Budget Lines | 预算线

📚 Budget Lines | 预算线

In microeconomics, a budget line (or budget constraint) illustrates all combinations of two goods that a consumer can afford given their income and the prices of the goods. It is a fundamental tool for understanding consumer choice and forms the backbone of the indifference curve analysis. By analyzing how budget lines shift and rotate, students can grasp the effects of income changes, price changes, taxes, and subsidies on consumer behaviour.

在微观经济学中，预算线（或预算约束）表示在给定收入和商品价格下，消费者能够负担得起的两种商品的所有组合。它是理解消费者选择的基本工具，也是无差异曲线分析的核心。通过分析预算线如何移动和旋转，学生可以掌握收入变化、价格变化、税收和补贴对消费者行为的影响。

---

1. Definition of Budget Line | 预算线的定义

A budget line shows the maximum combinations of two goods that a consumer can purchase with a fixed income, given constant prices. Every point on the line represents a consumption bundle that exactly exhausts the consumer's income. Points inside the budget line are affordable but leave some income unspent, while points outside are unattainable under current constraints.

预算线表示在固定收入和不变价格下，消费者能够购买的两种商品的最大组合。线上的每一点都代表一个恰好耗尽消费者收入的消费束。预算线内的点是可以负担的但并未花光所有收入，而线外的点在当前约束下是无法达到的。

---

2. The Budget Equation | 预算方程

The budget constraint can be expressed as an equation. If a consumer allocates their entire income M to buying two goods, X₁ and X₂, with prices P₁ and P₂ respectively, the budget equation is:

预算约束可以用方程表示。如果消费者将全部收入 M 用于购买两种商品 X₁ 和 X₂，价格分别为 P₁ 和 P₂，则预算方程为：

M = P₁X₁ + P₂X₂

This states that total spending on both goods must equal the consumer's money income. Solving for X₂ gives the equation of the budget line in slope-intercept form:

这表明购买两种商品的总支出必须等于消费者的货币收入。解出 X₂，即可得到截距式预算线方程：

X₂ = (M / P₂) – (P₁ / P₂) X₁

Here, M/P₂ is the vertical intercept when X₂ is on the vertical axis, and –P₁/P₂ is the slope.

这里，如果将 X₂ 放在纵轴，M/P₂ 就是纵轴截距，而 –P₁/P₂ 为斜率。

---

3. Slope of the Budget Line | 预算线的斜率

ich the consumer can trade one good for another in the market. In the equation X₂ = (M/P₂) – (P₁/P₂) X₁, the slope is – unity cost of consuming one more unit of good X₁ in terms of good X₂. For example, if P₁ = £2 and P₂ = £1, the slope is –2, meaning the consumer must give up 2 units of X₂ to obtain 1 extra unit of X₁.

预算线的斜率衡量消费者在市场中用一种商品交换另一种商品的比率。在方程 X₂ = (M/P₂) – (P₁/P₂) X₁ 中，斜率为 –P₁/P₂。这是多消费一单位商品 X₁ 的机会成本（以放弃的商品 X₂ 数量表示）。例如，如果 P₁ = 2英镑，P₂ = 1英镑，斜率为 –2，这意味着消费者必须放弃 2 单位 X₂ 才能获得额外 1 单位 X₁。

---

4. Changes in Income: Parallel Shifts | 收入变动：预算线的平行移动

When the consumer's income changes while prices remain constant, the budget line shifts in a parallel manner. An increase in income shifts the budget line outward, expanding the set of affordable bundles. A decrease in income shifts it inward. Both intercepts increase or decrease proportionally, so the slope remains unchanged.

当消费者收入变动而价格保持不变时，预算线会平行移动。收入增加使预算线向外移动，扩大了可负担的消费束集合。收入减少则使其向内移动。两个截距成比例地增加或减少，因此斜率保持不变。

Change	Effect on Budget Line	Slope
Income ↑	Outward parallel shift	Unchanged
Income ↓	Inward parallel shift	Unchanged

An increase in money income is like an increase in real income because consumers can afford more of both goods without any change in relative prices.

货币收入增加相当于实际收入增加，因为消费者能够在不改变相对价格的情况下购买更多两种商品。

---

广告 / Sponsored

5. Changes in Price: Rotation of the Budget Line | 价格变动：预算线的旋转

A change in the price of one good causes the budget line to pivot or rotate. If the price of good X₁ falls, the budget line rotates outward along the X₁-axis, increasing the maximum affordable quantity of X₁ while the intercept on the X₂-axis stays the same. If the price of X₁ rises, the line rotates inward. A change in the price of both goods in the same proportion is equivalent to a change in real income and causes a parallel shift.

