Assume that c = 0. Then there does not exist an equilibrium where every seller offers a high-quality good at price p < v.

Assume that such an equilibrium exists and consider a deviation where some seller asks price p′<p . He encounters queue length determined either by the high-quality buyers’ indifference

or by the low-quality buyers’ indifference

As v < 1 one immediately sees that if (25) holds the RHS of (26) is greater than the LHS. Consequently, only the low-quality buyers contact the deviating seller. From (26) we can solve ∂γ∂p′=−γ1−e−γ1−e−γ−γe−γ(v−p′) . Evaluating the derivative for the problem maxp′1−e−γp′ at p′=p we find that the derivative is positive if the price satisfies

But this is always the case in the tentative equilibrium as the RHS of (27) is the equilibrium price when every seller has a low-quality good.   □

Assume that c = 0. The proportion of sellers who produce a low-quality good, y(θ), determined by (28) is unique.

Expression (28) is equivalent to

It is immediate that f7(0)<0 as we have assumed that 12−e−θ2−θ2e−θ2<0 which is equivalent to θ<θ∼ . Similarly, it is immediate that f7(1)>0 . As f7′(y)=αe−αθ2y2+2βe−βθ2(1−y)2>0 we see that the solution y(θ) is unique.   □

Assume that c = 0. When the number of buyers increases the proportion of sellers who produce a low-quality good decreases, or ∂y∂θ<0 .

Totally differentiating (28) one easily determines

Again, from (28) we can determine

Let us see what the relationship between α and β would be if (32) were always zero, or by multiplying by θ/2 if it were the case that α2e−α−2β2e−β=0 . Totally differentiating we get

but as α > β (34) is strictly less than (33). This shows the claim.   □

Assume that c = 0. All equilibria feature perfect sorting.

Assume to the contrary. One possibility is that proportion z of high-quality buyers contact the low-quality sellers. In this case α=θ+zθ2y and β=(1−z)θ2(1−y) . Now the high-quality buyers have to be indifferent between the sellers, or e−β=e−αv which is equivalent to α=β−ln2 . Inserting this datum into (28) yields e−β=12ln2 . Inserting this back to the high-quality buyers’ indifference condition (30) gives e−α=1ln2 which is a contradiction.

The other possibility is that proportion z of the low-quality buyers contacts the high-quality sellers. In this case α=(1−z)θ2y and β=θ+zθ2(1−y) . Now the low-quality buyers have to be indifferent between the sellers, or e−α=−1+e−β+2βe−ββ which is equivalent to 1−e−β−2βe−β+βe−α=0 . But the LHS is decreasing in α and as we must have α > β the relation cannot hold.   □

Assume that the number of buyers satisfies θ≥θ∼ and the cost satisfies c<c∼(θ) , and assume further that both qualities are produced. Then the sellers’ indifference relation is equivalent to

There are two possibilities. First, some high-quality buyers contact low-quality sellers, or formally e−β=e−αv . But then we know from the construction of the equilibrium Section 5.1.3 that c>cˉ(θ) which is greater than c∼(θ) and consequently not compatible with our assumption.

Secondly, there is perfect sorting, or e−β>e−αv . It is straightforward to ascertain that the RHS of (35) is increasing in y, and the maximum y compatible with perfect sorting is given by β=α−ln12 . But this gives y=θ+ln12−θ2+ln1222ln12 , and inserting this into (35) one recovers function cˉ(θ) which is always larger than c∼(θ) and consequently our assumption is not valid.

Finally, it is straightforward to see that if everyone produces a low-quality good there is a profitable deviation to produce a high-quality good.   □

Assume that the cost satisfies c>c˜(θ) , and that both qualities are produced. Again from the sellers’ indifference condition we get (35). If there is not perfect sorting then

where α=(1+z)θ2y and β=(1−z)θ2(1−y) . To determine the largest possible value of c compatible with both qualities being produced we need to totally differentiate (36) to get

Then differentiating the RHS of (35) with respect to y one finds after some simplification that it is positive, or (β−α)βe−β−12αe−α>0 , because (36) implies that β > α.

This reasoning shows that the greatest value of c is attained when y approaches unity. Then α = θ, and β=θ−ln12 . Inserting these data into (35) one recovers c˜(θ) , and our assumption is not satisfied.

If there is perfect sorting or e−β>e−αv then we reason that for a fixed α the RHS of (35) is increasing with β, and the largest β compatible with perfect sorting is given by β=α−ln12 . Inserting this into (35) we recover cˉ(θ) and as this is always less than c˜(θ) our assumption is not valid.

