• No results found

Additional Proofs and Results from Section 3

Assumption 2: For any ? ∈ ?, define the set of strictly preferred allocations:

A.2 Additional Proofs and Results from Section 3

Proof of Lemma 1: Note that there must be values π‘žπ‘π‘‡ and π‘žπ‘π‘‡β€² such that:

Proof of Proposition 3: From lemma 1 and prices being bounded away from zero, we can always find a value of 𝑝̂𝑠 such that:

Proof of Theorem 2: From lemma 2 and rationalizability of the data:

65

∫ [𝑑𝑀𝑙(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–) + [𝑝𝑖𝑠 βˆ’ 𝑝̅] π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœπ‘–)]𝑑𝐹𝑝′′𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–

β‰₯ ∫ [𝑑𝑀𝑙(𝑝𝑖𝑠; πœƒπ‘–) + [π‘π‘–π‘ βˆ’ 𝑝̅] π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)] 𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

∫ [𝑑𝑀𝑙(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–) + 𝑝𝑖𝑠 π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœπ‘–)]𝑑𝐹𝑝′′𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–

β‰₯ ∫ [𝑑𝑀𝑙(𝑝𝑖𝑠; πœƒπ‘–) + 𝑝𝑖𝑠 π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)] 𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

∫ [𝑑𝑀𝑙(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–) + [𝑝𝑖𝑠 βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœπ‘–)]𝑑𝐹𝑝′′𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–

β‰₯ ∫ [𝑑𝑀𝑙(𝑝𝑖𝑠; πœƒπ‘–) + [π‘π‘–π‘ βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)] 𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

Rearranging yields:

∫ 𝑑𝑀𝑙(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–)𝑑𝐹𝑝′′𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–

β‰₯ ∫ 𝑑𝑀𝑙(𝑝𝑖𝑠; πœƒπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

βˆ’ ∫ [π‘π‘–π‘ βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝′′𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–

+ ∫ [𝑝𝑖𝑠 βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

We can show from lemma 1 and 𝑝𝑖𝑠 ∈ [𝑝̅, 𝑝̅ + π‘šΜ… 𝜏] = 𝒫 βˆ€π‘– that the last two terms total to a a non-negative value. Formally, for any 𝑝𝑖𝑠 ∈ (𝑝̅, 𝑝̅ + π‘šΜ… 𝜏), πœƒπ‘–, πœπ‘–, πœπ‘–β€²:

𝑝𝑖𝑠 > 𝑝̃𝑠 β‡’ 𝑝𝑏(𝑝𝑖𝑠) > 𝑝̃𝑠 β‡’ π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœπ‘–β€²) ≀ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–) 𝑝𝑖𝑠 ≀ 𝑝̃𝑠 β‡’ 𝑝𝑏(𝑝𝑖𝑠) < 𝑝̃𝑠 β‡’ π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœƒπ‘–, πœπ‘–β€²) β‰₯ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–) Either way:

66

[π‘π‘–π‘ βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœƒπ‘–, πœπ‘–β€²) ≀ [π‘π‘–π‘ βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–) Thus:

∫ [𝑝𝑖𝑠 βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

= ∫ [π‘π‘–π‘ βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

π‘π‘–π‘ βˆˆπ‘–π‘›π‘‘(𝒫),πœƒπ‘–,πœπ‘–

+ ∫ [𝑝𝑖𝑠 βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

π‘π‘–π‘ βˆˆπœ•π’«,πœƒπ‘–,πœπ‘–

β‰₯ ∫ [π‘π‘–π‘ βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝′′𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

π‘π‘–π‘ βˆˆπ‘–π‘›π‘‘(𝒫),πœƒπ‘–,πœπ‘–

+ ∫ [𝑝𝑖𝑠 βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

π‘π‘–π‘ βˆˆπœ•π’«,πœƒπ‘–,πœπ‘–

= ∫ [𝑝𝑖𝑠 βˆ’ 𝑝̃𝑠] π‘ž(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝′′𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–

∫ 𝑑𝑀𝑙(𝑝𝑏(𝑝𝑖𝑠); πœƒπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–

β‰₯ ∫ 𝑑𝑀𝑙(𝑝𝑖𝑠; πœƒπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

∎

We now proceed to proving Theorem 3, i.e. that deadweight loss is maximized given the available data and distribution πΉπœƒβˆ— by having a cutoff value Ξ” such that the ratio of

𝑑𝑀𝑙(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, β„Ž) to π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, 𝑙) βˆ’ π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, β„Ž) is greater (less than) Ξ” for those we assign a perceived price of 𝑝̅ + π‘šΜ… 𝜏 (respectively 𝑝̅). However, we must also consider the choice of conditional distribution of 𝜁. Toward that end, we note that deadweight loss is bounded by the product of the reduction in demand and π‘šΜ… 𝜏.

