (L = L max (I)). Technology investments are then given by (with R(I) = α(I).(1 + (1 + γ).L max (I) -γ.L max (I) 2 ) and C a constant):

∂ S I = 4.c H .S. C + (∂ I R -c.∂ I L max ) 2 θ 2 .(∂ I L) 2 + 4.S.c H .(R(I) - T.c θ.S -( θ.S.L max -T θ.S ).( ∂ I R ∂ I L max )) (17)
Finally external search activities are adjusted to account for both technology investments and the leverage structure (H(S)

= ∂ S I 2.c H + c 2.θ.c H -∂ I [(1+(1+γ).Lmax(I)-γ.Lmax(I) 2 ).α(I)] 2.c H .θ.∂ I Lmax
).

Example. Looking back at the 2018 US Public Accounting report, the analysis performed in sections (4) and ( 5) has shown that partners' productivity in small accounting partnerships (i.e. less than 50 partners) was decreasing as the partnership expands. The proposed theory therefore suggests that technology is not an available option for those type of firms. In larger partnerships, partners' productivity has been shown to increase, which is compatible with technology investments. However in the absence of more granular data, it not possible to further discuss to which extent does technology currently disrupt those firms. In summary, technology has the potential to change the way PSFs seek to improve their performance. When coordination can be neglected, this happens through a modification of their external search behavior. Otherwise, technology enable firms to over-come the limits entailed by labor coordination.

As technology starts to be more massively adopted in those practices, it would be interesting to witness how performance patterns change.

PSFs growth heterogeneity sources: a discussion

The model developed in this paper through section (2) to (5) has explained how PSFs currently maximize their performance. The associated examples have yet shown that proposed theory does not capture all of the PSFs heterogeneity in performance. This section therefore offers a few additional insights on why the model could yield different performance outcomes for different firms.

Service nature & PSFs performance. The professional services typology of [Løwendahl, 2005] was used in this paper to model a standard professional services delivery infrastructure (i.e. θ, T sales , T prod , γ) and to derive an optimal performance strategy. However as the field covers a broad range of industrial sub-sectors (e.g. the legal, accounting, engineering ()...) services), each of those dimensions can heavily vary from one service to the other. According to [Maister, 2012] (which framework is summarized in 5), those differences lead to 3 services types: "brains" (type "B"), "gray hairs" (type "G") and "procedure" (type "P"). "Brains" are client relation-based services, offered to long-term clients. "Grey hairs" (type "G") are more creative problem solving services , which leverage an expertise in a given field to solve the most unique and difficult problems there are. Finally "Procedure" (type "P") services are offer "off the shelf" (i.e. commoditized).

Those differences can be used to further explain some differences amongst PSFs. First the more complex the problem at the heart of the service, the more time an individual requires to develop the required selling skills ( [Løwendahl, 2005]). Therefore the probability of a producer to become a seller is lower in brain (resp. gray hair) firms than in gray hair (resp. procedure) professional services (i.e. θ decreases). Second, the more complex the service, the lower the ratio of production over sales activities (i.e. L max decreases) and the more difficult it becomes to invest in technology (i.e. |∂ I T prod | decreases). Leveraging the superscript "B"/"G"/"P" to refer to the service type and the notations of the previous sections, this translates into the following service delivery model properties.

Lemma 7.1 The more complex the service, the slower its organic growth (i.e. brains [resp. gray hair] services promote less partners than gray hair [resp. procedure] ones).

∀S > 0 (θ.L) B (S) ≤ (θ.L) G (S) ≤ (θ.L) P (S)
This means that to grow at a set rate, more complex services must rely more heavily on external search than commoditized ones. Additionally, as PSFs grow, they diversify their activities to deliver on a portfolio of service (see [START_REF] Greenwood | Reputation, diversification, and organizational explanations of performance in professional service firms[END_REF] for a discussion and the US accounting report for empirical data on the mix of services accounting firms offer for an example). Their overall performance can therefore differ based on their portfolio mix. If known differences in services characteristics can be used to explain some of the observed heterogeneity in PSFs performance, markets, defined in terms of clients and competitors pools can be extremely different from one place to the next, which naturally calls for a extension of the discussion on performance heterogeneity.

