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Utility functions for Debreu's ‘excess demands’
Journal
of Mathematical
UTILITY
Economics
FUNCTIONS
13 (1984) l-9. North-Holland
FOR DEBREU’S
‘EXCESS
DEMANDS’
John GEANAKOPLOS* Yale University, New Haven, CT 06520, USA Received
January
1979, final version
accepted
September
1983
Given an arbitrary function x: Iw’-+Iw’ satisfying Walras law and homogeneity, Debreu decomposed x into the sum of 1 ‘individually rational’ functions x(p) =x = i Z’(p). Here we find explicit utility functions UL, constructed on the basis of a simple geometric intuition, which give rise to Debreu’s excess demands Z”(p).
In a series of papers by Sonnenschein (1973), Mantel (1974), Debreu (1974), McFadden et al. (1974) and Mantel (1976), it has been demonstrated that neoclassical microeconomic theory imposes almost no restriction on community excess demand functions other than Walras law and homogeneity, if the economy contains no more commodities than consumers. Unfortunately, many of the ideas in these important proofs are hidden by the extremely complicated nature of the constructions. Perhaps the most remarkable of these proofs is Debreu’s. He decomposes an arbitrary continuous function x(p) on &+ (the ‘candidate excess demand’) satisfying Walras law and homogeneity of degree zero into 1 functions X”(P), k=l,... ,l and he constructs 1 systems of convex, monotonic indifference curves in such a way that, subject to the budget constraint p’x SO, X”(p) lies on the highest indifference curve of the kth system, k = t, . . . ,I, (so long as p is not too close to the boundary where some price is zero). In this paper I write down explicit concave and monotonic utility functions uk, k = 1,. . . ,I, constructed on the basis of a simple geometric intuition, such that maximizing the kth utility function, uk, subject to the budget constraint p’x 50, gives exactly the kth individual excess demand X”(p) in Debreu’s decomposition (away from the boundary). In order to guarantee concavity and monotonicity of the utility functions I impose the restriction that the original x(p) be differentiable as well as continuous, but I hope thereby to bring the ideas lying behind Debreu’s decomposition into sharper focus. The plan of this article is as follows. First the arbitrary x(p) is decomposed, x(P) = x = 1zk(p) = 1: = 1 fik(P)g k(p) as in the Debreu paper, where only the scalar functions Bk depend on the original candidate excess demand x(p). *I would like to acknowledge helpful conversations with Heraklis Polemarchakis, Truman Bewley and Bob Anderson, as well as the very useful comments of an anonymous referee.
J. Geanakoplos,
2
Utility functions for Debreu’s ‘excess demands’
In Part 2 we observe that ak(p) is the closest point in the budget set {XE [w’1p’x50) to the kth unit vector ek. We also note that gk(p) and ik(p) are elements of zk = {x E Iw’1Xi < 0, i # k, xk > O> for all p E @+, and conversely that for any x ~2~ there is a unique p(x), namely p(x) =ek -xkx/Ix12, such that Zk(p(x)) and Xk(p(x)) are scalar multiples of x. It follows immediately that p(X”(p))=p and hence that the utility function uk defined on all of 8” by uk(x)= -(J~k(p(x))=ek]2+~~-~k(p(~))~2) is uniquely maximized over the set {xE[W’Ip’x~O} f-18~ at X”(p), for any PEW+. In order to assure monotonicity and concavity, we must restrict our attention in Part 3 to a compact set of prices P, for instance {p E R’ 1Pi 2 E, i=l , . . . , 1,JpI= l} for any E> 0. Then each individual excess demand X”(p) will lie in some large convex and compact subset Xk of R’. By perturbing the utilities and taking advantage of the differentiability assumption, it is shown that each utility uk can be taken to be monotonic and strictly concave on Xk and still give rise to the same Z”(p). Finally, in Part 4 of the argument, uk is extended to all of R’, preserving concavity, monotonicity and the excess demands for all PEP, but perhaps not for p outside of P. Observe that since Xk is bounded, we can choose wkE R:, the initial endowment vectors for k = 1,. . . ,l, large enough so that the net trade space R’+ - wk contains Xk for all k = 1,. . . , 1. Notation Letfi’+=(pER’)p,>O,i=l,..., r}. We denote by T(p) the set {xER’ Ip’x=O} for all PEA:, where p’x means p transpose. We let p I x mean p’x = 0. Let ek be the kth standard basis vector, k = 1,. . . , I and let ZITcpjy be the projection of y perpendicularly onto T(p), i.e., in the direction p. Then HTCpjY= CI-PPfl(P(21Y. We write x>y iff XizYi,i=l,..., I and x#Y and x%y iff xi>yi,i=l,..., 1. Let x(p) denote a function x:R$+R’. We call x(p) a candidate aggregate excess demand function if it satisfies: (1) Homogeneity (H): x(p) =x(lp) for all A>0 and all pal?+. (2) Walras law (W): p’x(p) =0 for all pEti:. (3) Twice continuous differentiability (C’): x(p) is C2 on R$. We define a rational agent as an ordered pair (u,X) satisfying: (1) X=R$-w, for some WER’+, that is X is the set of net trades. (2) u is a function U: X+R which is monotonic, x > y*u(x) > u(y) and concave, u(Ax + (1 - I)y) 2 Au(x) + (1 - I)u(y) for all x and y and 0 5 A5 1. (3) Given PE I?‘+, the agent always acts to maximize U(X) such that x f X, p’x i 0. If x(p) is the unique solution to a rational agent’s maximization problem for all PEP c h+, and if x(p) is C2, then we call x(p) a rational individual excess demand on P.
