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X(# a2Right ParRight-Aligned Paragraph Numbers C @` A. ` ` (#` a3DocumentgDocument Style Style B b  ?  1.  a3Right ParRight-Aligned Paragraph Numbers L! ` ` @P 1. ` `  (# 2    I a4Right ParRight-Aligned Paragraph Numbers Uj` `  @ a. ` (# a5Right ParRight-Aligned Paragraph Numbers _o` `  @h(1)  hh#(#h a6Right ParRight-Aligned Paragraph Numbersh` `  hh#@$(a) hh#((# a7Right ParRight-Aligned Paragraph NumberspfJ` `  hh#(@*i) (h-(# 2K $ + a8Right ParRight-Aligned Paragraph NumbersyW"3!` `  hh#(-@p/a) -pp2(#p a1DocumentgDocument Style StyleXqq   l ^) I. ׃  Doc InitInitialize Document Style  0*0*  I. A. 1. a.(1)(a) i) a) I. 1. A. a.(1)(a) i) a)DocumentgTech InitInitialize Technical Style. k I. A. 1. a.(1)(a) i) a) 1 .1 .1 .1 .1 .1 .1 .1 Technical2M[ a5TechnicalTechnical Document Style)WD (1) . a6TechnicalTechnical Document Style)D (a) . a2TechnicalTechnical Document Style<6  ?  A.   a3TechnicalTechnical Document Style9Wg  2  1.   2VJa4TechnicalTechnical Document Style8bv{ 2  a.   a1TechnicalTechnical Document StyleF!<  ?  I.   a7TechnicalTechnical Document Style(@D i) . a8TechnicalTechnical Document Style(D a) . 2<WX9cPleadingHeader for numbered pleading paperP@n   $] X X` hp x (#%'0*,.8135@8:{C(0)=0; C is continuous, increasing and convex} We consider two domains of individual preferences. The  _ (classical) domain  D  contains preferences defined over r0 __dd<<[0, omega`]x  _+x6X@8;X@x6X@8;X@x6X@8;X@_[_0_,-_]z_3_x_+r  ?  X where !0"88dd<<tomegax6X@8;X@x6X@8;X@x6X@8;X@83t is this agent's endowment of input (different agents may have different endowments of input; unbounded endowments are  ? possible). Preferences in A0X&88dd<<oDx6X@8;X@x6X@8;X@x6X@8;X@8Do are non increasing in the input share x, nondecreasing in the output share y, locally nonsatiated, convex and continuous.  ?$ Our two Theorems use the subdomain B of D  containing the  ?(& binormal preferences. Preferences are binormal  if they satisfy the(&0*((@@  _ following property: if Ta0@__dd << (x_0,y_0)x6X@8;X@x6X@8;X@x6X@8;X@_(+0R_,B+0_)_x_yT is in the demand set of the budget line px+qy  r (where p,q, are non negative and not both zero), and if r'>r, then the demand set on the budget line px+qyr'  _  contains at least a point T0J __dd << (x_1,y_1)x6X@8;X@x6X@8;X@x6X@8;X@_(+1R_,B+1_)_x_yT such that 90Y%J __Ydd << x_1` `x_0x6X@8;X@x6X@8;X@x6X@8;X@_x~_x+1+0_9 and 80-J __-dd << y_0  y_1x6X@8;X@x6X@8;X@x6X@8;X@_yR_y+0+1_8. An equivalent formulation of binormality is useful in the  ? subsequent proofs. Consider a preference in  D  and two points  _ 0MT88Mdd <<s!(x hat,y hat), (x tilde, y tilde)x6X@8;X@x6X@8;X@x6X@8;X@8(8,8)j8,8(8,8)88n8^88xz8yZ8xJ8ys in the consumption set w0=T__dd <<[0, omega`] CDOT  _+x6X@8;X@x6X@8;X@x6X@8;X@_[_0_,-_]z_3__\+w such that @!0$T88dd <<x hat  x tildex6X@8;X@x6X@8;X@x6X@8;X@&888x8x8@ and @A0 "T88dd <<y hat  y tildex6X@8;X@x6X@8;X@x6X@8;X@&888y8y8@.  ? If a0>88dd << sigma hatx6X@8;X@x6X@8;X@x6X@8;X@88% (resp 0 >88dd << sigma tildex6X@8;X@x6X@8;X@x6X@8;X@88%) is the slope (dy/dx) of a line supporting the  ? preference upper contour set at [0}88}dd <<(x hat, y hat)x6X@8;X@x6X@8;X@x6X@8;X@8(8,8)888xz8y[ (resp _0}u88}dd <<(x tilde, y tilde)x6X@8;X@x6X@8;X@x6X@8;X@8(8,8)888xz8y_) then we must  ?R have I088dd <<% hat % tildex6X@8;X@x6X@8;X@x6X@8;X@8 88% 8%8I. We let the reader check that these two formulations of  ? binormality are equivalent (for preferences in  D ). The second formulation is computationally easier to use. For instance if a preference is represented by a differentiable utility function u, the combination of binormality and convexity is equivalent to the  _ property that the marginal rate of substitution 00 __dd <<MRS=(u_x/u_y)x6X@8;X@x6X@8;X@x6X@8;X@_MRS_u_+x'_u+yz__j_(_/_)ߍ be nondecreasing in both x and y.  ?F Most familiar preferences are binormal (and belong to B ). Any  X preference representable by a utility function of the form  ?f u!0]%88]dd << u(y)v(x)x6X@8;X@x6X@8;X@x6X@8;X@8u8yj8vZ8x8(z8)8(8)8u, where u, v are increasing, u is concave and v is convex,  ?! is in B .4 Examples include any type of convex, homothetic preferences such as the CobbDouglas utility functions and the constant elasticity of substitution utility functions.  ?%  Definition 1 :  ?P' Fix the technology (represented by F or C) and the set N of  _( agents (A0p /__dd <<omega_ix6X@8;X@x6X@8;X@x6X@8;X@_3+i is agent i's endowment of input). In the average return( ,**@@  _ game each agent i chooses his input contribution a0@__dd <<x_ix6X@8;X@x6X@8;X@x6X@8;X@_x+i in H0z@__dd << [0, omega_i]x6X@8;X@x6X@8;X@x6X@8;X@_[_0_,g_]z_3+iH and  _ receives the output share 0*__dd <<y_ix6X@8;X@x6X@8;X@x6X@8;X@_y+i given by (1). In the average cost  _ game each agent i chooses his output share 0 __dd <<y_ix6X@8;X@x6X@8;X@x6X@8;X@_y+i in 0 __dd << _+x6X@8;X@x6X@8;X@x6X@8;X@_+ and must  _ contribute the input share 0X __dd <<x_ix6X@8;X@x6X@8;X@x6X@8;X@_x+i given by (2). Note that !0z __dd <<x_ix6X@8;X@x6X@8;X@x6X@8;X@_x+i may  _ exceed A0 __dd <<omega_ix6X@8;X@x6X@8;X@x6X@8;X@_3+i. By convention we attach a welfare level below any level  _ in a0U__Udd <<[0, omega_i`] x  _+x6X@8;X@x6X@8;X@x6X@8;X@_[_0_,}_]z_3+i_xm_+߅ to a consumption T0'__'dd << (x_i,y_i)x6X@8;X@x6X@8;X@x6X@8;X@_(W_,_)_x+i_yL+iT such that 90W,__Wdd << x_i > omega_ix6X@8;X@x6X@8;X@x6X@8;X@_x+i+i_>W_39. Note that in the definition of the average cost game, we did not exclude the possibility that an agent becomes bankrupt (this may happen in particular if agent i's expectations of other agents' demands are too low). Naturally, bankruptcy never occurs in equilibrium. The following result is critical in the sequel.   ?    Theorem (Watts [1994]).  ?l Consider a technology (meeting assumptions (3)) and a set N of agents with preferences in B . Moreover, assume that F is strictly concave, and/or individual preferences are strictly convex. Then the average return game and the average cost game both have a unique Nash equilibrium.  ?