WPS7128 Policy Research Working Paper 7128 Motivations, Monitoring Technologies, and Pay for Performance Antonio Cordella Tito Cordella Development Economics Vice Presidency Operations and Strategy Unit December 2014 Policy Research Working Paper 7128 Abstract Monitoring technologies and pay for performance con- non-monotonic in agents’ motivations and monitoring tracts are becoming popular solutions to improve public effectiveness; (iii) investments aimed at improving agents' services delivery. Their track record is however mixed. To motivations and monitoring quality are substitutes when show why this may be the case, this paper develops a prin- agents are motivated, complements otherwise; and (iv) if the cipal agent model where agents’ motivations vary and so agents’ “type” is private information, the more and less moti- does the effectiveness of monitoring technologies. In such vated agents could be separated through a menu of pay for a set-up the model shows that: (i) monitoring technologies performance/non pay for performance contracts, such that should be introduced only if agents’ motivations are poor; only the less motivated choose the pay for performance ones. (ii) optimal pay for performance contracts are nonlinear/ This paper is a product of the Operations and Strategy Unit, Development Economics Vice Presidency. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at a.cordella@lse.ac.uk and tcordella@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Motivations, Monitoring Technologies, and Pay for Performance Antonio Cordellayand Tito Cordellaz December 1, 2014 JEL Classification Numbers: D82, J33, J45, M52. Keywords: Pay for Performance, Public Sector Management, Information and Communication Technologies, Asymmetric Information, Agents’ Motivation, Optimal Contracts. We would like to thank David Rosenblatt and James Trevino for helpful comments and suggestions. The usual disclaimers apply. y ISIG-Department of Managment-London School of Economics and Political Science. E- Mail:a.cordella@lse.ac.uk z Development Economics (DEC), The World Bank, 1818 H St. NW, Washington, DC 20433, USA; E-Mail:tcordella@worldbank.org 1 Introduction In the last two decades, governments across the world have invested massively in monitoring and reporting technologies to improve the quality of public service delivery. The idea that such technologies promote e¢ ciency gained increasing consensus in managerial circles, and it quickly spread to private companies and multilateral organizations.1 But what are the channels through which monitoring and reporting technologies contribute to an im- provement in public sector performance and to the provision of better services? According to the New Public Management school (NPM hereinafter), the road to e¢ ciency is paved by the three “Ms” : markets, managers and measurement (Ferlie et al., 1996), and measurement is what markets and managers have to rely upon to be able to exert control and enforce pay for performance contracts2 (PFP hereinafter). While a lot has been written on the e¤ects of the introduction of PFP on the productivity of public sector organizations (Frey at al., 2013; Moynihan and Pandey, 2010; Weibel et al., 2010), much less has been written on the impact of investments made to increase measurability in public sector PFP schemes. This is quite surprising when many large ICT investments have been justi…ed on the premise that enhanced monitoring and reporting technologies are key elements to improve organizational performances (Brynjolfsson et al., 2000; Dunleavy and Carrera, 2013; Garicano and Heaton, 2010). To better understand the trade-o¤s associated with measurability, this paper provides a simple theoretical framework to analyze the channels through which monitoring and reporting technologies may (or may not) increase the e¤ectiveness of PFP schemes. More precisely, we consider a stylized framework in which the measurement of outcomes is costly, and the alignment between the objectives of the agents and those of the principal is only partial. In such a set up, we show that (i) it is optimal for the principal to introduce a monitoring system only if the latter does not impose a too high burden on the agents, and/or if the agents are not su¢ ciently motivated; (ii) the design of an e¤ective PFP is complicated, and the optimal contract is highly non linear and/or non monotonic both in agents’motivations and in the “cost” of the monitoring system; (iii) investments aimed at improving agents’motivations and the quality of the monitoring are complements when agents are highly motivated and substitutes when they are not; (iv) if the agents’ “type” is private information, an e¤ective way for the principal to separate the more motivated from the less motivated agents is to o¤er a menu of contracts designed in a way that only the latter choose the PFP. The above …ndings may shed a new light on the …erce debate on public administration reforms and on the role played by e-government investments aimed at increasing the measurability and hence transparency and accountability of public sector organization (Barzelay, 2001; Bertot et al., 2010; Dunleavy et al., 2005; Pina et al., 2007). On one side, NPM advocates argue that investments in technologies that increase measurability boost organizations’productivity by facilitating the alignment of public servants’motivations with prede…ned organizational objectives (Aral et al., 2012; Ba et al., 2001). NPM advocates also point at the increasing popularity of PFP and e-government projects around the world as a measure of their success.3 On the opposite side, NPM critics argue that the increasing reliance of government programs on PFP reforms is a fad driven by consulting …rms, which by no means is justi…ed by the actual record of PFP or of e-government solutions.4 Our own reading of the literature is that, overall, the adoption of PFP and the di¤usion of e-government programs in the public sector has delivered mixed outcomes. Our model, suggesting that no one-size-…ts-all solution exists, may thus provide a clear rationale for why this may be the case. Of course, we are not the …rst who have looked at measurability in a principal agent framework; our model builds upon Holmstrom and Milgrom (1991),5 which …rst suggested that if agents have to perform 1 For instance, at the World Bank increased attentions is being paid on “deliverology,” that is, on how to maximize the developmental impact of the di¤erent programs by taking into account the incentives of the di¤erent stakeholders. 2 See Picot et al. (1996). 3 See, for instance, OECD (2005). 4 See, among others, Francois (2000), Moynihan and Pandey (2007), and especially Perry et al. (2009) who o¤er a comprehen- sive review of the e¤ects of PFP schemes on public sector organizations and come to the conclusions that performance-related pay consistently fails to deliver on its promises. 