两大部类扩大再生产中的广义拉格朗日乘子

在一个带有非负和不等式约束的优化问题有最优解的情形下，存在着广义拉格朗日乘子即资源的影子价格．本文探索给出马克思两大部类扩大再生产中的影子价格，为经典的马克思扩大再生产理论增添新的重要内容．首先使用“价值系数法”替代单纯形法，简便地求得了扩大再生产优化问题的最优解．然后运用库恩一塔克条件，确立了关于最优解与广义拉格朗日乘子的互补松弛条件的三个不等式组．进而利用这些不等式组和已知的最优解，简便地解出广义拉格朗日乘子，即两大部类扩大再生产中的影子价格．最后引用和借鉴《资本论》中的两个举例，对所获得的影子价格和目标函数最优值做了计算验证．

The Generalized Lagrange Multipliers in the Two-Sector Expanded Reproduction

The Generalized Lagrange Multiplier is the shadow price of resource. It exists when an optimal problem with constraint of no negative and inequation can get its optimal solution. This paper tried to give the shadow price in Marx＇s two-sector expanded repro- duction so as to add new important content to classical reproduction theory of Marx. Firstly, a ＂value coefficient method＂ instead of Sim- plex Method was used to get the optimal solutions to the optimization problem of expanded reproduction. Secondly, by use of Kuhn-Tucker Condition, three systems of inequations about complementary slackness condition of the optimal solutions and the Generalized Lagrange Multipliers were set. By use of the three systems of inequations and the optimal solutions known, the Generalized Lagrange Multipliers which are the shadow prices in the two-sector expanded reproduction are easily obtained. Finally, two examples in Marx＇s ＂Das kapital＂ were taken to verify the shadow prices and the optimal value of the target function.