< Recall that the above result breaks down when preferences are  ? in  D : it is easy to construct examples where either game has an arbitrary large number of equilibria.  ?"  3. The Tragedy of the Commons: A Global Statement In this section we state and prove a general formulation of the tragedy, namely the fact that the equilibrium of the average return and of the average cost games involve inefficient' 0*((@@ overproduction. First we recall two easy (and wellknown) formulations of the tragedy. We start with a local statement. Assume for simplicity that F(and C) are differentiable and preferences are represented by  _@ differentiable utility functions 0( __dd <<u_ix6X@8;X@x6X@8;X@x6X@8;X@_u+i. Consider an interior (that  _* is, 40*j__*dd <<x_i^* >0x6X@8;X@x6X@8;X@x6X@8;X@_x:i_>n_04 for all i) Nash equilibrium 05*!j__5*dd <<A(x_1^*,...,x_n^*)x6X@8;X@x6X@8;X@x6X@8;X@_(:1n_,_.^_._.N_,_)_x_xC:nZA߾ of the average return game. The first order conditions for the Nash equilibrium are:  _ #xddddd *dd xQO# ? `o (4) ă!_u_{ix} (x_i^* ,y_i^*)+u_{iy}(x_i^* ,y_i^*).(lambda_i .F'(x^*)+(1lambda_i).AF(x^*))=0~for~all~ix6X@8;X@x6X@8;X@x6X@8;X@_u+ix_x:i_y{:i_uJ+iyb_x:i_y;:i +iM _F _x<+i|_AF_x_for_all_i*_(_,_)_(F_,_) _. _( _. _(i _)Y_(_1_)_.l_(_)@_)0_06Z_R   _I_x_ __!$(#(# (#(#X!'#$where we set !0* 6__*dd <<lambda_i =x_i^*/x^*x6X@8;X@x6X@8;X@x6X@8;X@_+iU_x:i_x_E9_/߂, A0k88dd << AF(x)=F(x)/xx6X@8;X@x6X@8;X@x6X@8;X@8AFz8x8F8x:8x8(8)Z8(J8)8/j8ߙ, and a0mk__dd <<"x == x_1+,..+x_n x6X@8;X@x6X@8;X@x6X@8;X@_x_x"_x+n___z+1B_,_.2_."ߟ. Consider a small proportional reduction of the equilibrium  _ input profile 0*@ __*dd <<0(x_1^*,..,x_n^*)x6X@8;X@x6X@8;X@x6X@8;X@_(:1n_,_.^_._,2_)_xN_x:n0߭. In other words consider the profile  _. j0 *n__ *dd <<((1t)x_1^*,...,(1t)x_n^*)x6X@8;X@x6X@8;X@x6X@8;X@_(_(_1j_)Z:1_,>_._.._._,_(_1_)Z _)z_v_ _t_x_tv_x:nj, where 00. This, combined with the interiority of the equilibrium, and the property  _% $0-B+__-dd <<u_{iy}>0x6X@8;X@x6X@8;X@x6X@8;X@_u+iy*_>_0$ guarantee '0 B+__dd << delta_i > 0x6X@8;X@x6X@8;X@x6X@8;X@_ +i_>U_0': that is to say, the (small) proportional reduction of input shares is a Pareto improving move. A similar computation shows that a small proportional reduction of anyP|( \+))@@1'#3 R '## %'#( P equilibrium profile of the average cost game is a Pareto improving move (note that, in both cases, convexity of preferences is not essential). A global version of the tragedy is a statement of the following form: a Pareto improving move from the equilibrium to a Pareto superior outcome results in a lower overall production. Such a statement does not follow from the local argument given above. The global statement is easy to prove under the assumption of quasilinear preferences. Suppose for all i agent i's utility  _ takes the form x0 __dd << u_i(y_i)x_ix6X@8;X@x6X@8;X@x6X@8;X@_u+iW_y+i_x+i_($_)_x, with !0__dd <<u_ix6X@8;X@x6X@8;X@x6X@8;X@_u+i a differentiable, concave function, and let the cost function C be strictly convex (assume  _ also unbounded endowments of input: BA0zZ__zdd <<omega_i =+ infinityx6X@8;X@x6X@8;X@x6X@8;X@_3+i_w__B for all i). There is a unique Pareto efficient level of production  _ a01*__1*dd <<j0y^* == y_1^*+...+y_n^*,~where~(y_1^*,...,y_n^*) x6X@8;X@x6X@8;X@x6X@8;X@_yn_y_y':n^_where _y _y? :n_R_2_> V :1_.B_._._, _( :1j _, _.Z _. _.J _, _)j is characterized by the first order conditions  _x #0" * __" *dd <<u_i' (y _i^*)=C'(y^*)~for~all~ix6X@8;X@x6X@8;X@x6X@8;X@_u:iW_y:i+_Cc_y_for_all _i__(;_)_(G_)# On the other hand the first order conditions for the Nash  _\ equilibrium 0P $__dd << (y_1,...,y_n)x6X@8;X@x6X@8;X@x6X@8;X@_(+1R_,_.B_._.2_,w_)_y_y'+nߘ of the average cost game imply at once:  _F 0O*&__O*dd <<Fu_i' (y_i)  C'(y)x6X@8;X@x6X@8;X@x6X@8;X@_u:iW_y+i_CL_y__($_)_(_)F for all i  ?" The equilibrium involves underproduction if #0P(88dd <<y < y^*x6X@8;X@x6X@8;X@x6X@8;X@8y8y8<z#. In that case,  ?|$ we have 0Yp*88Ydd <<. C'(y) Note that the Nash equilibrium outcome (of either game) is never Pareto optimal if marginal cost strictly exceeds average cost (as will result if C is strictly convex): the proof follows essentially from the local argument at the beginning of this section (we omit the details). On the other hand, if marginal cost is constant up to the equilibrium output level, then the equilibrium outcome might be Pareto optimal (see, for instance, the numerical example in Section 1).  ?$  Proof of Theorem 1 :  _N& Let uA 0 ,__ dd <<%z hat _j=(x hat_j,y hat _j),j=1,...,nx6X@8;X@x6X@8;X@x6X@8;X@&__(__z+j_xL+j_y+j_j_n_I_W_(_,_)Y_,_19_,_.)_._._,u, be the Nash equilibrium outcome ofN& 0*((@@  _ the average cost game. We assume that a 0*@__*dd<<?z_j^*,j=1,...,nx6X@8;X@x6X@8;X@x6X@8;X@_z:jn_j._n__,^_1_,N_._.>_._,?߼ is a Pareto  _T optimal outcome Pareto superior to  0*__*dd<<0z hat_j,j=1,..,nx6X@8;X@x6X@8;X@x6X@8;X@&__z+jW_j_n_,G_1_,7_._.'_,_0߭. We suppose  >  0`Q~ `Qdd<<%sum_j x hat_j=x hat < x^*=sum_j x_j^*x6X@8;X@x6X@8;X@x6X@8;X@9I9I+juxRjxx+jixajB>2< and we derive a contradiction.  _ We fix an agent i and let  08 __dd<< GAMMA _i x6X@8;X@x6X@8;X@x6X@8;X@_+i be agent i's opportunity set  _ through  0p __dd<<z hat_ix6X@8;X@x6X@8;X@x6X@8;X@&__z+i in the average cost game: #xddddd\Nddx+# ? `^ (5) ă|{(alpha_i,beta_i) epsilon Gamma_i~ < ~alpha_i= {beta_i} over {beta_i+sum_{j != i} y hat_j} .C(beta_i+sum_{j !=i} y hat_j)x6X@8;X@x6X@8;X@x6X@8;X@!4(m4,4)\ 4.L 4(4)4414 44# 4iiii iK iRiK +j +ir y Rj 4CH iji4yj4<p4 +c 4<c]o9II '4|$(#(# (#(#!'#$The Nash equilibrium property for agent i means that there is a  _, line through  0 l__dd<<z hat_ix6X@8;X@x6X@8;X@x6X@8;X@&__z+i with slope (dy/dx)! 0l88dd<<r== x6X@8;X@x6X@8;X@x6X@8;X@8rA 0?l__dd<< % hat_ix6X@8;X@x6X@8;X@x6X@8;X@__%+i, and separating a 0+l__dd<<Gamma_ix6X@8;X@x6X@8;X@x6X@8;X@_+i from agent  _ i's upper contour set at  0hV__dd<<z hat_ix6X@8;X@x6X@8;X@x6X@8;X@&__z+i.  _ Denote by  0 @88dd<<rphix6X@8;X@x6X@8;X@x6X@8;X@8-r the function defined by (5): > 0j@__dd<<:(alpha_i,beta_i)epsilon Gamma~ < ~alpha_i = phi(beta_i)x6X@8;X@x6X@8;X@x6X@8;X@_(^_,_)_(R_)__"_ _=__-~_+iZ+i+i+iL_<_>.  ? The assumptions on C guarantee that  0*88dd<<rphix6X@8;X@x6X@8;X@x6X@8;X@8-r is convex and increasing. The separation property implies  _$ #xddddddddx+# ? `   (6) ă;phi_'(y hat_i)``1 over % hat_i`~phi_+'(y hat_i)x6X@8;X@x6X@8;X@x6X@8;X@!-_%\-::(3)<1)(n)z_fyi +iyim߶$(#(#$(#(# !'#$(where the notations  0*"__*dd<<phi_+'x6X@8;X@x6X@8;X@x6X@8;X@_-: and ! 0*"__*dd<<phi_'x6X@8;X@x6X@8;X@x6X@8;X@_-: stand respectively for the right ? hand and lefthand derivative of the convex function A 0$88dd<<rphix6X@8;X@x6X@8;X@x6X@8;X@8-r). Note, in  _B passing, that the above inequality implies va 0L&__Ldd<<0<% hat_i<+ infinityx6X@8;X@x6X@8;X@x6X@8;X@_0_<_<__%+iI__v. Computing righthand derivatives in formula (5) gives  # !#x)ddddda ddx+# ? `\ (7) ăWphi_+'(y hat _i)= {y hat _i} over y hat C_+' (y hat)+ {y haty hat _i} over y AC(y hat)x6X@8;X@x6X@8;X@x6X@8;X@0-<(B)() ( )<8N<>< uyim<yi8yWCy:<y*<yi8y AC| yPTߧ$(#(##(#(# !!'#$(where we set  0J@ -Jdd<<y hat =sum_j y hat_j)x6X@8;X@x6X@8;X@x6X@8;X@&yy+jeyRj9I2)߄.P&0*((@@1'#md'#a")'#,!PԌ _ Therefore, the inequality  0%*@__%*dd<<"AC(y)C_+'(y)x6X@8;X@x6X@8;X@x6X@8;X@_ACz_y_C"_y_(_)_(_)j_vZ:"ߟ (a consequence of (3)) implies: %A#x$ dddddW*ddx+# ? `a  (8) ăphi_+'(y hat _i)C_+'(y hat)x6X@8;X@x6X@8;X@x6X@8;X@_-:_:_($_)_(_)k_h_W_y+i_CT_y%$(#(#(#(#X!A'#$Next we pick a line supporting agent i's upper contour set at  _  0 * __ *dd<<z_i^*x6X@8;X@x6X@8;X@x6X@8;X@_z:i and we denote its slope by  0* __*dd<< sigma_i^*x6X@8;X@x6X@8;X@x6X@8;X@_%:i. The Pareto optimality of  0  %88 dd<<z^*x6X@8;X@x6X@8;X@x6X@8;X@8zz implies Ya#xdddddddx+# ? ` (9) ăHC_'(y^*)``1 over %_i^*~ ~C_+' (y^*) ~(where~y^*=sum_j y_j^*)x6X@8;X@x6X@8;X@x6X@8;X@!Cay:iCy where yjyjoVDo(V` V o(E)q1x()(o)=( I_%Y$(#(# (#(#!a'#$We are now ready to prove the statement of the theorem.  ?( Assume first C is strictly convex. Then the inequality 8! 0!h88dd<< y hat +aYV(,c;S) () ,,w  _ (where the notations 0Z%__Zdd<<(d xi_i / dy_i)_+x6X@8;X@x6X@8;X@x6X@8;X@_(_/_)_dr+i:_dy/+i_+ and !0*:$__*dd<<C_+'x6X@8;X@x6X@8;X@x6X@8;X@_C: stand for the righthand derivative of convex functions). Clearly the average cost sharing rule satisfies the above inequality (this is precisely (8) in the above proof). The statement of the Theorem is adapted as follows.  ?% Let A0+88dd<<z hatx6X@8;X@x6X@8;X@x6X@8;X@&88z be a Nash equilibrium of the cost sharing game, and let a0 ]"+88 dd<<z^*x6X@8;X@x6X@8;X@x6X@8;X@8zz  ?x' be a Pareto optimal allocation, Pareto superior to 0-88dd<<z hatx6X@8;X@x6X@8;X@x6X@8;X@&88z. Then total  ?") production is at least as large at 0~/88dd<<z hatx6X@8;X@x6X@8;X@x6X@8;X@&88z as it is at 0 b/88 dd<<z^*x6X@8;X@x6X@8;X@x6X@8;X@8zz. The proof is"),**@@  ? identical, once we define the function 0@88dd<<rphix6X@8;X@x6X@8;X@x6X@8;X@8-r by  _ ?0__dd<<-phi ( beta_i)= xi _i (beta_i, y hat _{i}; C)x6X@8;X@x6X@8;X@x6X@8;X@_-____(_)_(_,g_;W_)+i;+i+iO_y+i_CS_+c_? and note that inequality (8) holds by assumption. Similarly in a general surplus sharing game each agent chooses  _ his or her input contribution !0 __dd<<x_ix6X@8;X@x6X@8;X@x6X@8;X@_x+i and total output F(x) is divided  _ according to a surplus sharing rule A0e__edd<<ny_i= zeta_i (x_i,x _{i} ; F)x6X@8;X@x6X@8;X@x6X@8;X@_y+i+i_x +i_x+ib_F_J+W_ _(Z_,_;_)n. Assume that a0u#__dd<<zeta_ix6X@8;X@x6X@8;X@x6X@8;X@_ +i  _ is continuous, non decreasing and concave w.r.t. 0__dd<<x_ix6X@8;X@x6X@8;X@x6X@8;X@_x+i and that imputed marginal return is not smaller than actual return   0Bdd<<fleft ( {d zeta_i} over {d x_i} right )_+ (x_i, x_{i};F)  F'(x)~for~all~i,~all~{x vert 40 horz 57 }, ~all ~concave~Fx6X@8;X@x6X@8;X@x6X@8;X@&h9i&o9p|p/dzVidxRixiKxiFCF{x for allC iallKxallYconcaveF  >+HQV(,c;S)() ,,f Then any Nash equilibrium of the surplus sharing game entails overproduction (same statement as Theorem 1). Note that uniqueness of the Nash equilibrium is no longer guaranteed even under binormal preferences (see Watts [1995]). The proof's technique can also be used to show that underproduction takes place when the slope of agent i's opportunity set is no larger than the marginal product F'(x). Formally, in the   cost sharing game if u0O "O dd<<4{d xi _ i} over {dy_i} (y_i, y_{i}, C) ~ ~C'(y)x6X@8;X@x6X@8;X@x6X@8;X@|?bcdJ/i\_dyQ+i6yi{yCi C#C[yc(,,)()S!u then 0QM#Qdd<<5" sum_j z_j^*  sum _j z hat _jx6X@8;X@x6X@8;X@x6X@8;X@9I9I+juzaj+j4zRj YH5߲. In the   surplus sharing game if r0%dd<<1{d zeta_i} over {dx_i} (x_i,x_{i}, F)  F'(x)x6X@8;X@x6X@8;X@x6X@8;X@|?ccdI/i\_dxQ+i6xi{xCi FsFxc (,,)3(#)!r then !0Q_B&Qdd<<2sum_j z_j^*  sum_j z hat _jx6X@8;X@x6X@8;X@x6X@8;X@9I9I+juzaj+j4zRj YH2߯. For instance, consider the surplus sharing game where output  8$ is divided equally, EA0H*Hdd<<&zeta_i(x_i,x_{i}, F)= 1 over n F (x) x6X@8;X@x6X@8;X@x6X@8;X@! iWxixdi,F8ndFTx($,,)1() E, so that a0bx*bdd<<B{d xi_i} over {dx_i}  F'(x)x6X@8;X@x6X@8;X@x6X@8;X@|?bcdJ/i\_dxQ+i6Fnxc!()B߿. Therefore there is underproduction in the sense that the only way -'0*((@@Ԍ ? ԙto make everyone better off is to increase total production. However, in the cost sharing game where input is divided equally,    E0J Jdd<<&xi_i(y_i , y_{i}, C) = 1 over n C(y)x6X@8;X@x6X@8;X@x6X@8;X@!iYyiyfi.C8nfCVy(&,,)1() E, we have 0b` bdd<<C {d zeta_i} over {dy_i} C'(y)x6X@8;X@x6X@8;X@x6X@8;X@|?ccdI/i\_dyQ+i6Cnyc !()C. Therefore, from the proof of Theorem 1, the only way to make everyone better off is to  ? decrease total production.  ?5  Remark 2  Theorem 1 breaks down if preferences are not strictly convex and the cost (production) function is not strictly convex (concave). For instance consider the technology C(y)=y and n agents with identical preferences such that an agent's demand, when the price of one unit of output is one unit of input, is the  _ interval from 0U @ __U dd<<4#(x_1,y_1)=(1,1)~\to~(x_2,y_2)=(2,2)x6X@8;X@x6X@8;X@x6X@8;X@_(+1R_,B+1_)_(_1r_,_1b_)z_(j+2_, +2 _) _(b _2 _,R _2 _)_x_y2_to_x2 _y _r _4߱. Here the outcome  _ 0__dd<<i(x hat_i,y hat _i)=(1,1)x6X@8;X@x6X@8;X@x6X@8;X@_(W_,_)_(_1|_,_1l_)___x+i_yL+i_i is a (Pareto optimal) Nash equilibrium (not unique),  _i and 0%*__%*dd<<d(x_i^*,y_i^*)=(2,2)x6X@8;X@x6X@8;X@x6X@8;X@_(n_,_)_(2_2_,"_2_)_x:i_yc:izB_d is an outcome Pareto indifferent to the above. An interesting variant of Theorem 1 replaces the "strictness" assumption on preferences or cost by the non Pareto optimality of the Nash equilibrium.  ?m  Theorem 1*  ? Consider a technology meeting assumptions (3) and a set N of  ? agents with preferences in B . Let !0&88dd<<z hat x6X@8;X@x6X@8;X@x6X@8;X@&88z be a Nash equilibrium outcome  ?7" of the average cost (resp. average return) game. Suppose there  ?# exists a Pareto optimal outcome A0 *88 dd<<z^*x6X@8;X@x6X@8;X@x6X@8;X@8zz of the economy, and Pareto  _% superior to our equilibrium (no agent j prefers `a04*0+__4*dd<<z hat _j`\to~z_j^*x6X@8;X@x6X@8;X@x6X@8;X@&__z+j_to=_z:j`: at least%0*((@@Ԍ _ ԙone agent prefers `0* __*dd<<z_j^*`\to`z hat _jx6X@8;X@x6X@8;X@x6X@8;X@_z:j _to_z+j&_`). Then the total production is not smaller at the equilibrium outcome 088dd<<z hatx6X@8;X@x6X@8;X@x6X@8;X@&88z than at the Pareto superior  ? outcome 0 p> 88 dd<<z^*x6X@8;X@x6X@8;X@x6X@8;X@8zz. The proof of Theorem 1* is postponed to the Appendix.  ?       Remark 3  ?p Theorem 1 (and Theorem 1*) break down if individual  ? preferences vary in the classical domain  D , but are not necessarily binormal. Consider an economy with two agents and cost function C such that: #xdddddddx%C(20)=20;C(22)=24.5;C'(20)=2;C'(22)=3x6X@8;X@x6X@8;X@x6X@8;X@8CJ8Cr 8Cz 8C8(8208)8208;8(:822*8)824 8.858;2 8( 820 8) 82 8;:8(8228)83j88 z 8z8$(#(# (#(#X!'#$(note that these numbers are compatible with assumptions (3)). Preferences of agents 1 and 2 have the following supporting slopes (or marginal rates of substitution dy/dx) at the following points  T ddx !ddx( X T e e  P  _  T0'__'dd (x_i,y_i)x6X@8;X@x6X@8;X@x6X@8;X@_(W_,_)_x+i_yL+iT u    0mmdd,stack {alignl MRS~sigma_i # alignl (dy/dx)} x6X@8;X@x6X@8;X@x6X@8;X@MRSQi7dy7dx%7(z7/7)ߌe 3   Agent 1  _  !0= __ddz hat _1=(15,15)x6X@8;X@x6X@8;X@x6X@8;X@&__z+1R_(_15_,2_15"_)_     A0= dd4 over 7x6X@8;X@x6X@8;X@x6X@8;X@ ?\4\873 #u   _   a0*`"__*dd!z_1^*=(23.5,20)x6X@8;X@x6X@8;X@x6X@8;X@_z_:1n_(_23_.N_5_,>_20._)!ߞJ     0`"dd1 over 3x6X@8;X@x6X@8;X@x6X@8;X@ ?\1\83#3   Agent 2   _S  z0$__dd z hat_2=(5,5)x6X@8;X@x6X@8;X@x6X@8;X@&__z+2R_(_5B_,_52_)_zH  S  0$dd4 over 5x6X@8;X@x6X@8;X@x6X@8;X@ ?\4\853 S >!  _v  x0*&__*dd z_2^*=(1,2)x6X@8;X@x6X@8;X@x6X@8;X@_z_:2n_(_1^_,_2N_)x!  v  0&dd1 over 3x6X@8;X@x6X@8;X@x6X@8;X@ ?\1\83S  ?# Note that these data are compatible with preferences in  D  (it is enough to check the axiom of revealed preferences) and with agent  _' i (strictly) preferring !0 *A-__ *dd<<z_i^*x6X@8;X@x6X@8;X@x6X@8;X@_z:i to A0v-__dd<<z hat_ix6X@8;X@x6X@8;X@x6X@8;X@&__z+i ia0-Cv-88-dd<< (for~i=1,2)x6X@8;X@x6X@8;X@x6X@8;X@8(:818,*828)8forJ8i8i.0'0*((@@'#0Ԍ _ By construction {0U*@__U*dd<<(z_1^*, z_2^*)x6X@8;X@x6X@8;X@x6X@8;X@_(:1n_,^:2_)_z_zz{ is a Pareto optimal outcome, as we  _T have 07*R__7*dd<<%_1^*=%_2^*=1/C'(y^*)x6X@8;X@x6X@8;X@x6X@8;X@_%Y_%_(_$\:1:2_1_/P_(_)_C_y. And ~0__dd<<(z hat_1,z hat_2)x6X@8;X@x6X@8;X@x6X@8;X@_(+1R_,B+2_)___z_z~ is a Nash equilibrium of the average cost game, because formula (7) implies: #xx ddddd ddxNstack {alignl phi'(y hat_1)= {15} over {20} .` 2` + 5 over {20} . `1 = 1 over {% hat_1} # ~ # alignl phi'(y hat_2)=5 over {20} .` 2` +` {15} over {20} .` 1`=1 over {% hat_2}}x6X@8;X@x6X@8;X@x6X@8;X@"-%-^%ds"O"" sOA"(1")&15&203"."2>520"."11G n1(2)b;5&^203.2;15^20%.1 ;1s *2G"G^3"y3y KKK N$(#(#8(#(#x!'#$ 4. Comparing the Average Cost and Average Return Equilibrium  ?  Outcomes  ?`  Theorem 2   ? Same assumptions as in Watts' theorem and in Theorem 1.  ? Denote by 0` 88dd<<z^cx6X@8;X@x6X@8;X@x6X@8;X@8zzc and 0 88dd<<z^rx6X@8;X@x6X@8;X@x6X@8;X@8zzr respectively the (unique) Nash equilibrium outcome of the average cost game and of the average return game.  ? Then total production is at least as large at !0@288dd<<z^rx6X@8;X@x6X@8;X@x6X@8;X@8zzr as it is at A0t!288dd<<z^cx6X@8;X@x6X@8;X@x6X@8;X@8zzc.  ?  Corollary 1   ?d Under the assumptions of Theorem 2, suppose all agents have  ? identical preferences. Then they all prefer the average cost equilibrium outcome over the average return equilibrium outcome (or  ? they are all indifferent between these two outcomes).  ?  Corollary 2   ?4" Under the assumptions of Theorem 2, it cannot be the case that  ?# the average return equilibrium outcome a0*88dd<<z^rx6X@8;X@x6X@8;X@x6X@8;X@8zzr is Pareto superior to the  ?% average cost equilibrium outcome 0(+88dd<<z^cx6X@8;X@x6X@8;X@x6X@8;X@8zzc. More precisely, either the0%0*((@@x '#0  _ two equilibria coincide (W0*h@__*dd<< z_i^c=z_i^rx6X@8;X@x6X@8;X@x6X@8;X@_zc:i_z(r:i _W for all i), or there is an agent i  _ who strictly prefers 0*__*dd<<z_i^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:i to 0*__*dd<<z_i^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:i. The key to Theorem 2 is the fact than in the average cost game the marginal cost imputed to an agent, although smaller than the actual marginal cost, is larger than the marginal cost imputed by the average return game. Thus every player is closer to taking into account the full social cost of his actions in the average cost game than in the average return one; in turn this implies less overproduction in the former game than in the latter one. Before proving this fact formally in Lemma 1, we give an intuitive explanation. Suppose we start at the outcome  _# 50 c__ dd<<z_i=(x_i,y_i), i=1,...,nx6X@8;X@x6X@8;X@x6X@8;X@_z+i_xL+i_y+i_i_n_I_W_(_,_)Y_,_19_,_.)