Accessible Unlicensed Requires Authentication Published by De Gruyter January 13, 2018

Consumer Heterogeneity and Surplus under Two-Part Pricing

Sreya Kolay and Rajeev K. Tyagi

Abstract

For many products, consumers need to sign a pricing contract with the seller under uncertainty about their future consumption needs. Recent empirical literature has consistently pointed out that consumers may not be good forecasters of their future consumption needs, and may suffer from overestimation or underestimation biases. This paper considers consumers who are heterogeneous in their expected demands arising from heterogeneity in their biases about their forecasted consumption needs. We show that the optimal menu of two-part pricing in this case leads to the lower-expected demand segment getting exactly the same surplus on the average as the higher-expected demand segment, or the higher-expected demand segment getting even lower surplus on the average than the lower-expected demand segment. We show directions of externalities these unbiased, positively-biased, and negatively-biased segments impose on one another, and how they can be different under no price discrimination, and second- and third-degree price discrimination.

Appendix

A Benchmark Case of Third-Degree Price Discrimination

Unbiased Consumer Segment. Consumers in this segment make their choice using the true utility and demand functions, Utrue and qtrue(w), as given in eqs (1) and (2). The seller’s pricing problem is to choose a fixed fee F and a per-unit price w that maximize its expected profit, F+wE(qtrue(w)), subject to the consumer participation constraint that its ex-ante expected surplus be nonnegative, i.e., E[Utrue(qtrue(w))]wE[qtrue(w)]F0, where E[Utrue(qtrue(w))]wE[qtrue(w)]=12[(a+δbw)22b]+12[(aδbw)22b] from eq. (12). The maximand in the seller’s profit maximization problem is increasing in F, hence it is optimal for the seller to select F such that the consumer participation constraint binds. Therefore, F^=E[Utrue(qtrue(w))]wE[qtrue(w)]. Substituting this in the seller’s profit function and solving for w, we get the optimal per-unit price as wu=0. Using that, we get the optimal fixed fee as Fu=a2+δ22b, the seller’s expected profit as π=F=a2+δ22b. Given that consumers are unbiased, their ex-post surplus, on an average, is the same as their expected ex-ante surplus. Using the expressions of wu and Fu in E[Utrue(qtrue(wu))]wuE[qtrue(wu)]Fu, we get the average ex-post consumer surplus as zero.

Negatively-Biased Consumer Segment. Consumers in this segment make their choice using the utility and demand functions Un(w) and qn(w), as given in eqs (5) and (6). The seller’s pricing problem is to choose a fixed fee F and a per-unit price w that maximize its expected profit, F+wE(qtrue(w)), subject to the consumer participation constraint that its ex-ante expected surplus be nonnegative, i.e., E[Un(qn(w))]wE[qn(w)]F0, where E[Un(qn(w))]wE[qn(w)]=(12τ)[(a+δbw)22b]+(12+τ)[(aδbw)22b]. Solving this as in the unbiased case, we get the optimal prices[5] as wn=2δτb,Fn=12b[(a2δτ)(a6δτ)+δ2], and the seller’s expected profit as π=12b[(a2δτ)2+δ2]. Given that consumers are biased, their ex-post surplus, on an average, is not the same as their expected ex-ante surplus. Using the expressions of wn and Fn in E[Utrue(qtrue(wn))]wnE[qtrue(wn)]Fn, we get the average ex-post consumer surplus as 2δτ(a2δτ)b\gt0.

Positively-Biased Consumer Segment. Consumers in this segment make their choice using the utility and demand functions Up(q) and qp(w), as given in eqs (3) and (4). The seller’s pricing problem is to choose a fixed fee F and a per-unit price w that maximize its expected profit, F+wE(qtrue(w)), subject to the consumer participation constraint that its ex-ante expected surplus be nonnegative, i.e., E[Up(qp(w))]wE[qp(w)]F0, where E[Up(qp(w))]wE[qp(w)]=(12+σ)[(a+δbw)22b]+(12σ)[(aδbw)22b]. Solving this as in the unbiased and negatively-biased cases, we get the optimal prices as wp=2δσb,Fp=12b[(a+2δσ)(a+6δσ)+δ2], and the seller’s expected profit as π=12b[(a+2δσ)2+δ2]. Given that consumers are biased, their ex-post surplus, on an average, is not the same as their expected ex-ante surplus. Using the expressions of wp and Fp in E[Utrue(qtrue(wp))]wpE[qtrue(wp)]Fp, we get the average ex-post consumer surplus as 2δσ(a+2δσ)b\lt0.

