Simple hi-lo equilibrium for n prices

We analyze here a more general variant of a simple hi-lo equilibrium where there are n > 1 prices in every store. In the hi-hi regime, the sellers have n monopoly prices, pⁱ₁=...=pⁱ_n−1= 1, and, in thehi-loregime,n−1 monopoly prices and just one discount price,pⁱ_n<1.

As before, in this kind of an equilibrium where a store has never more than one discount price, a buyer has an incentive to switch the seller immediately after she finds one. The buyer’s search problem is non-trivial only if that has not occurred yet. We only need to cover that case.

To simplify notation, we denote the buyers’ possible search outcomes att= 1 by:

ω₀= ’no price from store 1 nor from store 2’

ω_mⁱ = ’a monopoly price from store i, no price from store -i’

ω^1,2_m = ’a monopoly price from store 1 and from store 2’

ω_dⁱ= ’a discount price from store i, no price or a higher price from store -i’

ω^1,2_d = ’a discount price from store 1 and from store 2’

ω^1,2_A = ’all prices from store 1 and from store 2’

Note that, if a seller is in thehi-loregime, the probability that a buyer who findsp < n prices at this seller does not find a single discount price is ^n−p

n . If a seller is in thehi-hi regime, it is one. Thus, whenn−pprices remain in the start store andn in the other store, the probability of next finding a discount price is _n−p¹ in the start store and ¹

n in the other store if the stores are in thehi-lo regime. It is zero if the store is in thehi-hi regime. Therefore, assuming the buyer has so far found a total ofp monopoly prices in her start store but none from the other store, the expected gain of finding one more price in the start store is

n−p n b a+^n−p

n b 1

n−p(1−E(p|ωⁱ_d))

and the expected gain of finding one more price in the other store is

b1

n(1−E(p|ωⁱ_d)).

Obviously, the former exceeds the latter. The buyer is closer to finding a discount price is she remains in her start store than if she changes to the other store. By an inductive argument, it can be shown that it is this difference in the expected gain from the next price that drives the buyers’ search incentives. Therefore, the buyers switch the seller only when they find a discount price or nothing remains.

The seller’s profit is an immediate extension of the two-items-per-one-store case. To construct a tentative equilibrium, we have to derive the profit in four cases on the path (two regimes for two sellers) and in two cases off the path (the other store could be in either regime) since, as we have seen, the sellers might prefer to have more than one discount price. Generically, the sellers’ profit is given by

Πⁱ=P r(ω_mⁱ) +P r(ω_dⁱ)pⁱ_n+P r(ω^1,2_m)1/2 +P r(ω_d^1,2)

1−F_nⁱ(pⁱ_n) pⁱ_n,

If the best prices the buyers find are selleri’s monopoly price or selleri’s discount price, they pay them. Instead, if the buyers find some monopoly prices from selleriand some monopoly prices from seller −i, they purchase from a random seller. If they find two discount prices, they buy for the lower one.

If neither of the two sellers has a discount price, the profit to selleriis

Πⁱ=P r(ωⁱ_m) +P r(ω^1,2_m )1/2

= 1/2 n

i=1

B_i+ n i=1

B_n+i1/2.

If only sellerihas a discount price, the profit to selleriis

Πⁱ=P r(ω_mⁱ) +P r(ω_dⁱ)p+P r(ω^1,2_m)1/2

If only seller−ihas a discount price, the profit to selleriis

Πⁱ=P r(ωⁱ_m) +P r(ω_m^1,2)1/2

If both of the two sellers have a discount price, the profit to selleriis

Πⁱ=P r(ω_mⁱ) +P r(ωⁱ_d)pⁱ_n+P r(ω^1,2_d )

n−(p−1 1)is the probability of finding a discount on thep’th draw. If altogether iprices are found, the likelihood of not finding a discount at the second seller is ^n−(i−p)

n

or zero (forilarge) whereas the likelihood of finding a discount at the second seller is ^i−p

n

or one (forilarge).

Instead, the seller’s profit after a deviation to two identical discount prices is given by

Πⁱ=P r(ω_mⁱ) +P r(ωⁱ_d)pⁱ_n+P r(ω^1,2_m )1/2 if the other seller does not have a discount price and

Πⁱ=P r(ω_mⁱ) +P r(ω_dⁱ)p+P r(ω^1,2_d )

if the other seller does have a discount price.²¹ If a store has two discount prices, (n−i)(n−i−1)

n(n−1) =ⁿ⁻²

n−(p−1) is thus the probability of finding a discount from that store on thep’th draw (forp≤n). A sum over thep’s, from the 1’st draw to thei’th draw, is

2n−i−1

n(n−1) when the (deviating) store has two discount prices and_n¹ when the (non-deviating) store has one discount price.

It is now straightforward to confirm that this formulation is equivalent to the one derived earlier forn= 2. When each seller has two items, we know that the simple hi-lo equilibrium is sustained forθ < θ^o(2) ≈ 713. To see what is the effect of additional items in stock beyond two, we determine this boundary also for n = 3. When sellers

21Clearly, if a one-price deviation is not profitable, a two-price deviation is not profitable.

have three items, we find numerically that the simplehi-lo equilibrium is sustained for, θ < θ^o(3)≈719. This is more relaxed.

Proposition 12 If each sellers has three items andθ≤θ^o≈719, there exists a simple hi-loequilibrium.

Interestingly, if the sellers have two items in stock, a seller’s profit is

Πⁱ(1,1) =a⁽²⁾/2 (B₁+B₂+B₃+B₄) + (1−a⁽²⁾)/2 (B₁+B₂+ 1/4B₃)>1/2 (B₁+B₂) where

a⁽²⁾= B₂ B₂+ 3B₃+ 4B₄

whereas if the sellers have three items in stock, the profit to a seller is

Πⁱ(1,1,1) =a⁽³⁾/2 (B₁+B₂+B₃+B₄+B₅+B₆)

+ (1−a⁽³⁾)/2 (B₁+B₂+B₃+ 2/3B₄+ 1/3B₅)>1/2 (B₁+B₂+B₃) where

a⁽³⁾= B₂+ 3B₃+ 3B₄+ 2B₅ B₂+ 3B₃+ 9B₄+ 19/3B₅+ 9B₆.

In words, the seller’s profit is larger forn= 3, where it has two monopoly prices and one discount, than forn= 2, where it has one monopoly price and one discount. When there is one price in one store, the seller’s profit isB₁.

Although we are not sure what happens forn >3 exactly, this suggests that inventory expansion and in-store price variation might represent an avenue for the sellers to raise the expected price towards the monopoly price, as in the Diamond (1971) outcome, yet give the buyers a reason to search. The finiteness of items in stock can help the sellers to commit to tremble away from the monopoly price level so that the stay-home outcome can be avoided.

However, especially for comparisons with a larger number of items in stock, we think it is important to take into account the possibility that additional items in stock can increase or decrease search efficiency. We consider that next.