📚 Mathematical Approaches to the Provision of Social Welfare | 社会福利提供的数学方法
Governments and public bodies use mathematics to design, fund, and evaluate social welfare programmes. From predicting demand for housing benefit to assessing the impact of a job-training scheme, A-level mathematical tools help policymakers make evidence-based decisions. This article connects core topics from the Edexcel A-level Mathematics specification to the provision of social welfare.
政府和公共机构利用数学来设计、资助和评估社会福利项目。从预测住房福利需求到评估职业培训计划的效果，A-level 数学工具帮助政策制定者做出基于证据的决策。本文将 Edexcel A-level 数学大纲中的核心主题与社会福利提供联系起来。
1. The Role of Mathematics in Welfare Provision | 数学在福利提供中的作用
Welfare provision involves allocating limited public funds among competing needs such as unemployment support, disability benefits, pensions, and social housing. Mathematical modelling allows analysts to forecast demand, estimate costs, and compare policy options under uncertainty. Key techniques include probability distributions, expectation, hypothesis testing, regression, optimisation, and time series analysis.
社会福利提供涉及在失业支持、残疾津贴、养老金和社会住房等相互竞争的需求之间分配有限的公共资金。数学建模使分析人员能够预测需求、估算成本并在不确定条件下比较政策方案。关键技术包括概率分布、期望、假设检验、回归、优化和时间序列分析。
2. Probability Distributions for Welfare Demand | 福利需求的概率分布
Daily welfare claims often follow a Poisson distribution when events occur independently at a constant average rate. For example, if a job centre receives an average of λ = 12 new benefit claims per day, the probability of receiving exactly x claims is:
每日福利申请通常服从泊松分布，前提是事件独立发生且平均发生率恒定。例如，若某就业中心每天平均收到 λ = 12 份新的福利申请，则恰好收到 x 份申请的概率为：
P(X = x) = e⁻λ λˣ / x!
For a fixed group of n potential claimants, each with probability p of applying, a binomial model B(n, p) is more appropriate. If n = 200 and p = 0.08, then the number of applicants X has probability P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ.
对于固定数量的 n 个潜在申请人，每人申请概率为 p，则二项模型 B(n, p) 更合适。若 n = 200，p = 0.08，则申请人数 X 的概率为 P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ。
These distributions help welfare agencies plan staffing levels and budget reserves for peak periods.
这些分布帮助福利机构规划人员配置和高峰期的预算储备。
3. Expected Value and Cost-Benefit Analysis | 期望值与成本效益分析
The expected value E(X) of a random variable gives the long-run average outcome. For a discrete variable, it is calculated as:
随机变量的期望值 E(X) 给出长期平均结果。对于离散变量，其计算公式为：
E(X) = Σ x P(X = x)
Suppose a welfare-to-work programme costs £800 per participant and has a 0.6 probability of leading to employment, which saves the state £1500 in benefits. The expected net saving per participant is 0.6 × 1500 − 800 = 900 − 800 = £100. A positive expected value indicates the programme is cost-effective on average.
假设一个从福利到工作的项目每位参与者花费 800 英镑，有 0.6 的概率实现就业，从而为政府节省 1500 英镑福利支出。每位参与者的预期净节省为 0.6 × 1500 − 800 = 900 − 800 = 100 英镑。正的期望值表明该项目平均而言具有成本效益。
Expected value is essential for comparing different welfare interventions when outcomes are uncertain.
当结果不确定时，期望值对于比较不同的福利干预措施至关重要。
4. Hypothesis Testing for Policy Evaluation | 政策评估中的假设检验
Hypothesis testing is used to determine whether observed data provide enough evidence to reject a null hypothesis. For example, a government claims that the unemployment rate in a region is p = 0.08. After a new training scheme, a sample of n = 500 gives 30 unemployed people, so the sample proportion is p̂ = 30/500 = 0.06. We test H₀: p = 0.08 against H₁: p < 0.08 at the 5% significance level.
