Price Discrimination with Loss Averse Consumers: Supplementary Material Jong-Hee Hahn · Jinwoo Kim · Sang-Hyun Kim · Jihong Lee

S.1 Personal Equilibrium: The (IC) and (IR) Constraints To help the reader, we provide the full expressions of (IC) and (IR) constraints appearing in Section 3 of the main text. Optimal Menu with Symmetric Information: Suppose that θL v(qL ) ≤ θH v(qH ) and tH ≥ tL . Then, u(rH |θH , R) = θH v(qH ) − tH + p[θH v(qH ) − θL v(qL ) − λ(tH − tL )]

(IRH )

≥ u(∅|θH , R) = p[tL − λθL v(qL )] + (1 − p)[tH − λθH v(qH )] u(rL |θL , R) = θL v(qL ) − tL + (1 − p)[(tH − tL ) − λ(θH v(qH ) − θL v(qL ))]

(IRL )

≥ u(∅|θH , R) = p[tL − λθL v(qL )] + (1 − p)[tH − λθH v(qH )] Pooling Menu: Suppose qL = qH = q and tL = tH = t. Then, u(r|θL , R) = θL v(q) − t − (1 − p)λ(θH − θL )v(q) ≥ u(∅|θL , R) = p[t − λθL v(q)] + (1 − p)[t − λθH v(q)]

(IRL )

Screening Menu: Suppose qL < qH and tL < tH . Then, u(rH |θH , R) = θH v(qH ) − tH + p[θH v(qH ) − θL v(qL ) − λ(tH − tL )] ≥ u(rL |θH , R) = θH v(qL ) − tL + p(θH − θL )v(qL )

(ICH )

+ (1 − p)[(tH − tL ) − λθH (v(qH ) − v(qL ))] u(rL |θL , R) = θL v(qL ) − tL + (1 − p)[(tH − tL ) − λ(θH v(qH ) − θL v(qL ))] ≥ u(rH |θL , R) = θL v(qH ) − tH − λ(1 − p)(θH − θL )v(qH )

(ICL )

+ p[θL (v(qH ) − v(qL )) − λ(tH − tL )] Note that (IRH ) and (IRL ) are the same as in the case of symmetric information above and thus omitted. J-H. Hahn School of Economics, Yonsei University, Seoul 03722, Korea E-mail: [email protected] J. Kim School of Economics, Seoul National University, Seoul 08826, Korea E-mail: [email protected] S-H. Kim School of Economics, Yonsei University, Seoul 03722, Korea E-mail: [email protected] J. Lee School of Economics, Seoul National University, Seoul 08826, Korea E-mail: [email protected]

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Hahn, Kim, Kim and Lee

Reverse-Screening Menu: Suppose qL > qH and tL > tH . If θL v(qL ) > θH v(qH ), then u(rH |θH , R) = θH v(qH ) − tH + p[(tL − tH ) − λ(θL v(qL ) − θH v(qH ))] ≥ u(rL |θH , R) = θH v(qL ) − tL + p(θH − θL )v(qL )

(ICH )

+ (1 − p)[θH (v(qL ) − v(qH )) − λ(tL − tH )] u(rH |θH , R) = θH v(qH ) − tH + p[(tL − tH ) − λ(θL v(qL ) − θH v(qH ))]

(IRH )

≥ u(∅|θH , R) = p[tL − λθL v(qL )] + (1 − p)[tH − λθH v(qH )] u(rL |θL , R) = θL v(qL ) − tL + (1 − p)[(θL v(qL ) − θH v(qH )) − λ(tL − tH )] ≥ u(rH |θL , R) = θL v(qH ) − tH − λ(1 − p)(θH − θL )v(qH )

(ICL )

+ p[(tL − tH ) − λθL (v(qL ) − v(qH ))] u(rL |θL , R) = θL v(qL ) − tL + (1 − p)[(θL v(qL ) − θH v(qH )) − λ(tL − tH )]

(IRL )

≥ u(∅|θH , R) = p[tL − λθL v(qL )] + (1 − p)[tH − λθH v(qH )]; If θL v(qL ) ≤ θH v(qH ), then u(rH |θH , R) = θH v(qH ) − tH + p[(tL − tH ) + (θH v(qH ) − θL v(qL ))] ≥ u(rL |θH , R) = θH v(qL ) − tL + p(θH − θL )v(qL )

(ICH )

+ (1 − p)[θH (v(qL ) − v(qH )) − λ(tL − tH )] u(rH |θH , R) = θH v(qH ) − tH + p[(tL − tH ) + (θH v(qH ) − θL v(qL ))]

(IRH )

≥ u(∅|θH , R) = p[tL − λθL v(qL )] + (1 − p)[tH − λθH v(qH )] u(rL |θL , R) = θL v(qL ) − tL − (1 − p)λ[(tL − tH ) + (θH v(qH ) − θL v(qL ))] ≥ u(rH |θL , R) = θL v(qH ) − tH − λ(1 − p)(θH − θL )v(qH )

(ICL )