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
Recall that u is monotonic if Du(x) 90 y’D*u(x)y
and
strictly
quasi-concave
3
if
Let x(p) be a candidate excess demand function, x: 8: +R’, and let Theorem. Then there exist 1 rational agents {(uk,Xk), P be a compact set, Pea:. k=l ,, . .,I> giving rise to 1 rational individual excess demands X’(p) such that C:=lZk(p)=x(p)for all pEPcW+. -Part 1. Decomposition. By Walras law, p’x(p) =0 for all p E I?:, i.e., x(p) E T(p). Now for all p E I?+, we can find a scalar B(p) such that x(p) + &P)(P/llPll)90. If x(p) is C* and homogeneous, then 0 can be chosen so as well. Let ZITfpjy denote the projection of y perpendicularly onto T(p). (See fig. 1). If {el, e*, . . . , e’} is the standard basis for Iw’ and x(p) =(x,(p),. . . ,x,(P))~ =
6*(Pk2 1j,(P)+e(p)$$)e2= ‘_____________;;:x(p)
I
+e(p)
-
.* /--
_.*’
,
I 1
_.-’ I
Fig. 1
lhr
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
4
ck= xk(dek, thenby noting that n,,,,e(p)(p/IIpII) =O, I
C
X(P)= ~T(p)m = UT(p) X(P)+ O(P) p
we can set
1
IlJ4
Let bk(P) = xkb4+ e(p)(pk/l I# and 2 “(P) = Bk(dnT(p)eky
k =
1,.
. . ,l.
We
have
just shown that x(p) can be decomposed at all PE @+ into I functions x”(P) =bkb)nT(p)ek, k= 1,. . . , 1 satisfying H, W, and C2 and that &(p) >O for all pE@+, and k=l,..., 1.
x(P)
=
kil
x k(d
=
,$
Pk(P)nT(p)ek.
Observe that IZTo,)ek is a vector with a strictly positive kth coordinate and 1- 1 strictly negative components, that is, x:(p) CO, if k and 0
There
xE{XEXk~p’x~0}]
exist functions ul,. . . ,d such that max [uk(x) such that is uniquely attained at zk(p) for all p E I?+ and k = 1,. . . , 1.
As a first step, consider the special case where Pk(p) = 1 for all p. Then X”(p) = IITcP,ek. Lemma I. Let C,,(x)= -l)lx-y112= -_~=1(xk-yk)2. Then ii, is a monotonic strictly concave utility function on X = {x E R’ 1xi < yi, i = 1,, . . , I} whose derived rational individual excess demand z(p) is exactly ZIITlp,yfor all PER’ with p’y > 0. Proof:
u,(x) is exactly the negative of the square of the distance between x and y. Maximizing u”,(x) on {XE R’ I p’xg0) E B(p) is equivalent to minimiz-
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
ing the distance between y and B(p), which obviously ZIT,o,y so long as pry20. See fig. 2. Moreover, on X,u”, Dii,(x)=D[-If=, (xi-yi)‘]= -2(. Xl -yI,...,xl-yy,)9b uY is monotonic. Furthermore, Cj2;t&,(x)= -21, hence concave. Q.E.D. \‘
I
5
occurs uniquely at is differentiable and since x
T(pa)
Fig. 2
Since p’ek > 0 for all p E Is’+ and k = 1,. . . , I, it follows that ak(p) = ZITCp,ekis a rational individual excess demand function that can be derived from the utility function u”“(x)= -~~x-_$/~~, k= 1,. . .,l, once iik is properly extended beyond X. The difficulty that remains is to prove that we can modify iIk(x), getting u”(x) such that the solution to max u”(x) given x E {x E Xklpfx SO} is Xk(p)= ek rather than IITo,)ek. bk(dnT(p) The Debreu construction of the indifference curves can be thought of as the bending of the circles centered at ek until they are tangent to T(p) at Pk(p)IITo,)ek instead of at IITcP,ek. See fig. 3. Debreu proved that this is possible on IRg= {p E l@+1(p~J\p(() 2 E}. W e sh a 11sh ow it is actually possible on all of dB’by defining a utility function on all xk = {x E R’ 1xi < 0, i # k, xk > O}. The idea is that given any x E Wk we can find a unique price vector p(x) E @+ and a unique multiple A(x)x of x such that I(x)x = ZITCpCxJjek. From Pythagoras’ I?_-,, hav: yth this not;tpn i&(x)= -I!,--ekjj2= -(llnT(p(x))ek-ek)12+ r.(pCxjje11)=(111(x)x-e I +11x--~(x)xIl ). We can get the correct excess demand by defining ak(x) = -( IflTCpCxnek -ek112 + IIX-Bk(P(X))nTcp(x))ekl(‘).
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
T(P)
Note that Ilx-e'1/2 = Ile'-nT(p)e'll* + iknT(p)e'/f
Fig. 3
Proof of Part 2. The ray {AX(L > O> and the point ek not on the line (since Lxi ~0 and 4 =0 for if k) determine a plane. Hence there is a unique line perpendicular to x through ek, namely p(x) = ek -xk(x/ xl’). Observe that p(x)‘x =xk -x~(Ix[‘/Ix~~) =0 and that, letting A(x)=x,/(x 2, I(x)x +p(x) =ek, hence indeed Ii’Tu,CXuek=L(x)x. See fig. 4. It can be seen that the vector p is the vector of residuals derived from the regression of ek onto x and that the first term in the above expression is simply the mean squared error of that regression, namely -/I$*-$lJi=
--l~f$-$i[=
-(f$-Zf$+l)=f$-1.
Geometrically, as p varies through I?:, II Tfpjek traces out. the hemisphere with center at $ek. Given any x E%~, the line from the origin through x intersects that hemisphere at exactly one point, I(x)x. The line segment from 2(x)x to ek is p(x). Recall from elementary geometry that two lines connecting the two endpoints of the diameter of a (semi)circle to the same point on the semicircle must meet at a right angle. Observe that p is uniquely determined up to positive scalar multiples by the line x and the point ek.
.l. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
T(P)
Fig. 4
Note that in fact p is a differentiable function of x. Note that xk was chosen so that the uniquely defined p(x) is strictly positive for all xeTk. See fig. 4. Observe that if x =ZIr(p)ek, then p(x) is indeed p, and if x =/WTCP)ek, then also p(x) =p. Thus we can define
u”(x)= - Ilb(p(x)) ek~ekl(2~~~X-~k(~(X))nr(p(x))ekl~2~ From the above demonstration we know that the problem maxuk(x) such that x ~8~ and p’x 50 is solved uniquely by /Ik(p)flTo,)ek =X”(p) since p(Xk(p)j =p and any multiple of Il ro,)ek maximizes the first term of uk subject to the budget constraint and Xk(p) uniquely maximizes the second term by making it zero. Q.E.D. Part 3.
Now let
u”(x)= -
Il&(P~X~~ek-ekl12-&exp{~llx-~k(p(x))n~,,,~,,ekll>.
For E small enough and n big enough, this is monotonic and strictly quasiconcave on a compact, convex set Xk containing {X”(p)I pep}. Proof of Part 3.