Lemma 5: If 𝑝𝑖𝑠 ∈ [𝑝̅, 𝑝̅ + π‘šΜ… 𝜏], then 𝑑𝑀𝑙(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–) ≀ [π‘ž(𝑝̅; πœƒπ‘–, πœπ‘–) βˆ’ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)] π‘šΜ… 𝜏 βˆ€πœƒπ‘–, πœπ‘–. Proof: Using lemma 2:

0 = 𝑑𝑀𝑙(𝑝̅; πœƒπ‘–, πœπ‘–) β‰₯ 𝑑𝑀𝑙(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–) βˆ’ [π‘π‘–π‘ βˆ’ 𝑝̅] βˆ— [π‘ž(𝑝̅; πœƒπ‘–, πœπ‘–) βˆ’ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)]

67

𝑑𝑀𝑙(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–) ≀ [π‘ž(𝑝̅; πœƒπ‘–, πœπ‘–) βˆ’ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)] βˆ— [π‘π‘–π‘ βˆ’ 𝑝̅] ≀ [π‘ž(𝑝̅; πœƒπ‘–, πœπ‘–) βˆ’ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)] π‘šΜ… 𝜏 ∎ Proof of Theorem 3: The outline of the proof is as follows. First, we use lemma 5 to show that the maximal deadweight loss consistent with aggregate demand and πΉπœƒβˆ— comes from a data-generating process in which agents perceiving the price 𝑝̅ + π‘šΜ… 𝜏 choose the lowest quantity consistent with preference maximization, whereas the other agents choose the largest such quantity. Then, we show that distributions satisfying such a property yield deadweight loss no larger than the proposed distribution, which exists.

First, consider an arbitrary distribution 𝐹𝑝𝑠,πœƒ,𝜁 (yielding well-defined aggregate demand and deadweight loss) such that πΉπœƒ = πΉπœƒβˆ— and:

𝐹𝑝𝑠 = {

0, 𝑝𝑖𝑠 < 𝑝̅

𝐹𝑝𝑠(𝑝̅), 𝑝𝑖𝑠 ∈ [𝑝̅, 𝑝̅ + π‘šΜ… 𝜏) 1, 𝑝𝑖𝑠 β‰₯ 𝑝̅ + π‘šΜ… 𝜏

In words, the above expression says that the support of 𝑝𝑠 is contained in {𝑝̅, 𝑝̅ + π‘šΜ… 𝜏}. By Theorem 2, the maximal value of deadweight loss consistent with aggregate demand and πΉπœƒβˆ— must satisfy this property. Consider some value 𝜌 ∈ [0,1] such that:

𝜌 ∫ π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, 𝑙) π‘‘πΉπœƒ|𝑝𝑠≠𝑝̅(πœƒπ‘–)

πœƒπ‘–

+ (1 βˆ’ 𝜌) ∫ π‘ž(𝑝̅; πœƒπ‘–, β„Ž)π‘‘πΉπœƒ|𝑝𝑠=𝑝̅(πœƒπ‘–)

πœƒπ‘–

= ∫ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

(10)

Such a value of 𝜌 must exist by the Intermediate Value Theorem, since by the definitions of 𝑙 &

β„Ž and the CLD as expressed in lemma 1:

∫ π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, 𝑙)π‘‘πΉπœƒ|𝑝𝑠≠𝑝̅(πœƒπ‘–)

πœƒπ‘–

≀ ∫ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

≀ ∫ π‘ž(𝑝̅; πœƒπ‘–, β„Ž)π‘‘πΉπœƒ|𝑝𝑠=𝑝̅(πœƒπ‘–)

πœƒπ‘–

68

In words, we are constructing an alternative distribution that rationalizes aggregate demand such that 𝑝𝑠 = 𝑝̅ + π‘šΜ… 𝜏 & 𝜁 = 𝑙 with probability 𝜌, and otherwise 𝑝𝑠 = 𝑝̅ & 𝜁 = β„Ž. We now show that this alternate distribution yields at least as much deadweight loss, thus showing that the maximal value of deadweight loss consistent with aggregate demand and πΉπœƒβˆ— must arise from a distribution in which almost surely (𝑝𝑠, 𝜁) = (𝑝̅, β„Ž) or (𝑝𝑠, 𝜁) = (𝑝̅ + π‘šΜ… 𝜏, 𝑙).