5 For a comprehensive survey to the theoretical and empirical work on the provision of incentives in …rms, see Prendergast 1 multiple tasks, some monitorable and some not, incentive based contracts, which (necessarily) focus on the latter, may induce agents to reallocate e¤ort in an ine¢ cient way. Given that most of the goals associated with the actions of public sector organizations are by nature not univocal and cannot not always be planned and de…ned before their executions (Moore, 2005; Alford and Huge, 2008), it is di¢ cult to map them in performance indicators (Proper et al. 2003; Behn 2003). Baker (2002), Langbein (2010), and Grand (2010) provide comprehensive discussions of the costs and bene…ts of using PFP when goals are not univocal and/or quanti…able and performance indicators are di¢ cult to establish. However, to our knowledge, there is no contribution that discusses how investments in monitoring technologies a¤ect the enforcement of PFP schemes in such an environment. Our main contribution to this literature is in explicitly modeling the costs associated with the introduction of monitoring and reporting technologies– the costs of managerial attention, according to Halac and Prat (2014)– and in studying how the interaction between such costs and agents’motivations a¤ects the optimal PFP scheme. Agents’motivations, in our view, are indeed a critical factor to take into consideration when discussing PFP. In this dimension, we build upon Dixit (2002) who emphasizes that many public sector employees (judges, teachers, doctors, social workers) may share some “idealistic or ethic purpose served by the agency” (p. 715). Starting from such a premise, Delfgaauw and Dur (2008) show that a PFP system, o¤ering steep incentives to the more dedicated workers, may help attract them to the public sector. Our model shares some of Delfgaauw and Dur’ s (2008) features. However, in our set-up, performance assessment schemes detract resources from the ultimate goals of the agency, and this leads us to reach the opposite conclusions, that is, PFP may end up inducing the more motivated agents to leave the organization. This phenomenon may also be seen as a re‡ ection of tensions between intrinsic and extrinsic motivations, as in Kreps (1997), and Benabou and Tirole (2003). Another insight of this paper, namely the fact that optimal PFP and the associated investments in monitoring and reporting technologies have to be tailored according to output measurability, is also in line with empirical evidence. Hasnain and Pierskalla (2012), in an up to date and comprehensive survey of the (empirical) literature, …nd evidence that, when tasks are simple and outcomes observable, the use of PFP is more e¤ective; at the same time, monitoring and reporting technologies are also more valuable in supporting PFP schemes (Ciborra, 1996) both because of moral hazard (i.e., incentive) and adverse selection (i.e., sorting more able workers, Lazear, 2000) considerations. Instead, when tasks are complex and outputs di¢ cult to measure, the introduction of PFP schemes and the use of monitoring and reporting technologies can create distortions on incentives (e.g., discouraging the most motivated workers) or foster the wrong type of sorting. This means that the decision of whether to adopt PFP schemes, their optimal design, and the decision of how much to invest in monitoring and reporting technologies should take all these e¤ects into consideration. A less abstract description of the kind of problems we address, and a spicier ‡ avor of our main results, may be derived from the following two examples, one in health care and the other in education. Consider …rst a hospital director who wants to improve the quality of patients’care that, simplifying, depends on the number of hours doctors work and on the quality of the care they provide. Assume that hours are observable but quality not. In order to improve doctors’incentives, the director may consider linking their compensation not only to the hours they spend in the hospital, but also to the quality of the care they provide. Since the latter is not directly measurable, the hospital can set up a costly monitoring system based on the doctors’ record of how they take care of each patient, and make part of the doctors’pay linked to the quality of their respective records. Of course, …lling a detailed record detracts precious time from actual patients’care, so that the optimal PFP should carefully weigh the monitoring system’ s costs and bene…ts. Consider now the case of a school principal who cares about students’learning that, simplifying again, depends on the number of hours kids are taught and on the quality of teaching. As before, assume that hours are observable, but quality (of teaching) is not. In order to improve teachers’ incentives to teach well, the principal may consider linking teachers’ compensation to the results of a pro…ciency test that students are asked to take. Since the results of such a test are an imperfect measure of what kids have (1999). 2 actually learned at school, and the preparation of such tests is costly (in terms of hours subtracted from actual teaching), we are again in the presence of trade-o¤s. Clearly, the more committed doctors are to patients’care, the more committed teachers are to education, the costlier the monitoring schemes– both in terms of set-up expenses and administrative e¤ort– the larger is the deadweight loss associated with the PFP schemes. An additional cost to be taken into consideration is the one associated with the possibility that the more committed professionals may consider leaving workplaces where too much e¤ort is devoted to costly performance measurement.6 These are the kinds of real world issue that our stylized model tries to address. The remaining of the paper is organized as follows: the next section presents the basic model, solves it when “quality” is observable and unobservable, and then derives the main results for the case in which it is observable if a costly (for the agent) performance assessment system is adopted. Section 3, relaxes the assumptions that the principal cannot invest to improve the assessment technology, and that the introduction of the performance assessment scheme does not a¤ect the intrinsic motivations of the agents. In section 4, the analysis is extended to discuss the case where agents’motivations are heterogeneous, and they are not observable by the principal. Finally, section 5 concludes. 