**2015-present**

**"Reducing the dimensionality of incentive compatibility constraint"**

This paper provides a systematic approach to reduce the dimensionality of incentive compatibility (IC) constraints when the original IC constraints are infinite-dimensional. We show that under a set of mild conditions, the IC constraints or any infinite-dimensional constraints can be reduced to countable. 。many or even a finite number of constraints

**"Rosca meets financial market" (with Hanming Fang and Li-an Zhou),**

*NBER Working Paper 21683*.

Rotating Savings and Credit Association (Rosca) is an important informal financial institution in many parts of the world used by participants to share income risks. What is the role of Rosca when formal credit market is introduced? We develop a model in which risk-averse participants attempt to hedge against their private income shocks with access to both Rosca and a formal credit market and investigate their interactions. Using the gap of the borrowing and saving interest rates as a measure of the imperfectness of the credit market, we compare three cases: (i) Rosca without credit market; (ii) Rosca with a perfect credit market; (iii) Rosca with an imperfect credit market. We show that a perfect credit market completely crowds out the role of Rosca. However, when credit market is present but imperfect, we show that Rosca and the formal credit market can complement each other in improving social welfare. Interestingly, we find that the social welfare in an environment with both Rosca and formal credit market does not necessarily increase monotonically as the imperfectness of the credit market converges to zero.

**"Non-parametric identification of moral hazard problems: first order approach and statistical inference"**

This paper develops a non-parametric methodology for identifying moral hazard problem, based on the first order condition (known as the Mirrlees-Holmstrom Condition (MHC)) of contract optimality in standard principal-agent model (Holmstrom, 1979). I show that MHC is equivalent to the attainment of the Cramer-Rao Lower Bound (CRLB) of estimation of marginal incentive cost. Therefore, a non-parametric testing for contract optimality is a correlation coefficient test between inverse marginal utility and the score function with respect to the nuisance effort parameter. The test is non-parametric in a sense that the contractual form, monetary utility, cost function of effort or the distribution of output are unknown, but the score function is estimated based additivity assumption of production technology. We show the agent's inverse marginal utility can be identified up to an affine transformation, under the null hypothesis. Meanwhile, we also propose an estimator for the loss of profit, compared with the unobserved counterfactual. In addition, the present approach is applied to test optimality and estimate bounds on the loss of profit for a piece-rate contract adopted by a cotton weaving factory in Zhejiang Province, China.

**"A penalty function approach to necessary optimality conditions for non-convex bi-level optimization" (with Chris Ryan)**

We use a penalty function approach to derive necessary optimality conditions for non-convex bilevel programs. The existing literature has concentrated on two basic approaches and their hybrids: (a) the KKT approach that applies to convex problems, and (b) the value function approach that posits calmness conditions and employs nonsmooth analysis. Both methods reformulate the problem into a single level optimization, possibly with complementarity or non-smooth constraints. We explore an alternate approach based on exact penalization of a max-min reformulation of the problem. This allows for a parsimonious optimality condition that involves a single alternate best response of the follower rather than an enumeration of best responses that is common in other methods. We provide examples where our optimality conditions hold but fail the constraint qualifications of other approaches.