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第 2 章:预算约束
第 2 章:预算约束

自强不息知行合

我们在 课程中的哪个位置
我们在课程中的哪个位置

· 我们正在研究 消费者行为
·我们正在研究消费者行为

·消费者行为 三个要素是:
·消费者行为三个要素是:

·优化约束条件 ·偏好
·优化约束条件 ·偏好

·优化目标 ·选择
·优化目标 ·选择

优化结果
优化结果

消费选择集
消费选择集

·消耗选择集是 可用于
·消耗选择集是可用于的所有消耗选择的集合
的所有消耗选择的集合

消费者
消费者费束

·向量 x=[x,X,X₃,...,×n]T 称为 c消耗束,即潜在消费表示 n 种商品 (服务) 中每一种的数量

·消费选择集是所有可能的替代消费组合的集合,xdesignedbyX∈R“ 消费选择集

·什么制约了消费选择?

·预算、时间和其他 资源限制

预算约束

·考虑一个消费捆绑包(x,X,...,xn) ·设商品价格 p,P,...,Pn 

·Q:When is this bundle affordable at given prices?
·Q:这个捆绑包什么时候可以以 given价格负担得起?

·A:When pX + …+PnXn≤m,where m is the consumer's (disposable)income
·A:当pX+...+PnXn≤m 时,其中m消费者(一次性)收入

·The consumer's budget set is the set of all affordable bundles:
·消费者的预算集是所有负担得起的捆绑包的集合

B(p,…,Pn,m)={(x,…,xn)|x≥0,…,xn≥0 and pX + …+PnXn≤m}
B(p,...,Pn,m)={(x,...,xn)|x≥0,...,xn≥0 和 pX+...+PnXn≤m}

·The budget constraint is the upper boundary of the budget set,i.e.all the
·预算约束预算上限,即所有

combinations of commodities such that equality holds in the constraint equation
商品的组合,使得相等性在约束方程成立

自 强 不 息

Budget Set and Constraint for Two Commodities
预算约束f2种商品

·(1)Crosscuts at both axes;(2)The budget constraint line;(3)The area included
·(1)两个轴上的横切;(2)预算约束线;(3)包含的区域

X2

m/p
米/p

自强不息 知行合

Budget Set and Constraint for Two Commodities
两种商品预算集和约束

X2

PX1+PX2=m means X =-(p/p)x+m/pso the slope is -p/pBudget Set
PX1+PX2=m 表示X=-(p/p)x+m/p因此斜率为 -p/p预算设置

m/p X1
米/人X1

自强不息 · 知行合NEU

What Does the Slope Mean?
斜率是什么意思?

·In the case where n=2,the slope of the budget line is -p/p
·在n=2的情况下预算线斜率 -p/p

·The market is willing to "substitute"good 1 for good 2 at this rate
·市场愿意这个汇率“用好 1 代替”2

·Increasingxby 1 must reduce xby p/p,so mathematically
·将 x 增加1 必须将 x 减少p/p,因此在数学上

PAx+PAx=0 -pi/p=△x/△x
PAx+PAx=0-pi/p=△x/△x

机会成本

·This slope measures the opportunity cost of consuming good 1
·这个斜率衡量的是消费商品的机会成本 1

·That is,the opportunity cost of one unit of good 1 is p/punits of good 2
·也就是说,商品1e个单位的机会成本p/商品 2 p 个单位

自 强 不 息 知 行 合

How the Budget Line Changes with Prices and Incomes
预算线如何PricesInmoney 而变化

·When prices and incomes change,the set of goods that a consumer can afford changes as well
·当价格收入发生变化时,消费者能够负担得起的商品组合也会发生变化

·Let us first think about the case where income increases from m to m'
·让我们首先考虑一下 incomeincm增加到m' 的情况

考虑消费者收入水平变化

·Note that intercepts of the budget line at both axes move outwards proportionally ·And the slope of the budget line remains the same
·请注意预算线at的两个截距按比例向外移动·预算线斜率相同

·So an increase in income causes a parallel,outward shift of the budget line
·因此收入的增加会导致预算线平行向外移动

收入的增加会导致预算线向外平移

行 合

How the Budget Line Changes with Prices and Incomes
预算线如何随价格和收入而变化

X

Original and
Original

new budget
预算

constraints are parallel(same slope).
约束parallel(相同斜率)。

X1

自强不息知行合

How the Budget Line Changes with Prices and Incomes
预算线如何价格收入而变化

·How income changes affect the budget line and budget set?
·收入变化如何影响预算项目和预算集?