Finally, it is straightforward to see that if everyone produces a high-quality good there is a profitable deviation to produce a low-quality good.   □

It remains to show sufficiency only for the cases where both qualities are produced. But this is done in the proofs for low and high quality where necessity is proved for the equilibria where only one quality is produced.   □

To show that both qualities are not produced or only low quality is not produced one proceeds as in the above proof of proposition 2: Only low quality. To show that other possible equilibria do not satisfy the intuitive criterion assume that the equilibrium price p satisfies p>ph . If a seller deviates to price p′ , and the buyers believe him to be a high-quality seller the expected queue length γ is determined by

From this we can determine ∂γ∂p′=−γ(1−e−γ)(1−e−γ−γe−γ)(1−p′) . The derivative of the deviators profit function 1−e−γp′ with respect to p′ turns out negative at p′=p if p>1−e−β−βe−β1−e−β=ph . But this shows that there is a profitable deviation downwards. An analogous argument works if v<p<ph .   □

Necessity Let us first assume that the optimal deviation under perfect information p˜=1−e−γ−γe−γ1−e−γ is greater than v; the queue length is determined by the buyers’ indifference condition 1−e−γγ(1−p˜)=e−θv . For this deviation to be non-profitable 1−e−γ−γe−γ−c≤1−e−θ−θe−θv or

Assume next that the optimal deviation under perfect information is less than v. Then, v is the optimal deviation that cannot be imitated by low-quality sellers. It is not profitable if (1−e−γ)v−c≤1−e−θ−θe−θv where the expected queue length γ is determined by the buyers’ indifference condition 1−e−γγ(1−v)=e−θv ; we know that there exists a unique γ that satisfies the relation. Denote it by γ(θ), and note that γ(0) = 0, ∂γ∂θ=γ2θ1−e−γ−γe−γ>0 , and γ(θ) > θ for all θ > 0.

Then we study for which values of γ it is true that 1−e−γ−γe−γ1−e−γ<v . This is equivalent to 1−e−γ−2γe−γ<0 where the LHS is decreasing for γ<12 and increasing thereafter. It is immediate that there exists γˆ∈(1,2) such that 1−e−γˆ−2γˆe−γˆ=0 . Inverting γ(θ)=γˆ gives θˆ≈0.56 , and consequently for all θ<θˆ the restriction for the cost c is given by

Sufficiency For θ≥θˆ the proof is like in the case of perfect information. Let us first assume that both qualities are produced, and that the high-quality sellers ask price ph=v . Then it must be the case that the sellers are indifferent or (1−e−α−αe−α)v=(1−e−β)v−c . Now, if there is not perfect sorting α=(1+z)θ2y and β=(1−z)θ2(1−y) for z > 0. From the high-quality buyers’ indifference condition e−αv=1−e−ββ(1−v) , which is equivalent to e−α=1−e−ββ , we immediately see that α < β.

Totally differentiating the indifference condition we find that ∂z∂y=1+zy . Next we determine what happens to the difference of the sellers’ revenues (1−e−β)v−(1−e−α−αe−α)v . Differentiating this with respect to y one finds that the derivative is given by θ2ve−β−1+zy(1−y)+(1−z)(1−y)2−αe−α1+zyy−(1+z)y2−>0 where the first term in the curly brackets is positive because α < β, and the second term is zero.

Similarly, if there is perfect sorting α=θ2y and β=θ2(1−y) and the corresponding derivative is given by 2θvαe−α1y2+e−β1(1−y)2>0 . As the one seller deviation to a higher price establishes the threshold cost cˆ(θ) , and this corresponds to y = 1 we see that decreasing y makes the difference between the revenues smaller. Consequently, the costs c that are consistent with both qualities being produced must satisfy c<cˆ(θ) . But this is a contradiction. Above we assumed that the high-quality price is ph=v but the same technique applies to any price ph>v .

From the proof of Claim 1 we know that an equilibrium where only high-quality goods are produced does not exist either.   □

It is clearest to establish the situation when c = 0 since increasing c just reduces the number of high-quality sellers. Consider the case θ=θ∼ . The analysis preceding Proposition 1 shows that at this value every seller produces a high-quality good, and earns profit v. The price is given by ph=1−e−β−βe−β1−e−β>v where β=θ∼2 . Lowering θ everything is as under perfect information until value θ≈1.68 ^[15], where the high-quality price equals v under perfect information.

