📚 Core Mathematics for Economics Year 1 | 经济学一年级核心数学
As an economics student in your first year, you may wonder why you are studying mathematics. The answer is simple: economics uses mathematical models to describe behaviour, predict outcomes, and optimise decisions. This article brings together the essential topics from the Edexcel Year 1 Mathematics syllabus and shows you how they apply directly to economic concepts such as demand, supply, profit, elasticity, and consumer surplus.
作为一年级经济学学生，你可能会好奇为什么要学习数学。答案很简单：经济学用数学模型来描述行为、预测结果并优化决策。本文整合了 Edexcel 一年级数学大纲中的核心主题，并展示它们如何直接应用于需求、供给、利润、弹性和消费者剩余等经济概念。
1. Linear Functions and Demand-Supply Models | 线性函数与供需模型
Linear functions are written in the form y = mx + c, where m is the gradient and c is the y-intercept. In economics, a linear demand curve often takes the form P = a – bQ, where P is price, Q is quantity, a is the price intercept, and -b is the slope. A linear supply curve may be written as P = c + dQ. The gradient represents how strongly quantity responds to price changes.
线性函数写作 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。在经济学中，线性需求曲线常写成 P = a – bQ，其中 P 是价格，Q 是数量，a 是价格截距，-b 是斜率。线性供给曲线可写成 P = c + dQ。斜率表示数量对价格变化的敏感程度。
For example, if a demand function is P = 100 – 2Q, then when Q = 0, the maximum price consumers are willing to pay is £100. As Q increases by one unit, price falls by £2. Recognising the gradient and intercepts allows you to sketch the graph quickly and interpret the economic meaning of each parameter.
例如，若需求函数为 P = 100 – 2Q，则当 Q = 0 时，消费者愿意支付的最高价格为 100 英镑。当 Q 每增加一单位，价格下降 2 英镑。识别斜率和截距可以帮助你快速画出图形，并解释每个参数的经济含义。
2. Quadratic Equations and Profit Maximization | 二次方程与利润最大化
Quadratic functions f(x) = ax² + bx + c appear frequently in cost and revenue analysis. Total revenue (TR) is often a quadratic function of output, such as TR = 100Q – 2Q². Total cost (TC) may take a quadratic form too. Profit π = TR – TC is then a quadratic function whose maximum value occurs at its vertex.
二次函数 f(x) = ax² + bx + c 经常出现在成本与收益分析中。总收益（TR）常常是产量的二次函数，例如 TR = 100Q – 2Q²。总成本（TC）也可能呈二次形式。利润 π = TR – TC 便是一个二次函数，其最大值出现在顶点处。
To find the output level that maximises profit, you can complete the square or use the formula Q = -b / (2a) for the x-coordinate of the vertex. Suppose π = -2Q² + 80Q – 200. Then the profit-maximising output is Q = -80 / (2 × -2) = 20. This method is essential for optimisation problems in microeconomics.
为了找到使利润最大化的产量，你可以使用配方法或顶点公式 Q = -b / (2a)。假设 π = -2Q² + 80Q – 200，则利润最大化的产量为 Q = -80 / (2 × -2) = 20。这种方法是微观经济学优化问题的基础。
3. Simultaneous Equations and Market Equilibrium | 联立方程与市场均衡
Market equilibrium occurs when quantity demanded equals quantity supplied. Given a demand equation P = 20 – 2Q and a supply equation P = 4 + Q, you can solve them simultaneously. Setting 20 – 2Q = 4 + Q gives 16 = 3Q, so Q = 16/3 ≈ 5.33. Substituting back gives P = 4 + 5.33 = 9.33.
市场均衡发生在需求量等于供给量的时候。给定需求方程 P = 20 – 2Q 和供给方程 P = 4 + Q，你可以联立求解。令 20 – 2Q = 4 + Q，得 16 = 3Q，所以 Q = 16/3 ≈ 5.33。代回得 P = 4 + 5.33 = 9.33。
In Edexcel Year 1 mathematics, you learn to solve simultaneous equations by substitution or elimination. These skills allow you to analyse how taxes, subsidies, or shifts in demand alter the equilibrium price and quantity. Understanding the algebra also helps you solve more complex models with non-linear equations later.
