Deadweight Loss Calculator
From the quantity change and price wedge caused by a tax or control, compute the deadweight loss: DWL = 0.5 x quantity change x price change (welfare triangle).
輸入資料
計算結果
重點速覽：Deadweight loss (DWL) is the welfare that consumers and producers lose, captured by no one, when a distortion (tax, control, monopoly) pushes volume below the optimal. Triangle approximation: DWL = ½ x quantity change x price wedge. Example: volume falls 10, wedge HK$4 → 0.5 x 10 x 4 = 20. The more elastic supply/demand, the larger the DWL for the same wedge. WARNING: linear-curve approximation; education only, not advice.
計算公式
無謂損失 = 0.5 × 數量變化 × 價格楔子 (三角形面積)。
供需越有彈性，同樣扭曲造成的無謂損失越大。
$$DWL = \tfrac{1}{2} \times \Delta Q \times \Delta P$$使用說明
- Enter the reduction in trade volume.
- Enter the price wedge between buyer and seller.
- View the deadweight loss.
無謂損失計算範例 (福利損失三角形)
| 情境 | 交易量變化 ΔQ | 價格楔子 ΔP | 無謂損失 DWL |
|---|---|---|---|
| 小幅課稅 | 8 | 3 | 12.00 |
| 中度課稅 | 10 | 4 | 20.00 |
| 高度扭曲 | 20 | 5 | 50.00 |
DWL = ½ × ΔQ × ΔP。交易量減少越多、價格楔子越大 (供需越有彈性)，無謂損失越大。
理財情境案例
案例一：計算課稅造成的無謂損失
假設政府對某商品課稅，使買賣雙方之間出現 4 元的價格楔子 (買家付的價與賣家收的價相差 4 元)，且交易量因此比無稅時減少了 10 單位。
套用公式：無謂損失 = ½ × 交易量變化 × 價格楔子 = ½ × 10 × 4 = 20 元。
這 20 元代表：因課稅而『沒有發生的交易』所損失的社會福利 — 這些交易原本能為消費者與生產者共同創造價值，卻因稅收扭曲而消失，且這部分價值誰也沒有得到 (連政府稅收都不包含它，稅收是另計的矩形部分)。這就是為何經濟學家稱之為『無謂』損失。
案例二：彈性如何放大無謂損失
同樣打入 5 元的價格楔子，若市場供需『很有彈性』(對價格敏感)，交易量會大幅萎縮，例如減少 20 單位：DWL = ½ × 20 × 5 = 50 元;若供需『缺乏彈性』(不敏感)，交易量只減少 8 單位：DWL = ½ × 8 × 5 = 20 元。
同樣的稅、同樣的價格楔子，前者的無謂損失是後者的 2.5 倍 — 差別完全來自彈性。
這帶出一個重要的稅務設計原則：對『需求或供給缺乏彈性』的商品課稅 (如菸酒、必需品)，因交易量萎縮較少，造成的無謂損失較小，稅收效率較高 (這也是拉姆齊法則的直覺);反之對彈性大的商品課重稅，會嚴重壓縮交易、製造大量無謂損失。當然，實務上還要兼顧公平與外部性等考量。
常見問題
What is deadweight loss and why does it occur?
DWL (welfare net loss) measures the efficiency loss when a market deviates from the free-competitive optimum — the value that disappears and is captured by no one. In a well-functioning market, quantity and price reach an efficient equilibrium where all trades with 'buyer willingness ≥ seller cost' occur and total surplus is maximised. A distortion (tax, price cap/floor, subsidy, monopoly) shifts volume away from that optimum (usually down), so some mutually beneficial trades no longer happen. Their lost surplus — not consumer benefit, not producer profit, not government tax — is pure loss. That is the 'deadweight'. With a tax: the wedge between buyer-paid and seller-received prices cuts volume; the government collects a rectangle (tax revenue), but the triangle of lost trades is DWL — net loss nobody receives.
How is DWL calculated, and why a triangle?
On the standard graph, DWL is a triangle: area = 0.5 x base x height. This tool's form: DWL = 0.5 x quantity change x price wedge. 'Quantity change' is the base (how much volume fell from equilibrium); 'price wedge' is the height (the buyer-seller price gap, i.e. the tax). Why a triangle, not a rectangle? Near the original equilibrium, the lost trades have buyer-seller gaps near zero (tiny loss); farther away, the gap grows. Summing losses from zero to the max wedge gives a triangle. Example: tax cuts volume by 10 and creates a HK$4 wedge → DWL = 0.5 x 10 x 4 = 20. This is a linear-curve approximation; real curves may bend, but it gives a good estimate.
What is DWL's policy meaning, and how does elasticity affect it?
DWL is the key tool for a policy's 'efficiency cost' — a tax raises revenue but also distorts and cuts volume, creating DWL; good tax design minimises DWL for needed revenue (hence taxing inelastic goods like tobacco/fuel has lower efficiency loss). Same logic applies to price controls, subsidies, tariffs, monopoly. Elasticity rule: the MORE elastic supply/demand (more price-sensitive), the LARGER the DWL for the same distortion; the less elastic, the smaller. Intuition: sensitive buyers/sellers slash volume on a price change, killing many mutually beneficial trades (long base); insensitive ones barely reduce volume (small base). This explains why governments tax inelastic goods.
How is DWL different from tax revenue?
A tax creates two different effects. One is tax revenue (what government gets) — a rectangle = remaining volume x per-unit tax; this is a transfer (consumers/producers lose, government gains, society's total unchanged), NOT DWL. The other is DWL — a triangle = the trades that vanished (buyer value exceeded seller cost but didn't happen); nobody gets it (not even government, since the trade didn't occur). So revenue is 'transferred', DWL is 'evaporated'. Evaluating a tax's efficiency hinges on its DWL — smaller is better. Pair with the consumer-surplus calculator.
Besides taxes, what else causes DWL?
Any force pushing volume away from the free equilibrium creates DWL; tax is the classic case. Others: (1) price controls — a binding price ceiling (e.g. rent control) cuts supply below optimum; a price floor (e.g. minimum wage above equilibrium, price supports) creates surplus; both shrink volume; (2) quotas/quantity limits — cap trade below optimum; (3) monopoly/market power — a monopolist restricts output and raises price above the competitive optimum, profiting (transfer) but losing the compressed trades (DWL) — the economic rationale for antitrust; (4) subsidies — push volume ABOVE optimum, the extra high-cost/low-value trades are also DWL (opposite direction); (5) uncorrected externalities — negative (pollution) or positive (vaccines, education) externalities leave volume off the social optimum, creating DWL. DWL is the universal yardstick for the efficiency cost of market interventions and failures.
相關工具
參考資料
內容審核：香港計算器財經團隊。計算邏輯與公式參考香港金融管理局（HKMA）及投資者及理財教育委員會（IFEC）之個人理財計算指引，結果僅供參考，實際以相關機構公佈為準。