Proof of Proposition 1.

I decompose the proof in 5 steps.

In Step 1, I characterize the equilibrium when only bidders with costs lower than ci+1 have a positive demand. In Step 2, I state necessary and sufficient conditions for demand schedules to form a “Demand” Function Equilibrium (DFE) tracing through ex-post optimal points when the financial constraints of i types of bidders are non-binding and that the financial constraints of the other m−i types are binding for prices in pi_,pi‾. Finally, in Step 3 and 4, I characterize the lower bound pi_ and the higher bound pi‾, that allow me to completely describe the ex-post optimal equilibria of the game.

Let Q be the quantity of good put for sale, Ni the total number of risk neutral bidders with asymmetric fixed costs ci,

xi(p)

the equilibrium strategy for type-i bidders, (piecewise continuously differentiable and downward sloping), and α the rationing scheme. The market clearing price is determined such that αNx(p∗(α)=Q, when N bidders participate in the auction.

1. I consider the case in which only bidders with costs lower than ci+1 have a positive demand.

In this case, I only consider degree of rationing in the interval αi,αi+1 for which the number of active bidders is exactly N1+...+Ni+1 where αi is defined such that the profits of bidders with costs ci is zero when α=αi.

I thus only consider prices p such that type-i bidders’ profits, i=1,...,i+1, are non-negative.

For these prices, type-1 to i+1 bidders’ demand schedules are identical, x1(p)=...=xi+1(p)=x(p) because their asymmetric costs are fixed costs and do not modify the bidders’ first order conditions.

Let suppose that all bidders except bidder j play their equilibrium strategy x(.), and that bidder j plays the strategy y(.).

The market clearing condition is:

Given α, the bidder j′ s residual supply is, then,

Bidders being risk neutral, a type-i bidder’s profits are:

A type-i bidder submits a demand schedule x(.). The first order condition with respect to the price is^[28]

The market-clearing condition is:

Differentiating (3) with respect to α gives

Putting this into (2):

This implies that dp∗(α)dα>0.

The equilibrium is thus ex-post optimal as p∗(.) is strictly increasing in α.

The solution of the differential eq. (5) is:

where ki+1 is the corresponding integration’s constant when only bidders with costs lower than ci+1 have a positive demand.

Second order conditions are also satisfied because the aggregate demand schedule is downward sloping on the interval of prices that can be sustained as equilibrium prices (see Step 2 of the Proposition).

The equilibrium price is completely determined for all degree of rationing in the interval αi,αi+1 where αi=kici(N1+...+Ni)1N1+...+Ni−1, i.e. defined such that the profits of bidders with costs ci is zero when α=αi.

I can now determine the form of the demand schedules when exactly i+1 bidders participate. Differentiating (6) gives:

From the previous eqs. (4) and (3), we have:

If we put this and (3) in (2), we get:

Moreover, inverting pi+1‾(.) in (6)

together with (7) gives the equilibrium demand schedule when exactly i+1 bidders participate:

Moreover, the positivity of the equilibrium price imposes an upper bound for each ki. As the slope of the demand schedules decreases when ki increases, this implies that there exists a lower bound for the slope of the demand schedules. Bidders are thus induced to bid more aggressively.

1. The demand schedules x(.) form a symmetric “Demand” Function Equilibrium tracing through ex-post optimal points if and only if (1)x(.) satisfies the first order condition of the problem together with the Market Clearing condition on each interval pi_,pi‾ such that the financial constraints of i bidders are non-binding and the other N−i are not, (2) the aggregate demand is non-increasing on the interval of prices that can be sustained as equilibrium prices, (3)x(.) is non-increasing for all i and (4) no bidder has incentives to deviate at each pi_ or pi‾ for all i.

2. Sufficiency: Assume that the financial constraints of i bidders are non-binding and that the financial constraints of the other N−i are binding for prices in pi_,pi‾. There are only i bidders who participate actively in the auction.

Since the aggregate demand is non-increasing for all realized equilibrium price, it intersects the fixed supply at a unique point for each α. Moreover, the demand schedules satisfy first order condition for ex-post profit maximization when the other firms choose their equilibrium strategy. Both conditions together implies that the second derivative of a bidder’s profit, Π′′, is negative for all p∗(α).

