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RESERVE PRICE VERSUS ENTRY FEE IN STANDARD AUCTIONS.

I. INTRODUCTION

Rent extraction from "higher type" buyers in the monopoly seller's problem often involves lesser allocation for "lower types," and in some situations, screening out low-type buyers below a certain cut-off called the screening level. In standard auctions like first-price auctions (FPA) and second-price auctions (SPA), (1) this screening level is implemented by using reserve prices and entry fees. (2) Many online auction sites like eBay give sellers the option to set reserve prices while listing their items. Live bidding at auction houses often starts from a reserve price. Entry fees are less common, and often take indirect forms like registration fee, charge for bidding form, and cover charge. Does an auctioneer have reason to prefer one tool for rent extraction over the other?

In the benchmark setting with bidders having independent private values and quasilinear utility the expected revenue maximizing (optimal) auction is implemented by FPA and SPA rules. In that environment, reserve price and entry fee play equivalent roles from revenue point of view (see Harris and Raviv 1981; Myerson 1981; Riley and Samuelson 1981). In a departure from the benchmark assumptions, Milgrom and Weber (1982) show that under all standard auction formats the seller's expected revenue in an affiliated values setting is the largest if the seller uses only an entry fee with zero reserve price over all reserve price and entry fee combinations. However, that does not explain why in reality so many auctions involve reserve prices. Waehrer, Harstad, and Rothkopf (1998) show that if the seller is risk averse and the bidders are risk neutral then she prefers to screen bidders using only a reserve price in a FPA. In this paper we consider the diametrically opposite scenario where the bidders are risk averse and the seller is risk neutral. (3)

Maskin and Riley (1984) consider the case of risk averse bidders and risk neutral seller and show that in the optimal auction the seller benefits by making losing bidders pay. This is a feature that has also been observed in auctions with budget constrained bidders (4) that, too, give rise to nonquasilinear utilities (see Che and Gale 1996, 1998; Laffont and Robert 1996). We consider only standard auctions and ask whether a risk neutral seller would prefer reserve price or entry fee while selling to risk averse bidders. This basic question with practical relevance has been surprisingly overlooked for this setting. We find that a seller prefers a reserve price in a SPA but an entry fee in a FPA. We also discuss some interesting consequence of our analysis on some known results on efficiency and bidders' ranking of auction formats.

II. THE MODEL

A revenue maximizing seller has K [greater than or equal to] 1 indivisible objects for which she has zero value. There are n ([greater than or equal to] K +1) symmetric bidders. Bidder i (i = 1, .., n) has a privately known value/type [v.sub.i] [greater than or equal to] 0 for a single object only and when the bidder receives the object for a payment p his utility is U(v,p)= 1 - [e.sup.-[rho](v-p)]. Thus the bidders are risk averse with constant absolute risk aversion. (5) If a bidder loses the auction and pays p his utility is U(0,p). [v.sub.i]-s are realizations of i.i.d. random variables [V.sub.i]-s with distribution function F(*) and density f(x) > 0 over [a, b] where 0 [less than or equal to] a < b < [infinity]. [v.sub.i]-s are private information but F is common knowledge. For m i.i.d. random variables with distribution F(*) the kth highest random variable is denoted by [V.sub.k.m]. The distribution and density of Vk:m are denoted by [F.sub.k:m](x) and [f.sub.k:m](x), respectively.

The auction unfolds as follows:

t = 0: The seller announces the auction rule (FPA or SPA), a reserve price r [greater than or equal to] 0, and an entry fee [phi] [greater than or equal to] 0. Each bidder i privately observes his value [v.sub.i].

t = 1 If bidder i chooses to bid he submits a sealed bid [b.sub.i] [greater than or equal to] r and the entry fee. Bids that are smaller than r or submitted without entry fee are discarded.

t = 2: Objects are allocated to the highest K bidders who clear the reserve price. Ties are broken randomly. Under FPA a winning bidder pays a price equal to his bid, while under SPA the winning bidder pays the higher of the K + 1th highest bid (6) and r (7).

When the reserve-entry fee combination in the SPA and FPA are set at their (revenue) optimal levels we call them optimal SPA and optimal FPA, respectively. The reserve and entry fee then are called optimal reserve and optimal entry fee.