+ p[(tL − tH ) − λθL (v(qL ) − v(qH ))] u(rL |θL , R) = θL v(qL ) − tL − (1 − p)λ[(tL − tH ) + (θH v(qH ) − θL v(qL ))]

(IRL )

≥ u(∅|θH , R) = p[tL − λθL v(qL )] + (1 − p)[tH − λθH v(qH )] S.2 Optimal TPE Menu with a Continuum of Types This section provides a detailed analysis of the continuum type case presented in Section 3.2 of the main text, including a proof of Theorem 2 and numerical examples. Given (21) in the main text, the seller’s revenue can be expressed as follows: letting Gθ (θ, λ) := ∂G(θ,λ) , ∂θ ! Z Z Z θ

θ

θ

t0 (s)ds + t(θ) dF (θ)

t(θ)dF (θ) = θ

θ

Z

θ θ

Z

=

!

θ 0

(v(q(s))) G(s, λ)ds dF (θ) + t(θ) θ

Z

θ θ

Z v(q(θ))G(θ, λ) −

= θ

Z

v(q(s))Gθ (s, λ)ds dF (θ) + t(θ) − v(q(θ))G(θ, λ) θ

θ

= θ

Z

!

θ

  1 − F (θ) (1 + λ) v(q(θ)) G(θ, λ) − Gθ (θ, λ) dF (θ) + t(θ) − θv(q(θ)) f (θ) 2

θ

v(q(θ))J(θ, λ)dF (θ) + t(θ) − θv(q(θ))

= θ

(1 + λ) , 2

(S1)

Price Discrimination with Loss Averse Consumers: Supplementary Material

where J(θ, λ) := G(θ, λ) −

3

1 − F (θ) Gθ (θ, λ), f (θ)

and the second equality follows from integration by parts as does the third equality along with the fact that G(θ, λ) = ( 1+λ 2 )θ. Let us refer to J(θ, λ) as “gain-loss adjusted virtual value,” which boils down to the usual virtual value J(θ) if λ = 1. The next result says that the transfer must be designed in such a way that the participation constraint is binding at the lowest type θ. Lemma S1 At the optimal menu, it must be that t(θ) =

1+λ θv(q(θ)), 2

which implies that the (IR) constraint (19) in the main text is satisfied and, moreover, binding at θ = θ. Proof Note first that the lowest type θ participates only if θ

Z U (θ) = θv(q(θ)) − t(θ) +

Z

θ

Z ≥

θ

θ

(t(s) − t(θ))dF (s) − λ θ

Z t(s)dF (s) − λ

θ

(sv(q(s)) − θv(q(θ)))dF (s) θ

V (s)dF (s)

(S2)

θ

or θv(q(θ))(1 + λ) − 2t(θ) ≥ 0, which implies that (S1) is maximized by setting t(θ) = ( 1+λ 2 )θv(q(θ)) for any q(θ) chosen. Given this, it is easy to verify that the participation constraints for all other types are also satisfied since, by the envelope theorem, we have ∂U (θ0 ; θ) 0 U (θ) = 0 = v(q(θ))(1 + F (θ) + λ(1 − F (θ))) ≥ 0 ∂θ θ =θ or the equilibrium payoff U (·) is increasing while the outside payoff is constant irrespective of type realization, as shown in (S2). Thus, the seller’s problem, denoted by [P c ], can be written as Z q(·)

θ

[v(q(θ))J(θ, λ) − cq(θ)] dF (θ),

max

[P c ]

θ

subject to q(·) being non-decreasing.1 The shape of the optimal quality schedule will depend on the behavior of J(·, λ), the gain-loss adjusted virtual value. While J(·, λ) may behave in a complicated way depending on the distribution F (·), it can be decreasing over some or entire range for high enough λ, which implies that the optimal menu must involve some pooling. To see this, we can obtain Jθ (θ, λ) :=

∂J(θ, λ) 1 − F (θ) = J 0 (θ)Gθ (θ, λ) − Gθθ (θ, λ), ∂θ f (θ)

(S3)

2

where Gθθ (θ, λ) := ∂G(θ,λ) . Two important terms here are Gθ (θ, λ) and Gθθ (θ, λ). As mentioned above, ∂2θ it is possible to have Gθ (θ, λ) < 0 as λ increases. Given this, (S3) implies that J(θ, λ) can decrease if Gθθ (θ, λ) > 0. Note that the last term of (S3) concerns the impact of loss aversion on the information rent.2 While Gθθ (θ, 1) = 0, having Gθθ (θ, λ) > 0 with λ > 1 means that loss aversion aggravates the information rent problem, which may cause the gain-loss adjusted virtual value to decrease and, hence, some pooling to arise. Indeed, this can happen if λ is high enough. We next turn to a formal proof of Theorem 2 of the main text. 1 Note

that global incentive compatibility is ignored here. 1−F (θ) the standard screening model, the expression f (θ) represents the information rents that have to be given up to the types above θ if an extra unit of good is to be sold to θ. 2 In