2 The Model Assume that an organization (the principal hereinafter) wants to maximize the success of a speci…c activity, the quality of education in our last example, which depends upon the contribution of two distinct but complementary components y and q . Let y be the component which is easy to verify/contract upon, and q the one which is not. One can think at y as a quantitative component, the hours taught in the same example, and q as a more qualitative one, the quality of teaching. We further assume that the success of the project is given by qy , and that the principal, who cares about the success of the project: (i) has a limited budget T , and (ii) he has to delegate the implementation of the activity to a “partially motivated” agent. By this, we mean that the agent does care about the principal’ s objective, but she also cares about her remuneration, and she receives negative utility from the e¤ort she devotes to the di¤erent components of the project. In what follows, we …rst present the two extreme cases, in which q is either perfectly observable or totally unobservable, and discuss the optimal remuneration schemes that the principal can o¤er in either case. This allows us to get a clear understanding of the distortions associated with incomplete contracting, and it facilitates the discussion of the third, most interesting case, in which q is observable, but only at a cost. As standard in the literature, we restrict our attention to linear contracts; in addition, to spare the reader the tedious algebra, in the main text we present the results in an intuitive way, and we refer the reader to the Appendix for the technical details. 2.1 Observable “quality” To set an e¢ ciency yardstick, we assume that both y and q , are monitorable/contractible. When this is the case, the principal,7 who is interested in maximizing qy , o¤ers a compensation package fw; k g, where w is the compensation8 per unit of y , and k the compensation per unit of q . Denoting by subscript O the observable case, the problem of the agent can be written as: M ax UO = qy + (wy + kq ) (q + y )2 , (1) y;q 2 where > 0 is a measure of the alignments between the objectives of the principal and those of the agent, y )2 s remuneration, which enters linearly in the objective function, and c(q; y ) = (q+ wy + kq the agent’ 2 is 6 This may explain why US private schools, where there is no testing, may not only attract highly quali…ed teachers, but can also pay them lower wages than in the public schools where test preparation is becoming an increasing burden. 7 Throughout the model, we assume that the principal moves …rst and o¤ers a non-renegotiable contract to the agent. 8 Here we remain vague about what compensation exactly means. In a market environment, it can be the price/salary paid for each of the activities delivered. In a non market envirionment, it can be the budget allocated to di¤erent teams. 3 the cost of e¤ort. We further assume that:9 2 ( ; ): (A.1) 2 Solving for (1), we obtain: k( )+w yO (w; k ) = ; (2) a) (2 w( )+k qO (w; k ) = ; (3) (2 ) where yO (w; k ), and qO (w; k ) denote the optimal choice of the agent for any given compensation package fw,q g. The principal’s problem can now be written as: M axVO = qO (w; k )yO (w; k ); such that: qO (w; k )k + yO (w; k )w T, (4) w;k s budget constraint. The solution of the problem is given by: the latter expression denoting the principal’ r T (2 ) wO = kO = ; (5) 2 s T yO = qO = ; (6) 2(2 ) T VO = : (7) 2(2 ) Since the objective functions (of the principal and of the agent) are symmetric in y and q , when both activities are contractible, the optimal compensation scheme rewards them equally, so that the e¤ort devoted to each of them is also equalized at equilibrium. 2.2 Unobservable “quality” We now move to the situation in which only y is observable/contractible. Designating by subscript U the unobservable case, the problem of the agent can now be written as: M ax UU = qy + wy (q + y )2 : (8) y;q 2 Solving for (8), one obtains w yU (w) = ; (9) (2 a) w( ) qU (w) = ; (10) (2 ) where qU (w) and yU (w) denote the optimal choice of the agent for any given w. The problem of the principal can now be written as: M ax VU = qU (w)yU (w); such that wyU (w) T; (11) w but, since the budget is given, and only y is contractible, the only option for the principal is to set wU = T =E [yU (w)]; (12) 9 > =2 insures that the objective function is concave and that a maximum exists. > , instead, insures the existence of an interior maximum, with q > 0. The last assumption is not necessary, but it allows us to reduce the number of cases, making the analysis simpler, without great loss of generality. 4 where E denotes the expectation operator. Assuming rational expectations, the solution of the problem is given by: s T (2 ) wU = ; (13) p T yU = p ; (14) (2 ) p ( ) T qU = p ; (15) (2 ) T( ) VU = : (16) (2 ) Comparing these results with the ones in the previous section, it is immediate to verify that yU > yO , and qU < qO . When compared with the situation in which both activities are contractible, the agent now overdelivers on the measurable activity y , and underdelivers on the non measurable one q . Of course, the utility of the principal is lower than when both activities are contractible, and the cost of contractual incompleteness decreases with . In fact, when increases, the agent puts additional e¤ort on the non measurable activity even if the latter is not remunerated; this results in an increase in qU , relative to yU , and thus in a reduction in the distortions. This is the reason why, when increases, given (12), the remuneration of the observable activity also increases (its “supply” decreases). 2.3 Costly observable “quality” After having brie‡ y discussed the cases in which “quality”is either perfectly observable or totally unobserv- able, we are now in a position to analyze the interesting case in which, while the principal cannot observe the agent’s choice of q , he can nonetheless introduce a costly (for the agent) performance assessment scheme, which allows him to extract information about her e¤ort. More precisely, we assume that the principal relies on a performance assessment scheme, s = s(q; e), that requires some additional e¤ort, e, on the part of the agent. To keep the analysis simple, we posit s(q ) = minfq; eg. (17) This means that the e¤ort that the agent devotes to q is “scored” in the performance assessment scheme only if she devotes an additional e¤ort equal to e, with > 0. is thus a measure of the e¢ cacy of the assessment scheme, the larger is , the more e¢ cient the scheme is. Notice that if e= can be interpreted literally at the additional cost that the agent should bear to have its performance properly assessed, it can be more generally thought of as a measure of how e¤ectual the performance assessment system is. More precisely, e= can be the cost of inducing the agent to invest in a (suboptimal) but contractible technology rather than in the optimal but not contractible one. We further assume that, in this case, the cost function becomes c(q; y; e) = (q + y + e)2 . Denoting by subscript C the costly observable quality case, the problem of the agent can be written as: M ax UC = qy + (wy + k minfq; eg) (q + y + e)2 ; (18) y;q;e 2 which yields w(1 + )2 k yC (w; k ) = ; (19) (k w ) qC (w; k ) = 2 ; (20) (k w ) eC (w; k ) = 2 2 ; (21) 5 with (1 + ) , and 2(1 + ) . As before, yC (w; k ), qC (w; k ), and eC (w; k ) denote the optimal choice of the agent given a compensation package fw; k g. The problem of the principal can