[论文求助] 求助论文下载The use of village agents in rural credit delivery [推广有奖]

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Sunz 发表于 2013-7-25 23:12:34

APPENDIX

Proof of Proposition 4.1: The claim is that constraints 1 and 3 are binding. The proof proceeds by showing that the Lagrange multipliers associated with constraints 1 and 3, respectively, are positive. The set of payments that are derived from this sub-programme are then shown not to violate the remaining constraint (constraint 2) from the full programme. Let [[Lambda].sub.1] and [[Lambda].sub.2] be the Lagrange multipliers associated with the constraints 1 and 3, respectively. Let [[Gamma].sub.[Theta]] represent the derivative of the Lagrangian with respect to the payment received in the [Theta]th state. The first order conditions for cost minimisation are the following:

[[Gamma].sub.r] = p([e.sub.H]) - [[Lambda].sub.1]p([e.sub.H])[V[prime].sub.r] - [[Lambda].sub.2]p([e.sub.H])[V[prime].sub.r] + [[Lambda].sub.2]p([e.sub.L])[V[prime].sub.r] = 0

[[Gamma].sub.n] = (1 - p([e.sub.H])) - [[Lambda].sub.1](1 - p([e.sub.H]))[V[prime].sub.n] - [[Lambda].sub.2](1 - p([e.sub.H]))[V[prime].sub.n] + [[Lambda].sub.2](1 - p([e.sub.L]))[V[prime].sub.n] = 0

In addition there are the complementary slackness conditions. The following solutions for the multipliers can be found.

[[Lambda].sub.1] = [p([e.sub.L])[V[prime].sub.n][Delta]p - (1 - p([e.sub.H]))(p([e.sub.L]) - p([e.sub.H]))[V[prime].sub.r]]/([V[prime].sub.n][V[prime].sub.r][ Delta]p)

[[Lambda].sub.2] = [(1 - p([e.sub.H]))p([e.sub.H])([V[prime].sub.n] - [V[prime].sub.r])]/([V[prime].sub.n][V[prime].sub.r][Delta]p)

Recall, [Delta]p [equivalent to] (p([e.sub.H]) - p([e.sub.L])). Since p(.) is positive, and the derivatives of the agent's utility function with respect to [w.sub.r] and [w.sub.n] are greater than zero and are not equal to each other, and since [w.sub.r] [greater than] [w.sub.n], the Lagrange multipliers can be shown to be positive. The payments derived from the binding constraints can be denoted, in utility terms, by V([[w.sub.r].sup.*]) and V([[w.sub.n].sup.*]). In order to complete the proof it must be shown that these proposed equilibrium payments do not violate the remaining constraint. For constraint 2 not to be violated requires that [e.sub.L] [greater than or equal to] V([[w.sub.n].sup.*]). It can be shown by algebraic manipulation that this is true. Hence, constraint 2 is not violated and therefore V([[w.sub.r].sup.*]) and V([[w.sub.n].sup.*]) are solutions to the problem.

Assistant Professor, Department of Economics, Loyola Marymount University, Los Angeles, CA (USA). The author thanks George Akerlof, Pranab Bardhan, Alain De Janvry, Claudio Gonzalez-Vega, Karla Hoff, Tom Micelli, and a referee of this journal for helpful comments. The author especially thanks Manny Esguerra, Raul Fabella, Ben Hermalin, and Michael Kevane for extensive discussions. In addition, he thanks seminar participants at Berkeley, the Board of Governors, Connecticut, Food Research Institute, George Washington, New York Fed, Ohio State, UC-Riverside, USC, and the University of the Philippines-Diliman.