Markets & PSFs performance. From a market standpoint, the development of a professional service occurs in 4 steps ( [START_REF] Carman | Growth strategies for service firms[END_REF]) through a mix of a customer segment and geography related activities. The first step in growth consists in competing to attract more and more clients in a given customer segment after a local introduction. The second step is about developing an offering by complementing the existing service in a peripheral fashion through sub-services that can increase competitiveness by boosting the overall productivity ( [START_REF] Skaggs | Strategic positioning, human capital, and performance in service organizations: A customer interaction approach[END_REF]). The third step relies on the service introduction in a new geography. The last step is about expanding the service to a new customer segment, which often leads to its redefinition. The choice of a given market as defined by its geography (for example choosing the market of the technology firms with more than 200 employees in London -United Kingdom versus the one in Berlin -Germany) has been empirically shown to drive growth differences. PSFs that introduce their services in large and fast growing metropolitan areas indeed experience a quicker growth than their counterparts [START_REF] Huallacháin | The location and growth of business and professional services in american metropolitan areas, 1976-1986[END_REF], [START_REF] Bryson | The creation, location and growth of small business service firms in the united kingdom[END_REF]). Looking back at the framework developed in section (3), the growth of a service is indeed capped by the overall market evolution as: Therefore the location choice of a PSF can boost (resp. reduce) its growth opportunities if the market grows quickly (resp. is or becomes saturated) (i.e. the constraint (18) becomes binding).

Note that the choice of the service extension to another customer segment has the equivalent effect as it relaxes the condition (18) for the firm. Therefore PSFs generally both expand geographically and across customer segment at the same time in order to maximize their growth potential ( [START_REF] Dhandapani | Two paths to diversification: Performance implications of related diversification across two dimensions in professional service firms[END_REF]). Client demand therefore drives partnership organic growth rates (i.e. θ = ∂tf (t,z)dz L(s)g(s)ds ).

Example. To illustrate those differences, let's assume that a given management consulting service is introduced by a firm in 2016. This service is offered to legal and accounting firms in France that have between 50 and 250 employees. This represents a client pool N (t) of 340 firms according the OECD Structural Business Statistics (ISIC Rev. 4) database. The same database reports that the segment grows by about 18 firms on average per year. For the sake of simplicity, all the firms that supply this service are assumed to have the same leverage ratio L max = 5 (which is on par with the benchmarks observed in the literature [START_REF] Kipping | Alternative pathways of change in professional services firms: The case of management consulting[END_REF]). Let us also assume that a partner can sell services to 8 to 10 clients when production is fully delegated (see ( [Zerni, 2012]) for benchmarks). This means that the market has room for 85 partners and can add an extra 2 new partners per year, which result in an overall promotion rate of θ = 1%.

Partners characteristics & PSFs performance. Outside of markets and service nature, partners characteristics are also known to generate heterogeneous PSFs performance patterns. First partners' productivity (i.e. α) fluctuates due to random exogenous shocks (see [START_REF] Lucas | Investment under uncertainty[END_REF]). This generates "noise" with respect to the overall performance signal of their partnership, notably in firms which services portfolio is not extensive. Second, the choices made by partners in terms of external search have also an impact on partnerships' performance. When PSFs engage in larger corporate acquisitions in order to extend their partnership, the amount of change that the operation entails has notable consequences on PSFs' growth rates (see [START_REF] Mckelvie | Advancing firm growth research: A focus on growth mode instead of growth rate[END_REF]). Acquisitions indeed entail lay offs as efficiencies of scope are identified during the merger/acquisition process (see [START_REF] Mckelvie | A resource-based view on organic and acquired growth[END_REF]) and reduces revenue growth speed ( [Coad, 2007]).Finally, the ambitions of constituent partners have of course an impact on the firm performance [START_REF] Mckelvie | Advancing firm growth research: A focus on growth mode instead of growth rate[END_REF]). Empirical work has indeed stressed that some leaders are not interested in maximizing business performance ( [START_REF] Wiklund | What do they think and feel about growth? an expectancy-value approach to small business managers' attitudes toward growth[END_REF]) as they focus on non financial questions such as the well-being of their employees or their independence ( [Quader, 2007]) and level of control in their partnership. Note that even for partners with a focus on financial objectives, preferences for short term over long term profitability can lead to different decisions.