·Increases in income m shift the constraint outward in a parallel manner,thereby enlargingthe
·收入的增加m平行的方式将约束向外移动,从而扩大

budget set and improving choice
预算设定和改进选择

·Decreases in income m shift the constraint inward ina parallel manner,thereby shrinking the
·收入m减少平行的方式将约束向内移动,从而缩小

budget set and reducing choice
预算设置和减少选择

·What are welfare implications of income changes?
·收入变化对福利有什么影响

·No original choice is lost and new choices are addedwhen income increases,so higher income cannot make a consumer worse off
·当收入增加时,不会失去原始选择并且会添加n选择,因此更高的收入不会使消费者的生活变得更糟

·An income decrease may (typically will)make the consumer worse off
·收入减少可能(通常会)使消费者的情况变得更糟

收入的增长不会让消费者的福利受损

How the Budget Line Changes with Prices and Incomes
预算线如何 价格收入而变化

考虑商品价格的变化

·Let us now consider price changes
·现在让我们考虑价格变化

·First,let us consider a decrease of pto p'
·首先,让我们 考虑 p p' 的减少

·The intercept of the budget line at the vertical axis remains the same
·预算线在垂直上的截距保持不变

·But the intercept of the budget line at the horizontal axis moves outwards ·The slope of the budget line changes (becomes flatter in particular)
·但预算线在横轴上的截距向外移动·预算线的斜率变化特别平坦)

·That is,the budget line pivots
·也就是说,预算线发生了转变

·Q:What happens if the price increases?
·Q:如果价格上涨怎样

自 强 不 息

How the Budget Line Changes with Prices and Incomes
预算线如何随价格和收入而变化

X₂ m/p

New affordable choices
新的经济实惠的选择

Budget
预算
constraint
约束

pivots;slope flattens
支点;坡度展平

from -p/pto -p/p
从 -p/p -p/p

m/p m/p' X1

自强不息·知行合NEU

How the Budget Line Changes with Prices and Incomes
预算线 C 如何价格和收入挂钩

·Reducing the price of one commodity pivots the constraint outward
·降低一种商品的价格 t约束向外

·No old choice is lost and new choices are added,so reducing one price cannot
·没有旧的选择丢失,新的选择被添加,所以降低一个价格是不能

make the consumer worse off
让消费者的处境变得更糟

·Similarly,increasing one price pivots the constraint inwards,reduces choice and
·同样,增加一个价格会使约束内转动,减少选择 和

may (typically will)make the consumer worse off
可能(通常会)使 CO消费者的情况变得更糟

商品1价格的下降,不会使消费者福利受损, 般来讲它会使消费者福利得到改进

自 强 不 息

How the Budget Line Changes with Prices and Incomes
预算线如何随价格和收入而变化

·The budget line is defined by two prices and one income,but one ofthese variables is redundant
·预算线两个价格和一个收入定义,但其中一个变量是多余的

·We could peg one of the prices,or the income,to some fixed value (often unity),
·我们可以其中一个价格或收入与某个固定价值(通常是统一挂钩

and adjust the other variables so as to describe exactly the same budget set
并调整其他变量,以便描述完全相同预算

·If we peg the income m=1,then the budget constraint can be rewritten as (p/m)x+(p/m)x=1
·如果我们收入 m=1 挂钩,那么预算约束可以改写为 (p/m)x+(p/m)x=1

·When we set one of the prices to 1,we often refer to that price as the numeraire price
·当我们将其中一个价格设置为1 时,我们通常价格称为数字价格

·In the case above,we can say good 2 is the numeraire commodity ·We can use the idea to study a particular kind of taxes
·在上面的例子,我们可以good2商品的数值 计·我们可以这个想法来研究一种p关节

自 强 不

税收对预算约束的影响

Uniform Ad Valorem Sales Taxes 单一从价销售税
Uniform Ad valorem Sales Taxes 单一从价销售税

·Think about a uniform and advalorem sales tax that increases all prices by a factor of t,from p to(1+t)p
·考虑一个统一从价销售,它使所有价格降低t,从p到 (1+t)p

·A uniform sales taxis applied uniformly to all commodities
·统一销售出租车适用于所有商品

m

P2
P2 系列

m

(1+t)p2
(1+吨)p2

PX1+PX2=m
PX1+PX2=米

PX +PX=m/(1+t)
PX+PX=m/(1+t)

