Financing professional partnerships

Abstract

Increases in net-debt obligations of profit sharing partnerships give these organizations a strong incentive to expand. This paper argues that when capital structure is transparent, partnerships can signal their hiring intentions to uninformed clients by their net-debt levels. Levin and Tadelis (2005) predicts that professional service firms with fewer informed clients will tend to choose to organize as partnerships rather than corporations. The present paper demonstrates that this prediction only holds when financial frictions are present.

Appendix

1.1 The inverse relationship between size and quality

Here we want to show that there is an inverse relationship between the size and the average quality of the workforce. To do this, first, we will show that firm size falls as the ability of the minimum ability employee rises. Then, we will demonstrate that average quality rises as the cut off ability level rises. Therefore, if the firm lowers (raises) its ability threshold, size increases (falls) and quality falls (rises).

Firm size is the fraction of the distribution of employees selected multiplied by the number of employees in the distribution. σ is a parameter that stands for the number of potential employees in the distribution G(a). For example, if the firm hired everyone, then firm size would be σ=$\sigma \left[ {G\left( {\overline a } \right) - G\left( {\underline a } \right)} \right]$. The firm will only select the highest ability subset of employees because all employees have the same wage. Therefore,

$$ N\left( {\widetilde{a}} \right) = \sigma \int\limits_{\tilde a}^{\bar a} {g(a)da} = \sigma \left( {1 - G\left( {\widetilde{a}} \right)} \right) $$

(33)

If we differentiate (33) by the ability of the marginal, lowest ability employee, then given that g(a) > 0 inside its support

$$ \frac{{\partial N\left( {\widetilde{a}} \right)}}{{\partial \tilde a}} = - g\left( {\widetilde{a}} \right) < 0,\quad \forall a \in \left[ {\underline a, \overline a } \right] $$

(34)

This is the first assertion that we wanted to prove. Q.E.D.

Secondly, let us turn to the average quality of the workforce.

$$ q\left( {N\left( {\widetilde{a}} \right)} \right) = q\left( {\widetilde{a}} \right) = \frac{{\int\limits_{\widetilde{a}}^{\overline a } {ag(a)da} }}{{G(\overline a ) - G\left( {\widetilde{a}} \right)}} $$

(35)

Differentiating (35) with respect to $\widetilde{a}$, we are left with the following:

$$ \frac{{\partial q\left( {\widetilde{a}} \right)}}{{\partial \widetilde{a}}} = \frac{{g\left( {\widetilde{a}} \right)}}{{G\left( {\overline a } \right) - G\left( {\widetilde{a}} \right)}}\left[ {\frac{1}{{G\left( {\overline a } \right) - G\left( {\widetilde{a}} \right)}}\left( {\int\limits_{\widetilde{a}}^{\overline a } {ag(a)da} } \right) - \widetilde{a}} \right] $$

(36)

If we substitute (35) into (36) it becomes easier to see why quality must rise in $\widetilde{a}$

$$ \frac{{\partial q\left( {\widetilde{a}} \right)}}{{\partial \widetilde{a}}} = \frac{{g\left( {\widetilde{a}} \right)}}{{1 - G\left( {\widetilde{a}} \right)}}\left[ {q\left( {\widetilde{a}} \right) - \widetilde{a}} \right] > 0,\quad \forall a \in \left[ {\underline a, \overline a } \right] $$

(37)

The change in average quality as the lowest quality rises is the hazard rate times the difference between the average quality and the lowest quality. The lowest ability employee, with ability $\widetilde{a}$, must have a lower ability than the average ability employee hired. Therefore, the square-bracketed term must be positive. Further, $g\left( {\widetilde{a}} \right) > 0$. The sign in (37) must be positive, which is the second part that we wanted to demonstrate. Q.E.D.

As the firm chooses a higher ability threshold, $\widetilde{a}$, for the lowest ability employee, size falls and quality rises.

1.2 Derivation of the comparative static, $\frac{{\partial {N^C}}}{{\partial \mu }} < 0$, in Eq. 12

The endogenous N ^C(μ) is entirely determined by the first order condition (FOC) in Eq. 9. We can use the first order condition to sign $\frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }}$. We can use the FOC to derive the comparative static using the implicit function rule. The implicit function rule requires us to find the cross partial of N and μ and the second order condition (SOC), respectively.

$$ \begin{array}{*{20}{c}} {{m_\mu } \equiv {{\left. {\frac{{{\partial^2}\left\{ {\pi - K} \right\}}}{{\partial N\partial \mu }}} \right|}_{N = {N^C}}} = {p_N}\left( {N^C} \right){N^C} + p\left( {N^C} \right) - p\left( {{N^e}\left( {\mathbf{z}} \right)} \right) + \left( {1 - \mu } \right){p_N}\left( {N^e} \right)\frac{{\partial {N^e}\left( {\mathbf{z}} \right)}}{{\partial \mu }}} \hfill \\ {{m_\mu } \equiv {{\left. {\frac{{{\partial^2}\left\{ {\pi - K} \right\}}}{{\partial N\partial \mu }}} \right|}_{N = {N^e} = {N^C}}} = {p_N}\left( {N^C} \right){N^C} + \left( {1 - \mu } \right){p_N}\left( {N^C} \right)\frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }}} \hfill \\ \end{array} $$