Conclusion.

Building upon a picture of professional services production mechanisms inspired by the seminal work of ( [Løwendahl, 2005]) and ( [Maister, 2012]), this paper provides performance optimal management policies for PSFs in terms of labor delegation, external partner search and technology investments.

The associated theory has been applied to explain the recent growth patterns of US accounting firms. Two new streams of research now naturally emerge from the proposed theory. First, a more granular perspective on the development of PSFs in terms of markets/location strategy appears of interest (see ( [Bodenman, 2000], [Harrington, 1995]) for empirical discussion elements). It would indeed prove useful to develop a business development as well labor cost optimization blueprint for PSFs considering an increasingly globalized service market (see [START_REF] Jensen | Tradable services: Understanding the scope and impact of services offshoring [with comments and discussion[END_REF] for a discussion on the impact of trade on professional services). This would notably expands on the high level work around competition presented in sections ( 3) and ( 7) of this paper. Second, it would be interesting to understand at a more granular level the incentive mechanisms at play in professional service firms. This would indeed provide a better idea of what cements such firms service delivery model and would potentially help understand why most PSFs start small, do not grow and die ( [Storey, 2016]).

Legal disclaimer. The findings and opinions expressed in this article are those of the author and do not reflect any positions from any company or institution.

Lemma 9.2 A partner can not delegate more production activities than available (i.e N delegated prod ≤ N ).

If

1 2.T coord (τ + T coord 2 ) 2 ≤ τ.T prod
T sales , there is no restriction on the maximum number of employees that a partner can leverage. Otherwise, the maximum leverage L max is (

τ T coord + 1 2 ).(1-1 - 4.τ.T prod T sales .(2.τ +T coord ) )
Proof. The condition N delegated prod ≤ N can be translated into the following condition on leverage:L.(τ +

T coord 2 ) -L 2 . T coord 2 ≤ τ.T prod
T sales . This means that, if it exists, the maximum number of employee a partner can leverage is solution of:

L.(τ + T coord 2 ) -L 2 . T coord 2 = τ.T prod T sales
This leads to the proposed solution and condition.

Proof of proposition (3.1) For clients to have an interest in purchasing the service, prices must be such that p(S) ≤ V . Additionally the market productivity maximization (eq.( 4)) leads to: .) ] in equation ( 7) leads to: 2.γ.L.α(S)+c-α(S).(1+γ)

∀S > 0 ∂ S p = -
θ.S = ∂ S Π. This means that the overall equation (7) translates into: ∂ t Π + L 2 .γ.α(S) + α(S)) = 0. As ∂ tS Π = 0, this can be differentiated against S to yield: ∂sα α = -2.L.γ 1+γ.L 2 .∂ S L, which then leads to eq. ( 8). If coordination can be neglected (i.e. γ = 0), the partner profit maximization problem (eq. 7) leads to L = L max = T prod T sales . The problem (eq. 7) indeed becomes linear in L and doesn't admit a strict maximum. This forces L to be set at L max . As partners define their optimal delegation scheme, the pace at which the partnership evolves becomes set. When coordination occurs, the partnership grows organically at rate of θ.L, therefore as the leverage increases, so does the partnership growth speed (∂ t ( ∂tS S ) = ∂ t S.θ.∂ S L = -θ 2.L . ).

Proof of proposition (6.1) As seen in the previous sections, the absence of coordination leads to L = L max (I) and the profit maximization problem becomes: ∂ t Π + max I(.),H(.) ((θ.

9

  Up to a constant C = c H .H 0 S 0 .(H0 + 2.θ.S0.Lmax(S0) -2.T (S0)) + (α(S0) -c).(Lmax(S0) + 1) 10 With M (S) = ( C. θ 2 .c H .S γ.α + θ.c H γ .(θ.S -T.(1 + γ)) + c 2 +α 2 .(1+γ) 2 4.γ.α-3.c.(1+γ) 2.γ

  Number of clients that can be served by partners≤ f (z)dzTotal number of clients available in the market (18)

D ( 19 )

 19 T D (S) ∂ S α = τ. ∂ S p.T D -∂ S T D .p T 2Proof of proposition (4.1) As pN (L, S) = α(S).(1 + (1 + γ).L -γ.L 2 ), the maximization problem [max L(

