UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN BOOKSTACKS B CENTRAL CIRCULATION BOOKSTACKS The person charging this material is re- sponsible for its renewal or its return to the library from which it was borrowed on or before the Latest Date stamped below. You may be charged a minimum fee of $75.00 for each lost book. Theft, mutilation, and underlining of books ore reasons for disciplinary action and may result in dismissal from the University. TO RENEW CALL TELEPHONE CENTER, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA CHAMPAIGN APR 2 9 1998 AUG 3 1998 When renewing by phone, write new due date below previous due date. L162 330 STX B385 1417 COPY 2 ^TlRtoJ BEBR FACULTY WORKING PAPER NO. 1417 THE LIBRARY OF THE APR ? 7 1983 LLINOJS Monotonicity of Second-Best Optimal Contracts George E. Monahan Vijay K. Vemuri WORKING PAPER SERIES ON THE POLITICAL ECONOMY OF INSTITUTIONS NO. 3 College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois, Urbana-Champaign Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/monotonicityofse1417mona BEBR FACULTY WORKING PAPER NO. 1417 College of Commerce and Business Administration University of Illinois at Urb ana -Champaign December 1987 WORKING PAPER SERIES ON THE POLITICAL ECONOMY OF INSTITUTIONS NO. 3 Monotonicity of Second-Best Optimal Contracts George E.Monahan, Associate Professor Department of Business Administration Vijay K. Vemuri, Graduate Student Department of Accountancy MONOTONICITY OF SECOND-BEST OPTIMAL CONTRACTS George E. Monahan Vijay K. Vemuri November, 1987 College of Commerce and Business Administration University of Illinois at Urb ana- Champaign 1206 South Sixth Street Champaign, Illinois 61820 We establish the monotonicity of second-best optimal contracts in the cost-benefit principal-agent model developed by Grossman and Hart (G-H). For a three-state, finite- action version of the model, we prove that the monotone likelihood ratio condition (MLRC) on the probabilities of outcomes guarantees that the optimal payment to the agent is monotonically increasing in output. We estabish a key result that exposes an error in an example that cause G-H to impose conditions stronger than MLRC in order to obtain monotonicity. This result is instrumental in our proof of the monotonicity of second-best optimal incentive schemes. Key Words: Principal-Agent, Monotonicity, Incentive Schemes, Monotone Likelihood Ratio Condition. MONOTONICITY OF SECOND-BEST OPTIMAL CONTRACTS By George E. Monahan and Vijay K. Vemuri 1. INTRODUCTION In a seminal paper, Grossman and Hart (hereafter denoted G-H)(1983) introduce the cost-benefit version of the principal-agent model and derive many interesting properties of optimal sharing rules. Some of their analysis relates to the monotonicity of second- best optimal sharing rules. In "first-order" principal-agent models, a "standard" condition guaranteeing that the optimal payment to the agent is monotonically increasing in output is the Monotone Likelihood Ratio Condition (MLRC). MLRC places restrictions on the probabilities of outcomes specified as functions of the agent's action. In first-order mod- els, the agents choose from a continuum of actions. To assure uniqueness of the action that solves the agent's incentive compatibility condition, a second requirement, called the Convex Distribution Function Condition (CDFC), is also placed on the output probabil- ities. See Rogerson (1985) for a discussion of first-order agency