ONLINE APPENDIX: TESTABLE IMPLICATIONS OF TRANSLATION INVARIANCE AND HOMOTHETICITY: VARIATIONAL, MAXMIN, CARA AND CRRA PREFERENCES

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1 ONLINE APPENDIX: TESTABLE IMPLICATIONS OF TRANSLATION INVARIANCE AND HOMOTHETICITY: VARIATIONAL, MAXMIN, CARA AND CRRA PREFERENCES CHRISTOPHER P. CHAMBERS, FEDERICO ECHENIQUE, AND KOTA SAITO In thi online appendix, we provide the complete proof of Theorem 4. We prove ome lemma. Lemma 1: A dataet D i maxmin rationalizable if and only if there are v k, λ k, = 1,2, k = 1,...,K, and π,π 0 with π π > 0, uch that: for all k uch that x k x k, πv k 1 = λ k p k 1 v k 2 = λk p k 2, where π = π when x k 1 < xk 2 and π = π when xk 1 > xk 2 ; for all k uch that x k = xk πv k 1 λk p k 1 πv k 1 λk p k 1 v k 2 = λ k p k 2. The number alo atify that v k v k when xk > x k. Proof: To prove ufficiency, let v k, λk, = 1,2, k = 1,...,K, and π,π 0 with π π beain thetatement ofthe lemma. Define µ,µ (S) afollow. Let µ 1 = π/(1+ π), µ 2 = 1/(1+ π), and µ 1 = π/(1+π), µ 2 = 1/(1+π). Since Chamber i affiliated with the Department of Economic, Univerity of California, San Diego. Echenique and Saito are affiliated with the Diviion of the Humanitie and Social Science, California Intitute of Technology, Paadena CA cpchamber@ucd.edu (Chamber), fede@caltech.edu (Echenique), aito@caltech.edu (Saito). Chamber gratefully acknowledge the upport of the NSF through grant SES We are very grateful to Simon Grant and Maimo Marinacci, who poed quetion to u that led to ome of the reult in thi paper. 1

2 2 CHAMBERS, ECHENIQUE, AND SAITO π π, µ 1 µ 1 and µ 2 µ 2. Define θ k = λ k /(1+ π) if x k 1 < xk 2 and θk = λ k /(1+π) if x k 1 > xk 2. Then we have that µ v k = θ k p k, with µ = µ when x k 1 < x k 2; and µ = µ when x k 1 > x k 2. Now conider k uch that x k 1 = x k 2. By the aumption, there exit π k uch that π π k π uch that πv k 1 = λk p k 1. Let µk 1 = π/(1+π), µk 2 = 1/(1+π), and θ k = λ k /(1+π). Since π π π, µ 1 π 1+π πk 1+π k π 1+π µ 1. Hence, there exit α k [0,1], µ k 1 = αk µ 1 +(1 α k )µ 1. Then, µ k 2 = 1 µk 1 = α k (1 µ 1 )+(1 α k )(1 µ 1 ) = α k µ 2 +(1 α k )µ 2. Giventhenumberv k itinowroutinetodefineacorrepondenceρuchthat if x x, y ρ(x) and y ρ(x ) then y y > 0, and with ρ(x k ) vk. Thi give a concave and increaing function u with u(c) = ρ(x). So θk p k µ u(x k ) for all and k uch that x k 1 x k 2. Moreover, for all k uch that x k 1 = x k 2 { (µ1 (θ k p k 1,θ k p k 2) co u(x k 1),µ 2 u(x k 2) ), ( µ 1 u(x k 1),µ 2 u(x k 2) )}. Hence the firt and econd order condition are atified for maxmin rationalization. We omit the proof of neceity. We will define matrice A, B, E uch that there exit number {v k},{λk }, π, π atifying the condition in Lemma 1 if and only if there exit a olution x to the ytem of inequalitie A x = 0, B x 0 and E x > 0. Let A be a matrix with 2K K + 1 column. The firt 2K column are labeled with a different pair (k,). The next 2 column are labeled π and π. The next K column are labeled with a k {1,...,K}. Finally the lat column i labeled p. For each (k,2) with k K 0, A ha a row with all zero entrie with the following exception. It ha a 1 in the column labeled (k,), among the firt group of 2K column. It ha a 1 in the column labeled k. In the column labeled p it ha log(p k ). For each (k,) with k K 1, A ha a row with all zero entrie with the following exception. It ha a 1 in the column labeled (k,), among the firt group of 2K column. It ha a 1 in the column labeled k. In the column