_._._,5 and player 1 decides he wants one more unit of output. If he is playing the average returns game then he knows  _ everyone else's level of input is fixed at 0__dd<<G(x hat _2,...,x hat _n)x6X@8;X@x6X@8;X@x6X@8;X@_(+2R_,_.B_._.2_,w_)___x_x'+nG. When player 1 increases his input contribution to receive more output,   all other players receive less output (as !0*`W*dd<<{x hat _j} over x F(x)x6X@8;X@x6X@8;X@x6X@8;X@?p<\<xj8xFF6x()߃ falls, due to the concavity of F). Since the others receive less output, the total  b level of production does not need to increase to bA0EJ#EJdd<<sum _j y hat _j + 1x6X@8;X@x6X@8;X@x6X@8;X@9I+juyRjB1b. Thus the total increase in input needed to give player 1 another unit of   output is smaller than ?a0Jx!'Jdd<<.C(sum _ j y hat _j + 1) C( sum _ j y hat _j)x6X@8;X@x6X@8;X@x6X@8;X@C+jeyRjC+jeyRj(1")(2)9I9Iyy2?. Suppose now that player 1 wants one more unit of output when he is playing the average cost game. Everyone else's demand is  _% fixed at 00,__dd<<G(y hat_2,..., y hat _n)x6X@8;X@x6X@8;X@x6X@8;X@_(+2R_,_.B_._.2_,w_)___y_y'+nG. If player 1 demands one more unit of output%0*((@@ the others will be forced to contribute more input (due to the concavity of F), but they will demand the same level of output. Thus the total increase in input needed to give player 1 another   4unit of output equals >0J Jdd<<-C( sum_ j y hat _j + 1) C(sum _ j y hat _j)x6X@8;X@x6X@8;X@x6X@8;X@C+jeyRjC+jeyRj(1")(2)9I9Iyy2>, and is larger than the 4total increase needed in the average returns game. Turning to the formal proofs, we start by defining two functions of which the respective graphs are the opportunity sets of a given agent in our two games. Given two nonnegative numbers  _} 20-__-dd<<x_0,y_0x6X@8;X@x6X@8;X@x6X@8;X@_xR_y+0_,+02 we define two functions 088dd<<rphix6X@8;X@x6X@8;X@x6X@8;X@8-r and 088dd<<rpsix6X@8;X@x6X@8;X@x6X@8;X@81r as follows:  _g #x+dddddcddx<stack {alignl phi(beta_i;y_0)= {beta_i} over {beta_i+y_0}.C(beta_i+y_0)~~~for~all~beta_i0 # ~ # alignl psi(alpha_i;x_0)={alpha_i} over {alpha_i+x_0} .F(alpha_i+x_0)~~~for~all~alpha_i0}x6X@8;X@x6X@8;X@x6X@8;X@I-IFII1b[^I(I;0I) 0wI.gI( 0 I)?I0(;00)*0.|( 0 )T0iSIyjiiyICci+ Iys Ifor3 Iallwiihx.i*i^xFxi@ x forH alliIII/^)r><߹$(#(#g(#(#@!'#$Thus !0__dd<<-beta_i  phi(beta_i;y_0)x6X@8;X@x6X@8;X@x6X@8;X@_^_-S_+i+i_y__('_;+0g_)-ߪ is agent i's opportunity set in the average cost   game when A05J` 5Jdd<<sum_{j != i} y_j=y_0x6X@8;X@x6X@8;X@x6X@8;X@9I+j+iyRjZy>+cR0߂. A similar interpretation holds for  _% a0e__dd<<0alpha_i  psi (alpha_i;x_0)x6X@8;X@x6X@8;X@x6X@8;X@_^_1h_+i+i_x__(<_;,+0|_)0߭. As noted in the proof of Theorem 1, the function 0"e88dd<<rphix6X@8;X@x6X@8;X@x6X@8;X@8-r  _ (resp. 0O!88dd<<rpsix6X@8;X@x6X@8;X@x6X@8;X@81r) is convex in 0O!__dd<<beta_ix6X@8;X@x6X@8;X@x6X@8;X@_+i (resp. concave in 0O!__dd<<alpha_ix6X@8;X@x6X@8;X@x6X@8;X@_+i). We denote by  _ 0*9#__*dd<<phi_'(beta_i;y_0)x6X@8;X@x6X@8;X@x6X@8;X@_-W_:_(+_;+0k_)+i_yߑ its left hand derivative w.r.t. !0n#__dd<<beta_ix6X@8;X@x6X@8;X@x6X@8;X@_+i (and similarly  _M A0 *%__ *dd<<psi_+'(alpha_i;x_0)x6X@8;X@x6X@8;X@x6X@8;X@_1l_:_(@_;0+0_)+i_xߒ is the righthand derivative of a03%88dd<<rpsix6X@8;X@x6X@8;X@x6X@8;X@81r w.r.t. 0%__dd<<alpha_ix6X@8;X@x6X@8;X@x6X@8;X@_+i).  ?!  Lemma 1   ?1# Fix an agent i and a technology satisfying (3). Given are  _$ four positive numbers 0+__dd<<x_i,x_0,y_i,y_0x6X@8;X@x6X@8;X@x6X@8;X@_x+iW_x_y+i_y_,+0_,d_,T+0ߚ such that 0&0*((@@'#50Ԍ ? #x@ddddd ddxk$ ? ` (11) ă&x_i=phi(y_i;y_0)~\and~y_i=psi(x_i;x_0)x6X@8;X@x6X@8;X@x6X@8;X@_x+iL_y+i_y)_and_yf+i8 _x +i} _x__W_-._1_(_; +0Y_)_( _; +0E _)ߛ$(#(#(#(#X!'#$Then we have   ?  !#x` dddddN ddxsH$ ? `  (12) ă-1 over {phi_'(y_i;y_0)}~~psi_'(x_i;x_0)x6X@8;X@x6X@8;X@x6X@8;X@?=q1)_(n_;^+0_)M(;0)\_-k1:o_y+i_yxBi xߤ$(#(# (#(#!!'#$Moreover  ? >A#xdddddddxsH  ? ` ăstack {alignl if~ AC(y_i+y_0) < C_' (y_i+y_0)~then~inequality~ (12)~ is~ strict # ~ # alignl if~AC(y_i+y_0)=C_'(y_i+y_0),~then~(12)~is~an~equality.}x6X@8;X@x6X@8;X@x6X@8;X@ifZACy?iyCy|iDy then  inequalityTisstrict^ifZ^AC^y?*i^y^C^y|*iD^yT ^then^is ^anT^equalityJ(0)G<(0 )(12)J^(*0^)^(*0 ^) ^, ^( ^12 ^)^.SU7^G^S79^>$(#(#(#(#!A'#$    The proofs of Lemma 1, Corollary 1 and Corollary 2 are postponed to  ? the Appendix.  ?   Proof of Theorem 2   ?0 We fix the technology, the preferences, and 20p88dd<<z^r,z^cx6X@8;X@x6X@8;X@x6X@8;X@8zzr8z(zc 8,2 as in the   statement of the theorem. We assume 0/QR/Qdd<<!y^r=sum_j y_j^r < sum_j y_j^c=y^cx6X@8;X@x6X@8;X@x6X@8;X@yr]+jyraaj/+jy]c3aj%yc 9IS9I< and we derive a contradiction. We can choose an agent i such that   a#xddddd#ddxk  ? `   ă F{y_i^r} over y^r``{y_i^c} over {y^c} ~\and~0`<` {y_i^c} over {y^c}x6X@8;X@x6X@8;X@x6X@8;X@]PzXy!r3iz8y!zrmXyc3im8yzcandXycm3i8yzc0'< ߧ$(#(#(#(#!a'#$As we assumed 0[Q@ "[Qdd<<#x^r=sum_j x_j^r`<`x^c = sum_j x_j^cx6X@8;X@x6X@8;X@x6X@8;X@xr]+jxraajx&c+jQxcaj v9I9I<, we have  _^ #x%ddddd #ddx\x_i^r=x^r. {y_i^r} over {y^r}~~x^r. {y_i^c} over {y^c}`<`x^c. {y_i^c} over {y^c} = x_i^cx6X@8;X@x6X@8;X@x6X@8;X@?x/rixUrXXyr3iX8yzrx;r>Xyc3i>8yzcx c XyG c 3i 8yG zc, x /c i6 ..h< .;! ߍ$(#(#^(#(#!'#$and by a similar argument, X!0*X)__*dd<< y_i^r`<`y_i^cx6X@8;X@x6X@8;X@x6X@8;X@_yr:i_yTc*:i_<X.  % We also assume that A0Q+Qdd<<)y_0^c=sum_{j != i} y_j^cx6X@8;X@x6X@8;X@x6X@8;X@yc]+j+iy+caja0 +c9I)ߦ is positive because when 3a0 *+__ *dd<<y_0^c=0x6X@8;X@x6X@8;X@x6X@8;X@_yc:0_0 _3, onep%0*((@@Q@'#`` '# !'#A'#b"a%'#(p checks easily that the equilibrium of the average cost game (where all agents other than i are inactive, and agent i alone uses the technology optimally) is also the equilibrium of the average return game. Thus we have from now on:  @ +#x ddddd *ddxk$ ? `  (13) ă:0x_i^r < x_i^c;~0_B_,_(A_/_)_(; _/ _)ߐ we have  _ A#xddddd!#ddxP{x_i^c} over x^c F_+'(x^r)+ {x_0^c} over x^c AF (x^r)~ ~ psi_+'(x_i^r;x_0^r)x6X@8;X@x6X@8;X@x6X@8;X@NkXxc3ik8xzcFxfrXxc8xzcAF]xrN x /r i xd /r/.$  / G()Y30(T) (E ;5 0 ) 1ߕ$(#(#(#(#!A'#$Finally, 0* __*dd<<F_+'x6X@8;X@x6X@8;X@x6X@8;X@_F: and AF are non increasing, and 20!88dd<<x^rz_i^c=(4.5,1.5)x6X@8;X@x6X@8;X@x6X@8;X@_zc:i __(_4q_._5a_,_1Q_._5A_)>߻ is an equilibrium of the average cost game, because   = *0^ = ?^ dd<<6phi'(1.5;1.5)= 1 over 2 C'(3)+ 1 over 2 AC(3)=9 over 2x6X@8;X@x6X@8;X@x6X@8;X@?- 0 (`1.P5;@1.05)182((3)C 1C 82 (@ 3 ) 9 82  &   hC ACڃ  < Thus production is higher in the average cost game, and the corresponding equilibrium is Pareto inferior to the average return equilibrium. We have been unable to determine whether or not in this example the (unanimous) preferences can be chosen in such a way that these two outcomes are respectively the unique equilibrium of each game.  ?E&  Remark 5   ?' Our final example is one where the preferences are in B , cost0'0*((@@;'#0 function is strictly convex and the average cost equilibrium outcome is not Pareto superior to the average return equilibrium outcome. As the reader may check by trying out a few numerical examples, this is not an easy example to build.  ?@ The example has two agents, and the cost function T0ax 88add<<C(y)=y^2x6X@8;X@x6X@8;X@x6X@8;X@8C8yj8y8(z8)z28T. The following table gives the MRS of both agents at the two  _ outcomes x0*__*dd<< z_i^c,`i=1,2x6X@8;X@x6X@8;X@x6X@8;X@_zc:i_i _,_1_,w_2_x and w!0*B__*dd<< z_i^r,i=1,2x6X@8;X@x6X@8;X@x6X@8;X@_zr:i_i _,q_1_,a_2_w. Of course, these MRS are chosen so as to make these outcomes respectively the average cost and average return equilibrium outcomes (that is to say: the MRS of  _& agent 1 at A0* f__*dd<<z_1^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:1 is precisely a09*f__9*dd<<!psi'(x_1^r;x_2^r)x6X@8;X@x6X@8;X@x6X@8;X@_1_(:1?_;/:2_)H_xr_x^r!ߞ and so on).  `   O A` ddxr44 a` ddxBh O p O     _  T0'L*__'dd (x_i,y_i)x6X@8;X@x6X@8;X@x6X@8;X@_(W_,_)_x+i_yL+iT    u0*ddMRS~%_i # ~(dy/dx)x6X@8;X@x6X@8;X@x6X@8;X@MRSi7dyJ7dxZ%j7(7/:7)uO  B  _  0*\ Y__*ddz _1^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:1C  (4.000,1.265)   .2259    _1  0*\ q__*ddz_2^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:2[  (6.000,1.897)[  .1977    _  0*\ #__*ddz_1^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:1   (4.017,1.252)   .2508   [  _  !0*\ __*ddz_2^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:2  (6.283,1.958)  .2166    `    ? Clearly we can choose agent 1's preferences in B  because their  _G MRS is imposed at two points A0*H$__*dd<<z_1^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:1 and a0*$__*dd<<z_1^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:1 such that 0*$__*dd<<z_1^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:1 is  _ automatically preferred to 0*X&__*dd<<z_1^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:1 (lower input contribution and higher output share). As for agent 2, we see on the table:  _$ i) that his MRS at 0**__*dd<<z_2^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:2 is larger that at 0**__*dd<<z_2^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:2 (as required by binormality of preferences);&0*((@@Ԍ _ ii) that  0*@__*dd<<z_2^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:2 lies above the line through ! 0*@__*dd<<z_2^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:2 with slope 1.977, so  _T that agent 2's preferences can be chosen in B with A 0*__*dd<<z_2^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:2 strictly  _ preferred to a 0* __*dd<<z_2^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:2.  ?  5. Concluding Comments  Given a common property technology, we can conceive of many different mechanisms to decentralize its use among equal owners. The two mechanisms we focus on, average return and average cost sharing, are probably the simplest ones for two reasons. First the message space is of dimension one and is a component of final consumption (agents choose either their actual output share or  ? their actual input share)5. Second the proportional sharing rule is the oldest and simplest equitable formula of distributive justice (usually attributed to Aristole, see e.g. BarHillel and Yaari [1988]). One possible application of our results is to mechanism design. Suppose a given set of agents own in common a certain  ?L technology, and can implement either the average cost game or the average return game. Our results suggest that they should choose the average cost game (Theorem 2). For instance, fishermen around a lake would be better off if each chose (non cooperatively) how many fish to catch and then fished as long as necessary to catch the desired amount, rather than each choosing (non cooperatively) how long to fish. The same applies to a production cooperative (Israelsen [1980]) where workers contribute different input levels (perhaps they work part'0*((@@ time on the cooperative's corn field). The workers are typically better off choosing their share of corn (and then working as long as requested by the manager) rather than deciding how long they wish to work and receiving a share of output at the end. Similarly, in the computer network example (Shenker [1990]), choosing individual demands makes more sense than choosing how long to wait in line (provided actual services are delivered in proportion to waiting time). Other examples include lobbying games (Tullock [1980], Riley [1991]) and models of patent races (Loury [1979], Posner [1992], Kitch [1977]): our results say that the participants would be better off selecting a probability of winning the prize rather than their level of effort (whether in lobbying or in research). Finally, examples could also come from the investment games ! la Bryant [1987], or the model of oligopoly with regulated price (Romano [1988]) where firms choose either the quantity they suppply (average return game) or the amount of revenue they wish to receive (average cost game). In the examples mentioned above the assumption of increasing marginal cost is natural. We show that under this assumption there is more overproduction in the average returns game. This conclusion depends crucially upon the assumption of increasing marginal costs: under decreasing marginal costs the presumption is the the comparison should be reversed. However, we have not found a systematical result similar to Theorem 2: this remains an area for future research.'0*((@@  ?  & APPENDIX ă  ?  Proof of Theorem 1*   _  Let  0` __dd<<Az hat_j,j=1,...,nx6X@8;X@x6X@8;X@x6X@8;X@&__z+jW_j_n_,G_1_,7_._.'