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Proof of Theorem 2 We first derive the following lemma. Lemma S2 A quality schedule solving [P c ] is constant whenever J(·, λ) is decreasing. Proof Following Myerson (1981) or Toikka (2011), let us first define the ironed-out function of J(·, λ) as follows: Letting Z θ K(θ, λ) := sup{h(θ) | h convex and h(θ) ≤ J(s, λ)ds}, θ

Rθ

i.e. the convex envelope of function

θ

J(s, λ)ds, define J(θ, λ) :=

∂K(θ, λ) . ∂θ

Letting Q denote the set of non-decreasing quality schedules, Toikka (2009) shows that Z

Z J(θ, λ)q(θ)dθ = sup

sup q(·)∈Q

θ

θ

q(·)

θ

J(θ, λ)q(θ)dθ. θ

Note that the RHS is an unconstrained optimization problem. With K being the convex envelope of Rθ J(s, λ)ds, we must have K being linear in θ in the range where J(θ, λ) is decreasing, which implies θ that J(·, λ) is constant in that range. This implies that the optimal quality schedule solving the above unconstrained problem, and thus [P c ], must be constant where J(·, λ) is decreasing. To make use of Lemma S2, it suffices to check that Jθ (θ, λ) is negative in the desired range. To do so, let us do some tedious calculation to obtain Jθ (θ, λ) = J 0 (θ)Gθ (θ, λ) −

1 − F (θ) Gθθ (θ, λ), f (θ)

where Gθ (θ, λ) =

(λ2 + 2λ)[F (θ)(1 − F (θ)) − θf (θ)] + 2λ + (1 + F (θ))(2 − F (θ)) + 3θf (θ) (2 − F (θ) + λF (θ))2

(S4)

and Gθθ (θ, λ) =

n (λ − 1)(λ + 3) λ[2θf (θ)2 − F (θ)(2f (θ) + θf 0 (θ))] (2 − F (θ) + λF (λ))3 o − 2θf (θ)2 − (2f (θ) + θf 0 (θ))(2 − F (θ)) .

(S5)

We proceed to prove each part of Theorem 2 in turn.

Part (a) First, we obtain the following lemma. Lemma S3 Consider any menu R with non-decreasing quality schedule q(·) and payment schedule t(·) satisfying (21) in the main text. If θ(1 + F (θ) + λ(1 − F (θ))) is non-decreasing in θ, R satisfies the (global) (IC) constraint. ˆ θ0 ). It suffices to prove that Proof To simplify notation, let θˆ denote θ(θ, 0 0 To that end, differentiate U (θ ; θ) with θ to obtain

∂U (θ 0 ;θ) ∂θ 0

≥ (≤) 0 if θ0 ≤ (≥) θ.

i h ∂U (θ0 ; θ) 0 0 ˆ + λ(1 − F (θ)) ˆ − t0 (θ0 ) [2 − F (θ0 ) + λF (θ0 )] = θ (v(q(θ ))) 1 + F ( θ) ∂θ0  i h  0 ˆ + λ(1 − F (θ)) ˆ = (v(q(θ0 ))) θ 1 + F (θ) − θ0 (1 + F (θ0 ) + λ(1 − F (θ0 ))) ,

(S6)

Price Discrimination with Loss Averse Consumers: Supplementary Material

5

where the second equality follows from substituting (21). For the optimality of θ0 = θ, it suffices to show that the expression in (S6) is non-negative (non-positive) if θ0 < (>)θ. If θ0 < θ, then θ0 ≤ θˆ ≤ θ and, since θ(1 + F (θ) + λ(1 − F (θ))) is non-decreasing,     ˆ + λ(1 − F (θ)) ˆ ˆ + λ(1 − F (θ)) ˆ θ 1 + F (θ) ≥ θˆ 1 + F (θ) ≥ θ0 (1 + F (θ0 ) + λ(1 − F (θ0 ))) implying that (S6) is non-negative. Also, if θ0 > θ, then θ0 ≥ θˆ ≥ θ and, thus,     ˆ + λ(1 − F (θ)) ˆ ˆ + λ(1 − F (θ)) ˆ θ 1 + F (θ) ≤ θˆ 1 + F (θ) ≤ θ0 (1 + F (θ0 ) + λ(1 − F (θ0 ))) , implying that (S6) is non-positive, as desired. Next, note that Jθ (θ, λ) = J 0 (θ)Gθ (θ, λ) so the result will obtain if Gθ (θ, λ) < 0 and thus, by continuity, Jθ is negative near θ. It is straightforward to check from (S4) that Gθ (θ, λ) < 0 is equivalent 2 +2λ−3 to requiring λ2(λ+1) > θf1(θ) . Part (b) We check that under the stated conditions, (i) Gθ (θ, λ) < 0, ∀θ if λ > λ1 for some λ1 ; (ii) Gθθ (θ, λ) > 0, ∀θ if λ > λ2 for some λ2 . For (i), note first that the coefficient for the quadratic term λ2 in the numerator of (S4) is negative since F (θ)(1 − F (θ)) ≤ F (θ) < θf (θ) , where the inequality is due to the condition θf (θ) > F (θ). This implies that one can find sufficiently large λ, say λ1 (θ), such that Gθ (θ, λ) < 0 if λ > λ1 (θ). Clearly, λ1 (θ) is continuous in θ so we can let λ1 = maxθ∈[θ,θ] λ1 (θ). For (ii), let us first see that the expression in the square bracket in (S5) is positive: 2θf (θ)2 − F (θ)(2f (θ) + θf 0 (θ)) ≥ 2θf (θ)2 − 2F (θ)f (θ) = 2f (θ)(θf (θ) − F (θ)) > 0, where the weak inequality follows from f 0 (θ) ≤ 0. Thus, one can find sufficiently large λ, say λ2 (θ), such that Gθθ (θ, λ) > 0 if λ > λ2 (θ). Also, λ2 (θ) is continuous in θ so we can let λ2 = maxθ∈[θ,θ] λ2 (θ). Letting λ = max{λ1 , λ2 }, Jθ (·, λ) is negative everywhere if λ > λ. Then, by Lemma S2, the optimal quality schedule must be constant everywhere. This schedule clearly satisfies global incentive compatibility, which completes the proof. Part (c) Consider any menu where q(θ) and t(θ) are (weakly) decreasing while θv(q(θ)) is increasing. Then, the utility of type θ from choosing the bundle of type θ0 is given by "Z ˆ 0 # Z 0 θ(θ,θ )