3 TRANSLATION INVARIANCE AND HOMOTHETICITY 3 labeled p it ha log(p k ). Finally, it ha a 1 in the column labeled π if and only if if = 1. For each (k,) with k K 2, A ha a row defined a above. The only difference i that when = 1 then it ha a 1 in the column labeled π intead of having a 1 in the column labeled π. Let B be a matrix with the ame number of column a A. The column of B are labeled like thoe of A. For each (k,1) with k K 0, B ha two row. In the firt row, B ha a row with all zero entrie with the following exception. It ha a 1 in the column labeled (k,), among the firt group of 2K column. It ha a 1 in the column labeled π. It ha a 1 in the column labeled k. In the column labeled p it ha log(p k ). In the econd row, B ha a row with all zero entrie with the following exception. It ha a 1 in the column labeled (k,), among the firt group of 2K column. It ha a 1 in the column labeled π. It ha a 1 in the column labeled k. In the column labeled p it ha log(p k ). In addition, B ha a row for each pair (x k,x k ) with xk > x k. The row for x k > xk ha all zeroe except for a 1 in column (k, ) and a 1 in column (k,). Finally, B ha one more row. Thi row a a 1 in the column for π and a 1 in the column for π. Let E be a matrix with the ame number of column a A, labeled a above, and a ingle row. The row ha all zeroe except for a 1 in column p. By Lemma 1, there i no rationalizing maxmin preference if and only if there i no olution to the ytem of inequalitie A x = 0, B x 0, and E x > 0. Suppoe that all log(p k ) are rational number. We hall ue the following verion of the Theorem of the Alternative, which can be found a Theorem in Stoer and Witzgall (1970). Lemma*: Let A be an m n matrix, B be an l n matrix, and E be an r n matrix. Suppoe that the entrie of the matrice A, B, and E belong to a commutative ordered field F. Exactly one of the following alternative i true. (1) There i u F n uch that A u = 0, B u 0, E u 0. (2) There i η F m, θ F l, and γ F r uch that η A+θ B+γ E = 0; θ 0 and γ > 0.

4 4 CHAMBERS, ECHENIQUE, AND SAITO Then the non-exitence of a olution to the ytem A x = 0, B x 0 and E x > 0 i equivalent to the exitence of integer vector η, θ and γ uch that θ 0, γ > 0, and η A+θ B +γe = 0. For a matrix D with 2K +2+K +1 column, let D 1 denote the ubmatrix correponding to the firt 2K column, D 2 correpond to the next 2, D 3 to the next K, and D 4 to the lat column. Note that, by contruction of A, B and E, η A+θ B +γe = 0 implie that η A 1 +θ B 1 = 0, η A 2 +θ B 2 = 0, η A 3 +θ B 3 = 0, η A 4 +θ B 4 +γ = 0. In fact, we can without lo aume that η, θ and γ take value of 1, 0 or 1. (Thi aumption i without lo becaue we can replace each row of matrice A, B and E with a many copie a indicated by the correponding vector η, θ or γ.) From the exitence of uch vector it follow that we can obtain a equence (x k i i ) n i=1 withxk i i i > x k i. The ourceofeach pair(x k i i i thecolumn i)ithat (k i, i )ofaimultiplied byη (ki, i ) > 0andthecolumn(k i, i)ofaimultiplied by η (k i, i ) < 0. The vector η mut then have η (ki, i ) > 0 and η (k i, i ) > 0, with a 1 in the firt column and a 1 in the econd. We hall prove that the equence (x k i i i) n i=1 atifie the propertie tated in the axiom. Firtly, η A 3 +θ B 3 = 0 mean that for each k, the number of i for which k = k i equal the number of i for which k = k i. Secondly, η A 2 +θ B 2 = 0 implie that: η (k,1) + θ (k,1) +θ π π = 0 k K 1 k K 0 η (k,1) θ (k,1) θ π π = 0, k K 2 k K 0 where θ π π i the nonnegative weight on the row aociated with π π; θ (k,1) and θ (k,1) are the nonnegative weight on the two row aociated with (k,1) with k K 0. (θ (k,1) and θ (k,1) correpond to the firt row and the econd row,