From the definition of deadweight loss:

∫ π‘šΜ… 𝜏 [π‘ž(𝑝̅; πœƒπ‘–, πœπ‘–) βˆ’ π‘ž(𝑝̅; πœƒπ‘–, 𝑙)] π‘‘πΉπœƒ,𝜁|𝑝𝑠≠𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

= ∫ [𝑑𝑀𝑙(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, 𝑙) βˆ’ 𝑑𝑀𝑙(𝑝̅; πœƒπ‘–, πœπ‘–)] π‘‘πΉπœƒ,𝜁|𝑝𝑠≠𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

From here, the definition of 𝑙, and using the fact that 𝑑𝑀𝑙(𝑝̅; πœƒπ‘–, πœπ‘–) = 0 βˆ€πœƒπ‘–, πœπ‘–, we have that 𝜌 β‰₯ 1 βˆ’ 𝐹𝑝𝑠(𝑝̅) implies that:

𝜌 ∫ 𝑑𝑀𝑙(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, 𝑙)π‘‘πΉπœƒ|𝑝𝑠≠𝑝̅(πœƒπ‘–)

πœƒπ‘–

+ (1 βˆ’ 𝜌) ∫ 𝑑𝑀𝑙(𝑝̅; πœƒπ‘–, β„Ž)π‘‘πΉπœƒ|𝑝𝑠=𝑝̅(πœƒπ‘–)

πœƒπ‘–

= 𝜌 ∫ 𝑑𝑀𝑙(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, 𝑙)π‘‘πΉπœƒ|𝑝𝑠≠𝑝̅(πœƒπ‘–)

πœƒπ‘–

β‰₯ [1 βˆ’ 𝐹𝑝𝑠(𝑝̅)] ∫ 𝑑𝑀𝑙(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, 𝑙)π‘‘πΉπœƒ|𝑝𝑠≠𝑝̅(πœƒπ‘–)

πœƒπ‘–

= [1 βˆ’ 𝐹𝑝𝑠(𝑝̅)] ∫ 𝑑𝑀𝑙(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, πœπ‘–)π‘‘πΉπœƒ,𝜁|𝑝𝑠≠𝑝̅(πœƒπ‘–)

πœƒπ‘–,πœπ‘–

+ 𝐹𝑝𝑠(𝑝̅) ∫ 𝑑𝑀𝑙(𝑝̅; πœƒπ‘–, πœπ‘–)π‘‘πΉπœƒ,𝜁|𝑝𝑠=𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

= ∫ 𝑑𝑀𝑙(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, πœπ‘–)π‘‘πΉπœƒ,𝜁(πœƒπ‘–)

πœƒπ‘–,πœπ‘–

69

The inequality follows from 𝜌 β‰₯ 1 βˆ’ 𝐹𝑝𝑠(𝑝̅) by assumption. This shows that whenever 𝜌 β‰₯ 1 βˆ’ 𝐹𝑝𝑠(𝑝̅), the proposed alternative distribution yields at least as much deadweight loss. Now suppose instead 𝜌 < 1 βˆ’ 𝐹𝑝𝑠(𝑝̅). From lemma 5:

∫ 𝑑𝑀𝑙(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, πœπ‘–)π‘‘πΉπœƒ,𝜁|𝑝𝑠≠𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

β‰₯ π‘šΜ… 𝜏 ∫ [π‘ž(𝑝̅; πœƒπ‘–, β„Ž) βˆ’ π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, πœπ‘–)] π‘šΜ… πœπ‘‘πΉπœƒ,𝜁|𝑝𝑠≠𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

In addition, we find it convenient to rewrite the aggregate demand-rationalizing equation as:

∫ π‘ž(𝑝𝑖𝑠; πœƒπ‘–, πœπ‘–)𝑑𝐹𝑝𝑠,πœƒ,𝜁(𝑝𝑖𝑠, πœƒπ‘–, πœπ‘–)

𝑝𝑖𝑠,πœƒπ‘–,πœπ‘–

= [1 βˆ’ 𝐹𝑝𝑠(𝑝̅)] ∫ π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘– , πœπ‘–)π‘‘πΉπœƒ,𝜁|𝑝𝑠≠𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

+ 𝐹𝑝𝑠(𝑝̅) ∫ π‘ž(𝑝̅; πœƒπ‘–, πœπ‘–)π‘‘πΉπœƒ,𝜁|𝑝𝑠=𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

And so, using equation (10) and rearranging terms:

𝜌 ∫ [π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘– , πœπ‘–) βˆ’ π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘– , 𝑙)]π‘‘πΉπœƒ,𝜁(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

= (1 βˆ’ 𝜌) ∫ π‘ž(𝑝̅; πœƒπ‘–, β„Ž)π‘‘πΉπœƒ|𝑝𝑠=𝑝̅(πœƒπ‘–)

πœƒπ‘–

βˆ’ [1 βˆ’ 𝐹𝑝𝑠(𝑝̅) βˆ’ 𝜌] ∫ π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, πœπ‘–)π‘‘πΉπœƒ,𝜁|𝑝𝑠≠𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