U (θ0 ; θ) = θv(q(θ0 )) − t(θ0 ) +

θ

(θv(q(θ0 )) − V (s))dF (s) +

(t(s) − t(θ0 ))dF (s)

θ

"Z

θ

θ 0

−λ

Z

(V (s) − θv(q(θ )))dF (s) + 0) ˆ θ(θ,θ

#

θ 0

(t(θ ) − t(s))dF (s) . θ0

By incentive compatibility, the first-order condition gives ∂ 0 0 U (θ ; θ) = [(θv(q(θ)))0 − t0 (θ)] [1 + F (θ) + λ(1 − F (θ))] = 0, ∂θ0 0 θ =θ which yields t0 (θ) = θ(v(q(θ)))0 .

(S7)

Given (S7), the seller’s expected revenue can be expressed as ! Z Z Z θ

θ

θ

t0 (s)ds + t(θ) dF (θ)

t(s)dF (θ) = θ

θ

Z

θ θ

= θ

Z

  1 − F (θ) v(q(θ)) θ − dF (θ) + t(θ) − θv(q(θ)) f (θ)

θ

v(q(θ))J(θ)dF (θ) + t(θ) − θv(q(θ)),

= θ

(S8)

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Hahn, Kim, Kim and Lee

where the second equality follows from (S7) and integration by parts. The participation constraint for the type θ, can be written as "Z

θ

U (θ) = θv(q(θ)) − t(θ) − λ

(sv(q(s)) − θv(q(θ)))dF (s) +

(t(θ) − t(s))dF (s)

θ θ

Z ≥

θ

θ

Z t(s)dF (s) − λ

θ

#

θ

Z

sv(q(s))dF (s), θ

or λ−1 t(θ) − θv(q(θ)) ≤ λ+1

Z

!

θ

t(s)dF (s) .

(S9)

θ

From (S8) and (S9), we obtain Z

θ

θ

Z t(s)dF (θ) ≤

θ

v(q(θ))J(θ)dF (θ) + t(θ) − θv(q(θ)) θ θ

Z ≤ θ

λ−1 v(q(θ))J(θ)dF (θ) + λ+1

Z

!

θ

t(s)dF (s) , θ

or Z θ

θ

λ+1 t(s)dF (θ) ≤ 2

!

θ

Z

v(q(θ))J(θ)dF (θ) . θ

Thus, the seller’s profit is bounded above by Z θ

θ



 λ+1 v(q(θ))J(θ) − cq(θ) dF (θ). 2

(S10)

We now show that (S10) is maximized by setting q(·) constant. To do so, consider any non-increasing Rθ q(·) and let q denote its expected value i.e. q = θ q(θ)dF (θ). Then, we must have Z θ

θ



! Z !  Z θ   θ λ+1 λ+1 v(q(θ))J(θ) − cq(θ) dF (θ) ≤ v(q(θ))dF (θ) J(θ)dF (θ) 2 2 θ θ Z θ − cq(θ)dF (θ) θ

! Z θ λ+1 ≤ v(q) J(θ)dF (θ) − cq 2 θ  Z θ λ+1 = v(q)J(θ) − cq dF (θ), 2 θ 



where the first inequality follows from the fact that v(q(·)) is non-increasing while J(·) is increasing, and the second from the Jensen’s inequality. Thus, (S10) is maximized by a constant q(·). Given the constant quality q that maximizes (S10), the upper bound of the seller’s profit can be achieved by setting t(θ) = λ+1 2 θv(q), ∀θ since Z θ

θ

λ+1 λ+1 v(q)J(θ)dF (θ) = θv(q) = 2 2

where the first equality follows from the fact that the (IR) constraint is satisfied.