假设检验用于确定观测数据是否提供足够证据来拒绝原假设。例如，某政府声称一个地区的失业率为 p = 0.08。实施新培训计划后，一个 n = 500 的样本中有 30 人失业，因此样本比例为 p̂ = 30/500 = 0.06。我们在 5% 显著性水平下检验 H₀: p = 0.08 对 H₁: p < 0.08。
The test statistic under the normal approximation is:
正态近似下的检验统计量为：
z = (p̂ − p) / √(p(1 − p)/n)
Substituting values gives z = (0.06 − 0.08) / √(0.08 × 0.92 / 500) ≈ −1.65. The critical value for a one-tailed test at 5% is −1.645, so we reject H₀, suggesting the training scheme has significantly reduced unemployment.
代入数值得到 z = (0.06 − 0.08) / √(0.08 × 0.92 / 500) ≈ −1.65。5% 单尾检验的临界值为 −1.645，因此我们拒绝 H₀，表明培训计划显著降低了失业率。
5. Regression Analysis of Welfare Spending | 福利支出的回归分析
Regression analysis models the relationship between a dependent variable and one or more independent variables. For welfare spending y and unemployment rate x, a simple linear regression has the form y = a + bx. The gradient b is estimated by:
回归分析对因变量和一个或多个自变量之间的关系进行建模。对于福利支出 y 和失业率 x，简单线性回归的形式为 y = a + bx。斜率 b 的估计公式为：
b = Sxy / Sxx
where Sxy = Σ(x − x̄)(y − ȳ) and Sxx = Σ(x − x̄)². The intercept a is given by a = ȳ − b x̄. If b = 0.4, then each percentage point increase in unemployment is associated with an additional £0.4 billion in welfare spending.
其中 Sxy = Σ(x − x̄)(y − ȳ)，Sxx = Σ(x − x̄)²。截距 a 由 a = ȳ − b x̄ 给出。如果 b = 0.4，那么失业率每上升一个百分点，福利支出将增加 4 亿英镑。
Regression helps policymakers identify key drivers of welfare costs and forecast future spending under different economic scenarios.
回归分析帮助政策制定者确定福利成本的关键驱动因素，并在不同经济情景下预测未来支出。
6. Optimisation in Resource Allocation | 资源分配中的优化
Linear programming is used to allocate a fixed budget between welfare programmes to maximise social benefit. Let x be the amount (in £ millions) spent on housing benefit and y the amount spent on child benefit. The objective function might be to maximise Z = 3x + 2y, where the coefficients represent the number of households supported per £ million.
线性规划用于在福利项目之间分配固定预算，以最大化社会效益。设 x 为住房福利支出（百万英镑），y 为儿童福利支出。目标函数可能是最大化 Z = 3x + 2y，其中系数表示每百万英镑支持的家庭数量。
Constraints include the total budget and non-negativity:
约束条件包括总预算和非负性：
x + y ≤ 100, x ≥ 0, y ≥ 0, 2x + y ≤ 160
Plotting the feasible region and evaluating Z at each vertex gives the optimal allocation. This technique ensures welfare funds are distributed efficiently rather than arbitrarily.
绘制可行域并评估每个顶点的 Z 值可得到最优分配。这一技术确保福利资金得到高效分配，而非任意分配。
7. Exponential Growth and Ageing Populations | 指数增长与人口老龄化
Pension spending grows as the elderly population increases. If the number of pensioners P grows at a constant annual rate r, after t years the population is:
养老金支出随着老年人口的增长而增加。如果养老金领取者人数 P 以恒定年增长率 r 增长，则 t 年后的人口为：
P = P₀(1 + r)ᵗ
For example, if P₀ = 2 million, r = 0.03, and t = 10, then P = 2 × (1.03)¹⁰ ≈ 2.69 million. Because pension costs are roughly proportional to the number of pensioners, this exponential model alerts governments to future funding gaps.
例如，若 P₀ = 200 万，r = 0.03，t = 10，则 P = 2 × (1.03)¹⁰ ≈ 269 万。由于养老金成本大致与养老金领取者人数成正比，这一指数模型提醒政府注意未来的资金缺口。
Logarithms can be used to find the doubling time: t = ln 2 / ln(1 + r).