_._,_A߾, be a Nash equilibrium outcome of the average cost game, and assume this outcome is Pareto inferior. We claim  _ that i) at least two agents are active (5 0p __dd<< y hat_j>0x6X@8;X@x6X@8;X@x6X@8;X@&__y+j_>W_05 holds for at least two   agents) and ii)  0%*0 __%*dd<<HAC(y hat)0,y hat _i>0:~then~ 1 over {C_+'(y hat)}``% hat_i`` 1 over {C_' (y hat)} # ~ # alignl if~x hat_i=y hat_i=0: ~~~~~~~~~~1 over {C_+'(y hat)}``% hat_i}x6X@8;X@x6X@8;X@x6X@8;X@ifZxXiy XithenC y Xi. CnyifZxiyi^C ^y in  n ^s '>0,\>0L:k 1W (G ) 1 ()0\:[ p1G ^(7 ^)#r# #b<# 2  e  2 'l 9  %p % ߊ$(#(# (#(#@!'#$The above system implies at once that our equilibrium is Pareto optimal.  ? We proceed now with the proof of Theorem 1*. We are given a  _ Pareto inferior equilibrium outcome #0__dd <<Az hat_j,j=1,...,nx6X@8;X@x6X@8;X@x6X@8;X@&__z+jW_j_n_,G_1_,7_._.'_._,_A߾ of the average cost game. By the claims i and ii we can assume that agents 1 and  _  2 are active and we know that $0%*L__%*dd <<IAC(y hat)_,.߫, Pareto superior to our equilibrium  ? outcome, and we assume 7A$0x"88dd << x hat 0:\{AC(t')=AC(t)\} => \{AC(t')=C_'(t')\}x6X@8;X@x6X@8;X@x6X@8;X@_for_all_t2_AC_tJ _AC _t _AC_t_C_t&._ _~"_.:nR_>_0B_:_{"_(Z_): _(* _) _} _> _{r_(_)b_(_)_}_$(#(#O(#(#X!!'#$Thus if *&0O*p __O*dd"<<AC(y_i+y_0)=C_'(y_i+y_0)x6X@8;X@x6X@8;X@x6X@8;X@_ACz_y+i_yw_C_y4+i_y_(7+0_)?_(t+0_)G__ :_*, equation (17) implies  _ '0*;__*dd"<<{@AC(F(x_i+x_0))=C_'(F(x_i+x_0))~ < ~AF(x_i+x_0)=F_'(x_i+x_0)x6X@8;X@x6X@8;X@x6X@8;X@_ACz_Fj_x+i_x_C_F_x+iT _xU _AF _x:+i_x_F_xw+i?_x_(_('+0w_)_)_(_( +0 _) _)E _(z+0_)_(+0_)7_g_sW:_d _<_B_N2:_{. In this case we have  ?O A#xddddd*dd"x7phi_'(y_i;y_0)=AC(y_i+y_0);psi_'(x_i;x_0)=AF(x_i+x_0)x6X@8;X@x6X@8;X@x6X@8;X@_-_1:__K K : _E__($_;+0d_)D_(y+0_)A_; _( _; +0 _)_(5+0_)W_y+i_yT_AC_y9+i_y _x +iX _x _AFx_x+i_xߎ$(#(#O (#(#X!A'#$and the announced equality follows at once. Q.E.D.  ?7  Proof of Corollary 1  Suppose all agents have identical preferences. The unique equilibrium of both games must be symmetrical, hence  _ a#x'ddddddd"x$z_i^c=z^c over n; z_i^r = z^r over nx6X@8;X@x6X@8;X@x6X@8;X@0zcizWc8nfz rizWrO8n'];$(#(#(#(# !a'#$Both !'0*G__*dd"<<z_i^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:i and A'0*| G__*dd"<<z_i^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:i are on the graph a'0|88dd"<<tgammax6X@8;X@x6X@8;X@x6X@8;X@8t of the convex function  _[ '0a__add"<<d!beta_i  alpha_i=C(n beta_i)/nx6X@8;X@x6X@8;X@x6X@8;X@_^__+i+i_C_n+i_n_2_"_(_)^_/d. By Theorem 2, '0* __*dd"<<z_i^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:i is below (or equal to) '0*e__*dd"<<z_i^rx6X@8;X@x6X@8;X@x6X@8;X@_zr:i on '0a"88dd"<<tgammax6X@8;X@x6X@8;X@x6X@8;X@8t.  _ The Nash equilibrium property of (0*( __*dd"<<z_i^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:i says that agent i's  _   upper contour at !(0* C#__*dd"<<z_i^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:i is supported by a line with slope A(0*,C#__*dd"<<%_i^cx6X@8;X@x6X@8;X@x6X@8;X@_%c:i, where: #xl%dddddG dd"x6?1 over {phi_+'(y^c over n; {(n1)} over n y^c)} ~ sigma_i^cx6X@8;X@x6X@8;X@x6X@8;X@ ?k1)(;( 1))\-t%yWc8nn8nyc ci6߳$(#(#W(#(#!'#$Moreover, formula (7) yields (#xl+ddddd' dd"xJ0 `H1 over {C_+'(y^c)}  1 over {phi_+'(y^c over n; {(n1)} over n y^c)} x6X@8;X@x6X@8;X@x6X@8;X@N Mk13()C1(~;1(1 ) )kCyR>cjyWc8nn!8n yM cW7gg!-($(#(#$(#(#1!'#$p"0*((@@Q '# !"'#A"''#a"%'#f)"+'#."pԒCombining these two inequalities we deduce: the slope of a  _ supporting line through a(0*__*dd#<<z_i^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:i is not smaller than the slope of  _ ;(03*$ __3*dd#<<gamma~at~z_i^cx6X@8;X@x6X@8;X@x6X@8;X@__at)_zc:i;. Hence agent i does not prefer any point on (0sY 88dd#<<tgammax6X@8;X@x6X@8;X@x6X@8;X@8t beyond (0*G"$ __*dd#<<z_i^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:i  _8 (such as (0*x __*dd#<<z_i^r x6X@8;X@x6X@8;X@x6X@8;X@_zr:i) to )0*\ x __*dd#<<z_i^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:i. Q.E.D.  ?  Proof of Corollary 2  Under the assumptions of Theorem 2, we suppose that no agent prefers the average cost equilibrium outcome over the average  t return one:  _ #! ddddd*dd#xg& ?  (18)  HZz_i^c` v _i z_i^r~for~all~i~(where~ PRECEQ_i~ is~ agent~ i's~ preference~relation) x6X@8;X@x6X@8;X@x6X@8;X@_zc:i+i_zrd:i6_for_all_i_where& +i _is _agent_i_sV_ preference^_relation_v _vZ_(_)H$d&d&d&d&X!c&$ t and we must show that W!)0*d__*dd#<< z_i^c=z_i^rx6X@8;X@x6X@8;X@x6X@8;X@_zc:i_z(r:i _W for all i. First we order the agents in such a way that   #xHddddd *dd#x$ ? ` (19) ăH4y_1^ry_1^c~~y_2^ry_2^c~...~y_n^ry_n^cx6X@8;X@x6X@8;X@x6X@8;X@_yr_y(c_yGr_yc _y r :n} _y$ c :n:1:1:2:2_.N_._. ___^_>_ _H$(#(#(#(#X!'#$By Theorem 2, A)0Q@ Qdd#Hsum_i(y_i^ry_i^c)0x6X@8;X@x6X@8;X@x6X@8;X@9I+iyrjai\ycaiu(S)C0H, therefore (19) implies:   #xB ddddd_ Qdd#!$ ? `eE  (20) ă1;sum_{i2} (y_i^ry_i^c)0~ < ~y_{1}^cy_{1}^rx6X@8;X@x6X@8;X@x6X@8;X@9I+iy4r aiycyaiyKc4 y r>+k<a a+2()0la1 a11$(#(#(#(# !'#$(with the notation a)0J b#Jdd#-y_{i} = Sum _{j != i} y_j)x6X@8;X@x6X@8;X@x6X@8;X@yRi~+j+iy"RjR*+c9Ir)-ߪ. Now we use the equilibrium  _ property of )0*P %__*dd#z_1^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:1:  _ " !#xJ(dddddZ*dd#!-I(phi(y_1^r;y_{1}^c);y_1^r)~ PRECEQ_1 (phi(y_1^c;y_{1}^c);y_1^c)=z_1^cx6X@8;X@x6X@8;X@x6X@8;X@_(_(:1v_;:1_)~_;n:1_)5+1_(z_(j :1 _;) :1y _) _; :1` _):1_-_-_y&r_yc_yr_y ca _y ci _y cP_zcf:_v : _-ߪ$(#(# "(#(#X!!'#$Note that l)0` *__dd#phi(beta_1;y_0)x6X@8;X@x6X@8;X@x6X@8;X@_-__(+1_;+0_)S_yl is non decreasing in )0V*__dd#y_0x6X@8;X@x6X@8;X@x6X@8;X@_y+0 (as AC is), hence by monotonicity of preferences and (20):`L&#0*((@@A c&a#H'##B '#"#J('#*!#`ԌA#x@ddddd*dd$!=Iz_1^r=(phi(y_1^r;y_{1}^r);y_1^r)~ PRECEQ_1 ((phi(y_1^r;y_{1}^c);y_1^r)x6X@8;X@x6X@8;X@x6X@8;X@_zr_yr]_yre_y r _y rH _y cP_yr:1_(v_(f:1_;%:1u_)_;:1\_)+1_(l _(a _(Q :1 _; :1` _) _;:1G_) _:,_v :_- _-=ߺ$(#(#(#(#X!A'#$Comparing the above two inequalities with our assumption (18) we conclude that agent 1 is indifferent between all three allocations  _x V)0* __*dd$ z_1^r,z_1^cx6X@8;X@x6X@8;X@x6X@8;X@_zr_z(c:1 _,:1V and *0x* __x*dd$(phi(y_1^r;y_{1}^c);y_1^r)x6X@8;X@x6X@8;X@x6X@8;X@_(_(:1v_;:1_)~_;n:1_)_-_y&r_yc_yrf:. In fact, it must be the case that  _ W!