Rθ θ

Z

θ

t(θ)dF (θ), θ

J(θ)dF (θ) = θ. It is straightforward to check that t u

Price Discrimination with Loss Averse Consumers: Supplementary Material

7

Numerical Examples The following examples demonstrate that λ need not be very high in order for some or full pooling to arise and also that a diverse pattern of pooling can emerge depending on the value distribution.3 Example S1 Suppose that θ is uniformly distributed on [1, 2] so F (θ) = θ − 1. Figure S1 draws the virtual value function J(·, λ) in the left panel and the ironed virtual value obtained using the technique of Myerson (1981) or Toikka (2011) in the right panel:4

Λ=2.6 2.0

Λ=2.3 1.5

Λ=2 1.0

Λ=1.7 Λ=1.4 1.2

1.4

1.6

1.8

2.0

1.4

1.6

1.8

2.0

2.0

1.5

Λ=2.6 Λ=2.3 Λ=2

1.0

Λ=1.7 Λ=1.4 1.2

Fig. S1 Gain-loss adjusted virtual value with uniform distribution

Pooling does not arise if λ = 1.4, and does arise in the interval [1.674, 2] if λ = 1.7, in the larger interval [1.236, 2] if λ = 2 and over the entire interval if λ = 2.3 or higher. Example S2 Suppose that θ is distributed on the interval [1, 2] with F (θ) = 1 − (2 − θ)n for n ≥ 1. Note that, as n grows, F (·) becomes more concave so that weights are shifting toward lower types. Figure S2 below illustrates J(·, 1.75) in the left panel and its ironing in the right panel:

1.5

n=1

1.5

n=1

1.4

n=1.4

1.4

n=1.4

1.3

n=3

1.3 1.2

1.2

n=3

1.1

n=6

1.1

n=6

1.0

1.0

1.2

1.4

1.6

1.8

2.0

1.2

1.4

1.6

1.8

2.0

Fig. S2 Gain-loss adjusted virtual value with distribution F (θ) = 1 − (2 − θ)n 3 In

thess examples, one can check for global incentive constraint by invoking Lemma S3. ironed virtual value is a modification of virtual value, which is monotone and whose pointwise maximization is equivalent to the maximization of expected virtual value subject to the constraint that the allocation rule is monotonic. 4 The

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Hahn, Kim, Kim and Lee

Thus, pooling occurs in the upper interval for n = 1, in the middle interval for n = 1.4, and in the lower interval if n = 3 or 6. As n grows, the pooling area shifts down as probability weights do. This can be understood from the fact that screening types with higher probability weights has a greater impact on the gain-loss utility.

S.3 Optimality of Reverse-Screening under TPPE We offer a numerical example to demonstrate the possibility of an optimal TPPE menu with reverse√ screening. Suppose that v(q) = q, θL = 2, θH = 3, p = 1/2, c = 1, and λ = 4. One can check that under these parameter values, λ > λS so there is no screening menu that can be a TPPE. We show that there is a reverse-screening TPPE menu that yields a higher profit than any pooling TPPE menu, which will imply that the optimal TPPE menu is reverse-screening. To do so, consider first a pooling menu R = {b = (q, t)}. For this to be a PPE, it is necessary to satisfy the PPE requirement with respect to the null menu. Under the given parameter values, this √ requirement is equivalent to U (R) ≥ U (∅) = 0, which yields t ≤ 47 q. By making this constraint binding

2 and plugging it into the seller’s profit, one can solve for the optimal profit equal to 14 74 , which sets an upper bound for the profit that the seller can achieve via any pooling menu. Let us next consider a reverse-screening menu R = {bL , bH } that satisfies a couple of constraints: u(bH |θH , R) ≥ u(bL |θH , R) and U (R) ≥ U (∅) = 0. The first constraint corresponds to (ICH ) while the second one to the requirement that R is (weakly) better than no participation. We solve for the optimal reverse-screening menu under these two constraints.5 At the optimum, the two constraints must be binding, which yields tL =

1 (−9v(qH ) + 58v(qL )) 28

and tH =

1 (39v(qH ) + 10v(qL )). 28

(S11)

Plug this into the profit function and maximize the resulting expression with qL and qH to obtain


 15
2
2

2 15 2 1 34 2 + 34 > 14 74 . Thus, qL = 28 and qH = 28 . The corresponding profit is equal to 2 28 28 we have obtained a reverse-screening TPPE menu that yields a higher profit than any pooling TPPE menu.

