Budget Lines | 预算线

📚 Budget Lines | 预算线

A budget line is a fundamental tool in consumer choice theory. It shows the maximum combinations of two goods that a consumer can afford given their income and the prices of the goods. Understanding budget lines is essential for analysing how consumers respond to changes in income, prices, and preferences.

预算线是消费者选择理论中的基本工具。它显示在给定收入和商品价格的情况下，消费者能够负担的两种商品的最大组合。理解预算线对于分析消费者如何应对收入、价格和偏好的变化至关重要。

1. Definition of a Budget Line | 预算线的定义

A budget line, also called the budget constraint, shows all combinations of two goods that a consumer can purchase by spending exactly all of their fixed money income, given the prices of the two goods.

预算线，也称为预算约束，显示在两种商品价格给定且花完全部固定货币收入时，消费者能够购买的所有两种商品组合。

It separates attainable bundles from unattainable bundles. Points on the line exhaust the entire budget, points inside the line do not use all income, and points outside are unaffordable.

它将可获得的消费束与不可获得的消费束分开。线上的点耗尽全部预算，线内的点没有花完全部收入，线外的点无法负担。

2. The Budget Equation | 预算方程

For two goods X₁ and X₂ with prices P₁ and P₂, and money income M, the budget equation is written as:

对于价格为 P₁ 和 P₂ 的两种商品 X₁ 和 X₂，货币收入为 M，预算方程可以写成：

M = P₁X₁ + P₂X₂

This states that total spending on X₁ plus total spending on X₂ must equal the consumer’s money income.

这表明在 X₁ 上的总支出加上在 X₂ 上的总支出必须等于消费者的货币收入。

Rearranging the equation for X₂ gives the budget line in slope-intercept form:

对 X₂ 求解该方程可得到预算线的斜截式：

X₂ = M/P₂ − (P₁/P₂)X₁

3. Slope of the Budget Line | 预算线的斜率

The slope of the budget line is −P₁/P₂. It measures the opportunity cost of consuming one more unit of X₁ in terms of the amount of X₂ forgone.

预算线的斜率为 −P₁/P₂。它衡量多消费一单位 X₁ 所放弃的 X₂ 数量的机会成本。

For example, if P₁ = 2 and P₂ = 4, the slope is −0.5. This means that to buy one extra unit of X₁, the consumer must give up half a unit of X₂.

例如，如果 P₁ = 2 且 P₂ = 4，斜率为 −0.5。这意味着为了多买一单位 X₁，消费者必须放弃半单位 X₂。

4. Drawing the Budget Line | 绘制预算线

To draw the budget line, first find the intercepts. If the consumer spends all income on X₁, they can buy M/P₁ units. If they spend all income on X₂, they can buy M/P₂ units.

要绘制预算线，首先求出截距。如果消费者将全部收入用于 X₁，可以购买 M/P₁ 单位。如果全部用于 X₂，可以购买 M/P₂ 单位。

Plot these two intercepts on the axes and join them with a straight line. The budget line slopes downward because buying more of one good requires buying less of the other.

在坐标轴上标出这两个截距，并用直线连接。预算线向下倾斜，因为多买一种商品就必须少买另一种商品。

5. Changes in Income | 收入变化

When money income increases and prices remain unchanged, both intercepts rise by the same proportion. The budget line shifts outward parallel to the original line.

当货币收入增加且价格保持不变时，两个截距按相同比例上升。预算线向外平移，与原线平行。

A fall in money income shifts the budget line inward, reducing the set of affordable bundles. The slope does not change because relative prices are unchanged.

货币收入下降使预算线向内平移，减少可负担的消费束集合。由于相对价格未变，斜率不发生变化。

6. Changes in Price | 价格变化

If the price of one good changes while the other price and income stay constant, only the intercept for that good changes. The budget line pivots or rotates around the other good’s intercept.

如果一种商品的价格变化，而另一种商品的价格和收入保持不变，则只有该商品对应的截距变化。预算线围绕另一种商品的截距旋转。

For example, a fall in P₁ increases the horizontal intercept from M/P₁ to M/P₁′ and makes the budget line flatter. The vertical intercept remains at M/P₂ because P₂ and M are unchanged.