Indeed, the first order condition of a bidder problem is

Differentiating this equation with respect to p and combining it with the second order condition of bidder i, I get

This is clearly non positive when and x(.) is non-increasing.

Global second order conditions for ex-post profit maximization are satisfied everywhere onpi_,pi‾. Thus, the demand schedules form a “Demand” Function Equilibrium tracing through ex-post optimal points on pi_,pi‾.

Moreover, as no bidder has incentives to deviate at each pi_ or pi‾ for all i, the demand schedules form a DFE tracing through ex-post optimal points on 0,v.

1. Necessity:

Satisfaction of the first order condition of the problem together with the Market Clearing condition is a necessary condition for a demand function to trace through ex-post optimal points. Moreover, if αix(p∗(α))≥0, then Π′′(p∗(α))≥0 . Therefore, the demand schedules x(.) cannot be a symmetric DFE.

1. I now determine the price pi‾ that is the highest price for which the financial constraints of i type of bidders are non-binding and the financial constraints of the other N−i type are binding.

pi‾

is the highest price such that bidders with costs c1,...,ci participate to the auction and bidders with costs ci+1,...,cN do not participate.

pi‾

is thus the price such that bidders with costs ci participate when p≤pi‾ and do not participate when p>pi‾.

Consequently, pi‾ is the price at which the profits of a bidder with costs ci when he participates to the auction is equal to the profits of the same bidder when he does not participate. This gives:

1. Let αi, the degree of rationing such that when α≤αi, bidders with costs ci+1 participate and when α>αi, only bidders with costs lower than ci participate.

As p∗(.), the equilibrium price, is strictly increasing, then αi exists for all i. Moreover, I have αi=kici(N1+...+Ni)1N1+...+Ni−1

The price pi_ is defined by pi_=pi+1‾(αi).

Remind that pi+1‾(α)=−ki+1αN1+...+Ni+1−1+v is the equilibrium price when the degree of rationing is α and only bidders with costs strictly lower than ci+1 participate. pi_ should satisfy the following equation in order to ensure that those bidders with costs strictly lower than ci+1 have no incentives to deviate when α=αi:

I finally get

Proof of Corollary 2.

When there are two types of bidders, from the previous Proposition, there exists α2 such that when α<α2, all bidders participate and when α>α2, only low cost bidders participate.

Moreover, as p∗(.) is increasing in α and α∈[α_,1], a necessary and sufficient condition to get the positivity of the equilibrium price is given by p∗α_≥0. That is

This gives the following lower bound for the set of equilibrium prices:

Using the result of Proposition 1 from Bourjade (2009), the lower bound for the set of equilibrium prices with no bidding costs is:

Consequently, when the fixed costs of high cost bidders are small enough,

then, the set of equilibrium prices is reduced in a case with N1 bidders with no costs and N2 bidders with costs c compared to a case with N1+N2 bidders with no costs. I can therefore conclude that the set of equilibrium prices is reduced when the difference in bidders’ costs is small enough.

Proof of Proposition 2

For an equilibrium to be monotonic, the demand schedule of a low cost bidder must be non-increasing in p‾. Indeed, high cost bidders’ demand schedules are non-increasing over 0,p∗(1) and low cost bidders’ ones are non-increasing over 0,p‾ and p‾,p∗(1). As N1+N2 bidders would participate on the interval 0,p‾ and only N1 onp‾,p∗(1) and as the bidders’ demand schedules are supposed to be symmetric, the market clearing condition allows us to characterize the bidders’ demand schedules. Consequently, the demand schedule of a low cost bidder is non-increasing in p‾ if and only if:

Notice that this inequality could only be satisfied when α2 goes to 0 which is impossible as α≥α_>0. Hence, there does not exist monotonic equilibria when α_>0.

Proof of Corollary 3

If the seller could use all degree of rationing in 0,1, the only monotonic equilibrium inverse demand schedules of low cost bidders is p∗(x)=v, for all x. Indeed their equilibrium inverse demand schedules is

for all x>0. As in all monotonic equilibria, one must have k1=v−p∗(1)Q=0, this proves the result.