now be written as: M ax VC = qC (w; k )yC (w; k ); such that: wyC (w; k ) + kqC (w; k ) T (22) w;k which has the following solutions: q q T wC = 2(1+ ); wC = wU = T (2 ) ; q kC = T (1+ 2 ) ; kC = 0;p q2 yC = T (1+ ) ; if > ; yC = yU = p T ; if < ; (2 ) 2 p (23) qC = p T ; or < C; qC = qU = p ( ) T ; or > c; 2(1+ ) (2 ) T eC = p ; eC = 0; 2(1+ ) VC = T VC P = VU = T(2( ) ); 2 ; p 2 ((1+ ) 1+ ) 4( ) where C , and (2 )2 . From a simple inspection of (23) it follows that Result 1 It is optimal for the principal to introduce a pay for performance scheme if, and only if: (i) the monitoring scheme is e¢ cient enough ( > ), or (ii) the agent is not su¢ ciently motivated ( < C ). To get a clear intuition of the results, it may be worth noticing that the observable and unobservable scenarios are the limit cases for the costly observable one, when tends to in…nity and zero, respectively. When is large, the distortions associated with performance assessment schemes tend to vanish, “quality” becomes easy to observe, and this is re‡ ected in the optimal contract. On the opposite, for su¢ ciently low values of , the cost of using the assessment system is so high that it is in the interest of the principal to treat quality as non contractible. In other words, e¤ort is costly for the agent and, thus, the introduction of an assessment scheme for incentive purposes is justi…ed only when it does not detract an excessive amount of resources from the other, productive, activity. Similarly, the distortions induced by the assessment scheme are justi…ed only if the agent is not already su¢ ciently committed to the goals of the organization. If she is, she will end up subtracting e¤ort, which she would have otherwise devoted to the productive activity, to score better in the performance assessment scheme. If these results are very intuitive, the characteristics of the optimal compensation contract are less so. In particular, using the expressions in (23), we have10 that: Result 2 The optimal compensation scheme has the following characteristics: (i) the remuneration w, as- sociated with the quantitative component y , is non linear in and , and non monotonic in ; (ii) the remuneration k , associated with the quantitative component q , is non linear in and , and non monotonic in . The characteristics of the optimal compensation scheme are illustrated in Figures 1 and 2 where, for arbitrary but reasonable values of the parameters,11 we plot the equilibrium value of compensation packages and the agent’ s activities for di¤erent values of the e¤ectiveness of the assessment system, , and of the agent’s motivation . Looking at …gure 1A, for a given value of , when the performance assessment scheme is highly distor- tionary (i.e, for small values of , < ) it is better not to rely on it and to set k = 0, as if e¤ort were not contractible. When the threshold level is reached, then it is optimal for the principal to adopt a per- formance assessment scheme. At this point, wC decreases sharply and kC jumps from zero to its maximum 1 0 See Appendix. 1 1 More precisely we assumed: T = 10; = :5; = :75 in Figure 1, and = 10 in Figure 2: 6 to then decrease for higher values of : The reason for this behavior is that when the assessment scheme is relatively costly (but not so much to discourage its use) substantial monetary incentives are needed to convince the agent to bear the burden of the scheme, a burden that decreases when the performance assess- ment system becomes more precise. Finally, when becomes large enough, the distortionary e¤ects of the performance assessment scheme tend to vanish, and the two inputs tend to be compensated in the same way. The corresponding levels of e¤ort devoted to the di¤erent types of inputs are plotted in Figure 1B. When the performance assessment system is highly distortionary, and only quantity is remunerated, the agents respond to the compensation scheme by overinvesting in the observable inputs y . When we reach , yC makes a discrete downward jump, qC and eC a discrete upward one. Notice that, for large values of , eC decreases and qC increases re‡ ecting the less distortionary nature of the performance assessment scheme. Moving now to Figure 2A, for any given value of , if is small, kC is large, and it decreases continuously with , until , when it drops to zero and only the quantitative component is remunerated. The reason, which we already mentioned, is that since the introduction of the assessment scheme is distortionary, such distortions are worth bearing only if the agent is not su¢ ciently motivated; in the case of highly motivated agents the assessment scheme creates an unnecessary burden. The fact that kC decreases with (for < ) just re‡ ects the fact that lower values of are associated with lower output levels and higher (per unit) remuneration (wC follows the same behavior). To get a better understanding of why this is the case, notice that: kC =wC = 0; if > ; yC =qC = kC =wC = 1+ ; if > ; which in turns implies that if the performance assessment scheme is introduced, its relative weight in the compensation package ( kC =wC ) is independent of the motivation of the agent, , and decreases with its e¤ ectiveness ( ). The reason is that, when the performance assessment system is used, as in standard production theory, 7 the best the principal can do is to equalize the marginal rate of technical substitution between q and y (y=q ) with the (constant) economic rate of substitution between the two factors, which is equal to 1 + 1= for the agent. To do so in our set up (where the di¤erent inputs have the exact same cost in terms of e¤ort), the principal will also have to …x the remuneration of the two activities according to the same proportions, which do not depend on . Instead, what does depend on is , the threshold value of for which it is worth to adopt the performance assessment system. Since @ =@ > 0, when the agent is more motivated, the less distortionary the assessment scheme should be for it to be worth implementing; this is, of course, because the more motivated the agent is, the higher is the output she produces when her e¤ort is not contractible. Finally, it is worth remarking that we …nd threshold levels of and below (or above) which the principal is better o¤ not using a performance assessment scheme in a model in which the principal bears no costs in introducing the performance assessment system. Of course, had we assumed a …xed cost associated with the introduction of the performance system our results would a fortiori hold true. 3 Extensions In order to verify the robustness of our results, we now relax two key assumptions in our model, namely the fact that the assessment technology is given, and that the introduction of the performance assessment scheme does not a¤ect the intrinsic motivations of the agents. 