Notice first that the derived demands are unaffected. Now
8
’
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
suppose P is compact in fir+; then (Zk(p)lp E P} is compact if Xk is Co, hence we can find a closed convex, bounded Xk such that (X”(p)lp E P} c Xk ~8~. Then if x is C’, so that &(p) is Cl, the gradient of IIX-Bk(p(x))nTcpcx))ek(12 is bounded from below on Xk. But it is obvious that the gradient of is proportional to p(x) 80 and hence is bounded from VP(xnek-ekl12 1117 below by a strictly positive constant in every coordinate on Xk. Hence for all sufficiently small E>O, uk(x) is monotonic on Xk and gives rise to the excess demand X”(p). The first term, v(x) = - llZITo,(+))ek-ekl12 =(xf/lxl”) - 1 is concave in the remaining l- 1 variables given any fixed value of xk and therefore the second derivative D2u is negative definite on [ek]‘. Since ek E b(x), x] it follows that D,” is negative definite on [p(x),x]‘. It is clearly identically 0 in the direction x. Furthermore, let f(x) = 1(x-P(p(x))n,,,,,,,e”lI. Since p(x) =p(lx) for all A> 0, it is easy to see that D2f2(x) and also D2w,(x) =O”[ -enf@)] are negative definite in the direction x. Thus for small enough E we know that D2uk(x) =D’v(x) + sD’w,(x) is negative definite on [p(x)]‘. In order to prove the quasi-concavity of uk on Xk, we must show that D2uk is negative definite on [Da”(x)]* = [Du(x) + sDw,(x)]‘- = [lp(x) + sDw,(x)]‘. Any y E S’ 1 can be uniquely written as y = al(p(x)/lp(x)l) + a2(x/(xl) + i = 3,. . . , n. Moreover, if a,(q,/lq,l) + . *. + cc,(q,/lq,l), where 4; E [p(x), xl’, wh ere the number M can be taken to be y I Duk(x), then 1~1~1 ~MME~DW,[, the same for all x in the compact set X k. Imagine expressing the matrix D2uk = D2v + sD2w, in the orthogonal basis
and then calculating y’D2aky. If czl= 0, then we would be done. If u2 = 0, then for small enough E we would also get y’D2uky<0, since D2v is negative definite on [p(x),x]’ and a,+0 and sD’w,+O as s+O. The only difficult case arises when a3 = ... = GI,=O. Then y’Duky = a:~,,, + 2cr1a2uP,+ E: *0 + a$sw;, + 2a,a,swz, + a$w~,. Now notice that as s--*0, so that ~~‘0, all the terms except 2c11a2vPx+ BREW:,are O(s’) and effectively disappear. We can assume that ~“2= 1 -cc:, that Iall 5 M.slDw,(, and that w!J, ~0. It follows that if Iw:,.l/l~w,,I is sufficiently big, then this last remaining sum is negative. And that is the point of introducing the exponential operator here. The reader can quickly verify that by taking it large, the ratio of second derivative to first derivative of ePnx can be made arbitrarily large. It follows similarly that
Iw:xl_ n
IDwnl - IDfo
can be made arbitrarily large.
large on the compact set Xk by taking II sufficiently
J. Geanakoplos,
Utility functions for Debreu’s ‘excess demands’
9
We can modify the utility functions uk one more time so that they are strictly concave and monotonic on Xk, k = 1,. . . ,I, and give rise to the same excess demands Xk(p) for PEP. We can then extend the uk to all of R’, preserving monotonicity and concavity. The X”(p) will again maximize nk subject to the budget constraint p’x 5 0, for all p E P, k = 1,. . . , 1. Part 4.
Proof of Part 4. Fortunately the proof of Part 4 is quite easy. Aumann (1975) demonstrated that it is always possible to monotonically transform a strictly quasi-concave C2 function into a strictly concave function on any compact set Xk. Simply define V”(x) = -IY~~‘@) for N big enough. Then and D2Vk = [ND2uk -N2Duk(Duk)‘]e-N”k which is DVk(x)=NDukeNUk negative definite on [Dak]* since D2uk is, and for N big enough is clearly negative definite on all of Xk. To extend Vk to all of R’, define Uk as the inlimum of all linear functions that lie above Vk on all of Xk. Formally, for every yeXk, let Ly be the linear function Ly(x) = Vk(y) +DVk(y)‘(x-y). Since Vk is C2 and concave on Xk, Ly(x) 2 Vk(x) for all x EX~ and Ly(y) = Vk(y). Let Uk(x)=inf {Ly(x) 1y E Xk}. Since the inf of concave (linear) functions is concave, tik is concave. Moreover, iik(x) = V”(x) for all x E Xk. Furthermore, since Xk is compact and Vk continuously differentiable, Uk is well-defined and finite on all R’. Q.E.D.
Finally, observe that it is possible to choose wkER$ such that wk%>Xk(p) for all p E P. In that case we can more traditionally restrict the feasible net trade space to R’+ - wk without disturbing the maximization of utility for p E P. If we had begun with an observable aggregate endowment w as well as x(p), this last argument would not in general be valid. There would indeed be restrictions on community excess demands [at least x(p) 2 w for all p]. References Aumann, R., 1975, Values of market games with a continuum of traders, Econometrica. Debreu, G., 1974, Excess demand functions, Journal of Mathematical Economics 1, 15-21. McFadden, Mas-Collel, Mantel and Richter, 1974, A characterization of community excess demand functions, Journal of Economic Theory 9, 361-374. Mantel, R.R., 1974, On the characterization of aggregate excess demands, Journal of Economic Theory 7, 348-353. Mantel, R.R., 1976, Homothetic preferences and community excess demand functions, Journal of Economic Theory 12, 197-201. Sonnenschein, H., 1973, Do Walras identity and continuity characterize the class of community excess demand functions?, Journal of Economic Theory 6, 3455354.
of Mathematical
UTILITY
Economics
FUNCTIONS
13 (1984) l-9. North-Holland
FOR DEBREU’S
‘EXCESS
DEMANDS’
John GEANAKOPLOS* Yale University, New Haven, CT 06520, USA Received
January
1979, final version
accepted
September
1983
Given an arbitrary function x: Iw’-+Iw’ satisfying Walras law and homogeneity, Debreu decomposed x into the sum of 1 ‘individually rational’ functions x(p) =x = i Z’(p). Here we find explicit utility functions UL, constructed on the basis of a simple geometric intuition, which give rise to Debreu’s excess demands Z”(p).
