Cross-Price Elasticity Calculator
Using the midpoint method, compute how the demand for X responds to a price change of Y — substitutes or complements.
輸入資料
計算結果
重點速覽：Cross-price elasticity Exy (midpoint) = (ΔQx / avg Qx) / (ΔPy / avg Py). Exy > 0 → substitutes (Y dearer, X demand up); Exy < 0 → complements (Y dearer, X demand down); ≈0 → unrelated. Larger |Exy| = stronger link. WARNING: arc elasticity over the range; sign matters more than size; other factors may move demand. Education, not advice.
計算公式
Exy = (商品X需求量變動 ÷ 平均X需求量) ÷ (商品Y價格變動 ÷ 平均Y價格)。
$$E_{xy} = \dfrac{\Delta Q_x / \bar{Q_x}}{\Delta P_y / \bar{P_y}}$$使用說明
- Enter X's quantity before and after Y's price change.
- Enter Y's price before and after.
- View Exy — positive = substitutes, negative = complements.
交叉價格彈性計算範例 (中點法)
| 情境 | X 需求量變化 | Y 價格變化 | Exy | 關係判定 |
|---|---|---|---|---|
| 替代品 | 500 → 600 | 20 → 25 | +0.82 | Y 漲價，X 需求增 (替代品) |
| 互補品 | 500 → 400 | 20 → 25 | −1.00 | Y 漲價，X 需求減 (互補品) |
| 無關 | 500 → 500 | 20 → 25 | 0.00 | Y 漲價，X 需求不變 (無關) |
Exy = (X 需求量變動比例) ÷ (Y 價格變動比例)，皆以中點法計算。正號為替代品、負號為互補品。
理財情境案例
案例一：判斷兩商品為替代品
假設商品 Y (例如某品牌咖啡) 價格從 20 元漲到 25 元，導致商品 X (另一品牌咖啡) 的需求量從 500 杯升到 600 杯。
以中點法計算：X 需求量變動比例 = (600 − 500) ÷ ((600+500)/2) = 100 ÷ 550 ≈ 0.1818;Y 價格變動比例 = (25 − 20) ÷ ((25+20)/2) = 5 ÷ 22.5 ≈ 0.2222。交叉價格彈性 Exy = 0.1818 ÷ 0.2222 ≈ +0.82。
Exy 為正，代表 Y 漲價時 X 的需求『增加』— 消費者從變貴的 Y 轉向 X，這正是『替代品』的特徵。彈性 +0.82 顯示兩者替代關係中等強度。對定價與競爭分析而言，這意味著競品調價會明顯影響自家銷量。
案例二：判斷兩商品為互補品
假設商品 Y (例如印表機) 價格從 20 漲到 25，導致商品 X (例如墨水匣) 的需求量從 500 降到 400。
中點法：X 需求變動比例 = (400 − 500) ÷ 450 ≈ −0.2222;Y 價格變動比例 = 5 ÷ 22.5 ≈ 0.2222。Exy = −0.2222 ÷ 0.2222 = −1.00。
Exy 為負，代表 Y 漲價時 X 的需求『減少』— 因為兩者要搭配使用 (印表機貴了、買的人少，連帶墨水匣也賣不動)，這是『互補品』的特徵。企業做搭售定價、產品組合或評估供應鏈連動時，交叉價格彈性是關鍵依據;而 Exy 接近 0 則表示兩商品需求互不相干。
常見問題
What does the sign of cross elasticity mean?
Positive = substitutes: Y dearer pushes consumers to X (coffee & tea). Negative = complements: Y dearer cuts X demand too (cars & petrol). Near zero = unrelated. For judging the relationship, the sign matters more than the number.
What does the magnitude tell us?
Larger absolute value = stronger linkage. A strong substitute (similar brands) has high Exy — a rival's small price cut steals many of your customers; weak substitutes/complements sit near zero. Firms use it to gauge competitor pricing impact and bundle pricing.
What is it used for?
Competition analysis (how much a rival's cut steals), bundle pricing (use a low-margin lead item to drive high-margin accessories), and antitrust/market definition (high cross elasticity = same relevant market). It reveals demand linkages between goods.
Why the midpoint method instead of start-value percentages?
The midpoint method gives direction-symmetric, unique results. Traditional 'change / start' percentages are asymmetric: 20→25 is +25%, but 25→20 is -20% — same move, different numbers, so elasticity differs by direction. Midpoint uses the average of start and end as denominator, so 20↔25 both give 5/22.5 ≈ 22.22%, symmetric and unique. This suits arc elasticity over a range. This tool applies midpoint to both quantity and price, so swapping initial/final yields the same absolute value.
How does it differ from own-price elasticity?
Both measure responsiveness to price, but own-price elasticity looks at a good's OWN price effect on its OWN demand (usually negative, for pricing/ revenue); cross-price elasticity looks at good Y's price effect on good X's demand, revealing the relationship between two goods. The sign of cross elasticity is economically meaningful (positive = substitutes, negative = complements), whereas own-price mainly uses the absolute value. Own answers 'if this good's price rises, does it still sell'; cross answers 'if another good's price rises, how does this good's sales move — friend or foe'. Use both for competition and portfolio decisions.
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。