This paper reviews rice procurement operations of Government of India from the standpoints of cost of procurement as well as effectiveness in supporting farmers’ incomes. The two channels in use for procuring rice till 2015, were custom milling of rice and levy. In the first, the government bought paddy directly from farmers at the minimum support price (MSP) and got it milled from private millers; while in the second, it purchased rice from private millers at a pre-announced levy price thus providing indirect price support to farmers. Secondary data reveal that levy, despite implying lower cost of procurement was discriminated against till about a decade back and eventually abolished in 2015 in favor of custom milling, better trusted to provide minimum price support. We analyze data from auctions of paddy from a year when levy was still important to investigate its impact on farmers’ revenues. We use semi-nonparametric estimates of millers’ values to simulate farmers’ expected revenues and find these to be rather close to the MSP; a closer analysis shows that bidder competition is critical to this result. Finally, we use our estimates to quantify the impact of change in levy price on farmers’ revenues and use this to discuss ways to revive the levy channel.
Hermite polynomials are a class of orthogonal polynomials that have support over the entire real line and the Gaussian function as the weighting function. The advantage of polynomial approximations using orthogonal (rather than ordinary) polynomials has to do with efficiency. 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 . 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 , , .
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 . 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 . The realized set of remaining (n − 1) bidders (all small) could be any one of different possibilities. The probability that one of these has valuation equal to w while the rest have valuations between r and w is . Finally, the probability that the valuations of all the (remaining) (p − n − 2) potential bidders are less than the reserve price is .
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 different combinations, and the probability that the (p − n − 1) remaining small bidders are inactive is . 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 ; 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 .
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 . 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 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 . Finally, the probability that (p − n − 1) small bidders’ valuations are less than r is .
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 (buyer), and (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 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 λ i (.) is the hazard-rate function, we have
Dividing throughout by 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
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 of β that is consistent and distributed normally asymptotically. Thus,
where V/T is the variance-covariance matrix of . Then , an estimator of r * solves
Expanding in a Taylor’s series about
Ignoring U, as it will be negligible in the neighborhood of , we obtain
In practice, we work with approximations, so let
We use the parameter estimates ( ) to compute the optimal reserve price ( ) and the value of m at . Then we construct 95% confidence intervals around 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
Second derivative of with respect to r is thus given by
Simplifying further, we get
Evaluating this at r = r * and using we get
To see how the curvature of Π(r) responds to changes in the number of bidders, we differentiate with respect to N (even though N takes integer values in actual fact)
Now , and
Along with , 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 ; it also implies first-order stochastic dominance (FOSD) of F 1 over F 2. For the FOC to hold with F 1 FOSD F 2, we must have while . So the third term on R.H.S. of Eq. (36) is also positive. QED.
I am grateful to Abhijit Banerji and J.V. Meenakshi for their guidance and for providing paddy auctions data. I thank K.L. Krishna, Anirban Kar and seminar participants at Delhi School of Economics, Lancaster University, Indian Statistical Institute – Delhi, Indian Institutes of Technology – Bombay and Guwahati for useful comments. I gratefully acknowledge help from Ashok Sinha in acquiring some of the procurement data.
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