In a series of papers by Sonnenschein (1973), Mantel (1974), Debreu (1974), McFadden et al. (1974) and Mantel (1976), it has been demonstrated that neoclassical microeconomic theory imposes almost no restriction on community excess demand functions other than Walras law and homogeneity, if the economy contains no more commodities than consumers. Unfortunately, many of the ideas in these important proofs are hidden by the extremely complicated nature of the constructions. Perhaps the most remarkable of these proofs is Debreu’s. He decomposes an arbitrary continuous function x(p) on &+ (the ‘candidate excess demand’) satisfying Walras law and homogeneity of degree zero into 1 functions X”(P), k=l,... ,l and he constructs 1 systems of convex, monotonic indifference curves in such a way that, subject to the budget constraint p’x SO, X”(p) lies on the highest indifference curve of the kth system, k = t, . . . ,I, (so long as p is not too close to the boundary where some price is zero). In this paper I write down explicit concave and monotonic utility functions uk, k = 1,. . . ,I, constructed on the basis of a simple geometric intuition, such that maximizing the kth utility function, uk, subject to the budget constraint p’x 50, gives exactly the kth individual excess demand X”(p) in Debreu’s decomposition (away from the boundary). In order to guarantee concavity and monotonicity of the utility functions I impose the restriction that the original x(p) be differentiable as well as continuous, but I hope thereby to bring the ideas lying behind Debreu’s decomposition into sharper focus. The plan of this article is as follows. First the arbitrary x(p) is decomposed, x(P) = x = 1zk(p) = 1: = 1 fik(P)g k(p) as in the Debreu paper, where only the scalar functions Bk depend on the original candidate excess demand x(p). *I would like to acknowledge helpful conversations with Heraklis Polemarchakis, Truman Bewley and Bob Anderson, as well as the very useful comments of an anonymous referee.
J. Geanakoplos,
2
Utility functions for Debreu’s ‘excess demands’
In Part 2 we observe that ak(p) is the closest point in the budget set {XE [w’1p’x50) to the kth unit vector ek. We also note that gk(p) and ik(p) are elements of zk = {x E Iw’1Xi < 0, i # k, xk > O> for all p E @+, and conversely that for any x ~2~ there is a unique p(x), namely p(x) =ek -xkx/Ix12, such that Zk(p(x)) and Xk(p(x)) are scalar multiples of x. It follows immediately that p(X”(p))=p and hence that the utility function uk defined on all of 8” by uk(x)= -(J~k(p(x))=ek]2+~~-~k(p(~))~2) is uniquely maximized over the set {xE[W’Ip’x~O} f-18~ at X”(p), for any PEW+. In order to assure monotonicity and concavity, we must restrict our attention in Part 3 to a compact set of prices P, for instance {p E R’ 1Pi 2 E, i=l , . . . , 1,JpI= l} for any E> 0. Then each individual excess demand X”(p) will lie in some large convex and compact subset Xk of R’. By perturbing the utilities and taking advantage of the differentiability assumption, it is shown that each utility uk can be taken to be monotonic and strictly concave on Xk and still give rise to the same Z”(p). Finally, in Part 4 of the argument, uk is extended to all of R’, preserving concavity, monotonicity and the excess demands for all PEP, but perhaps not for p outside of P. Observe that since Xk is bounded, we can choose wkE R:, the initial endowment vectors for k = 1,. . . ,l, large enough so that the net trade space R’+ - wk contains Xk for all k = 1,. . . , 1. Notation Letfi’+=(pER’)p,>O,i=l,..., r}. We denote by T(p) the set {xER’ Ip’x=O} for all PEA:, where p’x means p transpose. We let p I x mean p’x = 0. Let ek be the kth standard basis vector, k = 1,. . . , I and let ZITcpjy be the projection of y perpendicularly onto T(p), i.e., in the direction p. Then HTCpjY= CI-PPfl(P(21Y. We write x>y iff XizYi,i=l,..., I and x#Y and x%y iff xi>yi,i=l,..., 1. Let x(p) denote a function x:R$+R’. We call x(p) a candidate aggregate excess demand function if it satisfies: (1) Homogeneity (H): x(p) =x(lp) for all A>0 and all pal?+. (2) Walras law (W): p’x(p) =0 for all pEti:. (3) Twice continuous differentiability (C’): x(p) is C2 on R$. We define a rational agent as an ordered pair (u,X) satisfying: (1) X=R$-w, for some WER’+, that is X is the set of net trades. (2) u is a function U: X+R which is monotonic, x > y*u(x) > u(y) and concave, u(Ax + (1 - I)y) 2 Au(x) + (1 - I)u(y) for all x and y and 0 5 A5 1. (3) Given PE I?‘+, the agent always acts to maximize U(X) such that x f X, p’x i 0. If x(p) is the unique solution to a rational agent’s maximization problem for all PEP c h+, and if x(p) is C2, then we call x(p) a rational individual excess demand on P.