(38)

$$ {m_N} \equiv {\left. {\frac{{{\partial^2}\left\{ {\pi - K} \right\}}}{{\partial {N^2}}}} \right|_{N = {N^e} = {N^C}}} = \mu \left\{ {{p_{NN}}\left( {N^C} \right){N^C} + 2{p_N}\left( {N^C} \right)} \right\} < 0 $$

(39)

While we cannot immediately sign (38), some algebra allows us to isolate $\frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }}$.

$$ {\left. {\frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }} = - \frac{{{m_\mu }}}{m_N}} \right|_{N = {N^e} = {N^C}}} = - \frac{{\left( {{p_N}\left( {N^C} \right){N^C} + \left( {1 - \mu } \right){p_N}\left( {N^C} \right)\frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }}} \right)}}{{\mu \left\{ {{p_{NN}}\left( {N^C} \right){N^C} + 2{p_N}\left( {N^C} \right)} \right\}}} $$

$$ \begin{gathered} \frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }} = \frac{{ - {p_N}\left( {N^C} \right){N^C}}}{{\mu \left\{ {{p_{NN}}\left( {N^C} \right){N^C} + 2{p_N}\left( {N^C} \right)} \right\} + \left( {1 - \mu } \right){p_N}\left( {N^C} \right)}} < 0, \hfill \\ {\text{when }}\mu \in \left[ {0,\;1} \right). \hfill \\ \end{gathered} $$

(40)

The sign in (40) follows from the negative sign for the second order condition in (39) and the assumption that p _N  < 0.

1.3 Derivation of Eq. 13

We want to find out how the value of the firm’s equilibrium profit changes with a change in market monitoring. The envelope theorem allows us to differentiate the objective function without reference to the endogenous N ^C(μ). Consider the objective function in (8), evaluated at the optimum. If we differentiate it by μ:

$$ \begin{array}{*{20}{c}} {\frac{{\partial \left\{ {\pi - K} \right\}}}{{\partial \mu }}\left| {_{N = {N^e} = {N^C}}} \right. + \frac{{\partial \left\{ {\pi - K} \right\}}}{{\partial N}}\left| {_{N = {N^e} = {N^C}}} \right.\frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }}} \\ { = \frac{{\partial \left\{ {\pi - K} \right\}}}{{\partial \mu }}\left| {_{N = {N^e} = {N^C}}} \right.,\because \frac{{\partial \left\{ {\pi - K} \right\}}}{{\partial N}}\left| {_{N = {N^e} = {N^C}}} \right. = 0} \\ \end{array} $$

(41)

This is a restatement of the envelope theorem. Nevertheless, this problem is slightly complicated by the fact that we cannot ignore the μ contained in market expectations N ^e(r). Let us differentiate profits in Eq. 8 by μ:

$$ \begin{array}{*{20}{c}} {{{\left. {\frac{{\partial \left\{ {\pi - K} \right\}}}{{\partial \mu }}} \right|}_{N = {N^C}}}\quad = \,p\left( {N^C} \right){N^C} - p\left( {N^e} \right){N^C} + \left( {1 - \mu } \right){p_N}\left( {N^e} \right)\frac{{\partial {N^e}\left( {\mathbf{r}} \right)}}{{\partial \mu }}{N^C}} \hfill \\ {{{\left. {\frac{{\partial \{ \pi - K\} }}{{\partial \mu }}} \right|}_{N = {N^e} = {N^C}}} = \left( {1 - \mu } \right){p_N}\left( {N^e} \right)\frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }}{N^C}} \hfill \\ \end{array} $$

(42)

The second line relies on rational expectations shifting in response to exogenous stimuli in step with the endogenous N ^C according to the assumption in (5). Armed with the sign in Eq. 40, we now can conclude that Eq. 42 is, in fact, positive when 0 ≤ μ  < 1. That is,

$$ \begin{gathered} {\left. {\frac{{\partial \left\{ {\pi - K} \right\}}}{{\partial \mu }}} \right|_{N = {N^e} = {N^C}}} = \left( {1 - \mu } \right){p_N}\left( {N^e} \right)\frac{{\partial {N^C}\left( \mu \right)}}{{\partial \mu }}{N^C} > 0,~ \hfill \\ {\text{when }}\mu \in \left[ {0,\;1} \right). \hfill \\ \end{gathered} $$

(43)

This relationship is rewritten in Eq. 13.

1.4 Derivation of Eq. 23

Equation (23) is found by rearranging and combining the first order conditions in (15) and (22). Suppose that both first order conditions are satisfied by $N_i^*$ when a level of net-debt, $F_i^* \ne 0$ is chosen. First, Eq. 15 can be rearranged so that

$$ {p_N}\left( {N_i^*} \right){\left( {N_i^*} \right)^2} = - \frac{1}{\mu }\left( {K + F_i^*\left( {1 + {\theta_i}} \right)} \right) $$

(44)

If we multiply $N_i^*$by Eq. 22, then the first order condition can be rewritten as

$$ N_i^*{\left. {\frac{{\partial V_i^*}}{{\partial N}}} \right|_{N = N_i^*}} = {p_N}\left( {N_i^*} \right){\left( {N_i^*} \right)^2} + p\left( {N_i^*} \right)N_i^* - wN_i^* - \frac{{dc\left( {N_i^*} \right)}}{dN}N_i^* = 0. $$

