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Licensed Unlicensed Requires Authentication Published by De Gruyter January 5, 2019

Sorting into Contests: Evidence from Production Contracts

Zhen Wang and Tomislav Vukina

Abstract

In this paper, we investigate sorting patterns among chicken producers who are offered a menu of contracts to choose from. We show that the sorting equilibrium reveals a positive sorting where higher ability producers self-select themselves into contracts to grow larger chickens and lower ability types self-select themselves into contracts to grow smaller birds. We also show that eliciting this type of sorting behavior is profit maximizing for the principal. In the empirical part of the paper, we first estimate growers’ abilities using a two-way fixed effects model and subsequently use these estimated abilities to estimate a random utility model of contract choice. Our empirical results are supportive of the developed theory.

JEL Classification: M52; J43; D86

Appendix

Proof of Proposition 1:

Proof. Suppose there are three contracts in the pool of contract alternatives with Q¯1<Q¯2<Q¯3. The threshold ability which equates the expected utility from contract Q¯2 and contract Q¯3,

(19)a23=bNβ(N1)+a¯3Q¯3a¯2Q¯2Q¯3Q¯2+β(N1)(Q¯3+Q¯2)2γN

must be greater than a12 and smaller than amax. This is because when a23<a12, contract Q¯2 would be dominated by either contract Q¯1 or contract Q¯3 and would not be chosen by any grower, and when a23>amax, even the best grower would pick contract Q¯2 and no grower would choose contract Q¯3. Hence, it must be that amin<a12<a23<amax. With this condition, growers with abilities in (amin,a12) would choose contract Q¯1, growers with abilities in (a12,a23) would choose contract Q¯2 and growers with abilities in (a23,amax) would choose contract Q¯3. This sorting result for three contracts is illustrated in Figure 2b in the main text. The equilibrium in average abilities is calculated as the solution of the following system of equations:

(20)a¯1=amina12aig(ai)daiamina12g(ai)dai
(21)a¯2=a12a23aig(ai)daia12a23g(ai)dai
(22)a¯3=a23amaxaig(ai)daia23amaxg(ai)dai

with eqs. (9) and (19) used as the thresholds (integral limits). Same as in the two contracts case, here as well, the equilibrium average abilities are positively related to the contract expected outputs because a¯1<a12<a¯2<a23<a¯3, i. e.:

a¯1=amina12aig(ai)daiamina12g(ai)dai<amina12a12g(ai)daiamina12g(ai)dai=a12a¯2=a12a23aig(ai)daia12a23g(ai)dai>a12a23a12g(ai)daia12a23g(ai)dai=a12a¯2=a12a23aig(ai)daia12a23g(ai)dai<a12a23a23g(ai)daia12a23g(ai)dai=a23a¯3=a23amaxaig(ai)daia23amaxg(ai)dai>a23amaxa23g(ai)daia23amaxg(ai)dai=a23.

The same reasoning can be extended to K contract alternatives. Threshold abilities are expressed as

(23)ak1,k=bNβ(N1)+a¯kQ¯ka¯k1Q¯k1Q¯kQ¯k1+β(N1)(Q¯k+Q¯k1)2γNk=2,3,,K

and growers with abilities between ak1,kandak,k+1 would choose contract k. The equilibrium average abilities can be calculated from the following system of equations:

(24)a¯1=amina12aig(ai)daiamina12g(ai)dai
(25)a¯k=ak1,kak,k+1aig(ai)daiak1,kak,k+1g(ai)dai
(26)a¯K=aK1,Kamaxaig(ai)daiaK1,Kamaxg(ai)dai

and, same as before, in equilibrium, contracted average abilities are increasing with contract expected output.   □

Acknowledgement

We would like to thank Atsushi Inoue, Zheng Li, Walter Thurman, Xiaoyong Zheng and the participants of the Society of Labor Economists Conference in Raleigh, NC, May 5-6, 2017 for their helpful comments on an earlier version of this manuscript.

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Published Online: 2019-01-05

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