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127 Payment to outside mvestor E> Output FIGURE 8.12 The form of the optimal payment function equilibrium, each project has only a single outside investor. This allows us to avoid dealing with the complications that arise when there is more than one outside investor who may want to observe a projects output. Since outside investors are riskneutral and competitive, an entrepreneurs expected payment to the investor must equal (1 + r)(l  W) plus the investors expected spending on verifying output. The entrepreneurs expected income equals the projects expected output, which is exogenous, minus the expected payment to the investor. Thus the optimal contract is the one that minimizes the fraction of the time that the investor verihes output while providing the outside investor with the required rate of return. Given our assumptions, the contract that accomplishes this takes a simple form. If the payoff to the project exceeds some critical level D, then the entrepreneur pays the investor D and the investor does not verify output. But if the payoff is less than D, the entrepreneur pays the verihcation cost and takes all of output. Thus the contract is a debt contract. The entrepreneur borrows 1 W and promises to pay back D if that is possible. If the entrepreneurs output exceeds the amount that is due, he or she pays off the loan and keeps the surplus. And if the entrepreneur cannot make the required payment, all of his or her resources go to the lender. This payment function is shown m Figure 8.12. The argument that the optimal contract takes this form has several steps. First, when the investor does not verify output, the payment cannot depend on actual output. To see this, suppose that the payment is supposed to be Qi when output is ] and Q2 when output is Yz, with Q2 > Qiand that the investor does not verify output in either of these cases. Since the investor does not know output, when output is 2 ihe entrepreneur
The Equilibrium Value of D The next step of the analysis is to determine what value of D is specified in the contract. Investors are riskneutral and competitive, and the riskfree interest rate is r. Thus the expected payments to the investor, minus his or her expected spending on verification, must equal 1 + r times the amount of the loan, 1  W.To find the equilibrium value of D, we must therefore determine how the investors expected receipts net of verification costs vary with D, and then find the value of D that provides the investor with the required expected net receipts. To find the investors expected net receipts, suppose first that D is less than the projects maximum possible output, 2y. In this case, actual output can be either more or less than D. If output is more than D, the investor does not pay the verification cost and receives D. Since output is distributed *For formal proofs, see Townsend (1979) and Gale and Hellwig (1985). This analysis neglects two subtleties. First, it assumes that verihcation must be a deterministic function of the state. One can show, however, that a contract that makes verihcation a random function of the entrepreneurs announcement of output can improve on the contract shown in Figure 8.12 (Bernanke and Gertler, 1989). Second, the analysis assumes that the investor can commit to verification if the entrepreneur announces that output is less than D. For any announced level of output less than D, the investor prefers to receive that amount without verifying than with verifying. But if the investor can decide ex post not to verify, the entrepreneur has an incentive to announce low output. Thus the contract is not renegotiationproof. For simplicity, we ignore these complications. pretends ttiat it is Yi, and ttierefore pays Qi. Ttius ttie contract cannot make tfie payment wtien output is Yz exceed ttie payment wtien it is Y]. Second, and similarly, the payment with verification can never exceed the payment without verification, D; otherwise the entrepreneur always pretends that output is not equal to the values of output that yield a payment greater than D. In addition, the payment with verification cannot equal D; otherwise it is possible to reduce expected expenditures on verification by not verifying whenever the entrepreneur pays D. Third, the payment is D whenever output exceeds D. To see this, note that if the payment is ever less than D when output is greater than D, it is possible to increase the investors expected receipts and reduce expected verification costs by changing the payment to D for these levels of output; as a result, it is possible to construct a more efficient contract. Fourth, the entrepreneur cannot pay D if output is less than D; thus in these cases the investor must verify output. Finally, if the payment is less than aU of output when output is less than D, increasing the payment in these situations raises the investors expected receipts without changing expected verification costs. But this means that it is possible to reduce D, and thus to save on verification costs. Together, these facts imply that the optimal contract is a debt contract.
R{D) = 2yDD (D 2y y~C if D < 2y if D > 2y. (8.30) Equation (8.30) implies that when Dis less than 2 , (£) = [l(c/2y)]{D/2y). Thus R increases until D = 2y  and then decreases. The reason that R is eventually decreasing ui D is that when D is close to the maximum possible payoff, raising it further mainly means that the investor must verify output more often, and thus reduces his or her expected net receipts. At the maximum, the investors expected net revenues are R{2yc) = [(2y  c)/2y]y = R. Thus the maximum expected net revenues equal expected output when is zero, but are less than this when is greater than zero. Finally, R declines to  at D = 2y; thereafter further increases m D do not affect R(D). The R{D) function is plotted in Figure 8.13. Figure 8.14 shows three possible values of the investors required net revenues, (1 + r)(l  W). If the required net revenues equal Vjmore generally, if they are less than  there is a unique value of D that yields R{D) 2yc 2y FIGURE 8.13 The investors expected revenues net of verification costs uniformly on [0,2y], the probability of this occurring is (2y  D)/2y. If output is less than D, the investor pays the verihcation cost and receives all of output. The assumption that output is distributed uniformly implies that the probability of this occurring is D/2y, and that average output conditional on this event is D/2. If D exceeds 2y, on the other hand, then output is always less than D. Thus in this case the investor always pays the verification cost and receives all of output. In this case the expected payment is y. Thus the investors expected receipts minus verification costs are
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