(45)

Substituting the left hand side of Eq. 44 into (45), we get the following relationship:

$$ - \frac{1}{\mu }\left( {K + F_i^*\left( {1 + {\theta_i}} \right)} \right) + p\left( {N_i^*} \right)N_i^* - wN_i^* - \frac{{dc\left( {N_i^*} \right)}}{dN}N_i^* = 0 $$

(46)

Equation (46) can be rewritten as a function of $F_i^*$.

$$ F_i^* = \frac{1}{{1 + {\theta_i}}}\left[ {\mu \left( {p\left( {N_i^*} \right)N_i^* - wN_i^* - \frac{{d{c^{ - 1}}\left( {N_i^*} \right)}}{dN}N_i^*} \right) - K} \right] $$

(47)

Profits before fixed and financial costs when market monitoring is perfect and costly finance must be raised to hit hiring targets is

$$ p\left( {N_i^*} \right)N_i^* - wN_i^* \equiv \pi \left( {N_i^*;1,{\theta_i}} \right). $$

(48)

If (48) is inserted into (47), then we have a restatement of Eq. 23. Since this is what we wanted to derive, we are done.

1.5 Numerical example

Suppose that ability, a, is distributed uniformly on the continuous interval $\left[ {\underline a, \overline a } \right]$, where $\overline a$ is the talent of the highest ability individual in the distribution of potential employees. That is $a\tilde{}U{\left( {\underline{a} ,\,\overline{a} } \right)}$. The firm, which observes ability directly, will select professionals from the top of the talent distribution. That is, the firm will select individuals of abilities on the interval $a \in {\left[ {\widetilde{a},\,\overline{a} } \right]}$. The probability density function for the uniform distribution is as follows:

$$ g(a) = \left\{ {\begin{array}{*{20}{c}} {0,\quad \quad \quad {\text{when}}\;a < \underline a } \\ {\frac{1}{{\overline a - \underline a }},\quad {\text{when}}\;\underline a \leqslant a \leqslant \overline a } \\ {0,\quad \quad \quad {\text{when}}\;a > \overline a } \end{array} } \right. $$

A firm that only hires workers of ability $a \geqslant \widetilde{a}$ has an average ability of

$$ q\left( {\widetilde{a}} \right) = \int\limits_{\widetilde{a}}^{\overline a } {\frac{a}{{\overline a - \widetilde{a}}}} da = \left. {\frac{a^2}{{2\left( {\overline a - \widetilde{a}} \right)}}} \right|_{\widetilde{a}}^{\overline a } = \frac{{\overline a + \widetilde{a}}}{2} $$

(49)

Clients are risk neutral price takers who are randomly assigned employees from the pool of workers in the firm. They do not observe the workers’ ability directly. Instead, they are willing to pay for the expected quality of workers in the firm.

If the size of the potential workforce is given by the parameter σ  > 0, then the size of the firm is as follows:

$$ N\left( {\tilde a} \right) = \sigma \left( {\frac{{\bar a - \tilde a}}{{\bar a - {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}}}}} \right),\quad {\text{when}}\;a \sim U\left( {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}},\bar a} \right) $$

(50)

If we rearrange (50), we can solve for the ability cut off as a function of the size of the firm.

$$ \widetilde{a}(N) = \overline a - \frac{N}{\sigma }\left( {\overline a - \underline a } \right) $$

(51)

If we combine Eqs. 49 and 51, we can solve for average quality and price, respectively, as a function of firm size, N.

$$ \begin{gathered} p(N) = q(N) = \left( {\overline a - \frac{N}{{2\sigma }}\left( {\overline a - \underline a } \right)} \right) \hfill \\ {p_N}(N) = - \frac{1}{{2\sigma }}\left( {\overline a - \underline a } \right) \hfill \\ \end{gathered} $$

(52)

The first order condition from Eq. 21 can be combined with the price function in Eq. 52.

$$ {\left. {\frac{{\partial \left\{ {\pi \left( {N;1,0} \right) - K} \right\}}}{{\partial N}}} \right|_{N = N_0^*}} = \left( {\bar a - \frac{N_0^*}{{2\sigma }}\left( {\overline a - \underline a } \right)} \right) - \frac{N_0^*}{{2\sigma }}\left( {\overline a - \underline a } \right) - w = 0 $$

(53)

The implied first-best hiring rule is

$$ N_0^* = \sigma \left( {\frac{{\overline a - w}}{{\overline a - \underline a }}} \right) $$

(54)

Further, combining (54) and (52) the first best price is

$$ p\left( {N_0^*} \right) = \frac{1}{2}\left( {\overline a + w} \right) $$

(55)

If we insert Eq. 54 into our definition of profits before fixed costs, we can solve for this quantity. In this example, first-best profits before fixed costs is

$$ \pi \left( {N_0^*} \right) = \frac{{\sigma {{\left( {\overline a - w} \right)}^2}}}{{2\left( {\overline a - \underline a } \right)}}. $$

(56)

