Hermite Polynomials

Hermite polynomials are a class of orthogonal polynomials that have support over the entire real line and the Gaussian function exp ( − x 2 / 2 ) as the weighting function. The advantage of polynomial approximations using orthogonal (rather than ordinary) polynomials has to do with efficiency.^[26] The nth order Hermite polynomial is defined by (Paarsch and Hong 2006):

where d ⁿ /dx ⁿ refers to the nth derivative. The nth order normalized Hermite polynomial is defined by h n ( x ) = H n ( x ) n ! 2 π . As explained in Section 4, given the size of our sample, a Hermite series of order 2 is reasonable.

The first three Hermite polynomials are H ₀(x) = 1, H ₁(x) = x, H ₂(x) = x ²−1. The first three normalized Hermite polynomials are h 0 ( x ) = H 0 ( x ) 2 π , h 1 ( x ) = H 1 ( x ) 2 π , h 2 ( x ) = H 2 ( x ) 2 2 π .

Since we employ as weighting function, a normal density with mean zero and standard deviation 0.4, the Hermite polynomials must also be suitably modified so that they are orthonormal with respect to the weighting function exp ( − x 2 2 ( 0.4 ) 2 ) . The first three modified normalized Hermite polynomials are

Specifying the Joint Density of the Win Price and Winner’s Identity

The joint probability density of the win price and a specific small bidder winning is given by (Banerji and Meenakshi 2004):

The first term in this equation corresponds to the case where the large bidder’s valuation is less than the start price r. The probability that the large bidder’s valuation is less than r is F _V1(r); the probability that a specific small bidder’s valuation is more than w is ( 1 − F V 2 ( w ) ) . The realized set of remaining (n − 1) bidders (all small) could be any one of ( p − 3 n − 1 ) different possibilities. The probability that one of these has valuation equal to w while the rest have valuations between r and w is ( n − 1 ) f V 2 ( w ) ( F V 2 ( w ) − F V 2 ( r ) ) n − 2 . Finally, the probability that the valuations of all the (remaining) (p − n − 2) potential bidders are less than the reserve price is ( F V 2 ( r ) ) p − n − 2 .

The second term consists of two possibilities, the large bidder’s valuation being between r and w, and it being exactly w. In either case, the probability that a specific small bidder’s valuation is greater than w is (1 − F _V2(w)), the realized set of non-winning active small bidders can be one of ( p − 3 n − 2 ) different combinations, and the probability that the (p − n − 1) remaining small bidders are inactive is  ( F V 2 ( r ) ) p − n − 1 . Given this, the probability that the large bidder’s valuation is exactly w and that the (n − 2) small bidders valuations are between r and w is f V 1 ( w ) ( F V 2 ( w ) − F V 2 ( r ) ) n − 2 ; while the probability that the win price valuation belongs to one of (n − 2) small bidders and that the large bidder’s and (n − 3) small bidders’ valuations lie between r and w is ( n − 2 ) f V 2 ( w ) ( F V 1 ( w ) − F V 1 ( r ) ) ( F V 2 ( w ) − F V 2 ( r ) ) n − 3 .

The joint probability density of the win price and a specific large bidder winning is given by

The probability that a specific large bidder’s valuation is greater than the win price is ( 1 − F V 1 ( w ) ) . The realized set of the other (n − 1) active bidders who are all small is akin to a random draw from the set of (p − 2) small potential bidders and could be one of ( p − 2 n − 1 ) different combinations. The probability that one of these (n − 1) small bidders has valuation equal to the win price while the rest (n − 2) small non-winning active bidders’ valuations are between the reserve price and the win price is ( n − 1 ) f V 2 ( w ) ( F V 2 ( w ) − F V 2 ( r ) ) n − 2 . Finally, the probability that (p − n − 1) small bidders’ valuations are less than r is ( F V 2 ( r ) ) p − n − 1 .

Estimating Farmer’s Reservation Utility for a Lot of Paddy

In the event of a lot going unsold at the formal auction, it goes back to the shop of the katcha arhtia through whom the farmer sells his grain. Typically, it gets sold to some private miller later through mutual negotiations. The private miller is free to opt out of this possibility of a purchase and not get anything; the farmer is free to opt out of the sale and take his grain to some other market to sell. Thus it is useful to think of these negotiations in terms of bilateral bargaining with outside options.

We use a stylized model here, namely the complete information bargaining model of Rubinstein (1982), augmented with outside options (see for example Muthoo 1999 for a textbook exposition). Let the miller’s (buyer’s) valuation for the lot be v. We denote by s, the seller’s (farmer’s) use value for the lot, which is the worth that the farmer attaches to a lot that does not get sold anywhere and is eventually privately used. In the absence of outside options, the subgame perfect equilibrium shares of the players in Rubinstein’s model are r S r B + r S ( v − s ) (buyer), and r B r B + r S ( v − s ) (seller), where r _B and r _S are the buyer’s and seller’s respective discount rates. In the presence of outside options, the unique subgame perfect equilibrium division of the surplus (v − s) can be either the Rubinstein division, or a division in which one of the players gets a payoff equal to his outside option and the other gets the residual surplus. The idea is that if an outside option payoff is larger than what a player gets as his Rubinstein payoff, then the other player is forced to concede this payoff in the bargaining.