S.4 Proof of Proposition 6 Let us denote v = v(q ∗ ) and v 0 = v(q 0 ). We first show that the TPPE pooling menu M = {(q ∗ , t∗ )} must satisfy (27) in the main text. To do so, write the PPE constraints not to deviate to R = {∅, ∅}: t∗ ≤Φ v ∗ t 2 < θH v λ+1 t∗ 2 < θL . v λ+1

(U ) (F ICH ) (F ICL )

2 2 By Assumption 1-(ii), we have Φ > θH λ+1 > θL λ+1 , so the PPE constraint boils down to (U ), implying that (27) must be satisfied. Next, we prove that given M 0 = {b, b0 }, it is the PPE for both types to choose b. First, to show that {b, b} is a PE, the relevant (IC) and (IR) constraints are for the low type and can be written as follows:

t − t0 λ+1 ≤ θL v − v0 2 t λ+1 ≤ θL . v 2

(ICL ) (IRL )

With , δ, δ 0 = 0, these inequalities become Φ ≤ θL λ+1 2 , which is satisfied with slack due to Assumption 1-(ii). Thus, one can find sufficiently small , δ, δ 0 to satisfy the above inequalities. Next, we check the PPE requirements by going through all possible deviations. 5 We

show in the main text that a solution to this optimization problem satisfies all the other PPE requirements.

Price Discrimination with Loss Averse Consumers: Supplementary Material

9

Case 1: R = {∅, ∅} This menu cannot be a PE since u(∅|θH , {∅, ∅}) < u(b0 |θH , {∅, ∅}), which can be written as t0 < 2 v 0 . This inequality is satisfied by the construction of b0 . θH λ+1 Case 2: R = {∅, b} or R = {∅, b0 } This cannot be a PE since, with ˜b = b or b0 , u(∅|θL , {∅, ˜b}) < u(b0 |θL , {∅, ˜b}), which can be written as t0 1 + p + (1 − p)λ < θL . v0 1 + (1 − p) + pλ 0

2 < θL 1+p+(1−p)λ This inequality holds since vt 0 < θH λ+1 1+(1−p)+pλ due to Assumption 1-(iv). 0 0 Case 3: R = {b , b } This cannot be a PE since u(b0 |θH , {b0 , b0 }) < u(b|θH , {b0 , b0 }), which can be written as

t − t0 2 < θH . 0 v−v pλ + 1 2 With , δ, δ 0 = 0, this inequality becomes Φ < θH pλ+1 , which holds due to Assumption 1-(iii). Thus, one 0 can find sufficiently small .δ, δ to satisfy the above inequality. Case 4: R = {b0 , b} We show that this menus yields no greater ex-ante utility than {b, b} does, or U ({b0 , b}) ≤ U ({b, b}), which can be written as

[1 − (1 − p)(λ − 1)] (t − t0 ) ≤ θL [1 + (1 − p)(λ − 1)] (v − v 0 ). This inequality holds if 1 − (1 − p)(λ − 1) ≤ 0. In the case 1 − (1 − p)(λ − 1) > 0, with , δ, δ 0 = 0, this 1+(1−p)(λ−1) , which is satisfied with slack due to Assumption 1-(iii). Thus, inequality becomes Φ ≤ θL 1−(1−p)(λ−1) one can find sufficiently small .δ, δ 0 to satisfy the above inequality. Case 5: R = {b, b0 }. We show that this menu yields no greater ex-ante utility than {b, b} does, or U ({b, b0 }) ≤ U ({b, b}), which can be written as (1 − p) [θH (v − v 0 ) − (t − t0 )] [1 − p(λ − 1)] ≥ 0. Since λ < 1 + 0

t−t v−v 0 .

1 p 0

(S12)

or 1 − p(λ − 1) > 0, it suffices to show that θH (v − v 0 ) − (t − t0 ) ≥ 0 or θH ≥

With , δ, δ = 0, this inequality becomes θH ≥ Φ, which is satisfied with slack since θH − Φ = p [1 + (1 − p)(λ − 1)] (θH − θL ) > 0. Thus, one can find sufficiently small .δ, δ 0 to satisfy the desired inequality. Case 6: R = {b, ∅} or R = {b0 , ∅}. That these references cannot be PE can be shown using the same argument as in the proof of Lemma 6 of the main text. S.5 Alternative Reference Points S.5.1 Bundles as Stochastic Reference Point Proposition S1 Suppose that the buyer’s gain-loss utility is as given by (29) in the main text, and consider the case of a continuum of consumer types. If (q, t) : [θ, θ] → R+ × R is an optimal TPE menu then q(·) and t(·) are monotone increasing. Proof Let Θ = [θ, θ] with cdf F . Suppose that there are two types θ < θ0 for which q(θ) > q(θ0 ) and t(θ) > t(θ0 ). We draw a contradiction. First, it is straightforward to see that ˜ > q(θ0 )} = {θ˜ ∈ [θ, θ]|t(θ) > t(θ) ˜ > t(θ0 )}. ˆ ≡ {θ˜ ∈ [θ, θ]|q(θ) > q(θ) Θ The constraint for type θ not to mimic θ0 is given as  Z  Z     ˜ ˜ ˜ ˜ t(θ) − t(θ) dF (θ) θv(q(θ)) − t(θ) + θ v(q(θ)) − v(q(θ)) dF (θ) − λ ˜ ˆ ˜ Θ ˆ θ∈  θ∈ΘZ  Z     0 0 0 0 ˜ ˜ ˜ ˜ v(q(θ)) − v(q(θ )) dF (θ) + t(θ) − t(θ ) dF (θ) ≥θv(q(θ )) − t(θ ) + −λθ ˜ Θ ˆ θ∈

