第二章 利润最大化与利润函数

意值的x都是利润最大化的选择。当a<1时，可以用一阶 条件来解要素需求函数
w x( p, w) ap
供给函数由下式给出：
1 a 1
.
w y ( p, w) f ( x( p, w)) ap
a a 1
.
利润函数由下式给出：
1 a w ( p, w) py( p, w) wx( p, w) w a ap


Which can be characterized by the conditions
R(a* ) C (a* ) ai ai
i 1,2,, n
The basic constraints facing the firm: • technological constraints • market constraints
例子：CD技术的利润函数。 考虑f(x)＝xa (a>0)形式的生产函数利润最大化问题。 一阶条件是
pax
二阶条件简化成
a 1
w,
pa(a 1) x a2 0.
当a≤1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
A profit-maximizing firm wants to find a point on the production set with the maximal level of profits.
df x* dx
w p
In this two-dimensional case, it is easy to see the appropriate second-order condition for profit maximization,namely that the second derivative of the production function with respect to the input must be nonpositive.
There are several variants of profit function. The short-run profit function,also known as the restricted profit function:
( p, z ) max py
such that y Y ( z )
Profit maximization
Let’s 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
production function must 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， 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。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：
Chapter 2
Profit Maximization and Profit Function
Xiong Qiquan
2005/9/29
2.1 Profit Maximization
A basic assumption of most economic analysis
is that a firm acts so as to maximize its profit;that
出。不难看出，对于p>w而言，不存在最大化的
利润。当p>w时，如果你相最大化px-wx，你会选 择无穷大的x值。仅当p≤w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：
outputs of the firm. The profit maximization problem of the firm can be stated as
( p) max py
such that y Y
The function π(p) is called the profit function of the firm
(负半定) at the optimal point; that is, the second-order
condition requires that the Hessian Matrix(海塞矩阵)
2 f x* f11 D2 f x* xi x j f 21 f12 f 22
Profit-maximizing behavior can be characterized by calculus
f ( x* ) p wi xi
i 1,2பைடு நூலகம், n.
This condition simply says that the value of the marginal product of each factor must be equal to its price.
a1 , a2 ,an
max R a1 , a2 ,, an C a1 , a2 ,, an
A simple application of calculus shows that an optimal set of actions a*
* * * a* a1 , a2 ,, an
xi (tp, tw) i ( p, w)
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：
f ( x) p wi 0, 如果xi 0 xi f ( x) p wi 0, 如果xi 0 xi
所有的技术都有最大化的利润吗？
问题3：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
In two-dimensional case，profits are given by π
＝py－wx, isoprofit line： y w x
p p
py wx
slope
w p
y f x
p
Figure 2—1 Profit Maximization
The profitmaximizing amount of inputs occurs where the slope of the profit lines equals the slope of the production function.
must satisfy the condition hD 2 f x* ht 0 for all vectors h
Geometrically, the requirement that the Hessian matrix is negative semidefinite means that the
pf ( x ) wx 0.
* * *
假定我们以t>1的因子向上调整生产，现在的利润将是：
pf (tx* ) wtx* t pf ( x* ) wx* t * *.
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。
d 2 f x* dx
2
0
A 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
If the firm produces only one output,the profit function can be written as
( p, w) max pf ( x) wx
where p is now the (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。
is, a firm chooses actions (a1,a2, …, an), so as to maximize R(a1,a2,…,an)-C(a1,a2,…,an). The profit maximization problem facing the firm can be written as:


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。