*0* __*dd$ y_1^r=y_1^cx6X@8;X@x6X@8;X@x6X@8;X@_yr_y(c:1:1 _W because the separation between agent 1's upper contour at A*0*" __*dd$z_1^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:1 and his/her average cost opportunity set (namely the graph of  _ a*0*__*dd$d!beta_1  phi(beta_1;y_{1}^c))x6X@8;X@x6X@8;X@x6X@8;X@_^_-S_+1_(+1'_;g:1_)/_)_:_yFcd is strict: the two sets intersect only at *0* __*dd$z_1^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:1. To see why the separation is strict, recall that agent 1's preferences are strictly convex and/or C is strictly convex. If C is strictly  _$ convex, so is the function *0*Xd__*dd$phi(beta_1;y_{1}^c)x6X@8;X@x6X@8;X@x6X@8;X@_-__(+1_;:1k_)S_yc:ߒ (we omit the proof for brevity).  _ Once we know W*0* H__*dd$ y_1^r=y_1^cx6X@8;X@x6X@8;X@x6X@8;X@_yr_y(c:1:1 _W, we proceed by induction. Inequalities  _\ (19) now yield *0* __*dd$y_{2}^c``y_{2}^rx6X@8;X@x6X@8;X@x6X@8;X@_yc_yur:@_F::2:2߄and we show by the same argument that agent  _ 2 is indifferent between V+0*h__*dd$ z_2^r,z_2^cx6X@8;X@x6X@8;X@x6X@8;X@_zr_z(c:2 _,:2V and !+0x*K__x*dd$(phi(y_2^r;y_{2}^c),y_2^r)x6X@8;X@x6X@8;X@x6X@8;X@_(_(:2v_;:2_)~_,n:2_)_-_y&r_yc_yrf:. The separation  _ between agent 2's upper contour at A+0*D"__*dd$z_2^cx6X@8;X@x6X@8;X@x6X@8;X@_zc:2 and his/her average cost  _X opportunity set is strict, hence Wa+0*($__*dd$ y_2^r=y_2^cx6X@8;X@x6X@8;X@x6X@8;X@_yr_y(c:2:2 _W, again. The induction  _ argument now establishes W+0*h&__*dd$ y_i^r=y_i^cx6X@8;X@x6X@8;X@x6X@8;X@_yr:i_y(c:i _W for all i, and the proof of  ?# Corollary 2 is complete. Q.E.D. 0'$0*((@@@'#A$0Ԍ ? % REFERENCES ă Billera L. and D. Heath, 1982, Allocations of Shared Costs: a set  ?  of axioms yielding a unique procedure, Mathematics of  ?x  Operations Research 7, 3239. BarHillel, M. and M. Yaari, 1993, Judgments of Justice, in  ?  X  B. Mellers and J. Baron, Editors, Psychological Perspectives  ?(  on Justice, Cambridge: Cambridge University Press, Ch. 4, 5584.  ?  X Bryant, J., 1987, The Paradox of Thrift, Liquidity Preference and  ?H  Animal Spirits, Econometrica, 55, 5, 12311235.  ? Case, J.H., 1979, Chapter 2 in Economics and the Competitive  ?h  Process, New York: New York University Press. Dasgupta, P. and G. Heal, 1979, Economic Theory and Exhaustible  ?  Resources, Cambridge: Cambridge University Press.  ? Hardin, G., 1968, The Tragedy of the Commons, Science, 162:12438. Israelsen, L.D., 1980, "Collectives, Communes, and Incentives,"  ?8  Journal of Comparative Economics, 4, 99124. Kitch, E.W., 1977, The Nature and Function of the Patent System,  ?X  The Journal of Law and Economics, 20, 265290. Loury, G.C., 1979, Market Structure and Innovation: A  ?x  Reformulation, Quarterly Journal of Economics, 93,3, 395409.  ? Moulin, H., 1995, Chapter 6 in Cooperative Microeconomics, forthcoming, Princeton University Press.  ?(# Moulin, H. and S. Shenker, 1992, Serial Cost Sharing, Econometrica, 60,10091037. '%0*((@@ԌMoulin, H. and S. Shenker, 1994, Average Cost Pricing Versus Serial  ?  Cost Sharing: an axiomatic comparison, Journal of Economic  ?   Theory, 64,1, 178201.  ? Ostrom, E, 1991, Governing the Commons, New York: Cambridge University Press.  ? Posner, R.A., 1992, Chapter 3 in Economic Analysis of Law, Fourth  ?`  Edition, Boston: Little, Brown and Company. Riley, J.G., 1991, Asymmetric Contests, Mimeo, UCLA. Roemer J.E., 1989, A Public Ownership Resolution of the Tragedy of  ?  the Commons, Social Philosophy and Politics, 6, 2, 7492. Romano, R., 1988, Oligopolistic Competition for Market Share Via  ?0  Voluntary Excess Supply, International Journal of Industrial  ?  Organization, 6, 447468. Shenker, S., 1990, Making Greed Work in Networks: a game theoretic analysis of gateway service disciplines, mimeo, Xerox Palo Alto Research Center. Shubik M., 1962, Incentives, Decentralized Control, the Assignment  ?  of Joint Costs and Internal Pricing, Management Science, 32543.  ? Tullock, G., 1980, Efficient Rent Seeking, in Toward a Theory of  ?@  the Rent Seeking Society, J. Buchanan, R. Tollison and G. Tullock, Editors, Texas A & M Press. Watts, A., 1994, On the Uniqueness of Equilibrium in Cournot Oligopoly and Other Games, forthcoming in Games and Economic Behavior. '&0*((@@ԌWatts, A., 1995, Uniqueness of Equilibrium in Cost Sharing Games, mimeo, Vanderbilt University.'0*((@@  ?  & FOOTNOTES ă 1.  X @ +0_e_edd(<<w {dx_1 over dy_1} = y_1 over {y_1+66 {2 over 3}}  C'(y_1+66 {2 over 3})+ {66 {2 over 3 }} over {(y_1+66 {2 over 3})^ 2} C(y_1+66 {2 over 3}), x6X@8;X@x6X@8;X@x6X@8;X@sN^  P   {    kdxkdyy{yCy yCzy[1[11166283(za1B66m 2m 3 )m r66 2 3- ( 1 66283)2(a16623z),C^z m Bw    which equals 1 at ]+0% %dd(<<y_1=33 1 over 3x6X@8;X@x6X@8;X@x6X@8;X@!y1a33183 o] 2.    +0eedd(<<O {dy_1} over dx_1 = x_1 over {x_1+33 {1 over 3}} F'(x_1+33 {1 over 3})+{33 {1 over 3}} over {(x_1 + 33 {1 over 3})^2} F(x_1+33 {1 over 3}) x6X@8;X@x6X@8;X@x6X@8;X@s]m  -   X    zdyzdxxxFx xgFWxj1j11133183g(Wa133J 1J 3 )J r33u 1u 3 ( 1 33 1 83)2(a13313W)R;W J O    which equals ,0 dd(<<3 over 2x6X@8;X@x6X@8;X@x6X@8;X@ ?\3\82 for !,0z__dd(<<x_1x6X@8;X@x6X@8;X@x6X@8;X@_x+1 slightly larger than (A,02dd(<< 16 2 over 3x6X@8;X@x6X@8;X@x6X@8;X@!16L2L83 /( @3. When n identical agents (n3) with the same utility as above share our piecewise linear technology, the average cost equilibrium overproduces by 50(1(3/n))% and yields the joint surplus 150/n; compare with 50(1(3/2n))% overproduction and the joint surplus 75/n at the average return equilibrium. The efficient surplus is 50 for all n. 4. Notice that if either u(y) or v(x) is a linear function, we have a familiar quasilinear utility function. 5. Of course the two average mechanisms are not the only interesting mechanisms with this property. Several arguments, both strategic and normative, against the average cost and average return mechanisms are developed in Moulin and Shenker [1992], [1994] as well as Moulin [1995]. Yet these two average sharing rules have the enduring appeal of their great simplicity.