Abstract. In this Supplementary Material we provide the computation proce-dures to solve the examples of the paper “Endogenous bourse structures”. This material is not for publication.
Procedure to computeU1i(S) for examples 1, 2, and 3: With out loss of general-ity, consider any bourse S = S1; S2; S3; S4; S5:Given that markets are complete, we …nd the Radner equilibrium by solving the Arrow-Debreu equilibrium, where each consumer i maximizes ui(x1; x2; x3)subject to his budget constraintP
p (x !i) = 0;and markets clear, P
ixi =P
i!i, 8 :The procedure is standard. The steps are:
1) Take the …rst order condition of the Lagrangian function Li with respect to the consumption variables and shadow price i (one Arrow-Debreu restriction). We then obtain i(p!i) and xi(p!i); for all , where p!i p1!i1 +P
=1;2p( )!i( ): In general, we obtain the following expressions: xi1 = pi1p!i
1 i and xi( ) = p( )i( ) p!ii for = 1; 2;where
i = i1+P
=1;2 i( ):
2) Substitute these values in the Arrow-Debreu market clearing equilibrium condi-tions (see condition (D1.ii)) for bourse S and obtain the commodity prices that clear the markets. We obtain:
S1 S2 S3 S4 S5
p1 4 2 1.2941 8 8
p(1) 1 1 1 21 5
p(2) 1 1 1 5 21
Table SP-1
3) Find the equilibrium consumption values using these prices. We obtain x11 x1(1) x1(2) x21 x2(1) x2(2)
S1 2 8 0 2 0 8
S2 3 6 0 3 0 6
S3 4.0909 5.2941 0 4.0909 0 5.2941
S4 5.5 2.0952 0 n.a. n.a. n.a.
S5 n.a. n.a. n.a. 5.5 0 2.0952
Table SM-2
x31 x3(1) x3(2) x41 x4(1) x4(2) S1 n.a. n.a. n.a. n.a. n.a. n.a.
S2 4 4 4 n.a. n.a. n.a.
S3 n.a. n.a. n.a. 1.8182 4.7059 4.7059 S4 n.a. n.a. n.a. 2.5 1.9048 8
S5 n.a. n.a. n.a. 2.5 8 1.9048
Table SM-3
4)Substitute the equilibrium consumption values in the utility function to obtain the trader’s indirect utility in the second stage when he belongs to that bourse S. These values for bourses S1; S2 and S3 are given in Example 1. The indirect utility values for bourses S4 and S5, needed in Example 2, are given in the following table. Notice that the indirect utilities of the traders that do not belong to any bourse are obtained by evaluating the trader’s utility ui1 in his good endowments (there is only one good and all traders have their utilities strictly increasing in the consumption of this good).
S4 = (1; 4) S5 = (2; 4) U11(S4) = 2:4444
u21(!2) = 1:3862 u31(!3) = 2:4849 U14(S4) = 3:1819
u11(!1) = 1:3862 U12(S5) = 2:4444 u31(!3) = 2:4849 U14(S5) = 3:1819 Table SM-4
The following step is only needed for Example 1.
5)Given that we are in a complete markets framework, the Arrow-Debreu equilibrium coincides with the Radner equilibrium. Therefore, to compute traders’ portfolios we substitute the above Arrow-Debreu equilibrium consumption values in the corresponding Radner restrictions:
xi( ) !i( ) X
j=1;2
aj( )yij = 0, for = 1; 2:
Solving for each trader’s system of equations and unknowns, we get the asset trades of a trader in the bourse S: The values are given in Table 2 of the paper.
Procedure to compute examples 2 and 3: The non-anonymous membership fees are obtained by considering a welfarist agent that maximizes the weighted sum of indirect
utilities subject to individual budget constraints in period 0. That is, the maximization problem is max( i)i2S
P
i2S(1=2) (ln xi0) U1i(S) subject to xi0 = !i0 i(S); 8i 2 S; and P
i2S i(S) = z(s): Budget constraint in period 0 can be written with equality, as utility is increasing in the consumption of the single good. The membership fee formulas are obtained by taking the …rst order condition with respect to ( i(S))i2S:The memberships fees for bourses S1; S3; S4 and S5 are given in the text of Example 2, whereas the membership fees for bourse S2 are given in the text of Example 3.