  When coordination effects are neglected, profit maximization entails maximizing delegation. In this case, the condition ∂ S Π = 2.c H . H S holds and the problem becomes:∂ t Π+ c H .H S .(H +2.θ.S.L max -2.T (S))+(α -c).L max +α = 0. When differentiating against S, it comes that:[ c H .H S .(H + 2.θ.S.L max -2.T (S)) -c H .H 0 S 0 .(H 0 + 2.θ.S 0 .L max,0 -2.T (S 0 ))] = [(α 0 -c).L max,0 -(α t -c).L max,t + α(S 0 ) -α(S t)], which leads to the desired result. Proof of proposition (5.2) Proof. The problem (11) is equivalent to: ∂ t Π + max L(.),H(.) ((θ.S.L + H -T ).∂ S Π+p.N (L)-c.L-c H . H 2 S ) = 0. The maximization against H and against L yields that ∂ S π = 2.c H .H S = α(S).(2.γ.L-(1+γ))+c θ.S, which leads to the proposed solution for external hires (see eq. (13)). The problem (eq.11) then becomes: ∂ t Π+(γ.L 2 +1-T S.θ .(2.γ.L-(1+γ))).α(S)+ (α(S).(2.γ.L-(1+γ))+c)) 2 4.θ 2 .c H .S = 0. Differentiating against S and leveraging the fact that ∂ tS Π = 0, the proposed result appears as: ∂ S ((γ.L 2 + 1 -T S.θ .(2.γ.L -(1 + γ))).α) = -∂ S ( (α(S).(2.γ.L-(1+γ))+c)) 2 4.θ 2 .c H .S

  S.L max (I) + H -T )∂ S Π + p(I).τ T sales -c.L max (I) -(c H .H 2 +∂tI) S ) = 0. As ∂ t I = ∂ S I.(θ.S.L + H -T ), differentiating against H and I yields: ∂ S Π = 2.c H .H S = c+p θ.S + ∂ S I S . This leads to the desired formula for H(.) and I(.). Proof of proposition (6.2) When coordination is significant, partners profitability maximization leads to : ∂ t Π + max L(.),I(.),H(.) ((θ.S.L + H -T )∂ S Π + α(I).(1 + L.(1 + γ) -γ.L 2 ) -c.L -(θ.L + H-T S ).∂ S I) -c H .H 2 S = 0. Differentiating against I and assuming that delegation is L < L max (I), the condition: ∂ I α.(1 + L.(1 + γ) -γ.L 2 ) = 0 appears, which leads to the suggested result.

Figure 1 :

 1 Figure 1: Revenue evolution with delegation in US accounting firms.

Figure 2 :

 2 Figure 2: Partner productivity evolution with partnership size in US accounting firms.

Figure 3 :

 3 Figure 3: Partnership growth rates evolution in small US accounting partnerships (i.e less than 50 partners).

Figure 4 :

 4 Figure 4: Partnership estimated external hiring rates evolution in large US accounting partnerships (i.e more than 50 partners). g

Table 1 :

 1 2018 US accounting partners revenue sensitivity analysis.

	Parameter	Estimates	Estimates	Estimates
		across all partnerships across small partnerships across large partnerships
	φ	-0.048	0.078	-0.11
	α 0	0.14 (0.01)	0.22 (0.01)	0.11 (0.01)
	α 0 .γ	-0.0005 (0.0003)	0.0002 (0.0006)	0.0007 (0.0003)
	R 2	97%	97.1%	98%

table (1)) show that taking into account partner's productivity fluctuations into the proposed approach increases the explanatory power of the proposed model on the

Please refer to the US census website for high level statistics on the professional services sector, accessible within the North American Industrial Classification System (i.e. NAICS) through code 54

Within the professional services sector, some empirical evidences have even shown that some sub sector may not even follow Gibrat's law (see[START_REF] Galanter | Why the big get bigger: The promotion-to-partner tournament and the growth of large law firms[END_REF] for an example on law firms)

Expert services are indeed provided by partners who employs around 10 individuals (see[Zerni, 2012] for an accounting example). A firm that employs more than 500 individuals in a given location may therefore be considered as a portfolio of more than 50 specialties.