Let us consider the hiring decisions of the partnership. The FOC for the partnership is given by Eq. 15. Inserting in the values from Eq. 52, the first order condition becomes

$$ \begin{array}{*{20}{c}} {{{\left. {\frac{{\partial S}}{{\partial N}}} \right|}_{N = {N^e} = {N^P}}} = - \frac{\mu }{{2\sigma }}\left( {\overline a - \underline a } \right) + \frac{{K + F\left( {1 + {\theta _i}} \right)}}{{^2{{\left( {{N^P}} \right)}^2}}} = 0} \hfill \\ { \Rightarrow {N^P} = \sqrt {\frac{{2\sigma \left( {K + F\left( {1 + {\theta _i}} \right)} \right)}}{{\mu \left( {\overline a - \underline a } \right)}}} } \hfill \\ \end{array} $$

(57)

The equilibrium price for the partnership is

$$ p\left( {N^P} \right) = \bar a - \sqrt {\frac{{\left( {\overline a - \underline a } \right)\left( {K + F\left( {1 + {\theta_i}} \right)} \right)}}{{2\sigma \mu }}} . $$

(58)

The profit after fixed costs of the partnership as a whole is

$$ \pi ({N^P}) - K = (\bar a - w)\sqrt {\frac{{2\sigma (K + F(1 + {\theta_i}))}}{{\mu (\bar a - {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}})}}} - \frac{{K + F(1 + {\theta_i})}}{\mu } - K $$

(59)

Suppose that the firm has the following parameter values:

$$ \begin{array}{*{20}{c}} {\overline a = \$ 4} \hfill \\ {\underline a = 0} \hfill \\ {\sigma = 100} \hfill \\ {w = \$ 2} \hfill \\ {K = \$ 10} \hfill \hfill \\ \end{array} $$

(60)

The preceding parameter values mean that partner ability is uniformly distributed from zero to one; the maximum size of the firm is 100; the choke price where demand is zero is $4; and the firm cannot cover its variable costs if the price falls below the wage of $2. (Since all partners are paid their reservation wage in period 2, they will participate in period 4 even if they get a zero profit share.)

In addition we will assume that financial adjustment costs are zero.

$$ {\theta_0} = {\theta_d} = {\theta_e} = 0 $$

(61)

In the first-best, 100% market monitoring, benchmark case

$$ \begin{array}{*{20}{c}} {N_0^* = 50} \hfill \\ {p\left( {N_0^*} \right) = \$ 3} \hfill \\ {\pi \left( {N_0^*} \right) = \$ 50} \hfill \\ {\pi \left( {N_0^*} \right) - K = \$ 40} \hfill \\ \end{array} $$

(62)

We will consider the optimal capital structure decisions when market monitoring is high, μ = 0.8, and when market monitoring is low, μ = 0.1.

1.6 High levels of market monitoring (μ = 0.8)

A partnership with a transparent capital structure will hire the same number of employees, be able to charge the same price, and will realize the same profits before and after fixed costs as the first-best in Eq. 62. Nevertheless, to achieve this feat, the partnership will have to take on some debt.

Recall the formula for the optimal level of net-debt for the transparent partnership in Eq. 23. We can combine this with our formula for first-best profits before investment costs in Eq. 56 to get the following relationship:

$$ F_0^*\left( \mu \right) = \mu \left( {\frac{{\sigma {{\left( {\overline a - w} \right)}^2}}}{{2\left( {\overline a - \underline a } \right)}}} \right) - K $$

(63)

In this case this will be

$$ \begin{array}{*{20}{c}} {F_0^*\left( \mu \right) = {F^*}\left( {0.8} \right) =\$ 30} \hfill \\ {\alpha_0^*\left( \mu \right) = {\alpha^*}\left( {0.8} \right) = 0.} \hfill \\ \end{array} $$

(64)

If the partnership does not adjust its capital structure F = 0 as in Levin and Tadelis (2005), the size of the firm from Eq. 57, price from Eq. 58 profits after fixed costs from Eq. 59 are as follows when μ = 0.8:

$$ \begin{array}{*{20}{c}} {{N^P}\left( {F,\mu } \right) = {N^P}\left( {0,0.8} \right) = 25} \hfill \\ {p\left( {{N^P}\left( {F,\mu } \right)} \right) = p\left( {{N^P}\left( {0,0.8} \right)} \right) = \$ 3.50} \hfill \\ {\pi \left( {{N^P};F,\mu } \right) - K = \pi \left( {{N^P};0,0.8} \right) - K = \$ 27.50} \hfill \\ \end{array} $$

(65)

1.7 Low levels of market monitoring (μ  = 0.1)

The partnership in which clients observe the capital structure choices of the partnership can achieve the first-best profits in Eq. 62. Nevertheless, to do so when μ = μ _L  = 0.1 it must actually raise some cash by way of outside equity if it hopes to only hire N ^* = 50 partners. By combining the general formula for the outside equity stake in Eq. 25 with Eqs. 56 and 54 for this example, we can obtain the formula for a non-voting equity stake below:

$$ \alpha_0^*\left( \mu \right) = \left\{ {\begin{array}{*{20}{c}} {0,{\text{ when }}{F^*}\left( \mu \right) \geqslant 0} \\ {\frac{{K - \mu \left( {\frac{{\sigma {{\left( {\overline a - w} \right)}^2}}}{{2\left( {\overline a - \underline a } \right)}}} \right)}}{{\left( {1 - \mu } \right)\left( {\frac{{\sigma {{\left( {\overline a - w} \right)}^2}}}{{2\left( {\overline a - \underline a } \right)}}} \right) + w\sigma \left( {\frac{{\overline a - w}}{{\overline a - \underline a }}} \right)}},{\text{ otherwise}}.} \\ \end{array} } \right. $$