models. Milgrom (1981) discusses the general issues associated with monotonicity in contractual settings. Using a three-state, three-action numerical example, G-H claim to demonstrate that MLRC alone is not sufficient for monotonicity in their cost-benefit model. To establish monotonicity, they invoke fairly strong conditions. In this paper, we show that the G-H example contains a flaw that reopens the issue of the sufficiency of MLRC. Could a more judicious choice of parameter values rectify the example? One contribution of this paper is to prove that in the three-state version of the cost-benefit model, the answer to this question is no. The result we use to investigate the numerical example also plays a pivotal role in the proof that MLRC alone is indeed sufficient for monotonicity of second-best optimal contracts. Our analysis is done entirely within the general cost-benefit framework of G-H. We assume, however, that there are only three underlying states of the world. We do permit any finite number of actions are available to the agent. Even with the restriction to three states, the monotonicity result is surprisingly difficult to establish. Indeed, the entire paper is devoted to this single task. Our result suggests that MLRC is sufficient for monotonicity in the general, finite-state cost-benefit model. We are currently exploring ways in which the results established here can be used to prove this conjecture. The paper is organized as follows. We briefly review the cost-benefit model anc introduce the bulk of the notation in the next section. In Section 3, we discuss MLRC and the G-H numerical example. Next, we present and prove a result that illuminates th< fallacy in the G-H example. Section 5 contains preliminary results related to monotonicity Finally, Section 6 contains the proof that in the finite-action, three-state cost- benefit model, MLRC does indeed imply the monotonicity of second-best optimal contracts. A tedious proof of an intermediate result is relegated to an Appendix. 2. THE COST-BENEFIT PRINCIPAL-AGENT MODEL We briefly review the cost-benefit model, adopting the notation of G-H (1983). See G-H for complete details. Notation: q it i = l,...,n A = {a 1 ,...,a m } Ii U(a,I) = G{a) + K{a)V{I) u Real-valued outcomes ordered so that q x G{a 2 ) + K( G( ai ) +K{a l )V(I) > Gia?) + ^(oajVf/). 2. [U - G[a]]/K[a) eU = {v\v = V(I)} for some / G I for all a € A 3. For all a G A and i = 1, . . . , n, 7r< (a) > 0. 4. The principal is assumed to be risk neutral. Assumption 1 implies that the agent's utility of income be risk independent of action. The functional form of the utility function is more general than the additively separable utility functions assumed in much of the agency literature, and admits additively and multiplicatively separable functions as special cases. Assumption 3 eliminates the ability to infer actions from outcomes, thus establishing a "real" moral hazard problem. Assumption 4 follows the preponderance of the analysis in G-H. Risk-neutrality of the principal avoids having to make utility comparisons between the principal and agent. It also permits the principal-agent problem into two parts. First, the principal determines the least (expected) cost way of implementing each of the m actions in A. An output- based incentive scheme I Xi ... i I n is said to implement action a* £ A if I x , . . . , I n solve the following mathematical programming problem: Program 1: n min >^ -Ki{a)Ii »=i s.t. 2_\ TTi {&* )U (a* , Ii) > U (individual rationality) »= i n ^TTiia^UiaJt) > ^^(ajt/fa,/,) for all a € A. (incentive compatibility) Given the incentive schemes for implementing each action, the second phase of the cost-benefit approach is entered. The principal chooses the action that maximizes his/her net expected payoff. The optimal action is a solution to the following mathematical pro- gram: Program 2: maxf BfaJ-^TT^a)/, j . If a* G A is a solution to Program 2 and I lim .. t I n implements action a* , then /i, . . . , I„ is called a second-best optimal incentive scheme. Our objective is to identify conditions under which a second-best optimal incentive scheme is nondecreasing in output, or equivalently (given the ordering on output), is nondecreasing in t,t = 1, . . . ,n. Since V(-) is assumed to be both strictly increasing and strictly concave, h(-) is strictly increasing and strictly convex. Let v, = V(J<), for t = l,...,n. Given the assumptions on U(-, •), an important feature of the cost-benefit approach is that Program 1, with decision variables I x , . . . , /„ , can be written as an equivalent mathematical program with decision variables v x , . . . , v n . The new program has a convex objective function and a finite number of constraints that are each linear in the v^'s. To denote the explicit dependence of Program 1 on a* G A, we write the equivalent program as: Program l(a*): C(a') = min V 3r<(a*)fc(v<) • l, ...,»■ »'= 1 s.t. G(a* ) + if (a* ) ^ 7r € (a* ) Vi > U n n G(a*) + K{a)J2*ii a *) v i > G (°) + K(a) ^^(a)Vi for all a € A. »=1 i=X The Kuhn-Tucker conditions for Program l(a*) include the following: (1) h'M = a+ Yl ** aj€A _» m K(a')- Y, * K M aj€A _s • .7r t (a*) for all t, where A, \t x , . . . , p m are nonnegative Lagrange multipliers. 3. MLRC AND THE G-H NUMERICAL EXAMPLE There are two requirements for establishing the monotonicity of second-best optimal incentive schemes: the incentive scheme associated with some action must be nondecreasing in the state-descriptor and that action must be selected by the principal. Suppose the X and Y are nonnegative random variables with densities / and g, respectively. These random variables are said to have the Monotone Likelihood Ratio (MLR) property if f(x)/g(x) is nondecreasing in x. In the principal-agent context, MLRC holds if, for a, a' G A, a' •< a implies that w i (a')/ir i (a) is nonincreasing in i, where u < n denotes a complete ordering on A defined as: a' ■< a if and only if C FB (a 1 ) < C FB (a). Since the state-descriptor i enters the optimality conditions (l) only through 7r,-(ay)/7rj(a*), monotonicity of the incentive scheme that implements some action a* € A follows if 7r, (ay)/7Tj (a*) is decreasing in t for all actions a i for which /Zy > 0. If \ij is positive for some action a, that is "larger" than a*, 7r i (a,j)/'K i (a*) is increasing in i, thus confounding the demonstration that MLRC is sufficient for the monotonicity of second- best optimal sharing rules. In an attempt to demonstrate what can go wrong, G-H present the following numerical example: n = 3, m = 3, V(I) = (37) 1/3 , h{v) = -v\ Also U = ->/2+ — \/-, and Kia) = 1 for all a. 4 12V 4 v } i = l t' = 2 t' = 3 G(a 3 ) (2) ill "^333 -113 3 12 4 3 a, 2. i _i_ o 1 3 4 12 -y/2 _ i /l 12 4 V 4 -7 fr It is clear from (2), that MLRC holds. The solution to Program 1(^2) is: Vl =0, w 3 =\/2, v 3 = J 7 -, X = 1.25, fi x = 2.0, ^3 = 1.0, and C^) = 0.571. Also, C{a l ) = 0.0333 and C(a 3 ) = 