Inverse demand function

From Wikipedia, the free encyclopedia

In economics, an inverse demand function, P = f⁻¹(Q),is a function that maps the quantity of output demanded to the market price (dependent variable) for that output. Quantity demanded, Q, is a function of price; the inverse demand function treats price as a function of quantity demanded, and is also called the price function.^[1] Note that the inverse demand function is not the reciprocal of the demand function—the word "inverse" refers to the mathematical concept of an inverse function.

In mathematical terms, if the demand function is f(P),in which P is price, so the value of the function is the quantity demanded (Q), then the inverse demand function is f ^-1(Q), whose value is the highest price that could be charged and still generate the quantity demanded Q.^[2] This is to say that the inverse demand function is the demand function with the axes switched. This is useful because economists typically place price (P) on the vertical axis and quantity (Q) on the horizontal axis.

The inverse demand function is the same as the average revenue function, since P = AR.^[3]

To compute the inverse demand function, simply solve for P from the demand function. For example, if the demand function has the form Q = 240 - 2P then the inverse demand function would be P = 120 - 0.5Q.^[4]

The inverse demand function can be used to derive the total and marginal revenue functions. Total revenue equals price, P, times quantity, Q, or TR = P×Q. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². The marginal revenue function is the first derivative of the total revenue function or MR = 120 - Q. Note that in this linear example the MR function has the same y-intercept as the inverse demand function, the x-intercept of the MR function is one-half the value of the demand function, and the slope of the MR function is twice that of the inverse demand function. This relationship holds true for all linear demand equations. The importance of being able to quickly calculate MR is that the profit-maximizing condition for firms regardless of market structure is to produce where marginal revenue equals marginal cost (MC). To derive MC the first derivative of the total cost function is taken.

For example assume cost, C, equals 420 + 60Q + Q². then MC = 60 + 2Q.^[5] Equating MR to MC and solving for Q gives Q = 20. So 20 is the profit maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P.

The inverse demand function is the form of the demand function that appears in the famous Marshallian Scissors diagram. The function appears in this form because economists place the independent variable on the y-axis and the dependent variable on the x-axis. The slope of the inverse function is ∆P/∆Q. This fact should be kept in mind when calculating elasticity. The formula for elasticity is (∆Q/∆P) × (P/Q).

There is a close relationship between any inverse demand function for a linear demand equation and the marginal revenue function. For any linear demand function with an inverse demand equation of the form P = a - bQ, the marginal revenue function has the form MR = a - 2bQ.^[6] The marginal revenue function and inverse linear demand function have the following characteristics:

What is a partial correlation?

今天看Econometric Analysis，以前觉得自己懂一点，现在看了发现自己根本不懂。
贴上来吧：

 partial correlation

- Defined: Partial correlation is the relationship between two variables while controlling for a third variable. - Variables: IV is continuous, DV is continuous, and third variable is continuous - Relationship: Relationship amongst variables - Example Relationship between height and weight, while controlling for age - Assumptions: Normality. Linearity. 发现中文的还比较容易懂，英文的... 继续吧。

consumer surplus. argmax

在准备Dr. Fu布置的作业，paper里遇到了几个不懂的东西。
1.consumer surplus 消费者盈余

百度版：

用值与换值的差额就是消费者的盈余
假若一个苹果的市价（换值）是二元，你买五个。第五个（边际）的最高用值当然也是二元，否则你会多买一点或少买一点。这第五个的消费者盈余是零。然而，第一个苹果你愿意出十元之价（你的最高用值），第二个是八元，第三个是六元，第四个是四元，第五个才是二元。你每个须付之价只是二元。这样，你的消费者盈余是八元、六元、四元、二元、零，加起来是二十元。

还有网络版的：



Consumer surplus is a measure of the welfare that people gain from the consumption of goods and services, or a measure of the benefits they derive from the exchange of goods.
Consumer surplus is the difference between the total amount that consumers are willing and able to pay for a good or service (indicated by the demand curve) and the total amount that they actually do pay (i.e. the market price for the product). The level of consumer surplus is shown by the area under the demand curve and above the ruling market price as illustrated in the diagram below:

2.Numeraire good
一般等价物吧，如gold
commodity or good in terms of which all other prices r expressed.
3.argmax 
网上转的1：
最近在讀paper時看到了這個數學式子 y* = argmax f(t)，想當初數學這麼爛的我一定不懂argmax是什麼，找了好一會兒才找到，記錄一下避免忘記 :)

• y = f(t) 是一般常見的函式，代表給定一個t值，丟到f函式中會回傳一個值給y。

• y = max f(t) 代表：y 是f(t)函式所有的值中最大的output。

• y = arg max f(t) 代表：y 是f(t)函式中，會產生最大output的那個參數t。

看起來很模糊，舉個例子應該比較好理解：

假設有一個函式 f(t)，t 的可能範圍是 {0,1,2}，f(t=0) = 10 ; f(t=1) = 20 ; f(t=2) = 7，那分別對應的y如下：

• y = max f(t)  = 20
• y* = arg max f(t) = 1

這樣應該就很好理解了吧 :)


网上转的2

• y = f(t) 是一般常見的函数式，如果給定一個t值，f（t）函数式會赋一個值給y。

• y = max f(t) 代表：y 是f(t)函式所有的值中最大的output。

• y = arg max f(t) 代表：y 是f(t)函式中，會產生最大output的那個參數t。

看起來很模糊，舉個例子應該比較好理解：

假設有一個函式 f(t)，t 的可能範圍是 {0,1,2}，f(t=0) = 10 ; f(t=1) = 20 ; f(t=2) = 7，那分別對應的y如下：

• y = max f(t) = 20
• y= arg max f(t) = 1

实现：

ArgMax[f, x] 给出 f 最大化的坐标 xmax。

ArgMax[f, {x, y, ...}] 给出 f 最大化的坐标 {xmax, ymax, ...}。

ArgMax[{f, cons}, {x, y, ...}] 给出约束条件 cons 下 f 最大化的坐标。

ArgMax[{f, cons}, {x, y, ...}, dom] 给出域 dom 上 f 最大化的坐标，通常 Reals 或 Integers。