This article analyzes the optimal linear incentive contract to offer to would-be agents competing for a principal's project when the principal finds it costly to observe an agent's cost performance ex post and cannot commit to monitoring policy ex ante. Cost sharing reduces the winning bidder's informational rents but creates an incentive for the agent to pad costs ex post to slacken his effort. The optimal cost-sharing parameter generally differs from that when monitoring is costless, and it is higher the fewer the number of bidders and the larger the variance of their idiosyncratic cost.
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Dunne et al. (1995) studied this question.
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