0.7432. Notice that v 2 > v 3 so the incentive scheme that implements a? is not monotone. Notice also, that n x > and n 3 > 0, implying that the agent is indifferent between taking action a? and both actions a : and a 3 . G-H (page 24) remark, "The reason monotonicity breaks down ... is because, at the optimum 6 [in Program 1(02)], the agent is indifferent between a?, the action to be implemented, ai, a less costly action, and a 3 , a more costly action." The example is complete if it can be shown that it is optimal for the principal to choose action a 2 . G-H state (page 24) that u . . . it is easy to show that we can find q 1 < q 2 < (73 , such that (2a) 5(02) - 0(0*) > max{£(a 3 ) - C{a 3 ), 5( ai ) - C(a 1 )} n leading them to conclude that a . . . the optimal incentive scheme ... is not nondecreasing despite the satisfaction of MLRC." In the next section we demonstrate that this final claim is false. 4. A SUFFICIENT CONDITION Within the context of the numerical example in the previous section, the problem of finding q x < q 2 < q 3 that guarantee that a? will be optimal for the principal is expressed as the following problem: Find numbers q x , q 2 , q 3 that satisfy the following system of linear inequalities: 3 3 53 ^W* ~C( 53 ^W* ~C(a 3 ) »=i »=i 3 3 (3) 22 m (a^ )q 4 - C{o2 ) > 2J 7T, («i )ft ~ C(a x ) »=i »=i q-i > qi ?2 • Notice that in (3) tt >" replace a >" since any solution to (2a) necessarily satisfies (3). Our first result identifies sufficient conditions for there not to be a solution to a particular system of linear inequalities. We then show that the conditions are satisfied in the G-H example. Consider the following system of linear inequalities: 53 a »fc > A i i= 1 3 53 Aft >a 2 (4) 92 ~ Qi > ?3 - Q2 > q it i = 1, 2, 3 unconstrained in sign. We assume that the parameters A h A 2 , a*, ft, t = 1, 2, 3 satisfy: a x < a 2 < a 3 , ft > ft >ft, a x < 0, a 3 > 0, ft > 0, ft < 0, A 2 < and 53°^ = 53& =0. »=i »=i PROPOSITION 1. Suppose that (5) holds. If A ly /A 2 < min{a 1 /ft, a 3 /ft}, then there is no solution to (4). PROOF: Using Gale's Theorem of the Alternative [see, e.g., Mangasarian (1969), page 33], (4) does not have a solution if there are variables y x , y 2 , y 3 , y* that are each non-negative and that are all not zero, that satisfy the system of equations (6) where Ay=b, o=i 01 -1 °\ ~yi' -0- «2 (*3 ft 03 1 " ! y = y 2 V3 and b = At A 2 o7 -y 4 - .1. A = We will show that under the hypotheses, (6) always has a solution so that (4) does not. Since X^,= i <*» = ^,= 1 0* — 0> ^ ne sum °f the ^ TS ^ three rows of A is zero, and the rank of A is less than four. It is straightforward to determine that the rank of A is three and that there are an infinite number of solutions to (6). The general form of the solution is: y 2 ya y 4 = t (i-a 1 0/a 2 [ft -(A x ft -A 3ttl )t]/A a [-ft +(A x ft -A 2 a 3 )*]/A 2 where £ is any real number. Since ft > and A 2 < 0, y 3 can be positive only when A x ft > A 2 a x or, equivalently, when (?) Similarly, y 4 can be positive only when (8) A x ai A 2 ft Ai a 3 A 2 ft" If (7) and (8) are both satisfied, then there is a non-negative solution to (6) with at least one Vi > and there is no solution to (4). Q.E.D. We now use Proposition 1 to prove that within the context of the G-H example, there is no solution to (3), implying that action a? will not be chosen by the principal. Let (9) ai = 7^(02) - iti{a x ) and ft = ^(a?) - 7i\(a 3 ), i = 1, 2, 3. 