征税将使得预算线向内平移, 造成预算集的面积变小

m

(1+t)P1
(1+吨)P1

自强不息 · 知行合NEU

m P1
P1

Uniform Ad Valorem Sales Taxes
统一从价销售

·Auniform advalorem sales tax levied at rate t is equivalent to an income tax levied at rate t/(1+t)
·按税率 t 征收的统一从价销售税相当于税率t/(1+t) 征收的所得税

m

P2
P2 系列

m

(1+t)P2
(1+吨)P2 系列

Equivalent income loss
等值收入损失

is

m-1+-1+m
米-1+-1+米

m

(1+t)P1
(1+吨)P1

m

P1

X1

不 息

Other Taxes,Subsidies,and Rationing 其他税收、补贴和配额
其他税收补贴配给其他税收、补贴和配额

·In contrast,the government may impose quantity taxes 数量税
·相比之下,政府可能会征收数量

·For example,in the U.S.vou pav about 15 cents a gallon as a federal gasoline tax
·例如,在美国pav 大约每加仑 15 美分作为联邦汽油税

·A subsidy is the opposite of a tax
·补贴税收相反

·Similar to taxes,we also have quantity subsidies and advalorem subsidies
·税收类似我们也有 数量补贴从价补贴

·Another kind of tax or subsidy is a lump-sum tax or subsidy ·The government charges some fixed amount of money,regardless of the individual's behavior
·另一种税收补贴一次性税款补贴·政府收取一些固定数额钱,而不管个人的行为如何

·Thus a lump-sum tax meansthat the budget line of a consumer will shift inward because his money income has been reduced
·因此一次性税意味着消费者预算线向内移动,因为他货币收入减少了

·Similarly,a lump-sum subsidy means that the budget line will shift outward
·同样,一次性补贴意味着预算线向外移动

Other Taxes,Subsidies,and Rationing
其他税金、补助、配给

·Governments sometimes impose rationing constraints,such that the level of consumption of some good is fixed to be no larger than some amount
·政府有时会施加配给限制,以便将某些商品的消费水平固定为不超过一定数量

不 息

An Example:The Food Stamp Program
示例食品券计划

·What is the Food Stamp Program?
·什么是食品券计划

·Food stamps are coupons that can be legallyexchanged only for food
·食品券是只能合法兑换食品的优惠券

·Households who met certain eligibility requirements can be involved in this program
·满足特定资格要求的住户可以参与计划

Before 1979,the program provided an ad valorem subsidy on food $25购$153食品
1979 年以前该计划提供食品从价补贴$25购$153食品

·For example,a household was allowed to buy at most(which was determined according to the household economicstatus)$153 of food stamps for $25
·例如,一个家庭最多可以以25 美元的价格购买 153 美元的食品 st amps(根据家庭经济状况确定

·But for each dollar beyond $153 spent on food,no subsidy could be enjoyed
·但是,在食品上每花费超过 153 美元,就不能享受任何补贴

·After 1979,the program became a lump-sum subsidy $ 2 0 0 ·Ahousehold now receives agrant of $200 of food stamps a month,so it can consume$200 more food
·1979 年后,该计划成为一次性补贴的 $200·Ahousehold现在每月收到 200 美元的食品券补助金,因此它可以消耗 200 美元的食物

per month,regardless of how much it is spending on other goods
每月,无论其他商品SP结束金额是多少

不 息 知 行

An Example:The Food Stamp Program
示例:食品券 P

A

B

1979年以前 1979年以后

Shapes of Budget Constraints
预算结构的形状

·Q:What makes a budget constraint a straight line?
·Q:是什么使预算编制成为一条直线

·A:A straight line has a constant slope and the constraint is
·A:直线斜率是恒定c的斜

PX1+...+PnXn=m

so if prices are constants then a constraint is a straight line
因此如果价格是常数,那么约束就是一条直线

商品相对价格是常数,预算线的斜率就是一个常数

·But what if prices are not constants?
·但是,如果价格不是常数

·For example,bulk buying discounts,or price penalties for buying"too much" ·Then constraints will be curved
·例如,批量购买折扣,或购买“太多”的价格处罚 ·然后约束将是弯曲

·Suppose pis constant at$1 but that p=$2 for O≤x≤20 and p=$1 for x>20. Then the constraint's slope is
·假设 p $1 处恒定,但 O≤x₁≤20p=$2,x>20 p=$1则约束的斜率

and the constraint is
约束条件为

自 强 不

Shapes of Budget Constraints
预算约束的形状

多买优惠

自强不息 · 知行合NEU

Shapes of Budget Constraints
预算约束的形状

·What is the case for quantity penalties?
·数量罚款的情况是什么?

Budget
预算

Constraint
约束

X1

自强不息知行合

Shapes of Budget Constraints 预算线斜率为正的情况

·Negative prices
·负价格

·Commodity 1 is stinky garbage
·商品1垃圾

·You are paid $2 per unit to accept it:p=-$2.p=$1
·接受它时,您每单位需支付 2 美元:p=-2.p=1 美元

·Income,other than from accepting commodity 1,is m=$10 ·Then the constraint is-2x+x=10
·收入,除接受商品外1,ism=$10·则约束为 -2x+x=10

X1

自强不息知行合