VALORACIÓN DEL TRABAJO EN RED: VENTAJAS E INCONVENIENTES DETECTADOS

Lemma 1 For each _{i 2 I} and each _{n 2 N}, there exists a …nite partition, _Piⁿ, and a

…nite subset, ⁿ_i, of Pi, such that the following Assertions hold:

(i) 8P 2 Piⁿ; P 2 B(M) and i(P ) > 0, and ( ¹_i) 2 AS;

(ii) ⁿ_i ⁿ⁺¹_i , and, for every _{P 2 Pi}ⁿ,_{P \} ⁿ_i is a singleton;

(iii) 8! 2 Pi, _9!⁰ ₂ ⁿ_{i : k!}⁰ !k 6 L#S = n, and, hence, _{P i = [}n2N n i;

(iv) 9N 2 N; 8(Pi) (P_i), [( ^N_i ) (P_i) and #P_i 2 N, 8i 2 I] ) [Qc[(P_i)] 6= ?].

Proof Let_{i 2 I} be given and, for each _{n 2 N}, let:

Kn := fkⁿ:= (k¹_n; :::; k_n^L) 2 (N \ [1; 2ⁿ])^Lg;

P_s^kn:= Pi\ (fsg l2f1;:::;Lg]^kln₂n¹;^k₂^lnn]), for every (s; kn := (k¹_n; :::; k^L_n)) 2 Sⁱ Kn.

For each (s; n; k_n) 2 Si N Kn, such that P_s^kn 6= ?, we select a unique !^k_sⁿ 2 Ps^kn, and de…ne a set, ⁿ_{i := f!}^k_sⁿ_{2 Ps}^kn_{: s 2 S}i; k_n 2 Kn; P_s^kn 6= ?g, as follows:

for n = 1, we select one !^k_s¹ 2 Pi, for each _{s 2 S}i; we take !^k_s¹ 2 \^mi=1P_i 6= ?, whenever possible, and let ¹_{i := f!}^k_s¹ _{: s 2 S}ig;

forn > 1 arbitrary, given ^{n 1}_{i := f!}^ks^{n 1} 2 P^{skn 1} : s 2 Sⁱ; kn 12 K^{n 1}; Ps^{kn 1}6= ?g, we let, for every(s; kn) 2 Sⁱ Kn, such thatP_s^kn6= ?,³

!^k_sⁿ 8>

><

>>

:

be equal to !^ks^{n 1}, if there exists kn 12 K^{n 1}; such that !^ks^{n 1}2 ^{n 1}i \ Ps^kn

be set f ixed in P_s^kn; if ^{n 1}_i \ Ps^kn= ?

This yields, for each_{n 2 N}, a subset, ⁿ_{i := f!}^k_sⁿ _{: s 2 S}i; kn2 Kⁿ; P_s^kn 6= ?g, and a partition, _Piⁿ_{:= fPs}^kn_{: s 2 S}i; k_n 2 Kn; P_s^kn 6= ?g, ofP_i, satisfying Lemma 1-(i)-(ii)-(iii). We now prove Lemma 1-(iv), after noticing, from Lemma 1^-(i)-(ii), that ( ^N_i ) 2 AS.

3 Up to a shift in the upper boundary of Ps^kn, if required, we assume costlessly that i(Ps^kn) > 0when Ps^kn6= ?.

For each(i; n) 2 I N, we de…ne the vector spaceZ_iⁿ:= fz 2 R^J : V (!) z = 0; 8! 2 ⁿig

and derive a contradiction. Then, we de…ne N^o = max_i2IN_i and the compact set

Z := f(zi) 2 ^mi=1Z_i^?: k(zi)k = 1; Pm

i=1z_i2Pm i=1Z_ig.

Assume, by contraposition, that Lemma 1-(vi)fails. Then, from above, De…nition 3, and from [4] (De…nition 2.2, p. 397, and Proposition 3.1, p.401), for everyn> N^o, there exist an integer, Nn> n, …nite sets, P_i^Nn, de…ned for each_{i 2 I}, and portfolios,

(z_iⁿ) 2 Z, such that: ( ^N_iⁿ) (P_i^Nn) (Pi) and V (!i) zⁿ_i > 0, for every (i; !i) 2 I P_i^Nn, with one strict inequality. The sequence,_f(zⁿ_i)gn>N^o, may be assumed to converge in a compact set, say to(z_i) 2 Z. From the continuity of the scalar product and Lemma 1-(iii), the above relations on _f(ziⁿ_)gn>N^o, imply that V (!_i) z_i > 0 holds, for every

(i; !i) 2 I Pⁱ, with one strict inequality, since (z_i) 2 Z implies_{k(zi)k = 1}. We let the reader check (on asset prices), this contradicts the fact that (Pi)is arbitrage-free.

