Abstract
The estimation of principal agent models is a subset of inverse optimal problems. As of now, there is no consistent method of estimating all its parameters. In general, some proxies for the parameters have been utilized to test plausible economic implications of such models. This study develops a method of estimation for all the parameters using a very limited time series data for one contracting pair. Progress toward empirical reality, based on stylized facts, has been achieved by iteratively modifying the theoretical models and econometric methods. One of these results provides a theoretical justification for the econometric tools utilized in practice as well. However, a fundamental modification of the underlying assumptions is necessary. Given the emphasis on contracts in economic exchange, it is necessary to develop the methods further. The study also outlines some of the pertinent issues.
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Notes
- 1.
The recent book by Bolton and Dewatripont (2005) contains details of contract design. However, they did not deal with any econometric issues.
- 2.
- 3.
Note that vertical integration and circumventing the market lead to contracts within the firm. The labor market literature predominantly deals with this. Contracts with agents outside the firm, e.g., subcontractors, franchisees, etc., are equally dominant. The first set of studies is closer to the transaction cost argument of Coase while the latter is more concerned about agency costs. They also differ with respect to the duration of contracts. The present study concentrates mostly on the second type of contracts. The reader may refer to Masten and Saussier (2002) for a review of econometric studies dealing with the choice of market versus contracts.
- 4.
- 5.
Strictly speaking, there will be two types of randomness. First, there will be fluctuations in external market conditions. Neither the principal nor the agent can do much to neutralize this effect. Second, there can be moral hazard on the part of the agent. This may be quite unpredictable ex ante. A part of this randomness can be controlled if the principal exercises necessary control. In much of what follows, u contains the effect of randomness of both types. Unfortunately, neither the economic theory underlying the principal agent models nor the econometric methods have been able to deal with these two types of randomness separately.
- 6.
For the sake of analytical curiosity, consider the case where the agent chooses both y and p. This results in
$$ y=p\delta $$and
$$ p=y/2\lambda {\sigma}^2 $$These two equations can be satisfied simultaneously if and only if
$$ \delta =2\lambda {\sigma}^2 $$It may not be realistic to expect this a priori in any practical contracting situation. It is also unrealistic to believe that the principal does not keep any controls under his discretion for this is the only way he can safeguard his interest.
- 7.
In this formulation, u may be purely exogenous. In such a case, an agent that is made to share risk feels that he is being punished for something he did not do. There is then a possibility that he will reduce y if σ 2 increases. This was noted in Borenstein et al. (2007). Alternatively, Baker (2006) suggested the following. The agent may use more resources and incur greater costs, given his risk aversion, in order to be sure that he may deliver the promised output. This can also give rise to the possibility that y depends on λ and/or σ 2. However, note that in this formulation, it is independent of σ 2 ex ante given p. Ex post p will depend on σ 2 and hence the observed y is not independent of σ 2. By way of contrast, based on some experimental evidence, Sloof and van Pragg (2008) noted that y does not depend on σ 2 even ex post. It is difficult to replicate this result in principal agent models of this vintage.
- 8.
Sloof and van Praag (2008) and Aggarwal (2007) pointed out that the choice of p need not depend on δ and/or λσ 2. A similar argument can be found in Nichols (1983) and Svejnar and Smith (1984). The implications of such choices in franchise contracts have been explored in Lafontaine (1992) and Kaufman and Lafontaine (1994). However, it is rather difficult to develop satisfactory economic theory to justify such a choice by the principal in principal agent models of any vintage. Sharma and Rao (2007) reported one such attempt. Some aspects of this and other implications for econometric estimation will be considered in Sect. 13.5.
- 9.
In actual practice, it may appear that the firm is fixing the warranty. The consumer can then choose to buy or reject a product. When he does buy, he has the choice of the quantity purchased. This was the spirit of the KM model specification.
- 10.
Prominent studies include Asanuma (1989), Asanuma and Kikutani (1992), Yun (1999), and Okamuro (2001). Their studies differ mostly with respect to the proxies used. The econometric methodology is essentially the same. Related studies also include Allen and Lueck (1999), Gonzales-Diaz et al. (2000), Brickley (2002), Aggarwal (2007), Haubrich (1994), and Hernandez (2003).
- 11.
To the extent that I could understand these papers, it is quite clear that they did not really make any attempt to estimate δ from the observed data.
- 12.
Almost all subsequent empirical work on the principal agent models utilizes proxies. These studies did offer various useful insights into the economic mechanisms underlying such contracts. However, the estimation process rarely follows from the theoretical model specification.
- 13.
Ferrall and Shearer (1999) also voiced a similar concern.
- 14.
Note that
$$ \begin{array}{c}{y}_a=p\delta +u\\ {}={\delta}^2/\left(\delta +2\lambda {\sigma}^2\right)+u\end{array} $$Assume that u is normally distributed. Write the likelihood function. Since u is almost never identically zero, the first-order conditions with respect to δ and λ cannot be solved to obtain the corresponding estimators. Similarly, we may write
$$ {\pi}_a\left(\delta +2\lambda {\sigma}^2\right)/\delta ={\delta}^2/2\left(\delta +2\lambda {\sigma}^2\right)+u $$Once again, it is very easy to show that the first-order conditions for the maximization of the likelihood function cannot be solved for δ and λ.
- 15.
Strictly speaking, this is valid only when the data generation process corresponds to the optimization implicit in the KM formulation of the principal agent model. In practice, this may not be true. In such a case, an χ 2 test based on the deviations of π a from its estimated value can be utilized to test the validity of the model.
- 16.
