## 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.

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## Notes

We can apply the limiting arguments used in Prescott and Townsend (1984) to establish results when the commodity space is not finite.

Note that both

*k*and \(\omega \) are organization-specific variables, which affect simultaneously the production functions of all workers assigned to an organization.In the general equilibrium formulation, principals will be modeled as competitive intermediaries.

Madeira and Townsend (2008), for instance, assume that there is a cost related to adopting relative performance.

Under relative performance, one agent would consume \(c_{1}\) and the other would consume \(c_{2}\). The lottery mentioned gives each of these outcomes with probability 0.5.

Note that this consumption scheme would increase the payoff of lower effort for a given individual acting independently (as in relative performance), since a contingency in which he is the only one with the lower output would now generate higher consumption. So this scheme would not be feasible under relative performance.

Note that \(\lambda ^i\) represents ex-ante Pareto Weights, while \(\mu ^i\) represents ex-post inside-group Pareto weights. As it is shown in Prescott and Townsend (2002), they are not necessarily the same. For instance, it is possible that agents with different Pareto weights are assigned to different groups, and in each of these groups they have equal Pareto weights.

Technically, we solved Program 3, with 900 pairs of utility promises for each regime (groups with information, groups without information, relative performance with information, and relative performance without information). Wealth inequality is measured by Pareto weights \(\lambda \) in all of the examples.

Making the marginal utility of effort depend on \(\rho \) as above is convenient to study jointly different risk aversions since it makes effort and consumption comparable, which brings the regime areas to similar wealth ranges. Changes in the values of \(q^h\), \(q^l\), \(e^h\) and \(e^l\) have no impact in the general profile of solutions, although they change the wealth range where regimes are found.

There are some slightly upward sloping areas in some of these lines, possibly related to the fact that we have a finite (although large) grid.

We performed a larger set of computation than those presented in this session. we computed the general equilibrium for a grid of 20 values of \(\lambda \), 40 values of wealth, 5 levels of risk aversion and 4 specifications of the production function. In all of these specifications, when the cost of information is zero there is information acquisition with probability one, and group firms are the adopters whenever information is acquired.

This result may not be true in general, especially when investment in partial information about common shocks is possible. Partial information that bring small gains should be purchased by relative performance organizations if its cost is sufficiently low. However, in a region where relative performance dominates, if this gain is sufficiently small, relative performance will be dominant, and, in this case, not invest in information.

Note that wealth is sometimes negative, which can be interpreted as debt. Identical profiles with non-negative wealth would be observed if we assumed there is a fixed cost in terms of capital for the firms to produce.

In other words, consumption will be constant, regardless of output realizations.

In all of these production technologies, probabilities of high and low output conditional on effort (unconditional on \(\omega \)) are the same. The specifications depart from the benchmark case to make the effects of correlation clearer.

Without a loss of generality, an optimal group contract with different weights must randomize across \(\mu _i\), and it will do so only if advantageous. Therefore, the proof is valid.

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## Acknowledgments

We are thankful for helpful comments by Fernanda Estevan, Rafael Costa Lima, Ricardo Cavalcanti, Marcos Nakaguma, Ned Prescott, Robert Townsend, the associate editor, two anonymous referees and participants at LAMES (Buenos Aires, 2009), SAET (Paris, 2013), and academic seminars at FEA-USP, FEARP-USP, PUC-RJ, FGV-RJ, FGV-SP. Tee Kilenthong would like to thank the University of the Thai Chamber of Commerce for the financial support. Gabriel Madeira gratefully acknowledges financial support from Fipe.

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## Appendix: Proofs

### 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 symmetric^{Footnote 16}, 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

and

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:

The symmetry assumption implies the following:

Thus, the following is implied:

Substituting this equation into (45) yields the following:

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

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 \(e_i, \tilde{e}_{i}\):

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

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

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:

where

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**

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

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.

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Kilenthong, W.T., Madeira, G.A. Observability and endogenous organizations.
*Econ Theory* **63**, 587–619 (2017). https://doi.org/10.1007/s00199-016-0959-2

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DOI: https://doi.org/10.1007/s00199-016-0959-2

### Keywords

- Organizational design
- Observability
- Relative performance regime
- Group regime
- General equilibrium
- Value of information