This occurs either through specialization (for instance on an industrial sector or a specific offer) or through cross selling (selling an integrated package with multiple services to a client) (see[START_REF] Greenwood | Reputation, diversification, and organizational explanations of performance in professional service firms[END_REF] for a review)

Interestingly this means that every time that an employee has to coordinate with another, it takes about 0.2% of their time. As a point of reference, this is of a same order of magnitude as the coordination time observed in societies. This was assessed and reported by[Dunbar, 1993] to take about 0.3% of an individual time.

The accounting example developed in section (3) has indeed shown that labor coordination only occurs in certain specific cases (e.g. accounting partnerships with more than 50 partners).

For instance,[START_REF] Galanter | Why the big get bigger: The promotion-to-partner tournament and the growth of large law firms[END_REF] reports that it takes about 10 years for an individual to get a 5 to 20% change to get promoted, which means that promotion rates (θ) are in the [0.2%; 0.5%] range

With promotion rates θ averaging [0.2%,0.5%] per year and a leverage per partner of 10 employees, small partnerships (i.e. with less than 50 partners) only promote about 1 new partner per year

Note that this extends the classical literature around firm growth and technology investments (see[START_REF] Lucas | Investment under uncertainty[END_REF]) by providing a specific application to professional services firms.

1 2.T coord (τ + T coord 2 ) 2 ≥ (T prod + T sales -τ ) and a partner requires a minimum of L = ( τ

Proof. If a partner doesn't have enough bandwidth to service a client on its own (i.e. τ < T sales + T prod ), he/she needs to delegate the service production to a number of employees L > 0 solution of the following equation:

For this solution to exist, the following condition must be met 1 2.T coord (τ + T coord 2 ) 2 ≥ (T prod +T sales -τ ), which lead to the proposed solution.


M Abambres

How Not to Lead Academia

Keywords: Academia, Scholars, Higher Education, Schools, Universities, Academic Leadership, Academic, Passion in Work, Freedom, Trust

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Academic Leadership

In the recent past, a lot of attention has been devoted to student mental health on university 1 campi. Research has shown that a large portion of students experience prominent levels of depressive symptoms, and that a significant portion of scholars are at risk of burnout [START_REF] Gorczynski | More academics and students have mental health problems than ever before[END_REF]. Naturally, both previous facts are deeply related, since poor mental health have profound impact on the competence and productivity of academics.

1 From now on, 'university' might be used to refer to any type of higher education institution.

2

Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227

Causing people to supress parts of their identity is limiting a valuable source of creativity.

Managers should encourage identity integration (e.g., female engineers working in an environment where they don´t feel they have to dress like men) in order to boost innovation [START_REF] Amabile | Creativity and the Role of the Leader[END_REF]. Richard Ingersoll, professor of Education and Sociology at the University of Pennsylvania, has done extensive research on teacher retention and quality. His studies reveal that most teachers have little input into the decisions that affect their work, and this lack of control drives many out of the classroom [START_REF] Langhorne | The Teacher-Powered Schools Movement: Transforming Teachers from Industrial Workers to Professionals[END_REF]. A recent report from the Consortium for Policy Research in Education shows that schools with (i) higher levels of teacher leadershipthe amount of input teachers have in school decision-making, and (ii) higher focus on teaching and learning, produce greater student achievement [START_REF] Langhorne | The Teacher-Powered Schools Movement: Transforming Teachers from Industrial Workers to Professionals[END_REF]. Leadership is seen as the single key component in determining the success of a school [START_REF] Du Plessis | The meaning of out-of-field teaching for educational leadership[END_REF]. Teachers must be able to lead in the areas they are passionate about [START_REF] Maiers | Guidelines of Passion-Based Learning[END_REF]. According to teacherpowered.org, there are more than 120 teacher-powered public schools in USA, spanning at least 19 states and serving students from preschool to age 21. 85% of Americans agree teacher-powered schools are a good idea. These schools are not 'anti-principal'; instead, they are managed by true leaders [START_REF] Langhorne | The Teacher-Powered Schools Movement: Transforming Teachers from Industrial Workers to Professionals[END_REF].