3.1 Investing in technology Until now, we have assumed that the e¢ ciency of the performance assessment system is given, so that the only choice the principal has to undertake is whether to adopt it or not. Of course, these are quite strong assumptions, and it is interesting to see what happens in the more realistic scenario in which the principal can tailor the performance assessment scheme to his needs, by deciding how much to invest in its e¢ ciency. To keep the analysis simple, we assume a linear investment technology so that the cost g ( ) of a performance assessment technology with precision is , where is the (constant) marginal cost of improving precision. Since the principal moves …rst, the problem of the agent is the same as in the previous section, and the best response functions are still given by (19)-(20). The budget constraint of the principal now becomes: T = wyC (w; k ) + kqC (w; k ) ; (24) so that his problem can be written as: M ax VCT = qC (w; k )yC (w; k ); such that: (wyC (w; k ) + kqC (w; k ) T ): (25) w;k; Solving for (25), we obtain that if, in the previous case, the principal found it pro…table to use a performance assessment system only if the latter was precise enough, what now makes the di¤erence is how costly it is to have a precise enough performance assessment scheme. Accordingly,12 we found a threshold (T; ; ), with T (2 )3 ; (26) 16 ( ) such that if, and only if, < the principal …nds it optimal to use the performance assessment scheme. It is worth remarking that @ T (2 )2 (2 2 2 2 ) > =) = < 0; (27) 2 @ 16 2 ( )2 so that, also in this case, the more motivated the agent is, the more likely it is optimal for the principal not to adopt the performance assessment system. 1 2 See Appendix. 8 To illustrate the main results, we rely on Figures 3–5, which are plotted using the same parametrization as in the previous section.13 The …rst thing that is worth noticing is that the comparative statics with respect to (see Figure 3) is just the mirror image of the one with respect to (see Figure 1). This is not surprising at all; low marginal costs of improving the precision of the performance assessment scheme lead to a less distortionary scheme. The comparative statics with respect to (see Figures 2, and 4) is also not a¤ected by the introduction of the investment decision. We now turn our attention to the determinants of the investment in the performance assessment system. If, unsurprisingly, C decreases monotonically with and drops to 0, when reaches (see Figure 5A), the relation between the investment in technology and the agent’ s motivations may deserve a more thorough analysis. Indeed, as we prove in the Appendix and illustrate in Figure 5A, Result 3 The optimal investment in the precision of the performance assessment technology, , is non monotonic in the motivations of the agent: increases with until it reaches a point 0C (corresponding to ) where it drops to zero. The reason for such a behavior is that, as we know from the previous analysis, the more motivated the agent is, the less distortionary the performance assessment scheme should be for it to be worth implementing. This means that the more motivated an agent is the more the additional investment in the precision of the 1 3 In addition, we assumed = :1 in Figure 3A and 3B. 9 assessment scheme is worth. This is, of course, until we reach a strong enough level of motivation for which, no matter how precise, any feasible assessment system will just distract the agent from her duties. An interesting corollary of the result above is that Corollary 3.1 Investments aimed at improving agents’ motivations and the quality of the performance as- sessment system are substitutes when agents are highly motivated and complements when the opposite is true. 3.2 Intrinsic and extrinsic motivations A large psychology based literature (Kreps, 1997 and Perry et al., 2009, among others) argues that the deployment of PFP schemes can harm individual performance when agents are committed to the goals of the organization as extrinsic motivations may undermine intrinsic ones. This can be easily incorporated in our framework, for instance by assuming that should a performance based contract be o¤ered the problem of the agent becomes: M ax UM = qy + (wy + k minfq; eg) (q + y + e)2 ; (28) y;q;e 2 where 2 [1; 1) is a measure of the loss of intrinsic motivations due to the introduction of a PFP scheme, and subscript M stands for motivations. The solution of this problem14 in this case is very similar to (23). The only di¤erence (assuming that an interior maximum exists) is that M 2(1 + ) and that M > , and M < C : As expected, the general appeal of the pay for performance scheme decreases but the main trade-o¤s remain. 4 Heterogeneous agents From the previous analysis, we concluded that the design of optimal performance schemes should be tailored according to agents’motivations which we assumed to be known by the principal. However, usually principals do not know how motivated agents are, and, in addition, agents tend to di¤er with respect to motivations. To take this into account, in this section, we extend the analysis to discuss the case where agents’motivations are heterogeneous, and they are not observable by the principal. As standard in the literature, the latter, however, knows how motivations are distributed and can o¤er a menu of contracts to induce agents to 1 4 Available upon request. 10 reveal their type and compensate them accordingly. The kind of environment we describe here could be an organization where a fraction p of agents is of type H (high motivated), and a fraction (1 p) is of type L (low motivated) with H > L . We further assume that L < C < H so that a performance based contract is optimal for the low motivated agents but not for the high motivated ones. We then ask ourselves if there exists a pair of contracts fwH ; k H g and fwL ; k L g, such that type H prefers fwH ; k H g, type L prefers fwL ; k L g, and the principal prefers this menu of contract to any pooling contract fk P ; wP g that does not lead to a separation between types. Before moving to the discussion of the separating contract, it is useful to discuss how the introduction of two di¤erent types of agents may a¤ect the preferences of the principal for the di¤erent contracts. Assuming that the principal has a budget T , and he is forced to o¤er the same type of contract to all agents, we investigate the conditions under which he prefers to o¤er a performance based contract rather than a wage only (standard) one. If he o¤ers a standard contract, from (9) and (10) the agents optimal response, for any given w; is given by:15 i w ~ (w ) yU = i (2 ; (29) ai ) i w( i ) ~ (w ) qU = i (2 i) ; (30) with i 2 fL; H g. The problem of the principal can now be written as: L L H H L H ~ = (1 M ax VU ~ (w )yU p)qU ~ (w ) + p)qU ~ (w )yU ~ (w ); such that: (1 ~ (w ) + pyU p)yU ~ (w ) T: (31) w As in the unobservable case, since the budget is given, and only y is contractible, the only option for the principal is to to set L H wU = T =E [(1 p)yU ~ (w )]; ~ (w ) + pyU (32) where E denotes the expectation operator. Assuming rational expectations, from (29) and (32)we have that: s T L H (2 L )(2 H) wU~ = ; (33) (1 p) H (2 H ) + p L (2 L) substituting this expression in (9) and (10) we can solve for VU . Let now consider the performance based contract, for any given fw; k g, from (19)-(21), the agents’best responses are given by: i w(1 + )2 k ((1 + ) i ) e (w; k ) yC = ; (34) ( i (2 (1 + ) i ) i i k w ((1 + ) ) e (w; k ) qC = i ( i (2 (1 + ) i ) ; (35) i i k w ((1 + ) ) eCe (w; k ) = i ( i (2 (1 + ) i ) : (36) The problem of the principal can now be written as: L L H H M ax VC = (1 p)qC e (w; k )yC e (w; k ) + pqC e (w; k ); e (w; k )yC (37) w;k such that: L H L H T w((1 p)yC e (w; k ) + pyC e (w; k )) + k ((1 e (w; k ) + pqC p)qC e (w; k )): (38) 1 5 With a slight abuse of notation, in this section, we use subsript U ~ , to denote the wage only pooling contract (which is similar to the unobservable case discussed above), and subscipt Ce , for the performance based pooling contract (which is similar to the costly ovservable case discussed above). 