论微积分在经济分析中的应用(1)

娅蓉官乖

        
            摘要：微积分作为数学知识的基础,是学习经济学的必备知识,着重讨论了微积分在经济学中最基本的一些应用，计算边际成本、边际收入、边际利润并解释其经济意义,寻求最小生产成本或制定获得最大利润的一系列策略。 关键词：微积分；边际分析；弹性；成本；收入；利润；最大值；最小值      1导数在经济分析中的应用      1.1边际分析在经济分析中的的应用   1.1.1边际需求与边际供给   设需求函数Q=f（p）在点p处可导（其中Q为需求量，P为商品价格），则其边际函数Q’=f’(p)称为边际需求函数，简称边际需求。类似地，若供给函数Q=Q(P)可导（其中Q为供给量，P为商品价格），则其边际函数Q=Q(p)称为边际供给函数，简称边际供给。   1.1.2边际成本函数   总成本函数C=C(Q)=C0 C1(Q)；平均成本函数=(Q)=C(Q)Q；边际成本函数C’=C’(Q)．C’(Q0)称为当产量为Q0时的边际成本，其经济意义为：当产量达到Q0时，如果增减一个单位产品，则成本将相应增减C’’(Q0)个单位。   1.1.3边际收益函数   总收益函数R=R(Q)；平均收益函数=(Q)；边际收益函数R’=R’(Q)．   R’(Q0)称为当商品销售量为Q0时的边际收益。其经济意义为：当销售量达到Q0时，如果增减一个单位产品，则收益将相应地增减R’(Q0)个单位。   1.1.4边际利润函数   利润函数L=L(Q)=R(Q)-C(Q);平均利润函数;=(Q)边际利润函数L’=L’(Q)=R’(Q)-C’(Q).L’(Q0)称为当产量为Q0时的边际利润，其经济意义是：当产量达到Q0时，如果增减一个单位产品，则利润将相应增减L’(Q0)个单位。   例1某企业每月生产Q（吨）产品的总成本C（千元）是产量Q的函数，C(Q)=Q2-10Q 20。如果每吨产品销售价格2万元，求每月生产10吨、15吨、20吨时的边际利润。   解：每月生产Q吨产品的总收入函数为：   R（Q）=20Q   L（Q）=R（Q）-C（Q）=20Q-（Q2-1Q 20）   =-Q2 30Q-20   L’(Q)=(-Q2 30Q-20)’=-2Q 30   则每月生产10吨、15吨、20吨的边际利润分别为   L’(10)=-2×10 30=10（千元/吨）；   L’(15)=-2×15 30=0（千元/吨）；   L’(20)=-2×20 30=-10（千元/吨）；   以上结果表明：当月产量为10吨时，再增产1吨，利润将增加1万元；当月产量为15吨时，再增产1吨，利润则不会增加；当月产量为20吨时，再增产1吨，利润反而减少1万元。   显然，企业不能完全靠增加产量来提高利润，那么保持怎样的产量才能使企业获得最大利润呢？   1.2弹性在经济分析中的应用   1.2.1弹性函数   设函数y=f(x)在点x处可导，函数的相对改变量Δyy=f(x Δx)-f(x)y与自变量的相对改变量Δxx之比，当Δx→0时的极限称为函数y=f(x)在点x处的相对变化率，或称为弹性函数。记为EyEx
   
   












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