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We now specialize this

We now specialize this general framework to our empirical setting. We first assume that themarginal products of effort are all non-negative. This assumption is not too restrictive but helpsin interpreting the results of the model below. In addition, we assume that there are only twotypes of objective V1and V2and that the principal’s value is defined as:V = V 1+ V 2The difference between the two objective types V1and V2is that for objective V1, there exists aperformance measure P1such that the marginal products of effort in V1and P1are equal( p1= f1). For objective V2however, P2is both a noisy and possibly discordant measure of V2.Let σ1be the variance of ε1, σ2the variance of ε2, and σ12the covariance between ε1and ε2.When writing the incentive contract the principal maximizes:maxV+ V − α P − α Pα1, α 21subject to the agent choosing the effort level e that maximizes her expected utilityU ( α1P1( e)+ α2P2( e)− C(e))where C(e) is the cost of effort. For an agent with coefficient of risk aversion r and CARAutility preferences, this problem is equivalent to maximizing1V1 + V2− α1P1−α2P2− rVar(α1P1+ α2P2)2The solution to this problem (see proof in full paper) yields the following expressions for21122αi:r(fα =121r(f1+α =+ f )( f σ − pσ) + p'p(f ' f2212212( rσ+ p'p)(ff )( pσ−12112( rσ+ p'p)(f1' ff σ ) + f111' f' f111+ f ' f12+ rσ) + ( p'f11( p'f11+ p'f2) − p'f1+ rσ) + ( p'f+ rσ) − p'f1+ rσ1212)112( p'f( f)121' f1+ p'f12)+ f ' f )23. Implications and Comparative StaticsThe model in the previous section provides interesting insights about how incentives onperformance measures vary with different parameters. In this section, we examine some of itsimplications and derive comparative statics that will be useful for testing hypotheses about thecross-sectional variation in incentive strength for various performance incentives.First, consider the case when f = 20 . In this case, the buyer cares primarily about a singleobjective f 1for which a non-distorted performance measure ( p1= f1) exists. Notice that even inthis caseα 2is not necessarily zero: the principal may use additional performance measures eventhough a good performance measure exists. As can be seen from the expression for α1andα 2,the weight assigned to the second performance measure depends on three parameters: σ1, thenoise in the performance measure P1, σ12, the correlation between P1and P2, and the distortionin2P . The likelihood of using a second performance measure increases the noisier the firstperformance measure, the less noisy the second performance measure and the less distorted thesecond performance measure.3

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