We assume that the seller considers r and [phi] satisfying U(b, r + [phi]) > 0 (so that the highest type participates in the auction). Theorems 2 and 3 of Maskin and Riley (1984) on SPA and FPA bidding strategies extend to the case where the seller can set both reserve and entry fee and there are K units on sale. Accordingly, there is a screening level [v.sup.*] where b > [v.sup.*] [greater than or equal to] a such that bidders with values below v* do not bid in the auction. A bidder pays the entry fee and makes a valid bid if his value v [greater than or equal to] [v.sup.*]. The equilibrium bidding strategies under SPA and FPA are give for v [greater than or equal to] [v.sup.*] by

(1) [[beta].sub.SPA](v) = v.

and

(2)

[mathematical expression not reproducible]

The screening level [v.sup.*] is given by the following screening condition that makes type [v.sup.*] bidder indifferent between participation and nonparticipation under both auction formats

(3) [mathematical expression not reproducible]

Note that [v.sup.*] - r - [phi] [greater than or equal to] 0 and in particular [[beta].sub.SPA]([v.sup.*]) = [v.sup.*] [greater than or equal to] r. (8)

III. OPTIMAL RESERVE AND ENTRY FEE

Proposition 4 of Maskin and Riley (1984) on revenue ranking extends to our setting. When the reserve price and entry fee in both SPA and FPA are set at the SPA optimal levels [r.sup.*.sub.SPA] and [[phi].sup.*.sub.SPA] the expected revenue is higher under FPA than under SPA. Now if the reserve and entry fee in FPA are changed to the FPA optimal levels the FPA revenue cannot decrease. Thus the optimal FPA generates more expected revenue than the optimal SPA.

How necessary are reserves and entry fee in the auction? If a > 0 then the screening level [v.sup.*] > 0 (since [v.sup.*] [greater than or equal to] a). If a = 0 it can be verified that the optimal v* > 0 under both formats. The following proposition is a direct consequence of this observation. (9) Also, from this point we will consider [v.sup.*] > 0 without any further mention.

PROPOSITION 1. The optimal screening level [v.sup.*] > 0. Thus, it is optimal to set at least one of the two, reserve price and entry fee, at a positive level.

Risk averse bidders dislike payoff variability--a reason why bidders pay more in FPA than in SPA. At the same time, making losers pay can introduce a greater payment variability for the lower types which is exploited in the optimal auction (see Maskin and Riley 1984, among others). From an incentive point of view the increased payment variability for lower types discourages bidders with higher types from mimicking bidders with lower types. This "tightening" of "downward incentive" facilitates rent extraction from the bidders with higher values.

A bidder faces substantial payoff uncertainty in a SPA even conditional on winning. A bidder who has the highest value faces the maximum payoff uncertainty. For a fixed v*, as one replaces the entry fee by reserve the maximum payment v + [phi] by a bidder decreases and the minimum payment r + [phi] increases. The reduction in payoff uncertainty for the lower types may potentially "loosen" the downward incentive constraint for the higher types giving rise to a negative effect on the revenue. However, the reduction in payoff uncertainty for the high types also increases their payment. The following proposition shows that this positive effect on revenue makes reserve price an indispensable tool for surplus extraction in a SPA.

PROPOSITION 2. The optimal SPA involves only a positive reserve and a zero entry fee.

It is possible for the optimal entry fee to be positive for certain utilities. Mezzetti (2010) considers bidders with aversion to price risk (APR) with utility function U(v,p) = v - [gamma](p) where [gamma](0) = 0 and [gamma]', [gamma]" > 0. It can be checked that when there are two bidders with APR utilities and [gamma]' (0) = 0 then in a SPA for a single object it is optimal for the seller to set a positive entry fee along with a positive reserve price.

Under FPA a bidder faces no payoff uncertainty conditional on winning. The only uncertainty is from positive probabilities of winning and losing. For a fixed [v.sup.*], as the reserve price is replaced by an entry fee the payoff uncertainty to the highest type does not change, but it increases for the lower types who have a positive probability of losing. (10) The increased payoff uncertainty for the lower types has a negative effect on revenue, but the tightening of downward incentive constraint makes the highest type, who does not face a payoff uncertainty, pay more. This positive effect on the expected revenue makes entry fee a necessary tool in the optimal FPA.

PROPOSITION 3. The optimal FPA necessarily involves a positive entry fee.

It is possible for reserve price to be positive in the optimal FPA. For instance, if there are two constant absolute risk aversion (CARA) bidders with [rho] = 1, and the values are uniformly distributed over [0,1] then the optimal reserve and entry fee are given by [r.sup.*.sub.FPA] [approximately equal to] 0.156 and [[phi].sup.*.sub.FPA] [approximately equal to] 0.135. If, however, the bidders have APR utilities then it can be checked that the optimal FPA involves a positive entry fee and zero reserve price.

