第二章利润最大化与利润函数

1、Chapter 2 Profit Maximization and Profit Function Xiong Qiquan2005/9/292005/9/292.1 Profit Maximization A basic assumption of most economic analysis is that a firm acts so as to maximize its profit;that is, a firm chooses actions (a1,a2, , an), so as to maximize R(a1,a2,an)-C(a1,a2,an).The profit ma

2、ximization problem facing the firm can be written as:121212,max,nnna aaR a aaC a aaA simple application of calculus shows that an optimal set of actions a*2*1*,naaaaWhich can be characterized by the conditionsniaaCaaRii, 2 , 1)()(*The basic constraints facing the firm: technological constraints mark

3、et constraints Profit maximization Lets us consider the problem of a firm that takes prices as given in both its output and its factor markets. let p be a vector of prices for inputs and outputs of the firm. The profit maximization problem of the firm can be stated as Yythatsuchpyp max)(The function

4、 (p) is called the profit function of the firmThere are several variants of profit function.)(max),(zYythatsuchpyzpThe short-run profit function,also known as the restricted profit function:If the firm produces only one output,the profit function can be written as wxxpfwp)(max),(where p is now the (

5、scalar) price of output，w is the vector of factor prices, and the inputs are measured by the vector x=(x1,x2,xn), in this case,we can also define a variant of the restricted profit function,the cost function。).(),(.min),(zYxytswxzywcIn the short run,we may want to consider the restricted or short-ru

6、n cost functionProfit-maximizing behavior can be characterized by calculus ., 2 , 1)(*niwxxfpiiThis condition simply says that the value of the marginal product of each factor must be equal to its price. yVxtswxyxc. .min,Using the vector notation,we can also write these conditions as*pDfxwHere *1,nf

7、 xf xDf xxxis the gradient (梯度梯度)of f: the vector of partial derivatives （偏导数偏导数）of f with respect to each of its arguments.In two-dimensional case，profits are given by pywx, isoprofit line：wyxpp The profit-maximizing amount of inputs occurs where the slope of the profit lines equals the slope of th

8、e production function. yf xpywxpFigure 21 Profit Maximizationpwslope A profit-maximizing firm wants to find a point on the production set with the maximal level of profits.*df xwdxp In this two-dimensional case, it is easy to see the appropriate second-order condition for profit maximization,namely

9、that the second derivative of the production function with respect to the input must be nonpositive.2*20d f xdxA similar second-order condition holds in the multiple-input case.in this case, the second-order condition for profit maximization is that the matrix must be negative semidefinite (负半定) at

10、the optimal point; that is, the second-order condition requires that the Hessian Matrix(海塞矩阵海塞矩阵)2*11122*2122ijf xffD f xffx x must satisfy the condition for all vectors h2*0thD f xh Geometrically, the requirement that the Hessian matrix is negative semidefinite means that the production function mu

11、st lie below its tangent hyperplane(超平面).2.2 Factor Demand Function and the supply function 对每个价格向量（p,w）而言，一般都存在某个最优的要素选择集x*。这个给出我们最优的投入选择的以价格为自变量的函数被称作厂商的要素需求函数要素需求函数，可以表达成x(p,w)。类似地，函数y(p,w)=f(x(p,w)被称作厂商的供给函数供给函数。通常我们假定这些函数经过很好地定义并且性状良好，否则，引起的问题值得考虑：问题问题1 1：可能出现该技术不能由可微生产函数来描述；问题问题2 2： 上面得出的积分条件仅

12、当选择变量可以在最优点的一个开领域内变动时，才是有意义的2.2 Factor Demand Function and the supply function 在许多经济问题中，变量自然是非付的；如果一些变量在最优选择取零值，上面所描述的积分条件就可能是不适宜的。 如果在利润最大化问题中，我们限定x是非负的，相应的一阶条件就变成：0, 0)(0, 0)(iiiiiixwxxfpxwxxfp如果如果所有的技术都有最大化的利润吗？所有的技术都有最大化的利润吗？问题问题3：可能不存在利润最大化的生产计划可能不存在利润最大化的生产计划 1 对于生产函数f(x)=x来说，1单位x生产1单位产出。不难看出，

13、对于pw而言，不存在最大化的利润。当pw时，如果你相最大化px-wx，你会选择无穷大的x值。仅当pw时，这項技术的最大化的生产计划存在，但最优的利润水平是零。 2 规模报酬不变的技术不存在最大化的利润。为了说明这一点，假设我们可以找到某个（p,w），在这点上最优利润严格为正，以至于：. 0)(*wxxpf.)()(*twxxpftwtxtxpf假定我们以t1的因子向上调整生产，现在的利润将是： 这意味着，如果利润曾是正的，他们可以变得更大，因此，利润是不受约束的。因此，不存在利润最大化的生产计划。 对规模报酬不变的厂商而言，唯一重要的利润最大化的位置就是零利润。如果厂商正生产某个正的产出水平并

14、且它赚取零利润，那么它对其正在生产的产出水平是不感兴趣的。 问题问题4 4：即使当利润最大化的生产计划存在时，它也可能不是唯一的。如果对某种规模报酬不变的生产技术来说，（y,x）产生最大的零利润，那么（ty,tx）也将产生零利润，并且因而也是利润最大化的。在规模报酬不变的情况下，如果在某个（p,w）竟存在着利润最大化的选择，那么一般来说，就存在整个范围的利润最大化的生产计划。例子：CD技术的利润函数。 考虑f(x)xa (a0)形式的生产函数利润最大化问题。一阶条件是,1wpaxa. 0) 1(2axapa二阶条件简化成 当a1时，二阶条件才能满足，这意味着要让竞争性的利润最大化有意义，生产函

15、数必须是规模报酬不变或递减的。 如果a=1，一阶条件简化成p=w，因此，当w=p时，任意值的x都是利润最大化的选择。当a 1, we haveTherefore, could not possibly be a profit-maximizing bundle.*txpf txtwxt pfxwxtxx*x2.4 By profit maximization, we have1122.fxwTRSfwxNow,note that221 11212lnlnln.w xw xw wx x Therefore,221 112122121lnlnln1111lnlnlndTRSdw xw xdw wd

16、x xdxxdxx Move to Page 513.2 The FOCs are p/x=w, which gives us the demand function x=p/w,and the supply function y=ln(p/w). The profits from operating at this point are,maxln,0p wpp wp,lnlnppp wpwxppwwSince the firm can always choose x=0 and make zero profits,the profit function becomes3.4 The FOCs are1211 1210aapa xxw121212