在 Edexcel 一年级数学中，你会学习用代入法或消元法求解联立方程。这些技能让你能够分析税收、补贴或需求变动如何改变均衡价格和数量。理解代数运算也有助于你后来求解更复杂的非线性模型。
4. Exponential and Logarithmic Functions in Finance | 金融中的指数与对数函数
Exponential functions are written as y = a e^(kt) or y = a bᵗ. They describe growth and decay. In economics, compound interest is a classic example: if you invest P pounds at an annual interest rate r compounded continuously, the future value is A = P e^(rt). For discrete compounding, A = P(1 + r/n)^(nt).
指数函数写作 y = a e^(kt) 或 y = a bᵗ，用于描述增长与衰减。在经济学中，复利是一个经典例子：如果你投资 P 英镑，年利率为 r，连续复利时的未来值为 A = P e^(rt)。对于离散复利，A = P(1 + r/n)^(nt)。
Logarithmic functions are the inverse of exponentials. The natural logarithm ln x satisfies e^(ln x) = x. You may use logs to solve for t in the compound interest formula. For example, if A = 2P and r = 5%, then 2 = e^(0.05t), so t = ln 2 / 0.05 ≈ 13.86 years. This is the doubling time.
对数函数是指数函数的逆函数。自然对数 ln x 满足 e^(ln x) = x。你可以使用对数在复利公式中求解 t。例如，若 A = 2P 且 r = 5%，则 2 = e^(0.05t)，所以 t = ln 2 / 0.05 ≈ 13.86 年。这就是倍增时间。
5. Sequences and Series: Compound Interest | 数列与级数：复利
An arithmetic sequence has a common difference d, while a geometric sequence has a common ratio r. In finance, regular savings or loan repayments often form arithmetic or geometric series. The sum of a geometric series S_n = a(1 – rⁿ) / (1 – r) is used to value annuities.
等差数列有公差 d，等比数列有公比 r。在金融中，定期储蓄或贷款还款常构成等差或等比数列。等比数列求和公式 S_n = a(1 – rⁿ) / (1 – r) 用于计算年金的现值。
For example, if you save £100 at the end of each month and earn 1% interest per month, the future value after n months is a geometric series with first term 100 and common ratio 1.01. The sum is given by S_n = 100(1.01ⁿ – 1) / (0.01). This connects pure mathematics to real financial decisions.
例如，如果你每月末存入 100 英镑，每月获得 1% 利息，则 n 个月后的未来值是一个首项为 100、公比为 1.01 的等比数列。其和为 S_n = 100(1.01ⁿ – 1) / (0.01)。这将纯数学与真实金融决策联系起来。
6. Differentiation and Marginal Analysis | 微分与边际分析
Differentiation measures the rate of change of a function. The derivative dy/dx gives the slope of the tangent at any point. In economics, marginal cost (MC) is the derivative of total cost with respect to quantity: MC = dTC/dQ. Marginal revenue (MR) is the derivative of total revenue: MR = dTR/dQ.
微分用于衡量函数的变化率。导数 dy/dx 给出任意点处切线的斜率。在经济学中，边际成本（MC）是总成本对数量的导数：MC = dTC/dQ。边际收益（MR）是总收益的导数：MR = dTR/dQ。
If TR = 50Q – 0.5Q², then MR = 50 – Q. If TC = 20 + 3Q + 0.1Q², then MC = 3 + 0.2Q. Profit is maximised when MR = MC. Setting 50 – Q = 3 + 0.2Q gives Q = 47 / 1.2 ≈ 39.17. You also use the second derivative to confirm that this point is a maximum, not a minimum.
若 TR = 50Q – 0.5Q²，则 MR = 50 – Q。若 TC = 20 + 3Q + 0.1Q²，则 MC = 3 + 0.2Q。利润最大化条件为 MR = MC。令 50 – Q = 3 + 0.2Q，得 Q = 47 / 1.2 ≈ 39.17。你还要用二阶导数确认该点是最大值而非最小值。
7. Applications of Differentiation: Elasticity | 微分应用：弹性
Price elasticity of demand measures how quantity demanded responds to a price change. It is defined as ε = (dQ/dP) × (P/Q). Using differentiation, you can calculate elasticity at any point on a demand curve. When |ε| > 1, demand is elastic; when |ε| < 1, demand is inelastic.
需求价格弹性衡量需求量对价格变化的反应。其定义为 ε = (dQ/dP) × (P/Q)。使用微分，你可以计算需求曲线上任意一点的弹性。当 |ε| > 1 时，需求有弹性；当 |ε| < 1 时，需求缺乏弹性。
For a linear demand curve Q = 100 – 2P, dQ/dP = -2. At P = 20 and Q = 60, the elasticity is ε = -2 × 20/60 = -2/3, so demand is inelastic. At P = 40 and Q = 20, ε = -2 × 40/20 = -4, so demand is elastic. This shows that elasticity changes along a straight-line demand curve.