对数可用于求翻倍时间：t = ln 2 / ln(1 + r)。
8. Sampling and Confidence Intervals in Welfare Surveys | 福利调查中的抽样与置信区间
Surveys of welfare recipients produce sample statistics that estimate population parameters. The 95% confidence interval for a population mean μ when the population standard deviation σ is known is:
福利领取者调查产生的样本统计量用于估计总体参数。当总体标准差 σ 已知时，总体均值 μ 的 95% 置信区间为：
x̄ ± 1.96 × σ / √n
If a sample of n = 400 welfare recipients has mean weekly benefit x̄ = £120 and σ = £20, the confidence interval is 120 ± 1.96 × 20 / √400 = 120 ± 1.96, giving (118.04, 121.96). This interval means we are 95% confident that the true mean lies within this range.
如果一个 n = 400 的福利领取者样本的平均每周福利 x̄ = 120 英镑，σ = 20 英镑，则置信区间为 120 ± 1.96 × 20 / √400 = 120 ± 1.96，得到 (118.04, 121.96)。该区间意味着我们有 95% 的把握认为真实均值位于此范围内。
Confidence intervals allow policymakers to quantify uncertainty in survey-based estimates of welfare needs.
置信区间使政策制定者能够量化基于调查的福利需求估算中的不确定性。
9. Time Series Analysis of Welfare Claims | 福利申领的时间序列分析
Monthly welfare claim numbers form a time series with trend, seasonal variation, and random fluctuations. A 4-point moving average smooths the data to reveal the underlying trend. For consecutive monthly values y₁, y₂, y₃, y₄, the first centred moving average is (y₁ + y₂ + y₃ + y₄) / 4.
月度福利申请人数形成一个包含趋势、季节变动和随机波动的时间序列。4 点移动平均可平滑数据以揭示潜在趋势。对于连续月度值 y₁, y₂, y₃, y₄，第一个居中移动平均为 (y₁ + y₂ + y₃ + y₄) / 4。
Seasonal variation is calculated by subtracting the trend from the actual values. Forecasting future claims helps welfare agencies plan budgets and staffing. For example, if the trend increases by 2% per month and the seasonal index for December is 1.15, next December’s prediction is the trend value multiplied by 1.15.
季节变动的计算方法是实际值减去趋势值。预测未来申请人数有助于福利机构规划预算和人员配置。例如，若趋势每月增长 2%，12 月的季节指数为 1.15，则下一年 12 月的预测值为趋势值乘以 1.15。
10. Decision Trees and Risk Analysis | 决策树与风险分析
Decision trees evaluate welfare policy choices under uncertainty by calculating expected monetary values (EMV). Suppose a government must choose between a universal credit system and a targeted benefit system. The universal system has a 0.7 probability of saving £200 million and a 0.3 probability of costing £50 million more. Its EMV is 0.7 × 200 + 0.3 × (−50) = 140 − 15 = £125 million.
决策树通过计算预期货币价值 (EMV) 来评估不确定条件下的福利政策选择。假设政府必须在全民信用体系和定向福利体系之间选择。全民体系有 0.7 的概率节省 2 亿英镑，有 0.3 的概率多花费 5000 万英镑。其 EMV 为 0.7 × 200 + 0.3 × (−50) = 140 − 15 = 1.25 亿英镑。
The targeted system has a 0.8 probability of saving £150 million and a 0.2 probability of costing £20 million. Its EMV is 0.8 × 150 + 0.2 × (−20) = 120 − 4 = £116 million. Comparing EMVs shows the universal system has the higher expected saving, so it is preferred unless other risks are considered.
定向体系有 0.8 的概率节省 1.5 亿英镑，有 0.2 的概率多花费 2000 万英镑。其 EMV 为 0.8 × 150 + 0.2 × (−20) = 120 − 4 = 1.16 亿英镑。比较 EMV 可知全民体系具有更高的预期节省，因此除非考虑其他风险，否则应优先选择全民体系。
Decision trees combine probability, expectation, and optimisation to guide welfare policy.
决策树结合了概率、期望和优化来指导福利政策。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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