An interesting consequence of the above analysis is that it adds to some known results. Hu, Matthews, and Zou (2010) show quite generally that when bidders are risk averse and the seller can set only a reserve price the optimal FPA is more efficient ex post than the optimal SPA, that is, ([v.sup.*.sub.SPA] =) [r.sup.*.sub.SPA] > [r.sup.*.sub.FPA] (= [v.sup.*.sub.FPA]). Consider the effect of a small increment in screening level from [v.sup.*] to [v.sup.*] + [DELTA][v.sup.*] on expected revenue when the seller can use only a reserve price and [v.sup.*] is less than the optimal reserve. The optimality of positive entry fee as stated in Proposition 3 implies that this marginal revenue is higher in a FPA when the seller can also use an entry fee optimally to achieve this increment. In particular if [v.sup.*] is the optimal screening level in FPA when only reserve price can be used, this marginal revenue is approximately zero, so Proposition 3 implies that when the seller can also use an entry fee for the [DELTA][v.sup.*] increment the marginal revenue is positive. Thus, when entry fee may be used the seller benefits from increasing the screening level above what is optimal when only a reserve price may be used, that is, the optimal screening level [v.sup.*.sub.FPA] is higher when entry fee may be used along with reserve price. Can [v.sup.*.sub.FPA] exceed [v.sup.*.sub.FPA]? If there are two CARA bidders with [rho] = 1 and the values are uniformly distributed over [2, 4] then numerical computation gives [v.sup.*.sub.SPA] [approximately equal to] 2 while [v.sup.*.sub.FPA] [approximately equal to] 2.2. (11) In this case the optimal SPA is more efficient ex post than the optimal FPA.

While it is not easy to obtain necessary and sufficient condition for [v.sup.*.sub.SPA] < [v.sup.*.sub.FPA], simple sufficient conditions for [v.sup.*.sub.SPA] = a while [v.sup.*.sub.FPA] > a (so that SPA is more efficient than FPA) can be obtained relatively easily. For instance, when K = 1 and 1 [less than or equal to] af(a) < n, then if bidders are sufficiently risk averse we have [v.sup.*.sub.SPA] = a and [v.sup.*.sub.FPA] > a.

Given a screening level, a combination with lower entry fee and higher reserve reduces payoff uncertainty for the bidders so it is no surprise that in our setting the bidders prefer the combination with a higher reserve price. Matthews (1987) shows that given a common reserve price CARA bidders are indifferent between SPA and FPA. This result, combined with the result [r.sup.*.sub.SPA] > [r.sup.*.sub.FPA] in Hu, Matthews, and Zou (2010), implies that bidders prefer the optimal FPA to the optimal SPA when the seller can set only a reserve price. Suppose now the seller can set both the reserve price and entry fee. As we observed above, in that case it is possible to have [v.sup.*.sub.SPA] < [v.sup.*.sub.FPA]. Now if we set only reserves [r.sub.SPA] = [v.sup.*.sub.SPA] in the SPA and [r.sub.FPA] = [v.sup.*.sub.FPA] in the FPA, the bidders would prefer the SPA to the FPA. Since the optimal SPA involves only a reserve price and the optimal FPA involves a positive entry fee, it follows that the bidders prefer the optimal SPA to the optimal FPA whenever the former is more efficient than the latter. These observations are summarized in the following proposition.

PROPOSITION 4. For any two reserve-price-entry-fee combinations that implement the same screening level [v.sup.*] a bidder strictly prefers the combination with a higher reserve. Moreover, a bidder strictly prefers the optimal SPA to the optimal FPA whenever SPA is more efficient ex post than FPA.

IV. CONCLUDING DISCUSSION

Unlike previous studies, in this paper we allowed the seller to set both reserve price and entry fee in standard auctions with CARA bidders. Whether the reserve price or entry fee is an indispensable tool for rent extraction depends on whether the auction format is SPA or FPA. Our key result is driven by the concavity of utility in payment that is why the result also holds when bidders are averse to price risk. Utility functions are concave in payment when bidders are budget constrained, as well. It is possible to verify our key result on optimality of positive reserve price and entry fee in SPA and FPA, respectively, in examples of budget constrained bidders with both values and wealth being private information. Verifying these results more generally remains an interesting direction for future research.