B Second-Degree Price Discrimination

Optimal menu of two-part tariffs in the market with unbiased and negatively-biased consumers.

The seller’s profit maximization problem and the constraints are given in the paper. It is optimal for the seller to have constraints (IRn) and (ICun) bind, i.e. it is optimal to choose the fixed fee Fn to extract the entire ex-ante expected surplus from the lower-expected-demand negatively-biased consumers. The fixed fee Fu is selected to leave the higher-expected-demand unbiased consumers indifferent between choosing (wn,Fn) and (wu,Fu). Therefore,

(21)Fn=E[Un(qn(wn))]wnE[qn(wn)]=(0.5τ)((abwn+δ)22b)+(0.5+τ)((abwnδ)22b),
(22)Fu=E[Uu(qu(wu))]wuE[qu(wu)]E[Uu(qu(wn))]+wnE[qu(wn)]+Fn=[0.5((abwu+δ)22b)+0.5((abwuδ)22b)][0.5((abwn+δ)22b)+0.5((abwnδ)22b)]+(0.5τ)((abwn+δ)22b)+(0.5+τ)((abwnδ)22b).

Substituting these in the seller’s maximization problem and solving for wn and wu, we get[6]

(23)wu=0,wn=2δταb.

Using the expressions for wu and wn from eq. (23) in eqs (21) and (22), we get:

(24)Fu=a2α+δ2(α+8τ2)4aαδτ2αb,
(25)Fn=α2(a2+δ2)+4(1+2α)δ2τ24aα(1+α)δτ2α2b.

It remains to show that the remaining constraints (IRu) and (ICnu) also hold. We find that using eqs (23) and (24), E[Uu(qu(wu)]wuE[qu(wu)]Fu=2δτ[aα2δτ]αb\gt0, because, from Footnote 6, αα¯1=2δτ/[aδ]\gt2δτ/a. Therefore, (IRu) is satisfied. Next, we find that using eqs (23) and (24), E[Un(qn(wu)]wuE[qn(wu)]Fu=4δ2τ2αb\lt0.This along with the fact that (IRn) binds and hence E[Un(qn(wn)]wnE[qn(wn)]Fn=0 implies that (ICnu) is satisfied.

Alternatively, the seller can offer only (wu,Fu) as defined in eq. (7) to sell only to the unbiased segment. Comparing the seller’s profits, for αα¯1, it can be shown that the seller’s profit from the optimal menu of two-part tariffs is higher than its profit from serving only the unbiased segment. Therefore, for αα¯1, the seller offers the above menu to the two segments; and for α\ltα¯1, i.e. when the above menu is not feasible, the seller serves only the unbiased segment.

Claim in Section 5.1: Surplus of both segments2δτ[aα2δτ]αb\gt0.

The expression is positive when α2δτa. Now, 2δτa is smaller than α¯1=2δτ/[aδ]. From Footnote 6, αα¯1, and hence it follows that α2δτa.

Optimal menu of two-part tariffs in the market with unbiased and positively-biased consumers.

The seller’s profit maximization problem is given in the paper. The usual constraints are

(IRp):E[Up(qp(wp))]wpE[qp(wp)]Fp0,(IRu):E[Uu(qu(wu))]wuE[qu(wu)]Fu0,(ICpu):E[Up(qp(wp))]wpE[qp(wp)]FpE[Up(qp(wu))]wuE[qp(wu)]Fu,(ICup):E[Uu(qu(wu))]wuE[qu(wu)]FuE[Uu(qu(wp))]wpE[qu(wp)]Fp.