When θ≤1.68 the equilibrium price of the high-quality good remains at v, and the measure of high-quality sellers adjusts so that the sellers’ indifference condition is satisfied. For these values of θ we next demonstrate that the sellers’ indifference condition determines a unique y, and that the prices, pl and ph=v , constitute an equilibrium. It is clear that the price of the low-quality good is pl=1−e−α−αe−α1−e−αv , as the logic is the same as perfect information. The indifference condition of the sellers, for any c, is given by

or

At y = 0 the LHS of (41) is zero, at y = 1 it is e−θ2(1+θ2) , and its derivative with respect to y is easily seen to be positive. At y = 0 the RHS of (41) is e−θ2+2c , and at y = 1 it is 2c, and its derivative with respect to y is easily seen to be negative. Consequently, when there exists a solution it is unique, and it gives the division between low-quality and high-quality sellers. Increasing c shifts the RHS upwards and it immediate that there is a solution for any c≤e−θ2(1+θ2) .

Next we determine the part of the parameter space where there is perfect sorting. When the cost of production increases from zero the measure of sellers who produce a low-quality good grows. At some point high-quality buyers are indifferent between contacting the high-quality and low-quality sellers, or 1−e−ββ(1−v)=e−αv which is equivalent to

Note first that (42) determines a unique y. Consider function f4(y)≡1−e−β−βe−α=0 . At zero it is positive, or f4(0)=1−e−θ2 , and at unity it is negative, or f4(1)=1−∞ . To see that there is a unique zero observe that the derivative is given by f4′(y)=θ211−y11−ye−β−11−ye−α−1yαe−α . As e−α=1−e−ββ>e−β we see that at the zero of f4(y) the derivative is negative, and thus the solution is unique. Let us denote this by y(θ). It is clear that y(θ)>12 , as at value one half (42) becomes 1−e−θ−θe−θ which is always positive.

We have established that there is a unique y(θ)>1/2 that makes the high-quality buyers’ indifference condition (42) hold. To determine the threshold-c we first note that the thresholds under perfect and private information start to differ at θ≡θˇ≈0.78 which can be solved by equating y(θ) and (20). Then we solve the sellers’ indifference condition (40) for c, using the solution y(θ) (or alternatively e−β ) from (42).

It remains to show that price ph=v constitutes an equilibrium in the pricing subgame when θ∈0,θˇ . Any possible deviation is upwards. Consider p˜>v . The queue length γ the deviator expects is determined by the buyers’ indifference 1−e−γγ(1−p˜)=1−e−ββ(1−v) . The deviation is not profitable if 1−e−βv≥1−e−γp˜ . Utilising the indifference condition this is equivalent to

The LHS and RHS of (43) are equal when γ = β. The derivative with respect to γ of the LHS is smaller than that of the RHS if

but this holds for γ < β as we are in the part of the parameter space where 1−e−β−2βe−β<0 . This also shows that the equilibrium satisfies the intuitive criterion. Sufficiency follows from the proof of claims 1 and 2.    □

a) Perfect information

We restate the sellers’ indifference condition (17) and the high-quality buyers’ indifference condition (19),

and

where α=(1+z)θ2y and β=(1−z)θ2(1−y) . For notational simplicity we refer to f8(z,y,c) as f and to g1(z,y,c) as g in the reminder of part a) of the proof. The effect of an increase in c on on the sellers’ profit is given by.

By applying Cramer’s rule we get

and

where

Therefore

The numerator of the term in the brackets simplifies to

and is of the same sign as

The high-quality buyers' indifference condition g(z,y,c)=0, implies that β≥α , which means that 2y−1+z≥0. The sign of the denominator B is given by 1−zy−1+z1−y=2y−1+z and hence Dc(1−e−α−αe−α)12 is positive.

b) Private information

The sellers’ indifference condition and the high-quality buyers’ indifference condition condition are now

where α=(1+z)θ2y and β=(1−z)θ2(1−y) . For simplicity we refer to f9(z,y,c) as f and to g2(z,y,c) as g in the reminder of the proof. The effect of an increase in c on the sellers’ profit given by.

The partial derivatives are

By Cramer’s rule we get

and

Where

which is positive as g(z, y, z) = 0 implies that β≥α which means that e−α−e−β is positive.

Using the definitions of α and β , canceling terms that ad up to zero and simplifying we get

The denominator is positive. In addition β≥α implies that 2y−1+z≥0and that α+β≥2α , which implies that α+βe−β−αe−α≥e−β−12e−α , the right hand side of which we show (in the next proof) to be positive. Hence B is positive, and

which is positive as β≥α . We can now express

The numerator is positive as 1−zy−1+z1−y=2y−1+z≥0 . As also the denominator or B is positive we have shown that Dc(1−e−α−αe−α)12 is positive. Thus an increase in the cost leads to higher profits for the sellers.   □