9 It is straightforward to show that MLRC implies that these a's and /?'s satisfy (5). Let (10) Ax = 0(0-2) - C( ai ) and A 2 = C{a?) - C(a 3 ). Note that in the example, A 2 = 0.571 - 0.743 = -0.172 < and Aj = 0.571 - 0.033 = 0.538 > 0, as required. In the numerical example, a x /& = (1/3 - 2/3)/(l/3 - 1/12) = -4/3, a 3 / (3 3 = (1/3 - l/12)/(l/3 - 2/3) = -3/4. Since A x / A a = 0.538/ - 0.172 = —3.128 < —4/3 = min{a! / lt a 3 j /? 3 }, the principal would never pick action 02 in the G-H example. We will now show that this example cannot be rectified by a more judicious choice of parameter values. 10 5. MONOTONICITY: SOME PRELIMINARY RESULTS Denote the matrix of the probabilities of outcomes as functions of action as: (11) <*i Pi P2 P 3 Ch ?4 Ps Pe a 3 Pr Ps Ps )wi: ag relations hold: a. Pa - Pi < o e. Pi/ Pa > Pi/Ps b. p 4 - p 7 < f. Pi /Ps > Pz /Pe c. p 6 - p 3 < g. Pa/ Pi > Pb/Ps d. p 6 - p 9 < h. Ps /Pa > Pe /Pa (12) Without loss of generality, number the actions so that a x < a? < a 3 ; i.e., C FB (ai) < C FB (a^) < C FB (^3). Since a x is the least-costly action, it follows that C(a 1 ) = h[(U — G(a 1 )/K(a 1 )] = C FB (%). (Action a x can be implemented be setting 7, = C FB {a y ) for all It is easy to show that an optimal incentive scheme is monotone if C(a 3 ) < C(a 2 ). The principal prefers action a 3 to action a^, since MLRC implies that £(a t ), the gross expected benefit accruing to the principal, is nondecreasing in z. MLRC also implies that 7r, (a^/iTi (a 3 ) is decreasing in i for j = 1,2, and 3. Therefore, the incentive scheme that implements a 3 is monotonically increasing in output. Since the incentive scheme that implements action o x , the only other action that could possibly be second-best optimal, is constant, it also is monotonic. In light of the discussion above, if there is a non-monotone sharing rule, it is when a 2 is a second-best optimal action. From now on we will concentrate on the optimal incentive scheme that implements action a?. Let v* = {v[,v*,vl) denote such an incentive scheme; i.e., v* is a solution to Program 1(02). The incentive scheme v* is non-monotone if h'[v*) > h'(vl) for k > i. Suppose h'{v[) > h'(v*), which, in light of (l), implies that w> E**(%>3g}<£**(%>S|3$- y=i,3 v ' y=i,3 v ' Let n'. = HjKfo), j = 1,2,3. Using (11), write (13) as: (ia\ 1 P 1 , P? ^ , P2 , , Ps (14) n\ — -I- /z' 3 — < p' — + n' 3 —. P* Pa Pb Pb 11 The only other possible non-monotonicity is ^'(t;^) > h'(v^), which can be written as: (15) , Pi . , Pa . , Pa , Pq Mx — + M 3 — < Mi — + M 3 — • Pb Pb Pe Pe In summary, the optimal sharing rule is non-monotone if, and only if, either (14) or (15) hold. It will be convenient to write (14) in the following form: / Pi p 2 " - 1 Pa Pt" Ml — h'(v^) > h'{v^), which can never be optimal by Proposition 5 in G-H (which states that an optimal second-best incentive scheme cannot be nonincreasing in output for all states). If 7 < \i\f\i\ < 6, then h'(vl) < fc'(w') and h'(v* 2 ) > fc'(t/*). Finally, if 7 < n'Jn' 2 , then k'(v*) < V(w*) < h'(vl). Analogous statements hold when 6 > 7. Note that since h(-) is strictly increasing, h'{v') < (>) h'(v' k ) is equivalent to h{v') - h{v' k ) < (>)0. Let (16) x = h{vl) - h(v' 2 ), y = h(v;) - h{*l) % 9 = min{7,6}, = max{7, £}, and We summarize the discussion above in tabular form as follows: r <6 d6 (17) 7 < 6 NA + x < 0, y < x < 0, y >0 7 > 6 NA + x > 0, y > x < 0, y > + "NA" means tt ruled out by Proposition 5, G-H" 12 The next result provides an important linkage to Proposition 1, the sufficiency condi- tion in Section 4. We use this result to reduce the number of cases that must be examined in the proof of the main result given in the next section. Proposition 2. IfaJ^ < (>) a 3 //? 