We assume the buyers (millers) to have a discount rate equal to 15% per annum (which corresponds to the rate at which they could have borrowed from banks at that time), while the sellers (farmers) were able to borrow from the co-operative societies or the katcha arhtias at about 2% per month (i.e. 24% per annum). In the course of our interviews with them, farmers stated that if a lot goes unsold, then transporting it and selling it elsewhere (possibly at another market where auctions are not employed) can mean a discount of up to Rs 100 compared to the price obtainable in this market through auctions. We therefore estimate the outside option of the farmer, for each lot, as the expected second-highest valuation for that quality minus a penalty amount of Rs 100. On the other hand, if the miller opts out of the bargaining, his payoff is zero; in effect, the model then is one of alternating offers bargaining with an outside option for the seller.

The farmer’s equilibrium payoff in the bargaining model with an outside option available to him is the larger of his Rubinstein share s + r B r B + r S   ( v − s ) and his outside option. This equilibrium payoff is the x ₀ that we plug into Eq. (12). With v fixed at a small miller’s expected valuation for the lot and s being allowed to vary from zero to a small miller’s expected valuation discounted by Rs 100, we find that this reservation utility (x ₀) for the lot equals the farmer’s payoff from the outside option.

Derivation of Optimal Reserve Prices

Differentiating Π with respect to r

where

Thus

Since

where λ _i (.) is the hazard-rate function, we have

Dividing throughout by F 1 ( r ) N 1 − 1 F 2 ( r ) N 2 − 1 we get

The above equation gives for a lot of a specific quality, the first order condition for profit maximization of the seller.

Confidence Intervals for Optimal Reserve Prices

We use the Delta method to construct confidence intervals around the optimal reserve prices.

Recall that for a lot with quality vector z, the valuation of player i of type j is given by

The first-order condition for revenue-maximization (Eq. (13) in Section 5), which gives the optimal reserve price r ^* can be rewritten as an implicit function of r ^* and the parameter vector β

where

We now obtain the asymptotic distribution of the optimal reserve price based on the semi-nonparametric estimates.

Gallant and Nychka (1987) prove the consistency of SNP estimators for multivariate data. Fenton and Gallant (1996) specialize it to the univariate case. Wong and Severini (1991) establish root-n asymptotic normality and efficiency of semi-nonparametric maximum likelihood estimators.

Consider an estimator β ˆ of β that is consistent and distributed normally asymptotically. Thus,

where V/T is the variance-covariance matrix of β ˆ . Then r ˆ * , an estimator of r ^* solves

Expanding γ ( r ˆ * , β ˆ ) in a Taylor’s series about ( r * , β ˆ )

Ignoring U, as it will be negligible in the neighborhood of ( r * , β ) , we obtain

Thus,

In practice, we work with approximations, so let

We use the parameter estimates ( β ˆ ) to compute^[27] the optimal reserve price ( r ˆ * ) and the value of m at r ˆ * . Then we construct 95% confidence intervals around r ˆ * , β ˆ as follows.

Proof of Proposition 1

Differentiating the expected revenue (Eq. (12)) with respect to reserve price and setting N ₁ = 1 and N ₂ = N we get

where

Second derivative of Π ( r ) with respect to r is thus given by

where

Simplifying further, we get

Evaluating this at r = r ^* and using d π ( r ) dr | r = r * = 0 we get

To see how the curvature of Π(r) responds to changes in the number of bidders, we differentiate ( d 2 Π ( r ) d r 2 | r = r * ) with respect to N (even though N takes integer values in actual fact)

Now f ( r * ) > 0 , r * − x 0 > 0 and f i ′ ( r * ) ≥ − 2 f i ( r * ) r * − x 0

Along with 1 + N log  F 2 ( r * ) < 0 , this implies the first two terms on the R.H.S. of Eq. (36) are positive. Also, since F ₁ dominates F ₂ in the reverse hazard rate, we have f 1 F 1 > f 2 F 2 ; it also implies first-order stochastic dominance (FOSD) of F ₁ over F ₂. For the FOC d π ( r ) d r = 0 to hold with F ₁ FOSD F ₂, we must have 1 − F 1 ( r * ) − ( r * − x 0 ) f 1 ( r * ) > 0 while 1 − F 2 ( r * ) − ( r * − x 0 ) f 2 ( r * ) < 0 . So the third term on R.H.S. of Eq. (36) is also positive. QED.