Profit Function Definition
Profit is a residual — the signed remainder after all variable costs are subtracted from revenue. This residual structure is the mechanical source of elasticity amplification.
Profit = Revenue - COGS - Ad Spend - Platform Fees - Other Variable Costs
Elasticity(x) = (ΔProfit / Profit) ÷ (Δx / x)
Where x is any input variable (price, COGS unit cost, ad spend). An elasticity ratio of 3.0 means a 1% change in that input produces a 3% change in profit — in the same or opposite direction depending on the variable’s sign in the profit function.
Residual Amplification Mechanics
Thin margins create structural leverage on both sides. The amplification factor is inversely proportional to the profit margin rate.
Price Elasticity ≈ Revenue / Profit = 1 / Margin Rate
COGS Elasticity ≈ COGS / Profit
Ad Spend Elasticity ≈ Ad Spend / Profit
At a 20% net margin on $100 revenue (profit = $20):
| Input | Spend / Revenue | Elasticity Ratio |
|---|---|---|
| Price | $100 revenue base | ~5.0× |
| COGS | $30 | ~1.5× |
| Ad Spend | $15 | ~0.75× |
| Platform Fees | $13 | ~0.65× |
A 10% price increase on this structure adds $10 revenue with zero cost change — a 50% profit improvement. The same 10% applied to COGS ($3 increase) removes 15% of profit. The asymmetry is structural, not contextual.
Below 15% net margin, price elasticity exceeds 6.5×. At this threshold, a 5% involuntary price reduction (promotion, coupon, creator commission escalation) eliminates over 30% of profit. Model this range explicitly before running discount campaigns.
Lever Hierarchy
Three variable inputs dominate the profit sensitivity surface. Fixed costs (SaaS subscriptions, tooling, headcount) have zero per-order elasticity and are excluded from this analysis.
1. Price (highest elasticity)
- Every incremental revenue dollar above current price flows to profit with no corresponding variable cost increase
- Elasticity = 1 / margin rate (pure leverage)
- Direction: positive — price up → profit up, nonlinearly
2. COGS (second-order)
- Unit cost changes affect every order permanently; no volume dependency
- Elasticity = COGS / Profit at current volumes
- Direction: inverse — COGS up → profit down
- Compounding effect: a renegotiated supplier rate applies retroactively to all future volume
3. Ad Spend (most controllable, lowest standalone elasticity)
- Elasticity = Ad Spend / Profit at current volumes
- Direction: inverse for spend increases; but spend reductions carry a revenue dependency via ROAS
- Net elasticity on ad spend reduction = (Ad Spend / Profit) − (Ad Spend × (1/ROAS) × (1/Margin Rate))
- At ROAS < margin-rate reciprocal, cutting spend improves profit net of revenue loss
Multi-Variable Drift Compounding
Individual lever moves are additive in the profit function but interact multiplicatively when evaluated as percentage changes. Do not sum independent elasticity estimates when multiple variables shift simultaneously.
New Profit = (Revenue × (1 + ΔP)) - (COGS × (1 + ΔC)) - (Ad Spend × (1 + ΔA)) - Fixed Fees
Combined Elasticity ≠ E(price) + E(COGS) + E(ad spend)
Example — three simultaneous moves on the $100 / $42 profit base:
| Variable | Delta | Isolated Profit Impact |
|---|---|---|
| Price | −3% | −$3.00 → profit $39 |
| COGS | +5% | −$1.50 → profit $40.50 |
| Ad Spend | +10% | −$1.50 → profit $40.50 |
| Combined | — | −$6.00 → profit $36 |
The combined impact ($6.00) exceeds the sum of isolated impacts ($6.00 happens to be additive here but diverges with larger deltas). At ±10%+ moves across multiple variables, model the full profit equation — do not add elasticity ratios.
Fixed vs Variable Cost Elasticity
Only variable costs participate in per-order profit elasticity. Misclassifying cost types inflates or deflates sensitivity estimates.
| Cost Type | Variable? | Elasticity Participation |
|---|---|---|
| COGS / unit cost | Yes | Full — scales with every order |
| TikTok platform fee | Yes | Full — percentage of GMV |
| Creator commissions | Yes | Full — percentage of GMV |
| Shipping / fulfillment | Yes | Full — per-order |
| Ad spend (CPM/CPC) | Yes | Full — scales with campaign budget |
| SaaS subscriptions | No | Zero — fixed regardless of volume |
| Warehouse fixed lease | No | Zero — fixed regardless of volume |
Fixed costs affect absolute profit level but not the sensitivity ratios. Elasticity analysis operates on the variable cost stack only. At scale, fixed cost amortization reduces effective margin dilution — but this is a volume effect, not an elasticity effect.
Interactive Sensitivity Simulator
Base case: $100 revenue, $30 COGS, $15 ad spend, $13 platform and shipping fees. Starting profit: $42 (42% margin).
Try it
Sensitivity Thresholds by Margin Band
| Net Margin | Price Elasticity | COGS Elasticity | Break-even Price Drop |
|---|---|---|---|
| 30% | 3.3× | ~1.0× | −30% |
| 20% | 5.0× | ~1.5× | −20% |
| 15% | 6.7× | ~2.0× | −15% |
| 10% | 10.0× | ~3.0× | −10% |
At 10% margin, price elasticity reaches 10×. A 5% across-the-board discount event eliminates 50% of profit before accounting for COGS or fee escalation. Margin bands below 15% require explicit elasticity approval before any promotional pricing decision.
Operational Decision Rules
- Rank optimization targets by elasticity ratio before allocating effort — price renegotiation outperforms ad spend optimization at all margin levels
- Model all simultaneous variable shifts as a single profit equation, not summed elasticities
- Treat creator commission rate changes as COGS-equivalent for elasticity purposes — they are percentage-of-GMV variable costs
- Ad spend elasticity is net of ROAS impact; gross elasticity overstates the profit benefit of cuts
- Recompute elasticity ratios whenever cost structure changes by more than 5 percentage points — the ratios are margin-dependent, not static