例如，P₁ 下降使水平截距从 M/P₁ 增加到 M/P₁′，并使预算线变得更平坦。垂直截距仍为 M/P₂，因为 P₂ 和 M 未变。

7. The Budget Set and Affordable Choices | 预算集与可负担选择

The budget line is the boundary of the budget set. The budget set includes all bundles on or below the line that satisfy the inequality P₁X₁ + P₂X₂ ≤ M.

预算线是预算集的边界。预算集包括线上及线以下所有满足不等式 P₁X₁ + P₂X₂ ≤ M 的消费束。

Points inside the budget line are affordable but do not use up all income. Points outside the budget line are unaffordable given the current income and prices.

预算线内部的点是可以负担的，但没有花完全部收入。预算线外部的点在当前收入和价格下无法负担。

8. Consumer Equilibrium | 消费者均衡

A rational consumer maximises utility subject to the budget constraint. Equilibrium occurs where the highest attainable indifference curve is tangent to the budget line.

理性消费者在预算约束下实现效用最大化。均衡出现在最高可达到的无差异曲线与预算线相切的点。

At this tangency point, the marginal rate of substitution between the two goods equals the price ratio:

在这个切点处，两种商品之间的边际替代率等于价格比：

MRS = P₁/P₂

This condition means the rate at which the consumer is willing to trade one good for another equals the rate at which the market allows that trade.

这一条件意味着消费者愿意用一种商品交换另一种商品的主观比率等于市场允许交换的客观比率。

9. Shifts vs Rotations | 移动与旋转

A parallel shift of the budget line is caused by a change in money income, assuming constant relative prices. A rotation is caused by a change in one price while other conditions stay the same.

预算线的平行移动是由货币收入变化引起的，前提是相对价格不变。预算线的旋转是由一种商品价格变化引起的，其他条件保持不变。

If both prices and income change by the same percentage, the budget line does not move because real income and relative prices remain unchanged.

如果两种商品价格和收入按相同百分比变化，预算线不会移动，因为实际收入和相对价格保持不变。

10. Worked Example | 示例计算

Suppose a consumer has income M = 60, the price of good X₁ is P₁ = 3, and the price of good X₂ is P₂ = 2. The budget equation is:

假设消费者收入 M = 60，商品 X₁ 的价格 P₁ = 3，商品 X₂ 的价格 P₂ = 2。预算方程为：

60 = 3X₁ + 2X₂

The horizontal intercept is 60/3 = 20 units of X₁. The vertical intercept is 60/2 = 30 units of X₂. The slope is −3/2 = −1.5.

水平截距为 60/3 = 20 单位 X₁。垂直截距为 60/2 = 30 单位 X₂。斜率为 −3/2 = −1.5。

If income rises to 90 with prices unchanged, the new intercepts are 30 and 45. The budget line shifts outward in a parallel manner.

如果收入上升到 90 而价格不变，新的截距为 30 和 45。预算线向外平行移动。

11. Limitations and Extensions | 局限与扩展

The basic budget line model assumes only two goods, fixed money income, and constant prices. Real consumer choices involve many goods, saving, borrowing, and changing preferences.

基本预算线模型假设只有两种商品、固定货币收入和不变价格。现实消费者选择涉及多种商品、储蓄、借贷和偏好变化。

Extensions include intertemporal budget constraints that allow saving and borrowing over time, non-linear budget constraints from quantity discounts, and behavioural influences on spending.

扩展模型包括允许跨期储蓄和借贷的跨期预算约束、数量折扣导致的非线性预算约束，以及行为因素对支出的影响。

Despite these limitations, the budget line remains a powerful analytical tool for explaining consumer choice and the welfare effects of price and income changes.

尽管存在这些局限，预算线仍然是解释消费者选择以及价格和收入变化的福利效应的有力分析工具。

12. Exam Tips | 考试提示

When drawing budget lines in an exam, label both intercepts, use arrows to show shifts or rotations, and state clearly whether the slope changes.

在考试中绘制预算线时，标注两个截距，用箭头表示平移或旋转，并清楚地说明斜率是否变化。

For consumer equilibrium questions, show the tangency with an indifference curve and explain why a point inside the budget line is inefficient because utility can be increased by using more of the available income.

对于消费者均衡问题，画出与无差异曲线的切点，并解释为什么预算线内部的点是无效率的，因为可以通过使用更多可用收入来提高效用。

Published by TutorHao | Economics Revision Series | aleveler.com

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Budget Lines | 预算线