# 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,

**Negatively-Biased Consumer Segment**. Consumers in this segment make their choice using the utility and demand functions ^{[5]} as

**Positively-Biased Consumer Segment**. Consumers in this segment make their choice using the utility and demand functions

## 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

Substituting these in the seller’s maximization problem and solving for ^{[6]}

Using the expressions for

It remains to show that the remaining constraints

Alternatively, the seller can offer only (

*Claim in Section 5.1: Surplus of both segments*

The expression is positive when

**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

It is optimal for the seller to have constraints

Substituting these in the seller’s maximization problem and solving for ^{[7]}

Using the expressions for

We need to show that the remaining constraints

Alternatively, the seller can offer only (

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

The seller’s profit maximization problem is

subject to the following constraints:

It is optimal to choose the fixed fee

Substituting these in the seller’s maximization problem in eq. (31) and solving for ^{[8]}

Using the expressions for

It remains to show that the remaining constraints

Alternatively, the seller can offer only (

*Claim in Section 5.3: Surplus of lower-expected demand (negatively-biased) segment*

The sign of this expression is

*Claim in Section 5.3: Surplus of higher-expected demand (positively-biased) segment,***,***is positive if bias (**of the negatively-biased segment is sufficiently large or if bias (**is intermediate and size (**of the negatively-biased segment is sufficiently high, and negative otherwise*.

The sign of this expression is

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**Published Online:**2018-01-13

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