Aristotle (384 -322 B.C), a prodigious Greek researcher and writer (see Fig. 1), is amongst the greatest philosophers of all time. In areas such as logic, metaphysics and philosophy of mind, ethics, political theory, aesthetics and rhetoric, and empirical biology, Aristotle's theories have (i) provided interest and illumination, (ii) met with confrontation, and (iii) sparked debate [START_REF] Shields | Aristotle. The Stanford Encyclopedia of Philosophy[END_REF]. For Aristotle (Abambres 2019), a leader is the one caring about your happiness and creating the conditions for you to find your perfect role in the system (e.g., a company). According to Aristotle, if you are passionate at work, instead of fearful of (i) being fired, (ii) not getting a raise, (iii) not being promoted, or (iv) getting your values and opinions rejected or criticized (etc.), your performance will be way more effective. Throughout our careers we are taught to conform to the status quo, i.e. to the opinions and behaviors of others. Organizations and their workers both pay a price: decreased engagement, 4Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227 productivity and innovation. According to [START_REF] Gino | Let Your Workers Rebelthe big idea[END_REF], if you want engaged employees, you should encourage them to question the status quo and bring out their signature strengths.

"Those who are crazy enough to think they can change the World, are the ones who do", Steve Jobs

Out-of-Field Teaching/Researching

Hiring new professors is among the most consequential work done by academic departments, but the conduct of such tasks has often been left undiscussed [START_REF] Chambliss | Hiring Departmental Faculty[END_REF]. A solid consensus, based on research and common sense, recognizes that teachers must be knowledgeable about the subjects they teach if they are to help all students achieve high academic standards (Jerald and Ingersoll 2002). According to [START_REF] Chambliss | Hiring Departmental Faculty[END_REF], a well-tested rule to handle faculty searches, aimed at hiring the best possible person for the job, is to hire for demonstrated strength and not for lack of weaknesstoo often, candidates are eliminated for what they can´t do. If one hires for lack of weakness, what remains is one person who appears to be good at everything but great at nothing.

Unfortunately, some colleagues insist on requiring that all their faculty be proficient in multiple unrelated areas. Many ask (i) great teachers to be productive scientists, (ii) the most dedicated researchers to spend time meeting with assistant deans on faculty governance committees, or (iii) gifted student advisors to win grants from the National Science Foundation, and when they don´t they get fireda terrible waste [START_REF] Chambliss | Hiring Departmental Faculty[END_REF]. The standard issue "balance of teaching, scholarship, and service" is a recipe for mediocrity, in which each individual, rather than the organization as a whole, must include all the needed skills. The result will be everybody trying to shore up their weaknesses instead of being spectacular successes in their strong areas [START_REF] Chambliss | Hiring Departmental Faculty[END_REF]. In any case, the best hiring strategy is to hire someone who´s great at something and place 5Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227 him/her where that strength will do the most good. When hiring new talent, [START_REF] Finnigan | Hiring for Passion, Not Experience[END_REF] believes that passion can actually play a greater role than experience towards success. Kiesa Kelly (PhD), chair of the psychology department at Tennessee State University, aims to hire candidates who [START_REF] Abrams | Academic careers: 8 steps for hiring the faculty of your dreams[END_REF]) are passionate about (i) teaching, (ii) mentorship and (iii) cultural competence, instead of hiring for the (a) reputation of the institutions in their CV, and (b) publication record (a strongly criticized matter in [START_REF] Abambres | Research Counts[END_REF]Abambres et al. , 2019)). For [START_REF] Smoot | Passion and Awareness: What Great Teachers Have in Common[END_REF], great teachers are passionate about the content of their teaching.

Nevertheless, as most faculty know, it´s not uncommon to be asked to teach a course you don´t feel fully prepared to, especially if you are a junior faculty member. The issue of careless assignment of teachers is not new and is a universal problem, but in the past two decades it has become to be seen as one of the major hindrances to quality education. A major issue is knowing why something is important in an area outside your expertise [START_REF] Rhem | Tomorrow's Teaching and Learning -Teaching What You Don't Know[END_REF]), but the worst 6Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227 consequence is perhaps the creation of a vicious cycle (Sambe 2015)a teacher not satisfied with the job resigns, leaving an urgent vacancy that needs to be filled quickly by the administration, which most likely hires an out-of-field teacher (the easy way). Research has found that teaching out-of-field can compromise "teaching competence"e.g., ineffective teaching methods are likely to be used, teacher's identity and well-being are disrupted, and it's often associated with teacher strain and attrition [START_REF] Hobbs | Teaching Science Out-of-Field[END_REF], 2015[START_REF] Sambe | Out-of-Field Teaching -Consequences of teachers teaching out of their field of study[END_REF]. Education institutions are becoming too much like businessesteachers and students don´t really matter, which ignores the unique purpose of those organizationsproviding an effective teaching and learning environment.