APPENDIX: PROOFS

Proof of Proposition 1

Consider the SPA. The screening condition (3) can be rewritten as

[mathematical expression not reproducible]

from which we have

(A1) [mathematical expression not reproducible]

The SPA expected revenue is given by

[mathematical expression not reproducible]

Using Equation (A1) [[partial derivative]/[partial derivative]r]y[R.sup.SPA] can be written as

[mathematical expression not reproducible]

which is greater than

(A2)-n(1 - F([v.sup.*])) [F.sub.K.n-1]([v.sup.*]) + K[F.sub.k+1:n]([v.sup.*]) -[K.summation over (l=1)][F.sub.l:n]([v.sup.*])

for r < [v.sup.*] (since [mathematical expression not reproducible]). Taking derivative of Equation (A2) with respect to [v.sup.*] we have

nf([v.sup.*])[F.sub.K:n-1]([v.sup.*]) - n(1 - F([v.sup.*]))[f.sub.K:n-1]([v.sup.*]) + K[f.sub.K+1]([v.sup.*]) - [K.summation over (l=1)][f.sub.l]([v.sup.*])

which can be rewritten as

[mathematical expression not reproducible]

The terms in the last expression cancel out to 0. This together with the fact that at [v.sup.*] = b the expression (A2) is equal to zero implies that [[partial derivative].sup.RSPA]/[partial derivative]r > 0 for all r < [v.sup.*]. Therefore, the SPA expected revenue is maximized when the seller sets r = [v.sup.*] and [phi] = 0.

Proof of Proposition 2

Consider the FPA. We have, taking derivative of (2) and simplifying,

(A3) [mathematical expression not reproducible]

The expected revenue to the seller from the FPA is

(A4) [mathematical expression not reproducible]

Taking derivative with respect to r and, using Equations (Al) and (A3), we have

[mathematical expression not reproducible]

Evaluating at r = [v.sup.*] and rewriting this becomes

[mathematical expression not reproducible]

since [mathematical expression not reproducible]. Thus the derivative at r = [v.sup.*] is less than

(A5) [mathematical expression not reproducible]

Taking derivative of the bracketed expression with respect to [v.sup.*] we have

[mathematical expression not reproducible]

which is equal to zero. Furthermore, Equation (A5) is equal to 0 at [v.sup.*] = b, thus Equation (A5) is equal to 0 for [v.sup.*] [less than or equal to] b. Hence, we have [mathematical expression not reproducible] so that the optimal [phi] is positive.

Proof of Proposition 3

Given r and [v.sup.*], a bidder's expected utility in a FPA is equal to

[mathematical expression not reproducible]

which simplifies to

[mathematical expression not reproducible]

This is also the expected utility in a SPA.

Taking derivative of the expected utility with respect to r, substituting for [[partial derivative][phi]/[partial derivative]r] from Equation (A 1) and rewriting it becomes

[mathematical expression not reproducible]

We will show that this is positive. To do that, it is enough to show that the following expression is positive

[mathematical expression not reproducible]

Factoring out [e.sup.[rho]r][F.sub.K:n-1]([v.sup.*]), the bracketed expression simplifies to

(A6) [mathematical expression not reproducible]

The derivative of the expression with respect to v simplifies to

[mathematical expression not reproducible]

Consider the bracketed term. As v increases from [v.sup.*] to b it turns from being positive to negative, that is, Equation (A6) is increasing and then decreasing in v over [[v.sup.*], b]. The expression (A6) is equal to 0 at v = [v.sup.*] and at v=b it is equal to

[mathematical expression not reproducible]

Therefore, Equation (A6) is positive for [v.sup.*] < v < b. We have, thus, shown so far that the derivative of the expected utility to the bidder is increasing in r so that the expected utility is maximized at r = [v.sup.*]. Thus a bidder prefers a reserve price to an entry fee.

The last observation implies that in a FPA with screening level [v.sub.FPA.sup.*] a bidder prefers the optimal FPA (which necessarily involves an entry fee) to an FPA where r = [v.sup.*.sub.FPA]. Since a bidder is indifferent between an FPA and a SPA under a common screening level [v.sup.*.sub.FPA] and reserves set equal to the screening level, the bidder prefers a SPA with screening level [v.sup.*.sub.FPA] and r = [v.sup.*.sub.FPA] to the optimal FPA. Finally, since a bidder's expected utility in a reserve-only SPA is decreasing in the reserve we have that whenever [v.sup.*.sub.SPA] [less than or equal to] [v.sup.*.sub.FPA] the optimal SPA is preferred by the bidders over an optimal FPA.

ABBREVIATIONS

APR: Aversion to Price Risk

CARA: Constant Absolute Risk Aversion

FPA: First-Price Auctions

SPA: Second-Price Auctions

doi: 10.1111/ecin.12715

REFERENCES

Che, Y.-K., and I. Gale. "Financial Constraints in Auctions: Effects and Antidotes." Advances in Applied Microeconomics, 6, 19%, 97-120.