11 Since the problem is very complex, and closed form solutions are di¢ cult to …nd, we decided to carry out our analysis numerically, using the same parametrization as in the previous sections, and assuming that L = :6 and H = :9. In Figure 6, we compare the utility of the principal when he o¤ers a standard and a performance based pooling contract– as well as a separating contract that we will discuss next– as a function of the share p of motivated agents. For the very same reasons why, in our previous discussion, we found that a performance based contract is preferred by the principal if agents are not motivated enough, now we …nd that this is the case if the distribution of types is “bad enough.” More precisely we show that there is a threshold level of p, p, such that if p < p the principal prefers the performance based contract and the standard one otherwise. A pay for performance contract can however be costly– especially when there are enough motivated agents– and the principal may thus consider o¤ering separating contracts. Among these, the most natural choice would be a menu of two contracts, a performance based fwL ; k L g, and a wage only one fwH g, designed in a way that the low motivated agents choose the former and the more motivated ones the latter. Before proceeding, it is important to notice that, in our simple set up– see (16) and (23)–the utility of the principal is linear homogeneous in T . This means that, if the principal knew the agents’ type, he would confer the entire production to the the motivated agents, and the …nal outcome would not depend on their number.16 From (13), in this case,17 the “optimal” contract would be: s T H (2 H) b = w : (39) p We now move to the separating contract that we denote by subscript S . Focusing our attention on linear L L H compensation contracts, the optimal separating contract fwS ; kS ; wS g is the solution of the following 1 6 Such a result, of course, depends on the very speci…c functional forms we selected, but the intuition we derive here is quite general. Indeed, the smaller is the number of committed agents, the more the principal is willing to pay them so that they would be in charge of a larger share of the production. 1 7 Using (29) and (32) and assuming y L (w ) = 0: U 12 problem: L L L L L L H H H H M ax VS = (1 e (wS ; kS )qS (wS ; kS ) + pyU p)yC ~ (wS ); ~ (wS )qU (40) L ;k L ;w H wS S S such that : L L L L L L L L H H T (1 p)(wS yC e (wS ; kS )) + pwS yU e (wS ; kS ) + kS qC ~ ; (41) H H H L L U (wS ) > U (wS ; kS ); (42) L L L L H U (wS ; kS ) > U (wS ); (43) where (42) and (43) represent incentive compatibility constraints of the more and less motivated agents, respectively. Of course, in this set-up, where the principal would like to pay the more motivated agents more (more so if their number is lower), the only binding incentive compatibility constraint is the one of the less motivated agent, and it is more binding the larger their number is. The solution of the problem is illustrated in Figure 7– while the utility level of the principal o¤ering a separating contract (vis-à-vis the pooling one) in Figure 6 above. From a simple inspection of the …gure, is it evident that, in order to satisfy the less motivated agents’ participation constraint, the best the principal can do is to reduce the wage it o¤ers to the motivated agent vis-à-vis the constrained optimal one given by (39). This distortion, which is equal to the di¤erence between the red dotted and the black line in Figure 7, decreases with p and tends to zero when the share of motivated agents tends to one. This also implies that the utility of the principal increases with p, but it is always higher than in the case of a pooling contract, see Figure 6. A the same time, the compensation of the less motivated agents increases with their number, as they have to make up for the lower production levels of the high motivated agents. 5 Conclusions Tight …scal constraints and increased awareness about “citizens’ rights” are pressing governments to …nd innovative solutions to reform the public administration, cutting on its costs and increasing its “value propo- sition.” Among these, the most popular one, advocated by NPM scholars— and NPM oriented consulting 13 …rms— is perhaps that of increasing investments in managerial and technological solutions and rationalize public sector organizations by increasing the accountability and transparency of their activities. The main driver of such an agenda is the belief that an increase in the measurability of public admin- istration activities allows one to address those incentive problems that contribute signi…cantly to its poor performance. To get a better understanding of whether (and under which conditions) this is indeed the case, this paper develops a simple model that looks at the cost and bene…ts of using monitoring technologies to design PFP in the presence of principal agents problems. Our analysis, establishing that the decision of whether to adopt, and how to design, PFP depends on the interaction between agents’ motivations and the quality of the available monitoring technology, warns against one-size-…ts-all solutions. More precisely, we show that managerial and technological solutions that allow measuring the e¤ort of poorly motivated agents at a reasonable cost, and paying them accordingly, are de…nitively part of the solution, as NPM advocates argue. However, we also show that these solutions become part of the problem, when the contribution of the di¤erent tasks to the creation of value is di¢ cult to measure and/or when agents are committed to the goals of the organization. In addition, our analysis contributes to the current debate by providing a framework that allows discussion of the impact that investments in monitoring technologies— such as those that drive many e-government projects— have on the e¤ectiveness of the PFP schemes and the associated trade-o¤s. These …ndings can help explaining the mixed results associated with NPM reforms, and they call for a more critical approach to the adoption of monitoring technologies which pays at least as much attention to agents’commitment to public service delivery as to the “measurability” of their daily activities. 