Okamuro (2001, p. 364) pointed out that this is not a good estimate. Suppose π a = .1(y + u). That is, profit is about 10 % of the sales revenue. Then, it follows that V(π a) = s2 = .01 σ 2. Consequently, p = .1. This indicates that there is some risk absorption. However, the following scenario is plausible. Suppose output and cost increase by α percent. Let the price of output remain invariant. Then, profit will also increase by α percent. But, clearly, there is no risk absorption. It is much more important to model systematic fluctuations in demand to get a reliable estimator of p. In other words, as Masten and Saussier (2002, p. 284) pointed out, risk sharing is not the major determinant of output sharing contract choice.
- 17.
Estimation of λ crucially depends on the choice of p as recorded in the basic model. This can be contested on two counts. First, suppose σ 2 is very high. A risk-averse agent may ask for an even higher fixed upfront payment and a lower p compared to what the model signals. Similarly, when σ 2 is fairly low, he may be emboldened to take the risk. He may then accept a higher p and a lower upfront payment. That is, there may be incongruence between the observed data and the predictions of the model at both extremes of market uncertainty. Second, the principals may have a tendency to underpay agents by substantial amounts for the principal may suspect that the agent is not revealing correct information on three counts. (a) The agent may reveal a lower probability of success hoping that a low output target will be set. (b) The agent may be more risk averse than what he is willing to reveal. (c) The agent exaggerates his level of skill in contract negotiations. Further, the principal, not being fully aware of how δ and λ contribute to his profit, may prefer to reduce p. This possibility implies that the estimated value of δ may be an underestimate and that of λ an overestimate. It is necessary to free the model from its excessive dependence on σ 2 as the key to the entire decision process. Borenstein et al. (2007) also voiced similar concerns.
- 18.
Two other studies, viz., Dubois and Vukina (2004) and Paarsch and Shearer (2000), utilize the method of moments in a somewhat different manner. Their methods are not useful in the present context because the error u in y a and the error pu in π a are strictly correlated and the variance covariance matrix of errors is singular.
- 19.
Observe that the p so defined is not a constant independent of x. Instead, it is somewhat similar to a bonus (or incentive) scheme.
- 20.
This approach to the estimation problem requires data on p. But this is not available. However, it is clear that pλ is estimable. Hence, this can be used in the regression without loss of any other information.
- 21.
Note that the random variable ε introduced here is in no way related to the ε in the KM estimation of λ. Further, it can be argued that the y a and p equations constitute a pair of seemingly uncorrelated regressions. This is a matter of detail and not fundamental to the issue of identification being considered here.
- 22.
The practical difficulty is the following. Much of the empirical literature suggests that the principal’s choice of p will not be a function of δ, λ, or any other exogenous variation of x over time for any contract for a specific product type. The variations in p would only be over a cross section of different products handled by the same principal and/or agents. Even empirical studies utilizing panel data could not come to grips with this issue so far. Consequently, there is a great deal of ambiguity about the appropriate economic model and the corresponding econometric methods of estimation. Given the paucity of data, the best empirical studies should depend on panel data. Economic theory must be developed with this constraint in perspective. For example, the x variables may refer to contracts for different products instead of a time series for the same product. The estimation of δ, λ, and σ 2 should be modified accordingly.
- 23.
Ferrall and Shearer (1999) pointed out that the payment to the agent may take the form fixed upfront fees + p(y a − y min) where the share is only a bonus when the realized output exceeds a stipulated minimum. The nature of the analytical details does not change materially even if this version is pursued.
- 24.
The following argument can also be structured by assuming that the agent has a desired minimum level of profit. In that case, we require
$$ E\left({\pi}_{\mathrm{a}}\right)={p}^2\delta /2\ge m $$It cannot be stated in terms of π a per se since it contains a random realization pu.
- 25.
Suppose the contract fixes a y and σ 2 increases ex post. The agent may renege. An upfront payment may then have the ability to elicit better compliance and loyalty over time in long-term contracts. A similar argument holds even for ex post changes in y given σ 2. For all practical purposes, the reason for reneging will be the nonfulfillment of minimum returns. Hence, there is no new information in this argument.
- 26.
Chiappori and Salanie (2003) acknowledged two other contingencies. (a) The agent may not have sufficient initial wealth to withstand the uncertainties due to minimum income requirements for survival. (b) The agent, who is very likely to face a liquidity constraint even in his efforts to raise finances needed to incur the cost y 2/2δ, has to depend on the principal to provide an upfront payment. These details will not alter the basic structure of the problem.
- 27.
An impression is usually created that ex ante rents will increase with δ. Hence, the δ defined is the minimum. All agents with larger δ earn ex ante rents. This is the implication in Kaufman and Lafontaine (1994) and Chisholm (1997). To the extent that I am aware of it, no principal agent model specification can sustain this result.
- 28.
Observe that this analysis assumed that only one agent is involved. Since the optimization does not depend on δ, it can be claimed that the result is fairly general. Otherwise, the rest of the analysis should be interpreted as valid only for any arbitrarily chosen p*. The other drawback is that the model cannot specify the number of agents that the principal would prefer to deal with. Coordination problems and the associated diminishing returns, if they are pertinent, have yet to be appropriately incorporated.
- 29.
Suppose the principal is specifying a p* without reference to any maximization. Then,
$$ V\left({\pi}_{\mathrm{a}}\right)=p{*}^2{\sigma}^2/k $$so that
$$ p{*}^2=V\left({\pi}_{\mathrm{a}}\right)/V\left({y}_{\mathrm{a}}\right) $$If the null hypothesis about the validity of the model is justified, this estimate of p* should be the same for all the agents. Perhaps the variation over the sample, if any, can be utilized as a test of the principal fixing p* a priori. However, if the null hypothesis is not valid empirically, there is no way of estimating p* for all the agents.
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Rao, T.V.S.R. (2016). Estimating the Parameters. In: Risk Sharing, Risk Spreading and Efficient Regulation. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2562-1_13
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