Given the good endowments at period 0 and the membership fees values, we obtain the consumption values and indirect utilities at period 0 for bourses S1; S3; S4 and S5 (the values for bourse S2 are not needed in our examples). These are39
x10 x20 x40
S1 4 4 5.9
S3 3.5111 3.5111 3.8778 S4 2.9978 7 3.9022 S5 7 2.9978 3.9022
V1 V2 V4
S1 1.9218 1.9218 2.0253 S3 1.9312 1.9312 2.3016 S4 1.3418 1.3488 2.1662 S5 1.3488 1.3418 2.1662
Table SM-5 Table SM-6
Procedure to compute example 4: The equilibrium of the second stage is com-puted by solving a Radner type economy (one budget constraint for each node). By writing each trader’s budget constraints for states 1 and 2 in period 2 in equality form, and substi-tuting the state-consumption expressions in the utility function of the second stage, we get the following objective functions: for trader 5, (1=2) ln(4 q(S6)y5(S6) q(S7)y5(S7))+
ln(2 + y5(S6) + y5(S7)); for trader 6, ln(2 q(S6)y6(S6))+ (1=2) ln(6 + y6(S6)); for trader 7, ln(2 q(S7)y7(S7))+ (1=2) ln(6 + y7(S7)). Portfolios parametrized in asset prices are then obtained by solving the system of …rst order conditions on these ob-jective functions. For traders 6 and 7, these are y6(S6) = (2 12q(S6))=3q(S6) and y7(S7) = (2 12q(S7))=3q(S7). The …rst order conditions for trader 5 determine a sys-tem of two equations and two unknowns: 6 q(S6)y5(S0:5q(S6) q(S6)7)y5(S7)+ 2+y5(S61)+y5(S7) = 0and
0:5q(S7)
6 q(S6)y5(S6) q(S7)y5(S7)+ 2+y5(S61)+y5(S7) = 0:These two equations imply that q(S6) = q(S7):
Denoting this price by q, and solving for y5(S6) + y5(S7) in one of these two equations, we get y5(S6) + y5(S7) = (12 2q)=3q:Now, using the asset market clearing equations for the multiple memberships bourse structure, y5(S6) + y6(S6) = 0 and y5(S7) + y7(S7) = 0;
39Numbers in italics indicate that trader i’s consumption xi0and utility ui are evaluated at the trader’s good endowments (as the trader does not belong to any bourse).
we get q(S6) = q(S7) = 8=13 (notice that there is only one good and also that traders 6 and 7 are symmetric in preferences and endowments in the second stage). Portfo-lios are (y5(S6); y5(S7)) = (35=12; 35=12); y6(S6) = 35=12 and y7(S7) = 35=12: On the other hand, it can be shown that for the bourse structure with a unique bourse S8 = (5; 6; 7), with market clearing equation y5(S8)+ y6(S8)+ y7(S8) = 0, portfolios are y5(S8) = 70=12; y6(S8) = 35=12 and y7(S8) = 35=12: Indirect utilities in the second stage are U15 = 2:4982; U16 = 1:8940 and U17 = 1:8940, for either the bourse structure with multiple memberships or the bourse structure with a unique bourse. However, member-ship fees are di¤erent in the two types of bourse structures, which will determine di¤erent traders’ indirect utilities in period 0. These are: 5(S6) = 5(S7) = 4:7619; 6(S6) =
7(S7) = 1:2380; 5(S8) = 10:2310; 6(S8) = 7(S8) = 5:3846: Then, consumptions at period 0 are obtained using the respective Radner restriction xi0 = !i0 P
S:i2S i(S):
These are x5(fS6; S7g) = 5:4762; x6(S6) = x7(S7) = 7:7620; x5(S8) = 4:769; x6(S8) = x7(S8) = 3:6154: The indirect utilities in period 0 are given in the text of Example 4.
Procedure to compute Example 5: Bourse S9 is characterized by incomplete markets and therefore the procedure to compute equilibrium is di¤erent that the one described above when markets are complete. For bourse S9 the equilibrium for the second stage is obtained by solving a Radner type economy (one budget constraint for each node).
The steps are:
1)Since there is only one good, we can make the price of the good equal to 1 in every node. Then, we write the Radner budget constraints in equality form and obtain the equilibrium consumption (xi1 = !i1 q1y1i and xi( ) = !i( ) + a1 yi( ), for = 1; 2).
Consumption in period 1 is then parametrized by the asset trades and good endowment, while consumption at node of period 2 is parametrized by the asset returns and good endowment.
2) Substitute these parametrized consumption functions in the utility function ui1(xi1; xi(1); xi(2)) and take the …rst order conditions with respect to yi(1) and yi(2) to obtain the asset trades as a function of asset prices.
3) Apply asset market clearing equations (P
i2S9yi = 0) to obtain the asset price:
q = 1. Then, substitute this price in the in the previous expressions to obtain asset trades: y8(S9) = 10=3and y9(S9) = 10=3:
In what follows we also indicate the values for the other bourses (with complete mar-kets) S10; S11 and S12:
4) Substitute the values of (yi)i2S into period 1 and period 2 budget constraints to
calculate the equilibrium consumption values (xi1; xi(1); xi(2)).
S9 S10 S11 S12
(x81; x8(1); x8(2)) (2.6666,5.3333,0) (4,4,0) n.a. (2.7964,6.4489,0) (x91; x9(1); x9(2)) (5.3333,2.6666,0) n.a. (2.909,4.8,0) (5.2389,3.0204,0) (x101 ; x10(1); x10(2)) n.a. (4,4,2) (1.0909 ,7.2,2) (1.9646,4.5306,3)
Table SM-7
These consumption values will determine the value of the indirect utility function:
S9 S10 S11 S12
U18( ) 2.1643 2.0794 n.a. 2.378 U19( ) 2.1643 n.a. 1.8521 2.2088 U110( ) n.a. 2.426 2.3641 2.3978
Table SM-8
5) Membership fees are obtained using the formulas (8) and (9) given in the paper.
The values are:
S9 S10 S11 S12
8( ) 3 3.3077 n.a. 2.9144
9( ) 3 n.a. 3.4857 3.2051
10( ) n.a. 2.6922 2.5142 2.8804 Table SM-9
6) Substitute the values of the membership fees and good endowments in budget constraint of period 0 and calculate consumption in period 0, xi0. Then, substitute the value of xi0 in ui0, and obtain Vi(S) = (1=2) ln x0 U1i(xi1; xi(1); xi(2)). Table 5 in the paper gives the values of traders’indirect utilities Vi(S)at S9; S10; S11 and S12: