Observability and endogenous organizations

Abstract

This paper establishes a relationship between the observability of common shocks and optimal organizational design in a multiagent moral hazard environment. We consider two types of organizations, namely relative performance and cooperative regimes, and show that, with sufficient information regarding common shocks, a cooperative organization can be optimal even if outputs are highly correlated. The model is then embedded in a Walrasian general equilibrium model in which choices regarding organizations and investment in information on common shocks are jointly determined. Numerical results reveal that both cooperative and relative performance regimes can coexist in equilibrium but only cooperative organizations invest in full observability of common shocks. Changes in the cost of information and aggregate wealth can affect substantially the types of organizations operating and the matching patterns of heterogeneous agents in these organizations. General equilibrium effects are key in determining how information costs impact the way production is organized.

Appendix: Proofs

Proof of Proposition 1

This proof is written for the general case in which there is a finite number of firm members, $N \ge 2$. Consider a relative performance contract $\pi ^r$ with $\bar{u}$. Feasibility requires that $\pi ^r$ satisfies conditions (10), (11), (12), (13), (14). Perfect observability implies that one individual’s output is not informative with regard to the outputs of other individuals in the firm. Hence, an individual’s consumption $c_i$ will depend only on individual output $q_i$ and on the underlying state $\omega$. Given this information and the symmetry assumption, we can conclude that the optimal consumption for agent i under a relative performance regime is $c^r_i = c\left( q_i, \omega \right)$.

We will now show that there is a feasible group contract $\pi ^g$ that can generate a weakly larger surplus. Symmetry, i.e., $\bar{u}_{i}=\bar{u}$ for all i, implies that there is no loss of generality if we consider only a symmetric solution. That is, the within-group weights $\mu _i = \frac{1}{N}$ are symmetric¹⁶, where N is the number of firm members. This symmetric solution also implies symmetric consumption allocation, i.e., $c^g_i = \frac{c_a\left( {\mathbf {q}}, \omega \right) }{N}, \forall i$. For each relative performance contract $\pi ^r\left( {\mathbf {c}}, {\mathbf {q}}, {\mathbf {e}}, s\right)$, we define a group contract $\pi ^g\left( c_a, {\mathbf {\mu }}, {\mathbf {q}}, {\mathbf {e}}, s\right)$ such that

$$\begin{aligned} u\left( \frac{c_a\left( {\mathbf {q}}, \omega \right) }{N}\right) = \sum _{\ell =1}^N \frac{u\left( c\left( q_{\ell }, \omega \right) \right) }{N} \end{aligned}$$

(43)

and

$$\begin{aligned} \pi ^g\left( c_a, {\mathbf {\mu }}, {\mathbf {q}}, {\mathbf {e}}, \omega \right) = \left\{ \begin{array}{ll}\pi ^r\left( {\mathbf {c}}, {\mathbf {q}}, {\mathbf {e}}, \omega \right) &{} {\text { when condition }} (43) {\text { holds and }} \mu _i = \frac{1}{N}, \\ 0&{} {\text { otherwise}}. \end{array}\right. \nonumber \\ \end{aligned}$$

(44)

The concavity of the utility function implies that the aggregate consumption specified in the group contract is no larger than the sum of the consumption allocations specified in the relative performance contract, i.e., $c_a\left( {\mathbf {q}}, \omega \right) \le \sum _{i=1}^N c\left( q_i, \omega \right)$ which holds with equality if none of the incentive constraints is binding (so there is full insurance).

It is not difficult to show that the probability, mother nature, and consistency constraints, (10), (11) and (14), respectively, for the relative performance contract $\pi ^r$, imply that the probability, mother nature, and consistency constraints, (20), (21) and (23), respectively, for the group contract $\pi ^g$, hold. We now only need to show that the participation and incentive constraints, (22) and (19), respectively, also hold under contract $\pi ^g$. The separability of the utility function implies that there is no loss of generality if we suppress the effort in the utility function for the participation constraints.

The participation constraints, (12), in case of the relative performance contract $\pi ^r$ can be rewritten as follows:

$$\begin{aligned} \bar{u}_{i}\le & {} \sum _{{\mathbf {c}},{\mathbf {q}},{\mathbf {e}},\omega } \pi ^r\left( {\mathbf {c}},{\mathbf {q}},{\mathbf {e}},\omega \right) u\left( c^r_{i}\right) \nonumber \\= & {} \sum _{{\mathbf {c}},{\mathbf {e}},\omega } \sum _{q_{-i}}\sum _{q_i} \pi ^r\left( {\mathbf {c}},q_{-i}, q_i,{\mathbf {e}},\omega \right) u\left( c(q_i, \omega )\right) \end{aligned}$$

(45)

The symmetry assumption implies the following:

$$\begin{aligned}&\sum _{q_{-i}}\sum _{q_i} \pi ^r\left( {\mathbf {c}},q_{-i}, q_i,{\mathbf {e}},\omega \right) u\left( c(q_i, \omega )\right) \\&\quad = \sum _{q_{-j}}\sum _{q_j} \pi ^r\left( {\mathbf {c}},q_{-j}, q_j,{\mathbf {e}},\omega \right) u\left( c(q_i, \omega )\right) , \quad \forall i, j \end{aligned}$$

Thus, the following is implied:

$$\begin{aligned} \sum _{q_{-i}}\sum _{q_i} \pi ^r\left( {\mathbf {c}},q_{-i}, q_i,{\mathbf {e}},\omega \right) u\left( c(q_i, \omega )\right)= & {} \sum _{\ell =1}^N \sum _{q_{-\ell }}\sum _{q_\ell } \pi ^r\left( {\mathbf {c}},q_{-\ell }, q_\ell ,{\mathbf {e}},\omega \right) \frac{u\left( c(q_\ell , \omega )\right) }{N} \\= & {} \sum _{{\mathbf {q}}} \sum _{\ell =1}^N\pi ^r\left( {\mathbf {c}},q_{-\ell }, q_\ell ,{\mathbf {e}},\omega \right) \frac{u\left( c(q_\ell , \omega )\right) }{N} \\= & {} \sum _{{\mathbf {q}}} \pi ^r\left( {\mathbf {c}},{\mathbf {q}},{\mathbf {e}},\omega \right) \sum _{\ell =1}^N \frac{u\left( c(q_\ell , \omega )\right) }{N} \end{aligned}$$

Substituting this equation into (45) yields the following:

$$\begin{aligned} \bar{u}_{i}\le & {} \sum _{{\mathbf {c}},{\mathbf {q}},{\mathbf {e}},\omega } \pi ^r\left( {\mathbf {c}},{\mathbf {q}},{\mathbf {e}},\omega \right) \sum _{\ell =1}^N \frac{u\left( c(q_\ell , \omega )\right) }{N} \\= & {} \sum _{c_a, \mu ,{\mathbf {q}},{\mathbf {e}},\omega } \pi ^g\left( c_a, \mu ,{\mathbf {q}},{\mathbf {e}},\omega \right) u\left( \frac{c_a\left( {\mathbf {q}}, \omega \right) }{N}\right) \end{aligned}$$

Substituting (43) and (44) into the above inequality yields the participation constraint, (22), for the group contract $\pi ^g$:

$$\begin{aligned} \bar{u}_{i}\le & {} \sum _{c_a, \mu ,{\mathbf {q}},{\mathbf {e}},\omega } \pi ^g\left( c_a, \mu ,{\mathbf {q}},{\mathbf {e}},\omega \right) u\left( \frac{c_a\left( {\mathbf {q}}, \omega \right) }{N}\right) \end{aligned}$$

This inequality proves that the candidate group contract, as defined by (43) and (44), satisfies the participation constraint for the group contract, (22).

Now consider the incentive constraints (13) for the relative performance contract $\pi ^r$ for each $e_i, \tilde{e}_{i}$:

$$\begin{aligned}&\sum _{{\mathbf {c}},{\mathbf {q}},{\mathbf {e}}_{-i},\omega } \pi ^r\left( {\mathbf {c}},{\mathbf {q}},e_i, {\mathbf {e}}_{-i} ,\omega \right) u(c^r_{i},e_{i}) \\&\quad \ge \sum _{{\mathbf {c}},{\mathbf {q}}, {\mathbf {e}}_{-i},\omega } \pi ^r\left( {\mathbf {c}},{\mathbf {q}},e_i, {\mathbf {e}}_{-i} ,s\right) \frac{Pr\left( {\mathbf {q}} | \tilde{e}_i, {\mathbf {e}}_{-i} ,\omega \right) }{Pr\left( {\mathbf {q}} | e_i, {\mathbf {e}}_{-i} ,s\right) } u(c^r_{i},\tilde{e}_{i}). \end{aligned}$$

