Second Derivatives
• The derivative of a derivative is called a second derivative
• The second derivative can be denoted by
d 2 dq 2
or
d 2f dq 2
or
f "(q)
Second Order Condition
• This must mean that, in order for q* to be the optimum,
d 0 for q q * dq
and
d 0 for q q * dq
• At q*, d/dq must be decreasing
– the derivative of d/dq must be negative at q*
– an increase from q1 to q2 leads to a rise in
* 2
1
q1
q2
q*
= f(q)
0 q
Quantity
Functions with One Variable

• If output is increased beyond q*, profit will
Chapter 2
Mathematics for Microeconomics
Nicholson and Snyder, Copyright ©2008 by Thomson South-Western. All rights reserved.
数学在经济学中的应用
现在经济学当中的模型大多是数学模型 数学的优势：
d df lim f (q1 h) f (q1)
dq dq h0
h
• The value depends on the value of q1
Value of a Derivative at a Point
• The evaluation of the derivative at the point q = q1 can be denoted
d dq
q q1
• In our previous example,
d 0
dq q q1
d 0
dq q q3
d 0 dq q q*
First Order Condition for a Maximum
• For a function of one variable to attain its maximum value at some point, the derivative at that point must be zero
• Simple example: Manager of a firm wants to maximize profits
f (q)
Maximum profits of
*
* occur at q*
= f(q)
Quantity
q*
Functions with One Variable
• Vary q to see where maximum profit occurs
– consumers seek to maximize utility – firms seek to maximize profit
• This chapter introduces the mathematics common to these problems
Functions with One Variable
• The second order condition to represent a (local) maximum is
d 2 dq 2
f "(q) 0 q q *
q q *
求导法则
1.If b is a constant, then db 0 dx
2.If b is a constant, then d[bf (x)] bf '(x) dx
decline
– an increase from q* to q3 leads to a drop in
*
0
= f(q)
q
3
Quantity
q*
q3
Derivatives
• The derivative of = f(q) is the limit of /q for very small changes in q
在进行推理时要给出明确的前提和假设， 推理要遵循严格的逻辑，对错容易判断。 文字表述的弱点： 文字的表述相对含糊，容易产生歧义， 对错不容易明确判断。
The Mathematics of Optimization
• Economic theories assume that an economic agent is seeking to find the optimal value of some function
• Then
6. d[f (x) g(x)] f '(x) g'(x) dx
If the profit function was u-shaped, the first order condition would result in q* being chosen and would be minimized
*
Quantity
q*
Second Order Conditions
3.If b is constant, then dxb bxb1 dx
4. d ln x 1 dx x
求导法则
5. dax ax ln a for any constant a dx
– a special case of this rule is dex/dx = ex
求导法则
• Suppose that f(x) and g(x) are two functions of x and f’(x) and g’(x) exist
df 0
dq q q*
Second Order Conditions
• The first order condition (d/dq) is a necessary（必要） condition for a maximum, but it is not a sufficient（充分） condition