˜ Θ ˆ θ∈

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since the gain-loss utilities measured against all types θ˜ ∈ / Θ are canceled out from both sides. This equation can be re-arranged to   Z     ˜ ˜ ˆ − v(q(θ0 )) 1 + λF (Θ) ˆ + (λ − 1) θ v(q(θ)) 1 + F (Θ) v(q(θ))dF (θ) (S13) ˜ Θ ˆ θ∈ Z     ˜ ˜ ˆ − t(θ0 ) 1 + F (Θ) ˆ − (λ − 1) ≥ t(θ) 1 + λF (Θ) t(θ)dF (θ). ˜ Θ ˆ θ∈

Analogously, the constraint for type θ0 not to mimic θ can be written (after re-arrangement) as Z     0 ˜ ˜ ˆ ˆ t(θ) 1 + λF (Θ) − t(θ ) 1 + F (Θ) − (λ − 1) t(θ)dF (θ) ˜ Θ ˆ θ∈   Z     0 0 ˜ ˜ ˆ ˆ ≥ θ v(q(θ)) 1 + F (Θ) − v(q(θ )) 1 + λF (Θ) + (λ − 1) v(q(θ))dF (θ) . (S14) ˜ Θ ˆ θ∈

Since the expression in (S13) should be weakly greater than that in (S14) and since θ0 > θ, the expression in the square bracket of (S13) and (S14) must be non-positive. This cannot be true, however, since Z     0 ˜ ˜ ˆ ˆ v(q(θ)) 1 + F (Θ) − v(q(θ )) 1 + λF (Θ) + (λ − 1) v(q(θ))dF (θ) ˜ Θ ˆ θ∈     0 ˆ − v(q(θ0 )) 1 + λF (Θ) ˆ + (λ − 1)F (Θ)v(q(θ ˆ ≥ v(q(θ)) 1 + F (Θ) ))   ˆ [v(q(θ)) − v(q(θ0 ))] > 0, = 1 + F (Θ) ˜ ∀θ˜ ∈ Θ. ˆ where the first inequality holds since q(θ0 ) < q(θ), S.5.2 Average Bundle Consider binary types and menu {bL , bH }. We set type-θ buyer’s gain-loss utility from bundle b = (q, t) to be µ [θv(q) − (pθL v(qL ) + (1 − p)θH v(qH ))] + µ [(ptL + (1 − p)tH ) − t] , (S15) where µ is the loss aversion indicator function as defined in in the main text. Proposition S2 Suppose that the buyer’s gain-loss utility is given as in (S15). Then, the optimal menu is a pooling menu if p ≥ pˆ and λ ∈ [λS , λR ], where pˆ, λS and λR are as in Theorem 1 of the main text. Proof Let us first consider screening menus. We show that given the alternative reference point, the (IRL ) and (ICH ) constraints imply the corresponding conditions (10) and (11) in the main text where we consider screening TPE menus. According to part (b) of Proposition 3 in the main text, the optimal screening menu under conditions (10) and (11) in the main text is dominated by a pooling menu if λ ≥ λS . This is also true here. Write first the (IRL ) constraint as u(bL |θL , R) = θL vL − tL − λ [(pθL vL + (1 − p)θH vH ) − θL vL ] + [(ptL + (1 − p)tH ) − tL ] ≥ u(∅|θL , R) = −λ [pθL vL + (1 − p)θH vH ] + [ptL + (1 − p)tH ] , which can be rewritten as θL (λ + 1)vL ≥ 2tL , yielding the same inequality as (10) in the main text. The (ICH ) constraint takes different forms depending on the relative sizes of θH vL , θL vH and pθL vL + (1 − p)θH vH . If pθL vL + (1 − p)θH vH ≥ θH vL , then (ICH ) is given by u(bH |θH , R) = θH vH − tH + [θH vH − (pθL vL + (1 − p)θH vH )] − λ [tH − (ptL + (1 − p)tH )] ≥ u(bL |θH , R) = θH vL − tL − λ [(pθL vL + (1 − p)θH vH ) − θH vL ] + [(ptL + (1 − p)tH ) − tL ] , while otherwise it is given by θH vH − tH + [θH vH − (pθL vL + (1 − p)θH vH )] − λ [tH − (ptL + (1 − p)tH )] ≥ θH vL − tL + [θH vL − (pθL vL + (1 − p)θH vH )] + [(ptL + (1 − p)tH ) − tL ] .