Multiplying this constraint by $\frac{1}{N}$ and summing over i yields the following:

$$\begin{aligned}&\sum _{{\mathbf {c}},{\mathbf {q}},{\mathbf {e}}_{-i},\omega } \pi ^r\left( {\mathbf {c}},{\mathbf {q}}, {\mathbf {e}},\omega \right) \sum _{i=1}^N \frac{u(c\left( q_i, \omega \right) ,e_{i})}{N} \\&\quad \ge \sum _{{\mathbf {c}},{\mathbf {q}},{\mathbf {e}}_{-i},\omega } \pi ^r\left( {\mathbf {c}},{\mathbf {q}},{\mathbf {e}},\omega \right) \frac{Pr\left( {\mathbf {q}} | \tilde{e}_i, {\mathbf {e}}_{-i} ,s\right) }{Pr\left( {\mathbf {q}} | e_i, {\mathbf {e}}_{-i} ,s\right) } \sum _{i=1}^N \frac{u(c\left( q_i, \omega \right) ,\tilde{e}_{i})}{N}. \end{aligned}$$

Substituting (43) and (44) into the above inequality yields the incentive constraint, (19), for the group contract $\pi ^g$: for each $e_i, \tilde{e}_{i}$,

$$\begin{aligned}&\sum _{c_a, {\mathbf {q}},\omega } \pi ^g\left( c_a, {\mathbf {\mu }},{\mathbf {q}},{\mathbf {e}},\omega \right) \sum _i \mu _i u\left( \frac{c_a}{N}, e_i\right) \\&\quad \ge \sum _{c_a, {\mathbf {q}},\omega } \pi ^g\left( c_A, {\mathbf {\mu }},{\mathbf {q}},{\mathbf {e}},\omega \right) \frac{Pr\left( {\mathbf {q}} | \tilde{{\mathbf {e}}}, k, s\right) }{Pr\left( {\mathbf {q}} | {\mathbf {e}}, k, s\right) } \sum _i \mu _i u\left( \frac{c_a}{N}, \tilde{e}_i\right) \end{aligned}$$

This inequality proves that the candidate group contract, as defined by (43) and (44), satisfies the incentive constraint for the group contract, (19).

In summary, the candidate group contract, as defined by (43) and (44), is a feasible group contract that generates a weakly larger surplus. Further, if incentive constraints are binding under $\pi ^r$, for some vector ${\mathbf {q}}$, consumption is not equal among agents. This implies that $c_a\left( {\mathbf {q}}, \omega \right) < \sum _{i=1}^N c\left( q_i, \omega \right)$. Under the assumption that for any pair effort/common shock $(e,\omega )$, all output levels have positive probability, such ${\mathbf {q}}$ happens under $\pi ^r$ with positive probability. Therefore, $\pi ^g$ generates a strictly larger surplus, with the same utility for all agents. $\square$

Proof of Proposition 2

One of the difficulties with this proof is that the incentive compatibility constraints, (19), are not well defined when the denominator in each of these expressions, $Pr({\mathbf {q}}|{\mathbf {e}},k,s) = 0$, for some value of $({{\mathbf {q}},{\mathbf {e}},k,s})$. That case is applicable when idiosyncratic shocks are absent and efforts are identical among agents. This case requires a restatement of the incentive constraints.

For a level of output ${\mathbf {q}}$ that can be reached only out of equilibrium, i.e., $Pr({\mathbf {q}}|e_{i}, {\mathbf {e}}_{-i},s)=0$, there is no loss of generality establishing corresponding consumption at the minimum level, $\underline{{\mathbf {c}}}$, i.e., $Pr(\underline{{\mathbf {c}}}|{\mathbf {q}},{\mathbf {e}}) = 1$, where $\underline{{\mathbf {c}}} = \left( \underline{c}, \underline{c}\right)$ is the minimum level of consumption in the grid. The incentive constraints for the relative performance regime can now be reformulated as follows:

$$\begin{aligned} \sum _{{\mathbf {c}}, {\mathbf {q}}, {\mathbf {e}}_{-i},s} \pi ^{r}({\mathbf {c}},{\mathbf {q}},e_{i},{\mathbf {e_{-i}}},s)u(c_{i},e_{i})\ge \sum _{{\mathbf {c}}, {\mathbf {q}}, {\mathbf {e}}_{-i},s} \varPhi ({\mathbf {c}},{\mathbf {q}},\tilde{e}_{i},{\mathbf {e}},s) \ \forall e_{i}, \tilde{e}_i \end{aligned}$$