--. "Standard Auctions with Financially Constrained Bidders." Review of Economic Studies, 65(1), 1998, 1-21.

Harris, M., and A. Raviv. "Allocation Mechanisms and the Design of Auctions." Econometrica, 49(6), 1981, 1477-99.

Hu, A., S. A. Matthews, and L. Zou. "Risk Aversion and Optimal Reserve Prices in First - and Second-Price Auctions." Journal of Economic Theory, 145(3), 2010, 1188-202.

Laffont, J.-J., and J. Robert. "Optimal Auction with Financially Constrained Buyers." Economic Letters, 52(2), 1996, 181-86.

Levin, D., and J. L. Smith. "Equilibrium in Auctions with Entry." American Economic Review, 84(3), 1994, 585-99.

Maskin, E., and J. Riley. "Optimal Auctions with Risk Averse Buyers." Econometrica, 52(6), 1984, 1473-518.

Matthews, S. A. "Comparing Auctions for Risk Averse Buyers: A Buyer's Point of View." Econometrica, 55(3), 1987, 633-46.

Mezzetti, C. "Sequential Auctions with Informational Externalities and Aversion to Price Risk: Decreasing and Increasing Price Sequences." The Economic Journal, 121, 2010, 990-1016.

Milgrom, P, and R. Weber. "A Theory of Auctions and Bidding." Econometrica, 50(5), 1982, 1089-122.

Moreno, D., and J. Wooders. "Auctions with Heterogeneous Entry Costs." The RAND Journal of Economics, 42(2), 2011, 313-36.

Myerson, R. "Optimal Auction Design." Mathematics of Operations Research, 6(1), 1981, 58-73.

Pitchik, C. "Budget-Constrained Sequential Auctions with Incomplete Information." Games and Economic Behavior, 66(2), 2009, 928-49.

Riley, J. G., and W. F. Samuelson. "Optimal Auctions." American Economic Review, 71, 1981, 381-92.

Waehrer, K., R. M. Harstad, and M. H. Rothkopf. "Auction Form Preferences of Risk-Averse Bid Takers." The RAND Journal of Economics, 29(1), 1998, 179-92.

SUPPORTING INFORMATION

Additional supporting information may be found online in the Supporting Information section at the end of the article.

Appendix S1. Additional Details and Proofs.

INDRANIL CHAKRABORTY, I acknowledge research support from Singapore MOE AcRF R-122-000-254-115. I thank two anonymous referees for several helpful comments that greatly improved the paper.

Chakraborty: Associate Professor, Department of Economics, National University of Singapore, Singapore, 117570. Phone +65 6516 3729, Fax +65 6775 2646, E-mail indro@nus.edu.sg

(1.) In the independent private values setting that we consider second-price auction and English auction are equivalent. Thus, all our statements about the second-price auction hold for English auction, as well.

(2.) A large literature examines the role of "value discovery" fee and cost prior to observing their valuations which is also referred to as entry fee and cost (see the literature starting with Levin and Smith 1994). In that context, Moreno and Wooders (2011) show that if bidders have privately known value discovery cost then the seller is better off charging a value discovery fee rather than setting a reserve price in the auction. We consider entry fee that is submitted with a bid after a bidder learns his value.

(3.) Usually we think of large auctioneers like the government to be risk neutral.

(4.) See Pitchik (2009) for a nice review of the literature.

(5.) The usual formulation of CARA utility is U(v,p) = (l - [e.sup.-[rho](v-p)])/[rho]. The use of our affine transformation is for notational convenience and it does not change any result.

(6.) If K or fewer bids are submitted in the auction then the price is equal to r.

(7.) The auction rules are popularly called discriminatory (or pay-as-bid) rule and uniform-price (or KTh price) rule instead of FPA and SPA, respectively, when K > 1. Nonetheless, we will continue to call them the FPA and SPA rules with a slight abuse of terminology.

(8.) Note also that in Equation (3) it is possible to set [v.sup.*] = a when a > 0 by setting r < 0 and [phi] > 0. Thus, setting r = a and [phi] = 0 is not the only way of obtaining [v.sup.*] = a for a > 0.

(9.) The proof is routine and skipped here.

(10.) The payoff-[phi] from losing decreases and v - [beta](v) - [phi] from winning increases as [phi] increases while holding [v.sup.*] fixed.

(11.) Here [r.sup.*.sub.SPA] [approximately equal to] 2, [[phi].sup.*.sub.SPA] = 0, [r.sup.*.sub.FPA] [approximately equal to] 0, [[phi].sup.*.sub.FPA] [approximately equal to] 0.0927.
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