(66)

The capital structure choices obtained from combining Eqs. 60, 63, 66, and the level of market monitoring, μ  = 0.1, are

$$ \begin{array}{*{20}{c}} {F_0^*\left( \mu \right) = {F^*}\left( {0.1} \right) = - \$ 5} \hfill \\ {\alpha_0^*\left( \mu \right) = {\alpha^*}\left( {0.1} \right) = {1 \mathord{\left/{\vphantom {1 {29}}} \right.} {29}}.} \hfill \\ \end{array} $$

(67)

In the partnership of Levin and Tadelis (2005) where F = 0, the size of the firm from Eq. 57, price from Eq. 58 profits after fixed costs from Eq. 59 are approximately the following when μ = 0.1:

$$ \begin{array}{*{20}{c}} {{N^P}\left( {F,\mu } \right) = {N^P}\left( {0,0.1} \right) \approx 70.71} \hfill \\ {p\left( {{N^P}\left( {F,\mu } \right)} \right) = p\left( {{N^P}\left( {0,0.1} \right)} \right) \approx \$ 2.59} \hfill \\ {\pi \left( {{N^P};F,\mu } \right) - K = \pi \left( {{N^P};0,0.1} \right) - K \approx \$ 31.42} \hfill \\ \end{array} $$

(68)

1.8 Proof of proposition 2

To prove this proposition we will differentiate the profit function at its maximum point with respect to the cost parameter θ _i . That is, we will sign $\frac{{\partial V_i^*\left( {N_i^*} \right)}}{{\partial {\theta_i}}}$. To do this we will use the envelope theorem. The objective function in (21) only depends on θ _i through the indirect financing cost equation $c\left( {N;\mu, {\theta_i}} \right)$. Therefore, using the definition of $c\left( {N;\mu, {\theta_i}} \right)$ in Eq. 20

$$ \frac{{\partial V_i^*(N)}}{{\partial {\theta_i}}} = \frac{{\partial c\left( {N;\mu, {\theta_i}} \right)}}{{\partial {\theta_i}}} = \frac{1}{{{{\left( {1 + {\theta_i}} \right)}^2}}}\left\{ {\mu {N^2}{p_N}(N) + K} \right\}. $$

(69)

The envelope theorem allows us to translate to this to the optimal N=$N_i^*$. The optimal level of net-debt is given by Eq. 20 evaluated at $N_i^*$. This means that Eq. 69 can be rewritten as

$$ \frac{{\partial V_i^*\left( {N_i^*} \right)}}{{\partial {\theta_i}}} = \frac{{\partial c\left( {N_i^*;\mu, {\theta_i}} \right)}}{{\partial {\theta_i}}} = - \frac{1}{{1 + {\theta_i}}}F_i^*\left( {N_i^*;\mu, {\theta_i}} \right). $$

(70)

This is what we wanted to derive. Q.E.D.

An interpretation of this can be found near Eq. 28.

1.9 Proof of proposition 3

There are two pieces to this proof. The first piece is to show that there must be a μ which we will denote, $\hat \mu_d^*$, defined in Eq. 31. For all μ ≥$\hat \mu_d^*$, the corporation will be weakly more profitable than the transparent partnership. For all $\mu \in \left[ {{\mu^P},\hat \mu_d^*} \right)$ the partnership must be more profitable than the corporation. The second piece involves showing that there is no μ at or below μ ^P where the corporation can earn non-zero profits higher than the transparent partnership can earn.

First, let us show that, when financial adjustments are costly but visible to clients, the corporate form must be weakly preferred for some 1 ≤ μ <$\hat \mu_d^*$. If the π(N) is concave and the corporation’s profits are maximized at μ  = 1, then it must be the case that the corporation is becoming increasingly profitable as μ rises. (This is what we found in Eq. 13 without the global concavity assumption.) In contrast, the comparative static result in (30) says that the value of the transparent partnership is strictly declining in μ > $_0\mu_d^*$, which is defined in Eq. 26, when it takes on some positive level of net-debt. Therefore, given that the partnership finds it optimal to take on some positive net-debt $F_d^*\left( \mu \right) > 0$, there must be some range of μ,1 ≤ μ < $\hat \mu_d^*$, where the corporation is more profitable than the transparent partnership.

Alternatively, suppose that the transparent partnership optimally takes on no debt for all $\mu \in \left[ {{\mu^P},\;1} \right].$ We know that profits in the partnership with no net-debt are maximized at N ^P(μ ^P), where μ ^P is defined in (27). Because the profit function is strictly concave any movement away from N ^P(μ ^P) = $N_0^*$ will lead to a fall in profits for the partnership with zero net-debt. A sufficient condition for total profits in the zero net-debt partnership to be falling from $\mu \in \left( {{\mu^P},\;1} \right]$ is $\frac{{\partial {N^P}}}{{\partial \mu }} < 0.$ The implicit function rule allows us to derive the comparative static from the first order condition for the period 3 partnership in Eq. 15. When net-debt is zero, the comparative static is

$$ \frac{{\partial {N^P}}}{{\partial \mu }} = {\left. { - \frac{{{\partial^2}S}}{{\partial N\partial \mu }}\frac{{\partial {N^2}}}{{{\partial^2}S}}} \right|_{N = {N^e} = {N^P}}} = \frac{{ - {p_N}\left( {N^P} \right)}}{{\mu {p_{NN}}\left( {N^P} \right) - {{2K} \mathord{\left/{\vphantom {{2K} {{{\left( {N^P} \right)}^3}}}} \right.} {{{\left( {N^P} \right)}^3}}}}} < 0 $$