(46)

where

$$\begin{aligned} \varPhi ({\mathbf {c}}, {\mathbf {q}},\tilde{e}_{i},{\mathbf {e}},s) = \left\{ \begin{array}{ll}Pr(A_{s})Pr({\mathbf {e}})Pr( {\mathbf {q}}|\tilde{e}_{i}, {\mathbf {e_{-i}}},s)u(\underline{c},\tilde{e}_{i}),&{} {\text {when}} \ Pr( {\mathbf {q}}|e_{i}, {\mathbf {e_{-i}}},s)=0, \\ \pi ^{r}({\mathbf {c}},{\mathbf {q}},e_{i}, {\mathbf {e_{-i}}},s)\frac{Pr\left( {\mathbf {q}} | \tilde{e}_i, {\mathbf {e}}_{-i} ,s\right) }{Pr\left( {\mathbf {q}} | e_i, {\mathbf {e}}_{-i} ,s\right) } u(c_{i},\tilde{e}_{i}),&{} {\text {otherwise}}. \end{array}\right. \nonumber \end{aligned}$$

The proof now shows that a contract that solves Program 4 below, which is a relative performance problem without the incentive constraints (46) and $\bar{u}_i = \bar{u}$ for all i, is also a feasible relative performance contract (solves Program 1). In particular, we show that, a solution to Program 4 satisfies the incentive constraints (46) when idiosyncratic shocks are absent, when arbitrarily high punishment is allowed, and when $\bar{u}_i = \bar{u}$ for all i. $\square$

Program 4

$$\begin{aligned} \max _{\bar{\pi }^{r}} \sum _{{\mathbf {c}},{\mathbf {q}},{\mathbf {e}},s} \bar{\pi }^{r}\left( {\mathbf {c}},{\mathbf {q}},{\mathbf {e}},s\right) (q_{1}+q_{2}-c_{1}-c_{2}) \end{aligned}$$

(47)

subject to constraints (10), (11), (14), and

$$\begin{aligned} \sum _{{\mathbf {c}}, {\mathbf {q}},{\mathbf {e}},s} \bar{\pi }^{r}\left( {\mathbf {c}},{\mathbf {q}},{\mathbf {e}},s\right) u\left( c_{i},e_{i}\right) \ge \bar{u}\ \forall i. \end{aligned}$$

(48)

Let $\bar{\pi }^{r}\left( {\mathbf {c}},{\mathbf {q}},\bar{{\mathbf {e}}},s\right)$ be a solution to Program 4 such that effort levels are identical across agents, i.e., $\bar{{\mathbf {e}}} = \left( \bar{e}_1, \ldots , \bar{e}_N\right)$ and $\bar{e}_1 = \ldots = \bar{e}_N = \bar{e}$. If the minimal level of effort is adopted in the solution, then the incentive constraints would be trivially satisfied. If any other level of effort $\bar{e}$ is adopted with positive probability, then the incentive constraints would be satisfied only because effort affects the distribution of output. Otherwise, the principal could recommend a lower effort level and lower consumption by holding utility constant and increasing the surplus. As a result, there must be some state $\bar{\omega }$ such that for any $\hat{e}<\bar{e}$, $E(q| \bar{e}, k, \bar{\omega })>E(q| \hat{e}, k, \bar{\omega })$. Given the absence of idiosyncratic shocks, it must be the case that $f(\bar{q}| e, k, \bar{\omega })=1$ and $f(\hat{q}| \hat{e}, k, \bar{\omega })=1$ for some $\bar{q}>\hat{q}$. Thus, if $u(\hat{c}_i,\bar{e}_i)$ can be set at a sufficiently negative level, then the incentive compatibility constraints (46) will be satisfied. Note that since incentive constraints are not present in Program 4, any group contract delivering the same utility pair without full insurance will generate strictly lower surplus.