It is optimal for the seller to have constraints (IRu) and (ICpu) bind, i.e. it is optimal to choose the fixed fee Fu to extract the entire ex-ante expected surplus from the lower-expected-demand unbiased consumers. The fixed fee Fp is selected to leave the higher-expected-demand positively-biased consumers indifferent between choosing (wp,Fp) and (wu,Fu). Therefore,

(26)Fu=E[Uu(qu(wu))]wuE[qu(wu)]=0.5((abwu+δ)22b)+0.5((abwuδ)22b),
(27)Fp=E[Up(qp(wp))]wpE[qp(wp)]E[Up(qp(wu))]+wuE[qp(wu)]+Fu=[(0.5+σ)((abwp+δ)22b)+(0.5σ)((abwpδ)22b)][(0.5+σ)((abwu+δ)22b)+(0.5σ)((abwuδ)22b)]+0.5((abwu+δ)22b)+(0.5)((abwuδ)22b).

Substituting these in the seller’s maximization problem and solving for wu and wp, we get[7]

(28)wu=2αδσ(1α)b,wp=2δσb,

Using the expressions for wu and wp from eq. (28) in eqs (26) and (27), we get:

(29)Fu=(1α)2(a2+δ2)4aα(1α)δσ+4α2δ2σ22(1α)2b,
(30)Fp=a2(1α)+4a(1α)δσ+δ2(1+12σ2α4ασ2)2(1α)b.

We need to show that the remaining constraints (IRp) and (ICup) also hold. We find that using eqs (28) and (30), E[Up(qp(wp))]wpE[qp(wp)]Fp=2δσ[a(1α)2αδσ](1α)b0 according as αaa+2δσ. Now, from Footnote 7, we know that αα¯2=(aδ)/[aδ+2σδ].Since α¯2\ltaa+2δσ, it must hold that E[Up(qp(wp)]wpE[qp(wp)]Fp\gt0 and hence, (IRp) is satisfied. Next, we find that using eqs (28) and (30), E[Uu(qu(wp)]wpE[qu(wp)Fp=4δ2σ2b(1α)\lt0.This along with the fact that (IRu) binds and hence E[Uu(qu(wu)]wuE[qu(wu)]Fu=0 implies that (ICup) is satisfied.

Alternatively, the seller can offer only (wp,Fp) as defined in eq. (7) to sell only to the positively-biased segment. Comparing the seller’s profits, for αα¯2, it can be shown that the seller’s profit from the optimal menu of two-part tariffs is higher than its profit from serving only the positively-biased segment. Therefore, for αα¯2, the seller offers the above menu to the two segments; and for α\gtα¯2, i.e. when the above menu is not feasible, the seller serves only the positively-biased segment.

Optimal menu of two-part tariffs in the market with negatively-biased and positively-biased consumers.

The seller’s profit maximization problem is

(31)MaxFp,wp,Fn,wnα[Fn+wnE(qtrue(wn))]+(1α)[Fp+wpE(qtrue(wp))],

subject to the following constraints:

(32)(IRp):E[Up(qp(wp))]wpE[qp(wp)]Fp0,(ICpn):E[U(qp(wp))]wpE[qp(wp)]FpE[Up(qp(wn))]wnE[qp(wn)]Fn,(IRn):E[Un(qn(wn))]wnE[qn(wn)]Fn0,(ICnp):E[Un(qn(wn))]wnE[qn(wn)]FnE[Un(qn(wp))]wpE[qn(wp)]Fp.

It is optimal to choose the fixed fee Fn to extract the entire ex-ante expected surplus from the lower-expected-demand negatively-biased consumers. The fixed fee Fp is selected to leave the higher-expected-demand positively-biased consumers indifferent between choosing (wp,Fp) and (wn,Fn). Therefore,

(33)Fn=E[Un(qn(wn))]wnE[qn(wn)]=(0.5τ)((abwn+δ)22b)+(0.5+τ)((abwnδ)22b),
(34)Fp=E[U(qp(wp))]wpE[qp(wp)]E[Up(qp(wn))]+wnE[qp(wn)]+Fn=[(0.5+σ)((abwp+δ)22b)+(0.5σ)((abwpδ)22b)][(0.5+σ)((abwn+δ)22b)+(0.5σ)((abwnδ)22b)]+(0.5τ)((abwn+δ)22b)+(0.5+τ)((abwnδ)22b).