14 References [1] Alford, J., and Hughes, O. (2008), “Public value pragmatism as the next phase of public management,” American Review of Public Administration, 38: 130-148. [2] Aral, S., Brynjolfsson, E., and Wu, L. (2012), “Three-way complementarities: performance pay, human resource analytics, and information technology,” Management Science, 58: 913-931 [3] Ba, S., Stallaert, J., and Whinston, A. B. (2001), “Research commentary: introducing a third dimension in information systems design— the case for incentive alignment,” Information Systems Research, 12: 225-239. [4] Behn, R.D. (1998), “The new public management paradigm and the search for government accountabil- ity,” International Public Management Journal, 2: 131-164. [5] Baker, G. (2002), “Distortion and risk in optimal incentive contracts,” Journal of Human Resources 37: 728-51. [6] Barzelay, M. (2001), The new public management: improving research and policy dialogue, University of California Press, Berkeley. [7] Benabou, R., and Tirole, J. (2003), “Intrinsic and extrinsic motivation,” The Review of Economic Studies, 70: 489-520. [8] Bertot, J. C., Jaeger, P. T., and Grimes, J. M. (2010), “Using ICTs to create a culture of transparency: E-government and social media as openness and anti-corruption tools for societies,” Government Infor- mation Quarterly, 27: 264-271. [9] Brynjolfsson, E., and Hitt, L. M. (2003), “Beyond computation: Information technology, organizational transformation and business performance,” The Journal of Economic Perspectives 14: 23-48. [10] Ciborra, C. U. (1996), Teams, markets and systems, Cambridge University Press. [11] Delfgaauw, J., and Dur, R. (2008), “Incentives and workers’ motivation in the public sector,” The Economic Journal, 118: 171-191. [12] Dixit, A., (2002), “Incentives and organizations in the public sector: an interpretative review,” Journal of Human Resources, 37: 696-727. [13] Dunleavy, P, and Carrera, L. (2013), Growing the productivity of government services, Edward Elgar Publishing. [14] Dunleavy, P., Margetts, H., Bastow, S., and Tinkler, J. (2005), “New public management is dead– long live digital-era governance,” Journal of Public Administration Research and Theory, 16: 467-94. [15] Ferlie, E., Ashburner, L., Fitzgerald, L., and Pettigrew, A. (1996), New public management in action, Oxford: Oxford University Press. [16] Francois, P. (2000), “Public service motivation as an argument for government provision,” Journal of Public Economics, 78: 275-299. [17] Frey, B. S., Homberg, F., and Osterloh, M. (2013), “Organizational control systems and pay-for- performance in the public service,” Organization Studies, 34: 949-72. [18] Garicano, L., and Heaton, P. (2010), “Information technology, organization, and productivity in the public sector: evidence from police departments,” Journal of Labor Economics, 28: 167-201. 15 [19] Grand, J. L. (2010), “Knights and knaves return: public service motivation and the delivery of public services,” International Public Management Journal, 13: 56-71. [20] Halac, M., and Prat, A. (2014), “Managerial attention and worker engagement,” CEPR Discussion Paper N. 10035. [21] Hasnain, Z., and Pierskalla H.N. (2012), “Performance-related pay in the public sector: a review of theory and evidence,” World Bank Policy Research Working Paper N. 6043. [22] Holmstrom, B., and Milgrom, P. (1991), “Multitask principal-agent analyses: incentive contracts, asset ownership, and job design,” Journal of Law, Economics, & Organization, 7: 24-52. [23] Kreps, D. M. (1997), “Intrinsic motivation and extrinsic incentives,” The American Economic Review, 87: 359-364. [24] Langbein, L. (2010), “Economics, public service motivation, and pay for performance: complements or substitutes?” International Public Management Journal, 13: 9-23. [25] Lazear, E. P. (2000), “Performance pay and productivity,” The American Economic Review, 90: 1346- 1361. [26] Moore, M. (1995), Creating public value: Strategic management in government, Cambridge, MA: Har- vard University Press. [27] Moynihan, D. P., and Pandey, S. K. (2007), “The role of organizations in fostering public service motivation,” Public Administration Review, 67: 40-53. [28] Moynihan, D. P., and Pandey, S. K. (2010), “The big question for performance management: why do managers use performance information?,” Journal of Public Administration Research and Theory, 20: 849-66. [29] OECD (2005), Performance-related pay policies for government employees, OECD Publishing. [30] Perry, J. L., Engbers, T. A., and Jun, S. Y. (2009), “Back to the future? Performance-related pay, empirical research, and the perils of persistence,” Public Administration Review, 69: 39-51. [31] Picot, A., Ripperger,T., and Wol¤, B. (1996), “The fading boundaries of the …rm: the role of information and communication technology,” Journal of Institutional and Theoretical Economics, 152: 65-79. [32] Pina, V., Torres, L., and Royo, S. (2007), “Are ICTs improving transparency and accountability in the EU regional and local governments? An empirical study,” Public Administration, 85: 449-472. [33] Prendergast, C. (1999), “The provision of incentives in …rms,” Journal of Economic Literature, 37: 7-63. [34] Proper, C., and Wilson, D. (2003), “The use and usefulness of performance measures,” Oxford Review of Economic Policy, 19: 250-67 [35] Weibel, A., Rost, K., and Osterloh, M. (2010), “Pay for performance in the public sector— Bene…ts and (hidden) costs,” Journal of Public Administration Research and Theory, 20: 387-412. 16 6 Appendix 6.1 Observable “quality” The problem of the agent is: M ax UO = qy + (wy kq ) (q + y )2 ; (44) y;q 2 with …rst order conditions: @UO = ( )q y + w = 0; (45) @y @UO = ( )y q + k = 0: (46) @q It is immediate to verify that k( )+w yO (w; k ) = ; (47) a) (2 w( )+k qO (w; k ) = ; (48) (2 ) @ 2 UO @ 2 UO @y 2 @y@q solve this system. Finally, if (A.1) holds, the Hessian matrix @ 2 UO @UO = is negative @q@y @q 2 de…nite, so that the second order conditions for a maximum are veri…ed. The principal’ s problem can now be written as: M ax VO = qO (w; k )yO (w; k ); such that: qO (w; k ) + yO (w; k ) T 0: (49) w;k; Substituting (47) and (48) into (49), the associated Lagrangian problem can be written as: (w( ) + k )(k ( )+w ) (a(2kw + T ) + ((k w)2 2T ) ) M ax$O = + : (50) w;k; ( 2 2 )2 ( 2 ) where denotes the multiplier associated with the budget constraint. The …rst order conditions of the problem are: 2 2 @ $O 2w ( (2 ) ) + k( 2 +2 2 (2 )( ) ) = 2( = 0; (51) @w 2 ) 2 2 @ $O 2k ( (2 ) ) + w( 2 +2 2 (2 )( ) ) = 2( = 0; (52) @k 2 ) @ $O (T + 2k! ) + ((k w)2 2T ) = = 0: (53) @ ( 2 ) It is easy to verify that: r T (2 ) wO = kO = ; (54) 2 1 O = ; (55) 2(2 ) solve this system. Substituting these values in (47) and (48), we have: 17 s T yO = q O = : (56) 2(2 ) Now, de…ning by gO , the budget constraint, (T + 2k! ) + ((k w )2 2T ) gO (qO (w; k ) + yO (w; k ) T) = ; (57) (2 ) @gO @gO 0 @k @w @gO @ 2 VO @ 2 VO we can check that the determinant of the bordered Hessian matrix @k @k2 @k@w is equal to @gO @ 2 VO @ 2 VO @w @w@k @w2 8kw 2 (2 )2 and is positive, which is a su¢ cient condition for an interior maximum. 6.2 Unobservable “quality” The problem of the agent can be written as: M axUU = qy + wy (q + y )2 ; (58) y;q 2 with …rst order conditions: @UU = (1 )q + y w = 0; (59) @y @UU = (1 a)y + q = 0: (60) @q It is easy to verify that w yU (w) = ; (61) (2 a) w( ) qU (w) = ; (62) (2 ) solve this system. The Hessian matrix is the same as in the previous case and is negative de…nite, so that the second order conditions for a maximum are veri…ed. The problem of the principal can now be written as: M axVU = qU (w)yU (w); such that: wyU (w) T; (63) w but, since the budget is given, and only y is contractible, the only option for the principal is to to set wN = T =E [yU (w)]: (64) Assuming rational expectations, that is, E [yU (w)] = yU (w), substituting (64) in (61) and solving we have that s T (2 ) wU = : (65) Substituting now this expression into using (61) and (62), we have that: p T yU = p ; (66) (2 ) p ( ) T qU = p ; (67) (2 ) T( ) VU = : (68) (2 ) 18 6.3 Costly observable “quality” The problem of the agent is: M ax UC = qy + (1 )(wy + k minfq; eg) (q + y )2 : (69) y;q;e 2 We should distinguish two cases: in the …rst e = q , in the second e < q . Case 1): e = q The problem of the agent can be written as: eC = qy + (1 q M ax U )(wy + kq g)) (q + y + )2 ; (70) y;q;e 2 with …rst order conditions: eP @U (1 + ) C = ( )q y + w = 0; (71) @y eP @U (1 + ) (1 + )2 C = ( )y + 2 q + k = 0; (72) @q which yield w(1 + )2 k ((1 + ) ) eC (w; k ) y = ; (73) ( (2 (1 + ) a ) k w ((1 + ) ) eC (w; k ) q = : (74) ( (2 (1 + ) a ) P @ 2 UA P @ 2 UA (1+ ) @y 2 @y@q Finally, the Hessian matrix P @ 2 UA @UA P = (1+ ) (1+ )2 is negative de…nite if its deter- 2 @q@y @q 2 minant ( 2(1+ ) ) is positive, which is always the case if condition (A.1) holds, so that the second order conditions for a maximum are veri…ed. Substituting (73) and (74) in (70), we obtain: 2 eC (w; k ) = 2kw + (w (w k ))2 U : (75) 2 (2(1 + ) ) The principal’s problem can now be written as: eC = q M ax V eC (w; k ); such that: (q eC (w; k )y eC (w; k ) eC (w; k ) + y T ): (76) w;k; Substituting (73) in (74), into (76), the associated Lagrangian problem can be written as: 2 ( (k w(1 + ) w )(k ( + (1 + )(w kb + wb) ) M ax$C = + w;k; 2 (k 2 + w2 (1 + )2 + T + 2kw ( (1 + ) ) + ; with …rst order conditions: 19 @$eC w(1 + )2 +k = 2 = 0; (77) @w @$eC k +w = 2 = 0; (78) @k eC @$ k2 2 + w2 (1 + )2 + T ( 2(1 + ) ) + 2kw ( (1 + ) = ; (79) @ where is the Lagrange multiplier associated with the budget constraint and 2 ((2(1 + ) ) + (1 + ) ); 2(1 + )2 2 + 2 3 3 + 2 ( 6(1 + ) ) + 2 (1 + ) (2(1 + ) ): ( 2(1 + ) )2 . It is then possible to verify that s T (2(1 + ) ) eC w = ; (80) 2(1 + ) s e T (1 + )(2(1 + ) ) kC = ; (81) 2 2 e = ; (82) C 4(1 + ) 2 solve the system (77)-(79). Substituting these values into (73), (74) and (76) we obtain: s (1 + )T eC = y ; (83) 2(2(1 + ) ) s T (2(1 + ) eC = q ; (84) 2(1 + )(2(1 + ) ) eC T V = : (85) 2(2(1 + ) ) Now, de…ning by gC , the budget constraint, 2 (w2 (1 + )2 + k 2 2k! ((1 + ) ) T (2(1 + ) ) gC (qC (w; k ) + yC (w; k ) T) = ; (2(1 + ) ) (86) @gC @gC 0 @k @w @gC @2VeC @2VeC we can check that the determinant of the bordered Hessian matrix @k @w2 @w@k is equal to @gC @2VeC @VeC @w @k@w @w2 2 8kw 2 (2(1+ ) )2 and is positive, which is a su¢ cient condition for an interior maximum. Case 2): e < q s problem is: The agent’ bC = qy + (1 M ax U )(wy + k e) (q + y + e)2 ; (87) y;q;e 2 20 and the necessary and su¢ cient …rst order conditions for a maximum are: @UbC = ( )q (y + e) + w = 0; (88) @y @UbC = ( )y (q + e) + k = 0; (89) @q @UbC = k (y + q + e) + k = 0; (90) @e which yield k bC (w; k ) y = ; (91) k w bC (w; k ) q = ; (92) w (2 )k bC (w; k ) e = : (93) Substituting(91)-(93) into (87), we obtain b (w; k ) = k (2w U (2 )k ) : (94) C 2 Using (75) and (94), we have that eC (w; k ) bC (w; k ) = (w(1 + ) + k ( (1 + 2 ) )2 U U ; 2 (2(1 + ) ) and thus that b (w; k ) () 2( 1 + 1) > ; e (w; k ) > U UC C which is always veri…ed if (A.1) holds. This, in turns, implies that (83)-(85) is the solution of the problem. It now remains to verify that the principal is better o¤ by o¤ering a compensation scheme fweC ; e kC g than by linking the compensation only to y , as in the non observable case. Using (68) and (85), we have that: eC ( ) V VU = T ; (2 ) 2(2(1 + ) ) p eC 4(a ) 2 ((1 + ) 1+ ) V VU > 0, > , or > : (2 )2 This, in turn implies that the solution of the problem is given by: q q T wC = 2(1+ ); wC = wU = T (2 ) ; q T (1+ ) kC = 2 ; kC = 0;p q2 (1+ ) if > 4( ) yC = yU = p T ; yC = T 2 ; (2 )2 ; p (2 ) if < ; p T 2 ((1+ ) 1+ ) ( ) T or > ; qC = p ; or < ; qC = qU = p ; 2(1+ ) (2 ) eC = p T ; eC = 0; 2(1+ ) VC = T VC P = VU = T(2( ) ); 2 ; 21 p 2 ((1+ ) 1+ ) 4( ) where C , (2 )2 , and 2(1 + ) . Finally, noticing that > =) > 0;it is immediate to verify that: @wC T @ = p < 0; 2 2T (1+ ) @wC T @ = 3p < 0; 2(1+p ) 2 2T @kC T (1+ ) @wC @ = p 2 2T < 0; @ = p T( ) < 0; @kC T (4(1+ ) ) T (2 ) = p < 0; @wC @ 2 2 2T (1+ ) p @ = 0; @yC T2 (1+ ) @yC T2 ( ) @ = (2T )3=2 > 0; if > ; @ = (T (2 ))3=2 > 0; if < ; @yC T or > ; @yC or < : @ = 3=2 p > 0; @ =0 2 2(1+ )T @qC 2 3 @qC T 2p 2 T 2 = > 0; @ = (T (2 ))3=2 > 0; @ 2(T )3=2 2(1+ ) @qC ; @qC =0 @ = 2T p (4(1+ ) 2T ((1+ ) )3=2 ) > 0; @ @eC T2 p @ = > 0; 2(T )3=2 2(1+ ) @eC @ = T (4(1+ p ) (1+2 ) 2 2T ((1+ ) )3=2 < 0; 6.4 Investing in technology The problem of the principal is to M ax VT = qC (w; k )yC (w; k ) (wyC (w; k ) + kqC (w; k ) T ); (95) w;k; ; where qC (w; k ), yC (w; k ) are given by (73)-(74). The …rst order conditions of the problem are quite cum- bersome and available upon request (as the details for this section). Using Mathematica c , we can show that: p (2( + T ) T )(2( + T ) + ' wT = p ; (96) 2+ 4( + T ) p (( + T ) + ') (' 2( + T ))(T 2( + T ) kT = p ; (97) 2T 2 + 4( + T ) 2 ' T = ; (98) ( 2 ) solve the system of …rst order conditions.18 Substituting these values into the principal’ s objective function, we get: 4 T( 2 ) ' VT = : (99) 2( 2 )2 Comparing now this solution with the solution for the unobservable case, (68), we have that T( 2 )2 + 4 2 VCT VU = ; (100) 2( 2 )2 1 8 Whilewe have not been able to verify analytically that the second order conditions for a maximum are veri…ed, through numerical simulations we con…rmed that that they are for the parametrization we use in Figures 3-5. 22 and T (2 )3 VCT VU > 0 () < : (101) 16 ( ) In addition, @ T (2 )2 (2 2 2 2 ) p = <0, >( 3 1) @ 16 2 ( )2 condition that is always veri…ed since > : p p @ ( 2(2(2 +T ) T 4 2( +T ) T ) Finally, to prove Result 4, it is enough to show that @ t = 2 p > 0, which 2(a 2 ) (2( +T ) T ) is always the case. 23