对于线性需求曲线 Q = 100 – 2P，dQ/dP = -2。在 P = 20、Q = 60 时，弹性为 ε = -2 × 20/60 = -2/3，因此需求缺乏弹性。在 P = 40、Q = 20 时，ε = -2 × 40/20 = -4，因此需求有弹性。这表明线性需求曲线上的弹性会变化。
8. Integration and Consumer Surplus | 积分与消费者剩余
Integration is the reverse of differentiation. The definite integral of a function between two limits gives the area under the curve. In economics, consumer surplus is the area between the demand curve and the price line up to the quantity traded. It represents the extra value consumers receive beyond what they pay.
积分是微分的逆运算。函数在上下限之间的定积分给出曲线下的面积。在经济学中，消费者剩余是需求曲线与价格线之间、直到交易数量为止的区域面积。它代表消费者在支付之外获得的额外价值。
Suppose demand is P = 100 – Q² and the market price is P₀ = 64, so Q₀ = 6. Consumer surplus is the integral from 0 to 6 of (100 – Q²) dQ minus the rectangle 64 × 6. Compute: ∫₀⁶ (100 – Q²) dQ = [100Q – Q³/3]₀⁶ = 600 – 72 = 528. Then subtract 384 to get CS = 144.
假设需求为 P = 100 – Q²，市场价格为 P₀ = 64，则 Q₀ = 6。消费者剩余等于从 0 到 6 对 (100 – Q²) 的积分减去矩形 64 × 6。计算：∫₀⁶ (100 – Q²) dQ = [100Q – Q³/3]₀⁶ = 600 – 72 = 528。然后减去 384，得到 CS = 144。
9. Coordinate Geometry and Cost Curves | 坐标几何与成本曲线
Coordinate geometry deals with graphs of lines and curves. In economics, you often need to find the distance between two points, the midpoint, or the gradient of a line connecting two data points. The equation of a line through two points uses the formula y – y₁ = m(x – x₁), where m = (y₂ – y₁) / (x₂ – x₁).
坐标几何涉及直线和曲线的图形。在经济学中，你经常需要求两点间的距离、中点或连接两个数据点的直线斜率。通过两点的直线方程使用公式 y – y₁ = m(x – x₁)，其中 m = (y₂ – y₁) / (x₂ – x₁)。
For example, if a firm’s total cost increases from £200 at Q = 10 to £320 at Q = 25, the gradient of the line is (320 – 200) / (25 – 10) = 8. This represents the variable cost per unit if the function is linear. You can then write the linear cost function as TC = 8Q + 120.
例如，如果企业的总成本从 Q = 10 时的 200 英镑增加到 Q = 25 时的 320 英镑，则直线的斜率为 (320 – 200) / (25 – 10) = 8。如果函数是线性的，这代表每单位可变成本。然后你可以写出线性成本函数 TC = 8Q + 120。
10. Binomial Expansion and Probability in Economics | 二项式展开与经济概率
The binomial expansion states that (a + b)ⁿ = Σ ₖ₌₀ⁿ C(n,k) aⁿ⁻ᵏ bᵏ, where C(n,k) is the binomial coefficient. This is used in probability when there are two possible outcomes, such as a consumer choosing to buy or not to buy. The binomial distribution gives the probability of exactly k successes in n trials.
二项式定理指出 (a + b)ⁿ = Σ ₖ₌₀ⁿ C(n,k) aⁿ⁻ᵏ bᵏ，其中 C(n,k) 是二项式系数。这用于具有两种可能结果的概率问题，例如消费者选择购买或不购买。二项分布给出 n 次试验中恰好 k 次成功的概率。
Suppose n = 5 consumers each have a probability p = 0.4 of purchasing a product. The probability that exactly 3 buy is C(5,3) × 0.4³ × 0.6² = 10 × 0.064 × 0.36 = 0.2304. This binomial calculation is central to decision analysis in business and economics.
假设 n = 5 位消费者每人有 p = 0.4 的概率购买某产品。恰好 3 人购买的概率为 C(5,3) × 0.4³ × 0.6² = 10 × 0.064 × 0.36 = 0.2304。这种二项计算是企业和经济决策分析的核心。
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