中级宏观经济学Suppose the production function is Cobb-Douglas: f(K,

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中级宏观经济学
Suppose the production function is Cobb-Douglas: f(K,L)=KaLb. The market for capital and labor are both perfectly competitive so that wage equals the marginal product of labor and rental price of capital equals the marginal product of capital. Prove firms receive zero profit when a+b=1. Do firms’ profit remains zero when a+b1?

不知道你问题读懂没有.
　　假设生产函数时科布道格拉斯生产函数,f=K^aL^b.资本和劳动市场都是完全竞争的,那么工资就等于劳动的边际产出,资本租金就等于资本的边际产出.证明：a+b=1,则厂商是零利润的.如果a+b1,那么厂商是否仍然获得零利润?
　　证明：
　　If：f=K^aL^b,the wage is MPL,that is w=MPL=df/dL=bK^aL^(b-1)
　　the rental price r is MPK,that is r=MPK=df/dK=aK^(a-1)L^b
　　the total wage is L*w=L*MPL=bK^aL^b=bf
　　the total rental price is K*r=K*MPK=af
　　then：L*w+K*r=bf+af=(a+b)f=f
　　在产出中,扣除总工资和总资本租金的支付后剩余为0,f-L*w-K*r=0,即利润为0
　　如果：a+bf,此时谈不上什么利润了,因为产出都不够支付总工资和总租金.
　　事实上：这个题目的本质是从生产函数的性质讨论利润函数的问题.利润函数的存在性要有赖于生产函数规模报酬非递增的性质.如果规模报酬递减,则必然存在利润函数；如果,规模报酬不变,而又要利润函数存在,那么利润必然为0；如果规模报酬递增则不存在利润函数.在CD函数中,规模报酬的具体情况就是参数a+b与1的大小情况.
　　仅从数学规划上讲,利润函数是下列规划的最优解
　　max：π（p,u）=py-u*x
　　st：y>=0
　　xi>=0
　　f(x)>=y
　　其中,y=f(x)是生产函数,p是价格,x是投入品向量,u是投入品价格向量
　　这个规划最优解存在,那么生产函数f(x)必须拟凹,在CD函数中a+

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