Time Management in Faculty Hiring or New Role Assignment

In my experience 2 (including testimonials from other academics I´ve met), virtual all higher education private institutions show terrible time management skills in faculty hiring or new role assignment (for faculty already employed by the institution). They simply don´t assign a new job early enough (i.e., at least 1 month per course prior the start of the semester) so that professors are able to thoroughly prepare their classes. They just don´t care! Not even when they get spontaneous applications way before the beginning of classes or are aware in advance about professors' aims to teach new courses. Authorities of higher education institutions do not understand that this bad time management jeopardizes education and their business. With lack of time to prepare lectures, professors (i) might get anxious, (ii) might miss important hours of sleep, (iii) won´t have enough time to update their classes based on the state of art, (iv) won´t have enough time to improve the teaching methods, and (v) might have to stop researching to manage preparing all lectures in time. Both professors and students are negatively affected.

2, 3 Already quite vast, although still a young professional (check my LinkedIn).

7

Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227

Higher Education Management: Abambres' proposal

Next I´m describing in an upfront manner which fundamental decisions I´d take if I could manage a higher education institution or academic department. That decision-making is crucial for the success of any higher education institution, and according to my experience 3 it has not been adopted by most private universities. For me the University should not be a place to prepare people for real life, but a place to teach them how to be happier and change real life for better.

Hiring

I´d hire only passionate and proactive people and I´d offer them jobs that would fully comply with their passions. Concerning faculty hiring or role assignment for teaching, every employee would have been assigned any course (and respective syllabus, if not to be created from scratch by the employee) at least 2 months before the beginning of classes. A contract would be signed as soon as any professor knew all the teaching duties for the new semester.

Social Engagement

Building relationships with students beyond the classroom can help drive learning, since they typically work harder with people they have empathy with [START_REF] Maiers | Guidelines of Passion-Based Learning[END_REF][START_REF] Brooks | Students Learn From People They Love[END_REF]. I have felt throughout my experience that there's a lack of social engagement with students and other faculty in private universities, especially in Portugal. In South America, people are more open minded and talkative, which helps even if no social engagement is promoted by the university. Creating noncompulsory ways of social engagement is always beneficial because (i) everyone is free not to participate, and (ii) it´s an opportunity to get more empathy with students and colleagues, which increases happiness in work. 8Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227

Student Assessment

In my opinion all tests or exams should be multiple-choice-based and carried out in institutional computers for that purpose. A software should be created so that those types of assessment are immediately classified, saving lots of time to professors (who could use it to rest more, improve their classes, or research). In my opinion (I might publish a paper about it soon), it´s possible to assess (i) theoretical questions, (ii) reasoning questions, and (iii) the ability to get the right numerical solution of a problem, via multiple-choice tests. Moreover, it can be perfectly guaranteed (via correction criteria and the amount of answer choices per question) no risk of cheating and very low chances of selecting right answers by luck. Furthermore, group or individual home assignments should be evaluated by oral presentation in classroom only, i.e. every assessment in any course would be based on student's performance in the class/assessment room.

Open Mindedness

Authorities (deans, directors, academic coordinators) in academia should be highly open minded. Open mindedness is key for the development of creative and innovative work and decisions. All authorities should be open to the proactiveness of their employees, who (like me) might constantly be thinking on ways to improve their working conditions and to attract more professionals and students to be part of the university. People like me straightforwardly email/text authorities to share negative facts about the university (e.g., noisy library, outdated/damaged hardware and software) or suggest improvements. Unfortunately, I´ve felt that most authorities don´t like nor encourage this type of communication. In my experience, most of them not even answer my emails.

Prompt Communication

Some academic authorities (deans, directors, academic coordinators) take too much time to reply to emails or any other type of digital communication. Some don´t even care about replying at all (even when we get to know that messages are received and reade.g., WhatsApp). The minimum that should be done would be replying within 24 (ideally)-36 hours to let the sender know his/her message was received and will be carefully analyzed sometime soon. If one is too busy to make it possible, that task should be assigned to someone else.

