Best Response Function Calculator
Analyze firm reactions with clear linear payoff inputs. Compare outputs, profits, and Nash equilibrium instantly. Download reports and charts for lessons, homework, and revisions.
Calculator Inputs
This page uses a linear Cournot duopoly setting for best response functions.
Plotly Graph
The graph shows both reaction curves and the equilibrium point.
Example Data Table
| a | b | c₁ | c₂ | BR₁ | BR₂ | q₁* | q₂* | Price |
|---|---|---|---|---|---|---|---|---|
| 120 | 2 | 20 | 30 | 17.5 | 16.5 | 18.33 | 13.33 | 56.67 |
| 150 | 3 | 25 | 35 | 15.83 | 12.17 | 15 | 11.67 | 70 |
| 100 | 1.5 | 18 | 22 | 17.33 | 18 | 19.11 | 16.44 | 46.67 |
Formula Used
Assume inverse demand is P = a - b(q₁ + q₂).
Assume constant marginal costs are c₁ and c₂.
Firm 1 profit is π₁ = (P - c₁)q₁.
Firm 2 profit is π₂ = (P - c₂)q₂.
Firm 1 best response function is:
BR₁(q₂) = max(0, (a - c₁ - bq₂) / 2b)
Firm 2 best response function is:
BR₂(q₁) = max(0, (a - c₂ - bq₁) / 2b)
For the interior Nash equilibrium:
q₁* = (a - 2c₁ + c₂) / 3b
q₂* = (a - 2c₂ + c₁) / 3b
These formulas come from first-order profit maximization conditions.
How to Use This Calculator
- Enter the market demand intercept a.
- Enter the demand slope b.
- Enter each firm’s constant marginal cost.
- Provide a rival quantity for each best response check.
- Click Calculate Best Responses.
- Review best responses, equilibrium outputs, price, and profits.
- Inspect the reaction curves on the graph.
- Export the result as CSV or PDF when needed.
Frequently Asked Questions
1. What does a best response function show?
It shows the output level that maximizes one firm’s profit for every possible rival output level under the chosen market assumptions.
2. What model does this calculator use?
It uses a linear Cournot duopoly model with inverse demand, constant marginal costs, and quantity competition between two firms.
3. Why are some best responses zero?
If the rival produces too much or costs are too high, profit-maximizing output may become negative. The economically meaningful response is then zero.
4. What is the Nash equilibrium here?
It is the intersection of both best response curves. At that point, neither firm can improve profit by changing output alone.
5. Can I use decimal values?
Yes. The calculator accepts decimal inputs for demand, costs, and rival quantities, which helps with classroom and business-style examples.
6. What does the graph display?
The graph displays Firm 1’s reaction curve, Firm 2’s reaction curve, and the equilibrium point based on the entered parameters.
7. What assumptions should I remember?
The model assumes linear demand, simultaneous quantity choice, constant marginal costs, and no capacity limits or product differentiation.
8. When is this calculator useful?
It is useful for microeconomics, game theory, industrial organization, homework practice, lecture demos, and quick strategy comparisons.