(71)

This is unambiguously negative since p _N  < 0 and the denominator is the second order condition of a maximum point. Therefore, the zero net-debt partnership will never return to the optimal hiring point as long as $\mu \in \left( {{\mu^P},1} \right].$ In contrast, we know that the corporation’s profits are strictly rising in μ. Therefore, if the partnership takes on no net-debt, there must be some $\hat \mu_d^* = \hat \mu \in \left[ {{\mu^P},1} \right]$, where $\hat \mu$ is defined in Eq. 31, at which corporate profits equal partnership profits. Then corporate profits must weakly exceed partnership profits for those values of $\mu \in \left[ {\hat \mu_d^*,1} \right].$

Let us turn to the second part of the proof. Namely, partnership profits are always weakly higher than corporate profits from $\mu \in \left[ {0,\hat \mu_d^*} \right].$ We have already shown that the transparent partnership’s profits must be higher than the corporation from $\mu \in \left[ {{\mu^P},\hat \mu_d^*} \right)$. We still need to prove that the transparent partnership with costly financial adjustments is weakly more profitable than the corporation from $\mu \in \left[ {0,{\mu^P}} \right)$. To do this we consider a partnership with zero net-debt. (The transparent partnership can always choose to have zero net-debt and behave like a partnership that cannot adjust its capital structure.)

The second part of the proof relies more heavily on Levin and Tadelis (2005)’s propositions 1, 2, and 3 and on Ward (1958). Nevertheless, we do not attempt to confirm the assertion in proposition 2 of Levin and Tadelis (2005) that there exists a lower bound level of informed clients, $\underline \mu$ defined in Eq. 32, where the transparent partnership no longer makes positive profits.

The partnership that does not adjust its capital structure, $F_i^* = 0$, is always more selective than a corporation, given that it makes positive profits. If there exists some μ in which neither organization is viable, they both will make zero profits by not operating. The transparent partnership always has the option to not adjust its capital structure, $F_i^* = 0$, and mimic these payoffs.

We need to show that the partnership is always more profitable than the corporation for all μ  < μ ^P, in which either organization makes positive profits. Consider the first order condition in Eq. 15 for the period 3 partnership when F = 0.

$$ \mu \,{p_N}\left( {N^P} \right) + \frac{K}{{{{\left( {N^P} \right)}^2}}} = 0 $$

(72)

We can rearrange this by and multiplying by N ^P and then adding p(N ^P) to both sides.

$$ \mu \,{p_N}\left( {N^P} \right){N^P} + p\left( {N^P} \right) = p\left( {N^P} \right) - \frac{K}{N^P} $$

(73)

In contrast, let us consider the first order condition of the corporation in Eq. 10

$$ \mu \,{p_N}\left( {N^C} \right){N^C} + p\left( {N^C} \right) = w $$

(74)

On the left hand sides (LHS) are the rational expectations marginal revenues, which we will denote MR(N; μ) $\equiv \mu \,{p_N}(N)N + p(N)$, of the partnership and the corporation, respectively. On the right hand side (RHS) is the marginal cost of an employee to each organization. Suppose that the RHS of (73) is at least as large as the RHS of (74).

$$ \begin{array}{*{20}{c}} {p\left( {N^P} \right) - \frac{K}{N^P} \geqslant w} \hfill \\ {\pi \left( {{N^P},0;\mu } \right) - K = p\left( {N^P} \right){N^P} - w{N^P} - K \geqslant 0} \hfill \\ \end{array} $$

(75)

This implies that the LHS of (73), the marginal revenue of the partnership, must meet or exceed the LHS of (74), the marginal revenue of the corporation whenever the partnership with zero net-debt makes non-negative profits. Namely,

$$ \begin{array}{*{20}{c}} {MR\left( {{N^P};\mu } \right) \geqslant MR\left( {{N^C};\mu } \right)} \hfill \\ {\forall \mu {\text{ where }}\pi \left( {{N^P},0;\mu } \right) - K \geqslant 0.} \hfill \\ \end{array} $$

(76)

Therefore, (76) implies that the partnership is weakly more selective than the corporation for all μ where the partnership earns non-negative profits. Since π(N) is concave and the maximum point the for the partnership does not occur until N ^P(μ ^P) = $N_0^*$, this weakly greater selectivity implied by Eq. 76 indicates that the partnership must also be weakly more profitable than the corporation for all $\mu \in \left[ {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }},\;{\mu^P}} \right]$, where $\underline \mu$ is defined in (32) as the point where the partnership makes zero profits. Concavity also implies that the partnership with no net-debt makes strictly negative profits for all $\mu \in \left[ {0,{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }}} \right)$. Therefore, a partnership with zero net-debt will not operate for all $\mu \in \left[ {0,{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }}} \right)$.

