[[2025-07-14|25-07-14-16]] responding to niaz's question on calculating elasciticity not satisfied with log-log's example (sail and nail would be in log-level i.e. multiplicative to additive; twice for one more) but it's what it is.. | Specification | β‐Interpretation | Elasticity / Semi–elasticity | Vivid Example | | :----------------- | :------------------------------------ | :---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | **1. Level–Level** | “β units of Y per unit X” |  (elasticity varies) | **Industry:** Boutique consulting<br><br>**Lifecycle:** Nail (proof‑of‑concept)<br>**Situation:** Collaborative pilot project (non–zero‑sum)<br>Every extra hour spent customizing your slides adds a fixed $5 K to your pilot contract value (β=5). Funding growth is additive, not percentage‑driven. | | **2. Level–Log** | “100·β % change in Y per unit X” |  (semi‑elastic) | **Industry:** Seed‑stage biotech startup<br><br>**Lifecycle:** Scale (Series A traction)<br><br>**Situation:** “FOMO” among VCs (mildly zero‑sum)<br>Each extra pitch‑prep hour yields ~20 % more funding (β=0.2), because early investors rush to get in on a hot round. | | **3. Log–Level** | “β units of Y per 100 % change in X” |  (semi‑elastic) | **Industry:** SaaS platform<br><br>**Lifecycle:** Scale (user acquisition)<br>**Situation:** Network effects (non–zero‑sum)<br>Doubling your developer headcount (100 % ↑ X) delivers an extra $200 K in monthly recurring revenue (β=200), as new features unlock new customers. | | **4. Log–Log** | “β % change in Y per 1 % change in X” |  (constant‐elasticity) | **Industry:** Social media ad network<br><br>**Lifecycle:** Sail (maturity)<br>**Situation:** Attention economy (zero‐sum)<br>A 1 % increase in user‑referral rate leads to a 0.5 % uptick in ad revenue (β=0.5), because attention is scarce and gains for you imply losses elsewhere. | --- **Key takeaways**: - **Level–Level** is additive: best when effort → outcome exhibits a fixed “per‐unit” return. - **Level–Log** converts increments in X into _percent_ changes in Y: ideal if each extra effort unit yields a proportional boost in outcome. - **Log–Level** converts _percent_ changes in X into additive outcome boosts: useful when “doubling inputs” gives fixed output gains. - **Log–Log** gives constant elasticity: the go‑to when you believe Y and X grow proportionally (e.g., Cobb–Douglas style). four combinations of (Y, lnY) by (X, lnX). imagine a founder preparing for pitch with investors where X is the amount of time prepared and Y is the fund raised. the most basic linear one indicates marginal funding increase is the same no matter how many hours you invested so far. the second case has1 unit increase lead to 100 beta % unit increase which is increasing marginal return ; which is possible, if there's a network and fear of missing out effect among investors, the third case where the more effort you put in the pitch itself (customizing the pitching point of each investor), the more funding you can get (which i'm not necessarily criticizing as satisfying investors with distinct taste might mean robust business model). the third case where 1% increase leads to 1 unit increase iis most likely where the marginal return of investment diminishes. the fourth case is 1% increase leading to 100 beta % increase and i don't know what the example might be. $ \begin{gathered} Y=\alpha+\beta X \\ \frac{\partial Y}{\partial X}=\beta \end{gathered} $ $ \begin{gathered} \ln Y=\alpha+\beta X \\ \frac{\partial \ln Y}{\partial X}=\frac{\partial Y}{\partial X} \times \frac{\partial \ln Y}{\partial Y}=\frac{\partial Y}{\partial X} \times \frac{1}{Y}=\beta \end{gathered} $ $ \begin{gathered} Y=\alpha+\beta \ln X \\ \frac{\partial Y}{\partial \ln X}=\frac{\partial Y}{\partial X} \times \frac{\partial X}{\partial \ln X}=\frac{\partial Y}{\partial X} \times X=\beta \end{gathered} $ $ \begin{gathered} \ln Y=\alpha+\beta \ln X \\ \frac{\partial \ln Y}{\partial \ln X}=\frac{\partial Y}{\partial X} \times \frac{\partial \ln Y}{\partial Y} \times \frac{\partial X}{\partial \ln X}=\frac{\partial Y}{\partial X} \times \frac{1}{Y} \times X=\beta \end{gathered} $ 1 unit increase in $X$ changes $Y$ by $\beta$ units 1 unit increase in $X$ changes $Y$ by $100 \beta \%$ units 1\% increase in $X$ changes $Y$ by $\beta / 100$ units 1\% increase in $X$ changes $Y$ by $\beta \%$ units