3 , then 7 < (>)<5. PROOF: Deferred to the Appendix. We are now in position to state and prove the main result of this paper. 13 6. THE MAIN RESULT In this section we prove that MLRC implies monotonicity of second-best optimal incentive schemes in the three-state cost-benefit model. The proof uses ideas developed in the previous sections: we show that if, for some action there is an incentive scheme that is not monotonic, then it will never be optimal for the principal to choose that action. In light of the discussion in Section 5, non-monotonicity can only occur if 0(02) < C(a 3 ). Therefore, we assume that this condition holds and examine the case where (18) Ai > and A 2 < 0, with A x and A 2 as defined in (10). We now state the main result of this paper. THEOREM. In the three-state, m-action cost-beneGt principal-agent model, MLRC im- plies Ii < I 2 < I 3 , where 7 X , I 2 , h is a second-best optimal incentive scheme. PROOF: Let v* = (v* , v\ , v* z ) and w* = (w{ , w^ , u> 3 ) denote optimal solutions to Program l(a 2 ) and Program l(a 3 ), respectively. Suppose that the incentive scheme that implements action a? is not monotonic. CLAIM, v* is feasible in Program l(a 3 ). PROOF OF CLAIM: The constraints of Program 1(02) are: 3 (19) GM + KMj^ViMvi >U 3 3 (20) G(a 2 ) + Kfa) J2*GM+K M ^2*iM*i 3 3 (21) GM + KM^Vifafa >G(a 3 ) + K{a 3 )J2*iM v i and the constraints of Program l(a 3 ) are: 3 (22) G(a 3 ) + tf(a 3 ) 5^(05 )u/< > U t=i 3 3 (23) G(a 3 ) + K{a z ) £ jr< (a 3 )w t > G{a, ) + K(a x ) £ tt, (a, )w t 3 3 (24) G(a 3 ) + tf(a 3 )£ 7^(03)^ > G^) + tf^) ^.(a,)™, ,. »=i »=i 14 Since v* is not monotonic, it follows that (21) also holds as an equality when v = v* . (MLRC implies that 7r t (c^ )/7r» (a? ) is decreasing in i and that 7i\ (a 3 J/^ (a? ) is increasing in i. The only way in which v* can not be monotonic is if /x 3 , the coefficient of 7r i (a 3 )/7r i {a 2 ) is positive, which, by complementary slackness, implies that (21) holds as an equality.) Since (21) holds as an equality, (24) is obviously satisfied, (22) follows from (19), and (23) follows from (22). Q.E.D From (10), Ai __ V i=l *i( a x j^ x . Then (/?i a 3 — a x ^ 3 )y < 0. Since y < 0, this implies that flia 3 — a l (3 3 > 0, which in turn implies that a 3 //? 3 < tti//?i 5 a contradiction. Therefore, using (26), A 1 /A 2 < iE < minfc*!//?! , a 3 //? 3 } and so by Proposition 1, (3) has no solution. Case 2: c^/ft > a 3 //? 3 . Again using Proposition 2, the only possible non-monotonicity is x > and y > 0. Suppose R > a 3 /fi 3 . Then {(3 3 a x — a. 3 (3i) > and that (3 3 a x — a 3 /?i > 0. Since a; > 0, this implies that a 1 /(3 1 < o: 3 //? 3 , which again is a contradiction. As in Case 1, A x /A 2 < R < minfai//?!, a 3 /(3 3 } and again invoking Proposition 1, (3) has no solution. Q.E.D. The proof does not hinge on the fact that we only consider three actions. Suppose there are m > 3 actions. Let a* denote some action for which there is a non-monotone incentive scheme. Let a { denote an action that is "smaller" than a* and has the property that the agent is indifferent between e^ and a* in Program l(a*); if a* is the least-costly action, the incentive scheme that implements a* is constant with respect to i and