Calculator inputs
The page stacks vertically, while the calculator controls use three columns on large screens, two on medium screens, and one on mobile screens.
Example data table
This example uses the page defaults and shows how the economic inputs shape the optimization result.
| Input item | Example value | Meaning |
|---|---|---|
| Demand intercept (a) | 150 | Starting selling price before quantity expansion. |
| Demand slope (b) | 0.8 | Price declines by 0.8 per added unit. |
| Fixed cost | 1200 | Constant cost paid even at zero output. |
| Unit variable cost | 35 | Direct cost per produced unit. |
| Quadratic cost coefficient | 0.25 | Rising cost pressure from scaling output. |
| Capacity limit | 120 | Upper production or sales boundary. |
| Tax rate | 20% | Applied to positive operating profit. |
Formula used
Demand function:
P(Q) = a - bQ
Revenue function:
R(Q) = P(Q) × Q = (a - bQ)Q = aQ - bQ²
Cost function:
C(Q) = F + vQ + cQ²
Profit function:
π(Q) = R(Q) - C(Q) = -(b + c)Q² + (a - v)Q - F
Marginal conditions:
MR(Q) = a - 2bQ
MC(Q) = v + 2cQ
The calculator checks the feasible interval, computes the vertex of the quadratic profit curve, tests boundary points, then returns the quantity with the highest profit.
How to use this calculator
- Enter the demand intercept and demand slope for your pricing model.
- Fill in fixed cost, unit cost, and quadratic cost values.
- Set the capacity limit so output stays realistic.
- Enter the tax rate to estimate after-tax profit.
- Choose the start, end, and step values for the schedule table.
- Press Optimize Profit to show results above the form.
- Review the summary cards, break-even values, and Plotly graph.
- Use the CSV and PDF buttons to export the output.
FAQs
1) What does this calculator optimize?
It maximizes modeled profit by testing the quadratic profit function inside the feasible quantity range. The result balances falling prices against rising cost pressure and capacity.
2) Why do I need a demand slope?
The demand slope shows how price changes as quantity increases. A steeper slope means price falls faster, which usually lowers the quantity that maximizes profit.
3) What is the quadratic cost coefficient for?
It models growing production difficulty, congestion, overtime, waste, or inefficiency. Larger values raise marginal cost faster and often reduce the best output level.
4) Does the tool handle taxes?
Yes. It estimates after-tax profit by applying the entered tax rate only when profit is positive. Losses remain unchanged in the after-tax view.
5) What are break-even points?
Break-even points are quantities where modeled profit equals zero. They can help you see where operations switch from loss to profit and back again.
6) Why can the optimal quantity sit at a boundary?
If capacity is tight, demand becomes zero early, or the profit curve does not peak inside the interval, the best result can occur at zero output or the upper feasible limit.
7) What does the graph show?
The Plotly chart compares revenue, cost, and profit across quantity levels. It makes the intersection patterns, peak profit zone, and feasible operating region easier to understand.
8) Can I use this for real pricing decisions?
Yes, for structured analysis. Still, you should validate the demand and cost assumptions with market data, operational limits, and sensitivity testing before acting on the result.