Finally, we must show that the corporation does not operate for $\mu \in \left[ {0,{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }}} \right)$. That is, when the partnership with zero net-debt is out of business, the corporation is also out of business. The number of employees hired by the corporation when market monitoring is zero is either N ^C(0) implied by Eq. 10 or the corporation hires all available employees, $\bar N$. Consider when μ  = 0, then the first-order condition for the corporation is (74) is $p\left( {{N^C};0} \right) = w$. This implies that the corporation cannot cover its fixed costs:

$$ \begin{array}{*{20}{c}} {\pi \left( {{N^C};0} \right) - K = - K < 0,} \hfill \\ {\because K > 0.} \hfill \end{array} $$

(77)

By the assumption that we imported from Levin and Tadelis (2005) if the firm hires all available employees profits are negative, $\pi \left( {\bar N} \right) - K < 0.$ Therefore, the corporation must make strictly negative profits when μ = 0. The concavity of π(N) assumption implies that that the corporation’s profits are strictly increasing from μ = 0 to $\mu = \underline \mu$. When the LHS of (73) and (74) are equal, profits for both the corporation and the partnership must be identically zero. By definition in Eq. 32, the μ at this level of profits is $\underline \mu$. In short, we can conclude that the corporation does not operate from $\mu \in \left[ {0,{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }}} \right)$.

This is what we wanted to show. Q.E.D.

In summary, there must be a crossing point $\hat \mu_d^*$, defined in Eq. 31, where the transparent partnership with financing costs and the corporation make the same profits. When $\mu \in \left[ {\hat \mu_d^*,1} \right]$, the corporation is the weakly more profitable organization. For all $\mu \in \left[ {0,\hat \mu_d^*} \right)$, the transparent partnership is the weakly most profitable organization, given that any organization is profitable.

1.10 Derivation of Figure 2

The graph in Fig. 2 plots the profit functions as a function of market monitoring, μ, for the partnership with costly financial adjustments, the partnership that takes on zero net-debt, and the corporation. We assume a uniform distribution of employee talent as in Section 6.5. We have already derived the profit function for the partnership that takes on zero net in Eq. 59. That leaves us to derive the profit functions for the partnership that adjusts financial structure with costly financial adjustments and the profit function for the corporation. All parameter values in Fig. 2 are identical to Eq. 60. Further, we will assume that there are financial transaction costs to debt and equity as follows:

$$ \begin{array}{*{20}{c}} {{\theta_d} = 0.02} \hfill \\ {{\theta_e} = - 0.07} \hfill \\ \end{array} $$

(78)

First, let us derive the profit function for the partnership that adjusts net-debt. If we rearrange the period 2 partnership’s demand for employees as a function of net-debt, ${N^P}\left( {F;\mu, {\theta_i}} \right)$, in Eq. 57 to the function F(N; μ, θ _i ) we are left with

$$ F(N) = \frac{1}{{1 + {\theta_i}}}\frac{{\mu \left( {\overline a - \underline a } \right)}}{{2\sigma }}{(N)^2} - \frac{1}{{1 + {\theta_i}}}K. $$

(79)

The cost of net-debt function is

$$ c(N) = {\theta_i}F = \frac{{{\theta_i}}}{{1 + {\theta_i}}}\frac{{\mu \left( {\overline a - \underline a } \right)}}{{2\sigma }}{(N)^2} - \frac{{{\theta_i}}}{{1 + {\theta_i}}}K. $$

(80)

When there are financial frictions, the partnership has to weigh the gains from a more profitable employment policy against the financing costs, given in Eq. 22. Suppose that in equilibrium the partners’ profit shares exceed their opportunity cost. The problem for the period 2 partnership that is attempting to determine the optimal hiring level with costly finance that draws from the distribution of talent, which was introduced earlier in the Appendix, is in Eq. 81 below.

$$ \begin{array}{*{20}{c}} {\mathop {\arg \max }\limits_{w.r.t.\quad N} \quad V_i^* = \left( {p(N) - w} \right)N - K - c(N)} \hfill \\ { = \left( {\bar a - \frac{N}{{2\sigma }}(\bar a - {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}}) - w} \right)N - K - \frac{{{\theta_i}}}{{1 + {\theta_i}}}\frac{{\mu \left( {\overline a - \underline a } \right)}}{{2\sigma }}(N){)^2} - \frac{{{\theta_i}}}{{1 + {\theta_i}}}K} \hfill \\ \end{array} $$

(81)

Equation (81) can be obtained by combining the objective function in Eq. 21, the definition of c(N) in either Eq. 20 or 80, and the inverse demand function for this example in Eq. 52. Differentiating this function with respect to N, we can solve for the optimal hiring level of the partnership which can credibly signal its capital structure.

$$ \begin{array}{*{20}{c}} {{{\left. {\frac{{\partial V_i^*}}{{\partial N}}} \right|}_{N = N_i^*}} = \left( {\overline a - w} \right) - N_i^*\frac{1}{\sigma }\left( {\overline a - \underline a } \right) - N_i^*\frac{{{\theta_i}}}{{1 + {\theta_i}}}\frac{\mu }{\sigma }\left( {\overline a - \underline a } \right) = 0} \hfill \\ {\frac{{{\partial^2}V_i^*}}{{\partial {N^2}}} = - \left( {\frac{1}{\sigma }\left( {\overline a - \underline a } \right)} \right)\left( {\frac{{1 + {\theta_i}\left( {1 + \mu } \right)}}{{1 + {\theta_i}}}} \right) < 0} \hfill \end{array} $$

(82)

Since the second order condition above is unambiguously negative, we can conclude that the stationary point $N_i^*$ below is a maximum.

$$ N_i^* = \frac{{\sigma \left( {\overline a - w} \right)}}{{\left( {\overline a - \underline a } \right)}}\left( {\frac{{1 + {\theta_i}}}{{1 + {\theta_i}(1 + \mu )}}} \right) $$

(83)

If we compare this to the case where financing costs were zero in Eq. 54, we can verify that zero financial costs is a special case of Eq. 83. The hiring level in (83) is identical to Eq. 54 when θ _i  = 0.