一种商品价格的变化会导致预算线旋转。如果商品 X₁ 的价格下降，预算线会沿着 X₁ 轴向外旋转，增加 X₁ 的最大可购买量，而 X₂ 轴上的截距保持不变。如果 X₁ 的价格上升，预算线向内旋转。两种商品的价格同比例变化相当于实际收入的变动，会导致平行移动。

Change	Effect	New Slope
P₁ ↓	Rotates outward on X₁-axis	Less steep (flatter)
P₁ ↑	Rotates inward on X₁-axis	Steeper
P₂ ↓	Rotates outward on X₂-axis	Steeper
P₂ ↑	Rotates inward on X₂-axis	Flatter

When the price of X₁ falls, the slope becomes flatter because less X₂ must be sacrificed to obtain each unit of X₁. Conversely, a rise in P₁ makes the line steeper.

当 X₁ 的价格下降时，斜率变得更平缓，因为获得每单位 X₁ 所需要放弃的 X₂ 变少了。相反，P₁ 上升会使预算线更陡峭。

---

6. Real Income and Relative Prices | 实际收入与相对价格

The budget line embodies two key concepts: real income and relative price. Real income is the purchasing power of nominal income, measured by the intercepts M/P₁ and M/P₂. The slope –P₁/P₂ represents the relative price of good X₁ in terms of X₂. Thus, changes in prices affect both real income and the relative price, while changes in money income only alter real income without affecting relative prices.

预算线体现了两个关键概念：实际收入和相对价格。实际收入是名义收入的购买力，由截距 M/P₁ 和 M/P₂ 衡量。斜率 –P₁/P₂ 代表以商品 X₂ 表示的商品 X₁ 的相对价格。因此，价格变动同时影响实际收入和相对价格，而货币收入的变动仅改变实际收入，不影响相对价格。

For instance, if P₁ decreases, the consumer can buy more X₁ with the same income, so real income in terms of X₁ rises. At the same time, X₁ becomes cheaper relative to X₂, altering the slope.

例如，如果 P₁ 下降，消费者用相同收入可购买更多的 X₁，因此以 X₁ 衡量的实际收入上升。同时，X₁ 相对于 X₂ 变得更便宜，从而改变了斜率。

---

7. Budget Line and Indifference Curves: Consumer Equilibrium | 预算线与无差异曲线：消费者均衡

In consumer theory, the budget line is combined with indifference curves to determine the optimal consumption bundle. The consumer aims to reach the highest possible indifference curve that is tangent to the budget line. At the point of tangency, the slope of the indifference curve (the marginal rate of substitution, MRS) equals the slope of the budget line (–P₁/P₂). This equilibrium condition is:

在消费者理论中，预算线与无差异曲线结合使用，以确定最优消费束。消费者力求达到与预算线相切的最高无差异曲线。在切点处，无差异曲线的斜率（边际替代率 MRS）等于预算线的斜率（–P₁/P₂）。这一均衡条件为：

MRS = P₁ / P₂

This ensures that the consumer's subjective trade-off between the two goods matches the market trade-off, so no other affordable bundle yields higher utility.

这确保消费者在两种商品间的主观权衡与市场权衡相匹配，因此没有其他可负担的消费束能带来更高效用。

If MRS > P₁/P₂, the consumer values X₁ more than the market does and should consume more X₁. If MRS < P₁/P₂, the consumer should consume less X₁. The tangency point is optimal only when preferences are well-behaved (convex).

如果 MRS > P₁/P₂，消费者对 X₁ 的主观评价高于市场，应该多消费 X₁。如果 MRS < P₁/P₂，应减少 X₁ 的消费。只有当偏好为良性（凸性）时，切点才是最优解。

---

8. Corner Solutions and Non-Convex Preferences | 角点解与非凸偏好

Sometimes the optimal bundle lies at a corner of the budget line rather than at an interior tangency. This occurs when the MRS is always greater or always less than the price ratio, leading the consumer to spend all income on only one good. Corner solutions are common with perfect substitutes or perfect complements, and when preferences are non-convex.

有时最优消费束位于预算线的角点处，而非内部切点。当 MRS 始终大于或始终小于价格比率时就会发生这种情况，导致消费者将所有收入花在一种商品上。角点解在完全替代品或完全互补品以及偏好为非凸的情形中很常见。

For example, if X₁ and

Published by TutorHao | A-Level Economics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）