Lemma 2 For each n > N along Lemma 1, the economy Eⁿ admits an equilibrium, Cⁿ, namely, a collection of prices, !ⁿ₀ := (pⁿ₀; qⁿ) 2 M⁰, at t = 0, and !ⁿ_s := (s; pⁿ_s) 2 M^s,

Moreover, the equilibrium, Cⁿ, satis…es the following Assertions:

(iv) 8(n; i; s) 2 Nnf1; :::; Ng I S⁰, xⁿ_is2 [0; E]^L, where E := max_(s;l)2S0 LPm i=1e^l_is;

(v) 9" 2]0; 1] : p^nls > ", 8(n; s; l) 2 Nnf1; :::; Ng S L.

Proof Let n > N be given along Lemma 1. From Lemma 1-(iv), the payo¤ and in-formation structure of the economy Eⁿ,[Vⁿ; (S_iⁿ)], is arbitrage-free, along [4]. Hence, from ([6], Theorem 1 and proof) it admits an equilibrium, or (changing notations) a collection of prices, !ⁿ₀ := (pⁿ₀; qⁿ) 2 M0, !ⁿ_s := (s; pⁿ_s) 2 Ms, for each _{s 2 S}, and strategies, (xⁿ_i; z_iⁿ) 2 Biⁿ([!ⁿ_s]), for each _{i 2 I}, which satisfy Lemma 2-(i)-(ii)-(iii). The rest of the proof is similar to that of Theorem 1-(i)(simpler) and left to the reader.

Lemma 3 For the above sequence, _fCⁿ_g, of equilibria, it may be assumed to exist:

(i) !_s= limn!1 !ⁿ_s 2 M^s, for each _{s 2 S}⁰;

(ii) (x_is) := limn!1 (xⁿ_is)i2I 2 R^Lm, such that ^P_i2I (x_is eis) = 0, for each _{s 2 S}⁰;

(iii) (z_i) = limn!1(z_iⁿ)i2I 2 R^{J m}, such that ^Pm_i=1z_i = 0.

Moreover, we de…ne, for each _{i 2 I} and each _{n 2 N}, the following sets and mappings:

* the mapping, _{! 2 P}i7! argⁿi(!) 2 ⁿi, such that one _{P 2 Pi}ⁿ satis…es (!; argⁿ_i(!) 2 P²;

* from Assertion (i) and Lemma 2-(v), the belief, _i := _1+#S¹ ( _i+ P

s2S s), where s is (for each _{s 2 S}) the Dirac’s measure of !_s;

* P_i = Pi[ f!sg^s2S, the support of _{i 2 B};

* Bi(!; z) := f x 2 R^L+: ps(x eis)6 V (!) z g, for every ! := (s; ps) 2 M; z 2 R^J. Then, the following Assertions hold, for each _{i 2 I}:

(iv) fargⁿi(!)gⁿ2N converges to ! uniformly on Pi;

(v) 8s 2 S, _fxisg = arg max ui(x_i0; x), for _{x 2 B}i(!_s; z_i), along Assertions (i)-(ii)-(iii); we denote by x_i!

s := x_is2 R^L+ a related consumption decision contingent on !_s2 Ms;

(vi) the correspondence _{! 2 Pi} 7! arg max ui(x_i0; x), for _{x 2 B}i(!; z_i), is a continuous mapping, denoted by _{! 7! xi!}. The mapping, x_i : ! 2 f0g [ Pi 7! xi!, de…ned from Assertions (ii), (v) and above, is a consumption plan, that is, x_i 2 X( i);

(vii) U_i( _i; x_i) = lim_n!1uⁿ_i(xⁿ_i) 2 R+.

Proof Assertions (i)-(ii)result from Lemma 2-(iv)and compactness arguments.

Assertion (iii) For each _{i 2 I}, we let Z_i := fz 2 R^J : V (!) z = 0; 8! 2 Pⁱg and recall, from Lemma 1’s proof, that Z_i = fz 2 R^J : V (!) z = 0; 8! 2 ⁿig, for all n>N

(assumingN >N^o). The sequence_f(ziⁿ)i2Ig^n>N is bounded. Indeed, let := maxi2Ikeⁱk. The de…nition of _fCⁿ_gn>N yields, from budget constraints and clearance conditions:

(I) [Pm

i=1 z_iⁿ= 0 and V (!_i) z_iⁿ> , _{8(i; !}i) 2 I ⁿi], for every n > N.

Assume, by contradiction, _f(zⁿ_i)g is unbounded, i.e., there exists an extracted sequence, _f(zi^'(n)_)g, such that _{n < k(z}^'(n)_{i )k 6 n+1}, for every n > N. From (I), the portfolios (zⁿ_i) := ¹_n(z^'(n)_i )meet the relations _{1 < k(z}ⁿ_{i)k 6 1+n}¹, for every n > N, and:

(II) Pm

i=1 zⁿ_i = 0and V (!i) zⁿ_i > _n, _{8(i; !}i) 2 I ⁿi.

From (II), Lemma 1-(ii), the continuity of the scalar product and above, the sequence _f(zⁿ_i)g may be assumed to converge, say to (z_i), such that_{k(zi)k = 1}and:

(III) Pm

i=1z_i = 0 and V (!i) z_i > 0, _{8(i; !}i) 2 I ^Ni .