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The two inequalities can be simplified as [1 + p + (1 − p)λ] θH (vH − vL ) − p(λ − 1)(θH − θL )vL ≥ [1 + (1 − p) + pλ] (tH − tL ), and 2θH (vH − vL ) ≥ [1 + (1 − p) + pλ] (tH − tL ), respectively. It is straightforward to check that the LHS of these inequalities is smaller than [1 + p + (1 − p)λ]θH [vH − vL ]. Thus, it implies (11) in the main text. The analysis of comparing pooling and reverse-screening menus is analogous and hence omitted. One can show that the (IC) and (IR) constraints under the alternative reference point are identical to the same constraints in the analysis of optimal reverse-screening TPE menu. Therefore, it follows from Proposition 4 in the main text that reverse-screening is dominated by pooling if λ ≤ λR . S.5.3 Additive Separability We show that if gain-loss utility only applies to the gross utility, the monopolist’s problem is no different from the standard one. The following analysis modifies the binary type model described in Section 2 of the main text. First, we formulate the (ICH ) constraint. There are three cases to consider: 1. When θH v(qH ) − tH ≥ θH v(qL ) − tL , (ICH ) is θH v(qH ) − tH + p[θH v(qH ) − tH − (θL v(qL ) − tL )] ≥ θH v(qL ) − tL + p(θH − θL )v(qL ) − (1 − p)λ[θH v(qH ) − tH − (θH v(qL ) − tL )], which reduces to [θH v(qH ) − tH − (θH v(qL ) − tL )][1 + p + (1 − p)λ] ≥ 0. This amounts to (ICH ) in the standard case without loss aversion. 2. When θH v(qH ) − tH < θL v(qL ) − tL , (ICH ) is θH v(qH ) − tH − pλ[θH v(qH ) − tH − (θL v(qL ) − tL )] ≥ θH v(qL ) − tL + p(θH − θL )v(qL ) + (1 − p)[θH v(qL ) − tL − (θH v(qH ) − tH )], which cannot be satisfied since the intrinsic utility of LHS if smaller than that of RHS by assumption, and the gain-loss utility of LHS is negative while that of RHS is positive. 3. When θL v(qL ) − tL ≤ θH v(qH ) − tH < θH v(qL ) − tL , (ICH ) is θH v(qH ) − tH + p[θH v(qH ) − tH − (θL v(qL ) − tL )] ≥ θH v(qL ) − tL + p(θH − θL )v(qL ) + (1 − p)[θH v(qL ) − tL − (θH v(qH ) − tH )], which can be rearranged to [θH v(qH ) − tH − (θH v(qL ) − tL )][1 + p + (1 − p)] ≥ 0. But, this contradicts the assumption. We therefore focus on the case θH v(qH ) − tH ≥ θH v(qL ) − tL for (ICH ), which in turn implies θH v(qH ) − tH > θL v(qL ) − tL . Furthermore, (IRH ) can be ignored since u(rH |θH , R) > u(rL |θL , R) ≥ u(∅|θL , R) = u(∅|θH , R), where the second inequality follows from (IRL ). We next formulate (IRL ). There are again three cases to consider: 1. When θH v(qH ) − tH < 0, (IRL ) is θL v(qL ) − tL − (1 − p)λ[θH v(qH ) − tH − (θL v(qL ) − tL )] ≥ p[tL − θL v(qL )] + (1 − p)[tH − θH v(qH )] which reduces to [θL v(qL ) − tL ][1 + p + (1 − p)λ] ≥ [θH v(qH ) − tH ](1 − p)(λ − 1). But, this contradicts the assumption that θL v(qL ) − tL < θH v(qH ) − tH < 0.

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2. When θL v(qL ) − tL ≥ 0, (IRL ) is θL v(qL ) − tL − (1 − p)λ[θH v(qH ) − tH − (θL v(qL ) − tL )] ≥ −pλ[θL v(qL ) − tL ] − (1 − p)λ[θH v(qH ) − tH ], which implies that [θL v(qL ) − tL ](1 + λ) ≥ 0. 3. When θL v(qL ) − tL < 0 ≤ θH v(qH ) − tH , (IRL ) is θL v(qL ) − tL − (1 − p)λ[θH v(qH ) − tH − (θL v(qL ) − tL )] ≥ p[tL − θL v(qL )] − (1 − p)λ[θH v(qH ) − tH ], which implies that [θL v(qL ) − tL ](1 − p)(λ − 1) ≥ 0. Since λ > 1, the latter two cases imply that (IRL ) is θL v(qL ) − tL ≥ 0, just as in the standard case without loss aversion. Finally, consider (ICL ). Since θH v(qH ) − tH > θL v(qL ) − tL , the form of (ICL ) depends on whether (i) θL v(qL ) − tL ≥ θL v(qH ) − tH , or (ii) otherwise. In case (i), (ICL ) is θL v(qL ) − tL − (1 − p)λ[θH v(qH ) − tH − (θL v(qL ) − tL )] ≥ θL v(qH ) − tH − (1 − p)λ(θH − θL )v(qH ) − pλ[θL v(qL ) − tL − (θL v(qH ) − tH )] which reduces to [θL v(qL ) − tL − (θL v(qH ) − tH )][1 + λ] ≥ 0,

(S16)

yielding the same (ICL ) as in the standard model. In case (ii), the resulting (ICL ) is identical to (S16) except that the multiplier in the LHS becomes 1 + p + (1 − p)λ instead of 1 + λ.

REFERENCES Myerson, R. (1981), “Optimal Auction Design,” Mathematics of Operations Research, 6, 58-73. Toikka, J. (2011), “Ironing without Control,” Journal of Economic Theory, 146, 2510-2526.