hence is monotone. Let a } denote any action that is "larger" than a* and also has the property that the agent is indifferent between a 3 and a* in Program l( by (12d) 6 = p 6 - p 3 > by (12c) c — P4P9 — PeP7 > by (12g) and (12h) ^ = PiPe — P3P4 > by (12e) and (I2f) e = Pa ~ Pi > by (12b) / = Pi - P 4 by (12a). Note that in the derivation leading to (A2), 6 — d = p y p b — p 2 p 4 > by (12e) and that f - d = p 2 p 6 - p z p b > by (I2f). We use the following results. LEMMA 1. Suppose that a',6',c',<2',e', and /' are positive numbers such that (i) a' /&' > c'/d' > e'/f, (ii) b' > d', and (iii) /' > d'. Then (A4) o' - c' € - c' If -d' > f -d! PROOF: Multiply both sides of a'/b' > c'/d' by b'/c' to obtain a'/c' > b'/d'. Therefore, (o' - c')/c' > (6' - d')d', which implies that (a' - c')/(b' - d') > c'/d', since V - d' > 0. Similarly, c'/d' > e'/f implies that e'/f > f'/d', so that (e' - c')/c' < (/' - d')d'. Therefore, (e' - c')/(f - d') < c'/d', since f - d' > and (A4) is obtained. Q.E.D. LEMMA 2. If o",6",c", and d" are positive numbers such that a"/b" > c" /d", then a"/b" > {a" + c")/(b" + d") > c"/d". PROOF: Similar to the proof of Lemma 1 and is omitted. 18 Note that with the definitions in (A3), (A2) becomes sgn(7 - 6) = -sgn a — c e — c b-d f-d The definition of the primed variables in Lemma 1 depend upon the sign of (o^ //? x — a 3 //3 3 ). We assume the following: 1. Ifajfii a 3 //? 3 , then a' = e, b' = /, c' = c, d' = d, e' = a, f = b. In the inequalities that follow, the first relation holds under Condition 1, while the paren- thetical relation prevails under Condition 2. Using (A3), hypothesis (i) of Lemma 1 is (AS) ?L^Pl > (<) ** - *«> > (<) *Z». Pe ~ Pz PaPz "PiPe Pi — Pi From the comment following (A3), hypotheses (ii) and (iii) of Lemma 1 are satisfied. If we show (A5) holds, Lemma 1 implies that sgn{^ — 6) = —sgn(a 1 //3 i — a 3 //? 3 ), completing the proof. The hypothesis that ai//?i < (>) a 3 /(3 3 implies that Pi (Po ~ Pe ) / % Pa(P4 - Pi) Pi(Pe ~ Pz) Pa(Pi -P4) or, invoking Lemma 2, ( A6) Pa ~Pe / x P1P9 ~ PiPe + P3P4 - P3P7 Pe - Pz PiPe - P1P3 + P1P3 - P3P4 Now (p 9 - p 6 )/(p 6 - Pz) > (<) (P4 - Pt)/(Pi - P4) implies that ( A7 ) P1P9 "PiPe +P3P4 -P3P7 > (<)p 4 P9 -PeP7- Substituting (A7) into (A6), yields P9 - Pe / v P4P9 ~PeP7 __ PeP7 -P4P9 Pe~P3 P1P6-P3P4 P3P4-P1P6 and the first inequality in (A5) is satisfied. Using a similar argument, a x jp x < (>) a 3 /fi 3 implies that P?(P9 ~Pe) / x P 9 (P4 -P7) P 7 (Pe -Ps) P 9 (Pi ~P 4 ) 19 or / Ag x PtPq - PrPe , * P9P4 "PoP? P7P8 - P7P3 P9P1 - P9P4 Since the numerator and denominator in (A8) are both positive, it follows from Lemma 2 that /.gN P4P9 -P6P? / x P9P4 -P9P7 __ P4 - Pi P?Pg - P7P3 + P9P1 - P9P4 P9P1 - P9P4 Pi - P4 From (A7), we see that the denominator of the left-side of (A9) is greater than (less than) Pi Pe - Pa P4 • Therefore, (A10) P

(<) ?^i. Pi Pe - P3 P4 Pi - P4 Multiplying the top and bottom of left-side of (A10) yields the second inequality in (A5). Q.E.D. 20 Papers in the Political Economy of Institutions Series No. 1 Susan I. Cohen. "Pareto Optimality and Bidding for Contracts." Working Paper # 1411 No. 2 Jan K. Brueckner and Kangoh Lee. "Spatially-Limited Altruism, Mixed Clubs, and Local Income Redistribution." Working Paper #1406 No. 3 George E. Monahan and Vijay K. Vemuri. "Monotonicity of Second-Best Optimal Contracts." Working Paper #1417 HECKMAN BINDERY INC. JUN95 « a t„ pi..,^ N. MANCHESTER, I Bound -To -Hum | ND | ANA 46962 J