Passion

The way to maximize any company's performance is to hire passionate people only. Research shows that the best teachers are passionate about teaching and driven by a deep desire to help others [START_REF] Hattie | Identifying accomplished teachers: A validation study. Assessing teachers for professional certification: The first decade of the National Board for Professional Teaching Standards[END_REF]. I wouldn´t hire people who work for other companies outside academia.

It turns out that those employees are not passionate enough about teaching or researching. Those who do, think their time is too short to work, get new knowledge, and improve their skills. Working outside academia typically makes those employees perform crucial academic duties in a rush, which jeopardizes education. The real passionate people I´d hire would be the ones who don´t have many passionsthose who like to focus on a few areas so they can excel in all of them.

According to [START_REF] Murray | Teaching Strategies to Help Students Find Their Passion[END_REF], passionate teachers are those committed to students' success, i.e.

showing students how education can be used to achieve their goals/passionssometimes called passion-based learning, instead of focusing on the achievement of good grades. By doing what they love, students learn to try hard, take risks, and 'never' quit. Being around passionate people is the best way to become passionate [START_REF] Hatfield | Emotional Contagion[END_REF][START_REF] Maiers | Guidelines of Passion-Based Learning[END_REF]), which can drive success.

"The lecturer should give the audience full reason to believe that all his/her powers have been exerted for their pleasure and instruction", Michael Faraday (in AMM 2019). -Asking faculty not to present power point slides with contents not in the national language, even though the class is fully given in the national language and the secondary languages used in slides are the two most popular in the western world (Spanish and English). Moreover, one should bear in mind that faculty might have no other choice than using other languages (e.g., from textbooks used to prepare the classes, from presentations previously prepared for a foreign audience) to guarantee that classes are prepared on time, as consequence of the awful time management practiced by academic authorities (see section 3).

-Asking faculty to solve all the exercises on the classroom board, step-by-step, instead of explaining the step-by-step resolution in power point slides made available to every student. They argue that if we don´t do it on the board, students won´t follow all our reasoning and won´t have the courage to tell the professor they don´t understand some step (LoL).

3

  Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227

Fig. 1 .

 1 Fig. 1. Statue of Aristotle in Stagira, Greece (taken from URL).

Fig. 2 .

 2 Fig. 2. Out-of-field (or at least "out-of-passion") researching also occurs, inclusively with students, as reported in here.

10

  Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227 "Teaching is not a profession; it´s a passion.Without passion for your subject and a desire for your students to learn and be the best in the World, then we have failed as a teacher", John F.Podojil (in AMM 2019). "Don´t ask yourself what the World needs, ask yourself what makes you come alive and then go and do that. Because what the World needs is people who come alive", Howard Thurman (in Maiers 2017).4.7 Freedom / TrustMost of private universities I already worked for care more about student complaints than what hired faculty think to be the best teaching methods for a particular course. Regardless the number of complaints, those coordinators and directors believe that the best education is the one leading to immediate happiness during any class. According to opinions from Clóvis de Barros Filho, Kant (renowned philosophers) and mine, those academic authorities are fully wrong(Abambres 2019). Educating in that way makes Man's growth unfeasible, because an effective education process must include student discomfort, 'pain', mistakes, correction or warning. Acting in higher education like students are dumb or still children, hyper-protecting them, will definitely slow down the development of their intellectual capacity. Examples of such behavior by academic authorities are: ID: hal-02281227 © 2019 by Abambres M (CC BY 4.0) 11 Abambres M (2019). How Not to Lead Academia. HAL (Sept 6 th ), id: hal-02281227

"

  It doesn´t make sense to hire smart people and tell them what to do; we hire smart people so they can tell us what to do", Steve Jobs (in Schroeder 2018).

© 2019 by Abambres M. Open access publication under the terms and conditions of the Creative Commons Attribution 4.0 (CC BY 4.0) license (creativecommons.org/licenses/by/4.0).

Acknowledgements

I dedicate this paper to my beloved parents Ana Abambres and José Abambres, my cats Tiger and Cheetah, and all those who enjoy and support my work and mindset.