βˆ’ 𝐹𝑝𝑠(𝑝̅) ∫ π‘ž(𝑝̅ + π‘šΜ… 𝜏; πœƒπ‘–, πœπ‘–)π‘‘πΉπœƒ,𝜁|𝑝𝑠≠𝑝̅(πœƒπ‘–, πœπ‘–)

πœƒπ‘–,πœπ‘–

Using lemma 5 and plugging in yields:

ρ

Thus, we know that the maximal deadweight loss consistent with aggregate demand andFΞΈβˆ— is generated by a distribution in which with probability one either(ps, ΞΆ) = (Β―p, h) or (ps, ΞΆ) = (Β―p + Β―mΟ„, l). We refer to distributions of this sort as binary distributions.

Now, we show that the proposed distribution maximizes deadweight loss among all binary

70

distributions, and thus among all distributions, that rationalize aggregate demand such that FΞΈ = FΞΈβˆ—. Towards that end, we first show that the proposed distribution exists. Note by lemma 5 and the CLD as in lemma 1:

In words, aggregate demand is contained between when all agents perceive a high price &

have typeh and when all agents perceive a low price & have type l. Furthermore, one can confirm that for anyβˆ†, βˆ†0, Ξ³, Ξ³0such that0 ≀ βˆ† < βˆ†0 ≀ Β―mΟ„ and 0 ≀ Ξ³ < Ξ³0 ≀ 1:

Thus, we can pickβˆ† such that:

Z

If both sides hold with equality, we can defineΞ³ arbitrarily. Otherwise, we define Ξ³ so that the market clears:

We now have the valuesβˆ† and Ξ³ such that the market clears. Suppressing βˆ† and Ξ³ subscripts fromq, we can say that:˜

Finally, to show that the proposed distribution maximizes deadweight loss, consider ar-bitrary binary distribution Fps,ΞΈ,ΞΆ that rationalizes aggregate demand. Defining PF(ps 6=

Β―

p|ΞΈi) ≑ 1 βˆ’ Fps|ΞΈ=ΞΈi(Β―p + Β―mΟ„ ) as the probability that (ps, ΞΆ) = (Β―p + Β―mΟ„, l) conditional on ΞΈi, 71

rationalizing aggregate demand withFΞΈ = FΞΈβˆ—means that:

We can now write the difference in generated values of aggregate deadweight loss as:

Z

We complete the proof by showing the right-hand side of the last inequality is zero. Since both distributions rationalize the same aggregate demand:

Z

Subtracting both sides fromR

Finally, subtracting the right-hand side from the left-hand size and multiplying by zero yields the desired result. Thus:

Z

ΞΈi

 q(θ˜ i) βˆ’ q(Β―p; ΞΈi, h)

q(Β―p + Β―mΟ„ ; ΞΈi, l) βˆ’ q(Β―p; ΞΈi, h)βˆ’ PF(ps6= Β―p|ΞΈi)dwl(Β―p + Β―mΟ„ ; ΞΈi, l)dFΞΈβˆ—(ΞΈi) = 0

In words, deadweight loss from the proposed distribution is at least as great as the dead-weight loss from any binary distribution that also rationalizes aggregate demand and with the true distribution of preference types. From the first part of the proof, any distribution that rationalized aggregate demand and had the support of perceived prices contained inβˆ‚dP yielded deadweight loss no greater than what one could obtain with a binary distribution that rationalized aggregate demand withFΞΈ = FΞΈβˆ—. Theorem 2 noted that any distribution that rationalized aggregate demand withFΞΈ = FΞΈβˆ— yielded deadweight loss no greater than that one could obtain with a distribution that had the support of perceived prices contained inβˆ‚dP, rationalized aggregate demand, and had FΞΈ = FΞΈβˆ—. Therefore, any distribution that rationalizes aggregate demand and withFΞΈ= FΞΈβˆ—yields deadweight loss no greater than the proposed distribution.

Claim: Ifmit βŠ₯ Ξ²it,Ξ²it ≀ 0 with probability one, and mithas an exponential distribution, then:

m ≀ 0.5 β‡’ E[dwlΛ† it|Ο„ ] ≀ βˆ’1

2E[β˜it]Ο„2βˆ€Ο„

Proof : Since the variance of an exponentially-distributed random variable is its squared

ex-pected value:

E(m2it) = V ar(mit) + (E[mit])2= 2(E[mit])2

73

74

Also, from the independence of salience and sticker price responsiveness:

π‘šΜ‚ = 𝔼[𝛽̃𝑖𝑑]

Download now "Three Essays at the In..."

Outline

Related documents