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
Recall that u is monotonic if Du(x) 90 y’D*u(x)y
and
strictly
quasi-concave
3
if
Let x(p) be a candidate excess demand function, x: 8: +R’, and let Theorem. Then there exist 1 rational agents {(uk,Xk), P be a compact set, Pea:. k=l ,, . .,I> giving rise to 1 rational individual excess demands X’(p) such that C:=lZk(p)=x(p)for all pEPcW+. -Part 1. Decomposition. By Walras law, p’x(p) =0 for all p E I?:, i.e., x(p) E T(p). Now for all p E I?+, we can find a scalar B(p) such that x(p) + &P)(P/llPll)90. If x(p) is C* and homogeneous, then 0 can be chosen so as well. Let ZITfpjy denote the projection of y perpendicularly onto T(p). (See fig. 1). If {el, e*, . . . , e’} is the standard basis for Iw’ and x(p) =(x,(p),. . . ,x,(P))~ =
6*(Pk2 1j,(P)+e(p)$$)e2= ‘_____________;;:x(p)
I
+e(p)
-
.* /--
_.*’
,
I 1
_.-’ I
Fig. 1
lhr
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
4
ck= xk(dek, thenby noting that n,,,,e(p)(p/IIpII) =O, I
C
X(P)= ~T(p)m = UT(p) X(P)+ O(P) p
we can set
1
IlJ4
Let bk(P) = xkb4+ e(p)(pk/l I# and 2 “(P) = Bk(dnT(p)eky
k =
1,.
. . ,l.
We
have
just shown that x(p) can be decomposed at all PE @+ into I functions x”(P) =bkb)nT(p)ek, k= 1,. . . , 1 satisfying H, W, and C2 and that &(p) >O for all pE@+, and k=l,..., 1.
x(P)
=
kil
x k(d
=
,$
Pk(P)nT(p)ek.
Observe that IZTo,)ek is a vector with a strictly positive kth coordinate and 1- 1 strictly negative components, that is, x:(p) CO, if k and 0
There
xE{XEXk~p’x~0}]
exist functions ul,. . . ,d such that max [uk(x) such that is uniquely attained at zk(p) for all p E I?+ and k = 1,. . . , 1.
As a first step, consider the special case where Pk(p) = 1 for all p. Then X”(p) = IITcP,ek. Lemma I. Let C,,(x)= -l)lx-y112= -_~=1(xk-yk)2. Then ii, is a monotonic strictly concave utility function on X = {x E R’ 1xi < yi, i = 1,, . . , I} whose derived rational individual excess demand z(p) is exactly ZIITlp,yfor all PER’ with p’y > 0. Proof:
u,(x) is exactly the negative of the square of the distance between x and y. Maximizing u”,(x) on {XE R’ I p’xg0) E B(p) is equivalent to minimiz-
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
ing the distance between y and B(p), which obviously ZIT,o,y so long as pry20. See fig. 2. Moreover, on X,u”, Dii,(x)=D[-If=, (xi-yi)‘]= -2(. Xl -yI,...,xl-yy,)9b uY is monotonic. Furthermore, Cj2;t&,(x)= -21, hence concave. Q.E.D. \‘
I
5
occurs uniquely at is differentiable and since x
T(pa)
Fig. 2
Since p’ek > 0 for all p E Is’+ and k = 1,. . . , I, it follows that ak(p) = ZITCp,ekis a rational individual excess demand function that can be derived from the utility function u”“(x)= -~~x-_$/~~, k= 1,. . .,l, once iik is properly extended beyond X. The difficulty that remains is to prove that we can modify iIk(x), getting u”(x) such that the solution to max u”(x) given x E {x E Xklpfx SO} is Xk(p)= ek rather than IITo,)ek. bk(dnT(p) The Debreu construction of the indifference curves can be thought of as the bending of the circles centered at ek until they are tangent to T(p) at Pk(p)IITo,)ek instead of at IITcP,ek. See fig. 3. Debreu proved that this is possible on IRg= {p E l@+1(p~J\p(() 2 E}. W e sh a 11sh ow it is actually possible on all of dB’by defining a utility function on all xk = {x E R’ 1xi < 0, i # k, xk > O}. The idea is that given any x E Wk we can find a unique price vector p(x) E @+ and a unique multiple A(x)x of x such that I(x)x = ZITCpCxJjek. From Pythagoras’ I?_-,, hav: yth this not;tpn i&(x)= -I!,--ekjj2= -(llnT(p(x))ek-ek)12+ r.(pCxjje11)=(111(x)x-e I +11x--~(x)xIl ). We can get the correct excess demand by defining ak(x) = -( IflTCpCxnek -ek112 + IIX-Bk(P(X))nTcp(x))ekl(‘).