5 repectively). Note that Hence, TRANSLATION INVARIANCE AND HOMOTHETICITY 5 k K 1 η (k,1) = #{i : k i K 1, = 1} #{i : k i K 1, = 1} #I 1,1 #I 1,1, k K 2 η (k,1) = #{i : k i K 2, = 1} #{i : k i K 2, = 1} #I 2,1 #I 2,1, k K 0 θ (k,1) = #{i : k i K 0, = 1} #I 0,1, k K 0 θ (k,1) = #{i : k i K 0, = 1} #I 0,1. #I 1,1 #I 1,1 +#I 0,1 = #I 2,1 #I 2,1 +#I 0,1 0. Therefore the equence (x k i i i) n i=1 atifie the econd property tated in the axiom. Finally, ince η A 4 +θ B 4 +γ = 0, 0 > γ = η A 4 +θ B 4 = (k,) (K 0,2) (K 1 K 2,1) η (k,) ( logp k ) + k K 0 θ (k,1) ( logp k 1 )+ k K 0 θ (k,1) logpk 1 Hence = n i=1 log pk i i p k i i n i=1 p k i i p k i i > 1. The above proof aume that the log of price i rational. The proof of the theorem follow along the ame line a Echenique and Saito (2015). Specifically, we have hown the following.

6 6 CHAMBERS, ECHENIQUE, AND SAITO Lemma 2: If {(x k,p k )} i a dataet atifying SARMEU, in which logp k Q for all k, then the dataet i maxmin rationalizable. One can then prove the following Lemma 3: If {(x k,p k )} i a dataet that atifie SARMEU, and ε > 0 then there i a collection of price {q k } uch that logq k Q, p k q k < ε, and the dataet {(x k,q k )} atifie SARMEU. The proof of Lemma 3 i exactly the ame a in Echenique and Saito (2015). Lemma 2 etablihe the reult in dataet in which the log of price i rational. Conider an arbitrary data et {(x k,p k )}, with price that may not be rational. Suppoe toward a contradiction that the dataet atifie SARMEU, but that it i not maxmin rational. Specifically then, by Lemma 1, uppoe that there i no olution to the ytem A x = 0, B x 0 and E x > 0. Then by Lemma* there are real vector η, θ and γ uch that θ 0, γ > 0, and η A+θ B +γe = 0. Let{q k }bevectorofpriceuchthatthedataet{(x k,q k )}atifiesarmeu and logq k Q for all k and. (Such {qk } exit by Lemma 3.) Furthermore, the price q k can be choen arbitrarily cloe to p k. Contruct matrice A, B, and E from thi dataet in the ame way a A, B, and E above. Note that only the price are different in {(x k,q k )} compared to {(x k,p k )}. So E = E, B i = B i and A i = A i for i = 1,2,3. Since only price q k are different in thi dataet, only A 4 and B 4 may be different from A 4 and B 4, repectively. ByLemma3,wecanchooepriceq k uchthat η A 4+θ B 4 (η A 4 +θ B 4 ) < γ/2. We have hown that η A 4 + θ B 4 = γ, o the choice of price q k guarantee that η A 4 +θ B 4 < 0. Let γ = η A 4 θ B 4 > 0. Note that θ A i +η B i +γ E i = 0 for i = 1,2,3. Hence η A 4 +θ B 4 +γ E 4 = η A 4 +θ B 4 +γ = 0. We alo have that η 0 and γ > 0. Therefore θ, η, and γ exhibit a olution to the dual ytem for dataet {(x k,q k )}, a contradiction with Lemma 2.

7 TRANSLATION INVARIANCE AND HOMOTHETICITY 7 Reference Echenique, F. and K. Saito (2015): Savage in the Market, Econometrica, 83, Stoer, J. and C. Witzgall (1970): Convexity and Optimization in Finite Dimenion, Springer-Verlag Berlin.