Substituting these in the seller’s maximization problem in eq. (31) and solving for wn and wp, we get[8]

(35)wn=2δ(τ+σ(1α))αb,wp=2δσb,

Using the expressions for wn and wp from eq. (35) in eqs (33) and (34), we get:

(36)Fn=α2δ2+(aα2δσ(1α)2δτ)(aα2δσ(1α)2δτ(1+2α))2α2b,
(37)Fp=a2α+4aαδ(στ)+δ2(α(1+4σ(σ2τ))+8(σ+τ)2)2αb.

It remains to show that the remaining constraints (IRp) and (ICnp) also hold. We find that using eqs (35) and (37), E[Up(qp(wp)]wpE[qp(wp)]Fp=2δ(σ+τ)(aα2(1α)δσ2δτ)αb0 according as α2δ(σ+τ)a+2δσ. But 2δ(σ+τ)a+2δσ\ltα¯3=2δ(σ+τ)aδ+2δσ. Then, since by Footnote 8, αα¯3 and hence E[Up(qp(wp)]wpE[qp(wp)]Fp\gt0 and (IRp) is satisfied. Next, we find that using eqs (35) and (37), E[Un(qn(wp)]wpE[qn(wp)]Fp=4δ2(σ+τ)2αb\lt0.This along with the fact that (IRn) binds and hence E[Un(qn(wn)]wnE[qn(wn)]Fn=0 implies that (ICnp) is satisfied.

Alternatively, the seller can offer only (wp,Fp) as defined in eq. (7) to sell only to the positively-biased segment. Comparing the seller’s profits, for αα¯3, it can be shown that the seller’s profit from the optimal menu of two-part tariffs is higher than its profit from serving only the positively-biased segment. Therefore, for αα¯3, the seller offers the above menu to the two segments; and for α\ltα¯3, i.e. when the above menu is not feasible, the seller serves only the positively-biased segment.

Claim in Section 5.3: Surplus of lower-expected demand (negatively-biased) segment2δτ(aα2(1α)δσ2δτ)αb\gt0.

The sign of this expression is 0 according as α2δ(σ+τ)/[a+2δσ]. Since we have αα¯3=2δ(σ+τ)aδ+2δσ as noted in Footnote 8, and since α¯3\gt2δ(σ+τ)/[a+2δσ], it must hold that the expression is positive.

Claim in Section 5.3: Surplus of higher-expected demand (positively-biased) segment,2δ(aατ2δ(σ2+(2α)στ+τ2))αb,is positive if bias (τ)of the negatively-biased segment is sufficiently large or if bias (τ)is intermediate and size (α)of the negatively-biased segment is sufficiently high, and negative otherwise.

The sign of this expression is 0 according as α2δ(σ+τ)2(aτ+2δστ)=α^. Since 1\gtαα¯3, as described in Footnote 8, it follows that if α^ exceeds 1, then it must hold that α\ltα^ and the expression above is negative for all α. Similarly, if α^ is lower than α¯3, then it must hold that α\gtα^, and the expression is positive for all α. Otherwise, the expression is positive or negative depending on whether αα^. Now, it can be shown that α^\gt1 when τ\ltτ_=(a2δσ)4δ14δa24aδσ12δ2σ2, and similarly, it can be shown that α^\ltα¯3 when τ\gtτ¯=aσδσ+2δσ2δ. Therefore, we may conclude that when τ\gtτ¯ or when τ_\ltτ\ltτ¯ and αα^, then the expression above is positive. Otherwise, when τ\ltτ_ or when τ_\ltτ\ltτ¯ and α\ltα^, then the expression above is negative.

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Published Online: 2018-01-13

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