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
T(P)
Note that Ilx-e'1/2 = Ile'-nT(p)e'll* + iknT(p)e'/f
Fig. 3
Proof of Part 2. The ray {AX(L > O> and the point ek not on the line (since Lxi ~0 and 4 =0 for if k) determine a plane. Hence there is a unique line perpendicular to x through ek, namely p(x) = ek -xk(x/ xl’). Observe that p(x)‘x =xk -x~(Ix[‘/Ix~~) =0 and that, letting A(x)=x,/(x 2, I(x)x +p(x) =ek, hence indeed Ii’Tu,CXuek=L(x)x. See fig. 4. It can be seen that the vector p is the vector of residuals derived from the regression of ek onto x and that the first term in the above expression is simply the mean squared error of that regression, namely -/I$*-$lJi=
--l~f$-$i[=
-(f$-Zf$+l)=f$-1.
Geometrically, as p varies through I?:, II Tfpjek traces out. the hemisphere with center at $ek. Given any x E%~, the line from the origin through x intersects that hemisphere at exactly one point, I(x)x. The line segment from 2(x)x to ek is p(x). Recall from elementary geometry that two lines connecting the two endpoints of the diameter of a (semi)circle to the same point on the semicircle must meet at a right angle. Observe that p is uniquely determined up to positive scalar multiples by the line x and the point ek.
.l. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
T(P)
Fig. 4
Note that in fact p is a differentiable function of x. Note that xk was chosen so that the uniquely defined p(x) is strictly positive for all xeTk. See fig. 4. Observe that if x =ZIr(p)ek, then p(x) is indeed p, and if x =/WTCP)ek, then also p(x) =p. Thus we can define
u”(x)= - Ilb(p(x)) ek~ekl(2~~~X-~k(~(X))nr(p(x))ekl~2~ From the above demonstration we know that the problem maxuk(x) such that x ~8~ and p’x 50 is solved uniquely by /Ik(p)flTo,)ek =X”(p) since p(Xk(p)j =p and any multiple of Il ro,)ek maximizes the first term of uk subject to the budget constraint and Xk(p) uniquely maximizes the second term by making it zero. Q.E.D. Part 3.
Now let
u”(x)= -
Il&(P~X~~ek-ekl12-&exp{~llx-~k(p(x))n~,,,~,,ekll>.
For E small enough and n big enough, this is monotonic and strictly quasiconcave on a compact, convex set Xk containing {X”(p)I pep}. Proof of Part 3.