Combining p(N) in Eq. 52 and $N_i^*$ from (83) we have the equilibrium price,

$$ p(N_i^*) = \frac{{\bar a\left( {1 + {\theta_i}\left( {1 + 2\mu } \right)} \right) + \left( {1 + {\theta_i}} \right)w}}{{2\left( {1 + {\theta_i}\left( {1 + \mu } \right)} \right)}}. $$

(84)

Equation (84) is identical to Eq. 55 when θ _i  = 0.

The profits before fixed and financing costs are

$$ \pi \left( {N_i^*} \right) = \frac{{\sigma \left( {1 + {\theta_i}} \right)}}{{2\left( {\bar a - {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a}}} \right)}}{\left( {\frac{{\bar a - w}}{{1 + {\theta_i}\left( {1 + \mu } \right)}}} \right)^2}\left( {1 + {\theta_i}\left( {1 + 2\mu } \right)} \right). $$

(85)

Financial cost are obtained by combining Eqs. 80 and 83 below

$$ c\left( {N_i^*} \right) = \frac{{\mu \sigma {\theta_i}\left( {1 + {\theta_i}} \right)}}{{2\left( {\overline a - \underline a } \right)}}{\left( {\frac{{\overline a - w}}{{1 + {\theta_i}\left( {1 + \mu } \right)}}} \right)^2} - \frac{{{\theta_i}}}{{1 + {\theta_i}}}K $$

(86)

Net-debt raised can be obtained by dividing Eq. 86 by θ _i .

$$ \begin{array}{*{20}{c}} {F_i^* = {{c\left( {N_i^*} \right)} \mathord{\left/{\vphantom {{c\left( {N_i^*} \right)} {{\theta_i}}}} \right.} {{\theta_i}}} = \frac{1}{{1 + {\theta_i}}}\left[ {\mu \left( {\pi \left( {N_i^*} \right) - \frac{{dc\left( {N_i^*} \right)}}{dN}N_i^*} \right) - K} \right]} \hfill \\ { = \frac{{\mu \sigma \left( {1 + {\theta_i}} \right)}}{{2\left( {\overline a - \underline a } \right)}}{{\left( {\frac{{\bar a - w}}{{1 + {\theta_i}\left( {1 + \mu } \right)}}} \right)}^2} - \frac{K}{{1 + {\theta_i}}}} \hfill \\ \end{array} $$

(87)

The equilibrium value of the firm is obtained by combining Eq. 85,—K, and the financing costs in Eq. 86 into the equation below:

$$ \begin{array}{*{20}{c}} {V_i^* = \pi \left( {N_i^*} \right) - K - c\left( {N_i^*} \right)} \hfill \\ { = \frac{{\sigma \left( {1 + {\theta_i}} \right)}}{{2\left( {\overline a - \underline a } \right)}}\frac{{{{\left( {\bar a - w} \right)}^2}}}{{1 + {\theta_i}\left( {1 + \mu } \right)}} - \frac{K}{{1 + {\theta_i}}}} \hfill \\ \end{array} $$

(88)

Equation (88) was the function used to plot the top curve in Fig. 2.

On the other hand, a corporation will hire according to the first order condition in (10), which can be rewritten for this example as the following:

$$ \begin{gathered} {\left. {\frac{{\partial {V^C}}}{{\partial N}}} \right|_{N = {N^e} = {N^C}}} = {\left. {\frac{{\partial \left\{ {\pi - K} \right\}}}{{\partial N}}} \right|_{N = {N^e} = {N^C}}}\quad = \hfill \\ \left( {\overline a - \frac{N^C}{{2\sigma }}\left( {\overline a - \underline a } \right)} \right) - \frac{{\mu {N^C}}}{{2\sigma }}\left( {\overline a - \underline a } \right) - w = 0 \hfill \\ \end{gathered} $$

(89)

This implies that the corporation will hire

$$ {N^C} = \sigma \left( {\frac{{\overline a - w}}{{\overline a - \underline a }}} \right)\left( {\frac{2}{{1 + \mu }}} \right). $$

(90)

The equilibrium price for the corporation will be

$$ p\left( {N^C} \right) = \frac{1}{{1 + \mu }}\left( {\mu \bar a + w} \right). $$

(91)

The profits after fixed costs will be

$$ \pi \left( {N^C} \right) = \frac{{2\mu \sigma }}{{\left( {\overline a - \underline a } \right)}}{\left( {\frac{{\overline a - w}}{{1 + \mu }}} \right)^2} - K. $$

(92)

Equation (92) was used to plot the profit curve resembling a quarter circle in Fig. 2.