From relations(III), Lemma 1-(iv), ([4], Proposition 3.1) and above, the relation

(z_i) 2 ^mi=1Z_i holds and implies(z_i) = 0, from the elimination of useless deals of sub-Section 2.4, which contradicts the fact that _{k(zi)k = 1}. Hence, the sequence _f(ziⁿ_)g is bounded and may be assumed to converge, say to (z_i) 2 R^{J m}. Then, the relation

Pm

i=1z_i = 0 results asymptotically from the clearance conditions of Lemma 2-(iv). Assertions(iv) is immediate from the de…nition and compactness arguments.

Assertion (v) Let (i; s) 2 I S be given. For every tuple (n; ! := (s; ps); !⁰; z) 2 N nf1; :::; Ng Ms M^s R^J, we consider the following (possibly empty) sets:

B_i(!; z) := fy 2 R^L+: p_s(y e_is)6 V (!) zgandB_i⁰(!; !⁰; z) := fy 2 R^L+: p_s(y e_is)6 V (!⁰) zg.

For each n > N, the fact that _Cⁿ is an equilibrium of _Eⁿ implies, from Lemma 2:

(I) (!^{n 1}_s ; !ⁿ_s) 2 M²s and xⁿ_is2 arg maxy2Bi⁰(!ⁿ_s;!ⁿs ¹;zⁿ_i)ui(xⁿ_i0; y).

As a standard application of Berge’s Theorem (see, e.g., [8], p. 19), the corre-spondence(x; !; !⁰; z) 2 R^L+ Ms Ms R^J7! arg maxy2B⁰i(!;!⁰;z) u_i(x; y), which is actually a mapping (from Assumption A2 ), is continuous at (x_i0; !_s; !_s; z_i), since u_i and B_i⁰

are continuous. Moreover, the relation (x_i0; x_is; !_s; z_i) = lim_n!1(xⁿ_i0; xⁿ_is; !ⁿ_s; z_iⁿ) holds from Lemma 2-(i)-(ii)-(iii). Hence, the relations(I) pass to that limit and yield:

fx_i!sg := fxisg = arg maxy2Bi(!s;z_i)ui(x_i0; y).

Assertion (vi)Let _{i 2 I} be given. For every _{(!; n) 2 P}i N n f1; :::; Ng, the fact that

Cⁿ is an equilibrium of _Eⁿ and Assumption A2 yield:

(I) fxⁿiargⁿ_i(!)g = arg max ui(xⁿ_i0; y) for _{y 2 B}i(argⁿ_i(!); z_iⁿ).

From Lemma 2-(ii)-(iii)-(iv), the relation(!; x_i0; z_i) = lim_n!1(argⁿ_i(!); xⁿ_i0; z_iⁿ)holds, whereas, from Assumption A2 and ([8], p. 19), the correspondence (x; !; z) 2 R^L+ Pi R^J 7! arg maxy2Bⁱ(!;z) ui(x; y) is a continuous mapping, since ui and Bi are continuous. Hence, passing to the limit into relations (I) yields a continuous map-ping, _{! 2 P}i 7! xi! := arg max_y2Bi(!;zi)ui(x_i0; y), which, from Lemma 3-(v) and above, is embedded into a continuous mapping,x_i : ! 2 f0g [ Pi 7! xi!, i.e.,x_i 2 X( i).

Assertion (vii) Let _{i 2 I} and x_i 2 X( i) be given, along Lemma 3-(vi). Let '_i : (x; !; z) 2 R^L+ P_i R^J7! arg maxy2Bi(!;z) u_i(x; y)be de…ned on its domain. By the same token as for proving Assertion (vi), '_i and Ui : (x; !; z) 2 R^L+ Pi R^J 7! uⁱ(x; '_i(x; !; z))

are continuous mappings and, moreover, the relations ui(x_i0; x_i!) = Ui(x_i0; !; z_i) and

u_i(xⁿ_i0; xⁿ_{i arg}n

i(!)) = U_i(xⁿ_i0;argⁿ_i(!); z_iⁿ) hold, for every_{(!; n) 2 P}i N n f1; :::; Ng. Then, the uniform continuity of ui and Ui on compact sets, and Lemma 3-(ii)-(iii)-(iv)yield:

(I) 8" > 0; 9N^"> N : 8n > N^"; 8! 2 Pⁱ,

j ui(x_i0; x_i!) u_i(xⁿ_i0; xⁿ_{i arg}n

i(!)) j +P

s2S j ui(x_i0; x_is) u_i(xⁿ_i0; xⁿ_is) j < "^.

Moreover, we recall the following de…nitions, for every n > N:

(II) U_i( _i; x_i) := _1+#S¹ R Then, Lemma 3-(vii)results immediately from relations(I)-(II)-(III)above.

Proof of Lemma 4 It is similar to that of Lemma 3, hence, left to the reader.

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