Notice first that the derived demands are unaffected. Now
8
’
J. Geanakoplos, Utility functions for Debreu’s ‘excess demands’
suppose P is compact in fir+; then (Zk(p)lp E P} is compact if Xk is Co, hence we can find a closed convex, bounded Xk such that (X”(p)lp E P} c Xk ~8~. Then if x is C’, so that &(p) is Cl, the gradient of IIX-Bk(p(x))nTcpcx))ek(12 is bounded from below on Xk. But it is obvious that the gradient of is proportional to p(x) 80 and hence is bounded from VP(xnek-ekl12 1117 below by a strictly positive constant in every coordinate on Xk. Hence for all sufficiently small E>O, uk(x) is monotonic on Xk and gives rise to the excess demand X”(p). The first term, v(x) = - llZITo,(+))ek-ekl12 =(xf/lxl”) - 1 is concave in the remaining l- 1 variables given any fixed value of xk and therefore the second derivative D2u is negative definite on [ek]‘. Since ek E b(x), x] it follows that D,” is negative definite on [p(x),x]‘. It is clearly identically 0 in the direction x. Furthermore, let f(x) = 1(x-P(p(x))n,,,,,,,e”lI. Since p(x) =p(lx) for all A> 0, it is easy to see that D2f2(x) and also D2w,(x) =O”[ -enf@)] are negative definite in the direction x. Thus for small enough E we know that D2uk(x) =D’v(x) + sD’w,(x) is negative definite on [p(x)]‘. In order to prove the quasi-concavity of uk on Xk, we must show that D2uk is negative definite on [Da”(x)]* = [Du(x) + sDw,(x)]‘- = [lp(x) + sDw,(x)]‘. Any y E S’ 1 can be uniquely written as y = al(p(x)/lp(x)l) + a2(x/(xl) + i = 3,. . . , n. Moreover, if a,(q,/lq,l) + . *. + cc,(q,/lq,l), where 4; E [p(x), xl’, wh ere the number M can be taken to be y I Duk(x), then 1~1~1 ~MME~DW,[, the same for all x in the compact set X k. Imagine expressing the matrix D2uk = D2v + sD2w, in the orthogonal basis
and then calculating y’D2aky. If czl= 0, then we would be done. If u2 = 0, then for small enough E we would also get y’D2uky<0, since D2v is negative definite on [p(x),x]’ and a,+0 and sD’w,+O as s+O. The only difficult case arises when a3 = ... = GI,=O. Then y’Duky = a:~,,, + 2cr1a2uP,+ E: *0 + a$sw;, + 2a,a,swz, + a$w~,. Now notice that as s--*0, so that ~~‘0, all the terms except 2c11a2vPx+ BREW:,are O(s’) and effectively disappear. We can assume that ~“2= 1 -cc:, that Iall 5 M.slDw,(, and that w!J, ~0. It follows that if Iw:,.l/l~w,,I is sufficiently big, then this last remaining sum is negative. And that is the point of introducing the exponential operator here. The reader can quickly verify that by taking it large, the ratio of second derivative to first derivative of ePnx can be made arbitrarily large. It follows similarly that
Iw:xl_ n
IDwnl - IDfo
can be made arbitrarily large.
large on the compact set Xk by taking II sufficiently
J. Geanakoplos,
Utility functions for Debreu’s ‘excess demands’
9
We can modify the utility functions uk one more time so that they are strictly concave and monotonic on Xk, k = 1,. . . ,I, and give rise to the same excess demands Xk(p) for PEP. We can then extend the uk to all of R’, preserving monotonicity and concavity. The X”(p) will again maximize nk subject to the budget constraint p’x 5 0, for all p E P, k = 1,. . . , 1. Part 4.
Proof of Part 4. Fortunately the proof of Part 4 is quite easy. Aumann (1975) demonstrated that it is always possible to monotonically transform a strictly quasi-concave C2 function into a strictly concave function on any compact set Xk. Simply define V”(x) = -IY~~‘@) for N big enough. Then and D2Vk = [ND2uk -N2Duk(Duk)‘]e-N”k which is DVk(x)=NDukeNUk negative definite on [Dak]* since D2uk is, and for N big enough is clearly negative definite on all of Xk. To extend Vk to all of R’, define Uk as the inlimum of all linear functions that lie above Vk on all of Xk. Formally, for every yeXk, let Ly be the linear function Ly(x) = Vk(y) +DVk(y)‘(x-y). Since Vk is C2 and concave on Xk, Ly(x) 2 Vk(x) for all x EX~ and Ly(y) = Vk(y). Let Uk(x)=inf {Ly(x) 1y E Xk}. Since the inf of concave (linear) functions is concave, tik is concave. Moreover, iik(x) = V”(x) for all x E Xk. Furthermore, since Xk is compact and Vk continuously differentiable, Uk is well-defined and finite on all R’. Q.E.D.
Finally, observe that it is possible to choose wkER$ such that wk%>Xk(p) for all p E P. In that case we can more traditionally restrict the feasible net trade space to R’+ - wk without disturbing the maximization of utility for p E P. If we had begun with an observable aggregate endowment w as well as x(p), this last argument would not in general be valid. There would indeed be restrictions on community excess demands [at least x(p) 2 w for all p]. References Aumann, R., 1975, Values of market games with a continuum of traders, Econometrica. Debreu, G., 1974, Excess demand functions, Journal of Mathematical Economics 1, 15-21. McFadden, Mas-Collel, Mantel and Richter, 1974, A characterization of community excess demand functions, Journal of Economic Theory 9, 361-374. Mantel, R.R., 1974, On the characterization of aggregate excess demands, Journal of Economic Theory 7, 348-353. Mantel, R.R., 1976, Homothetic preferences and community excess demand functions, Journal of Economic Theory 12, 197-201. Sonnenschein, H., 1973, Do Walras identity and continuity characterize the class of community excess demand functions?, Journal of Economic Theory 6, 3455354.