Pricing with Substitute or Complementary Products
Compare optimal pricing under collusion vs. competition, with or without a common retailer. Enter demand parameters for two substitute or complementary products (or estimate them from data via regression), and the tool computes equilibrium prices, quantities, and profits across four scenarios: (1) single-owner direct, (2) single-owner with retailer, (3) Bertrand competition direct, and (4) Bertrand competition with retailer. Also includes demand prediction and own-price and cross-price elasticity calculators.
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1 Model Setup
Notation
Given prices P₁ and P₂, calculate the predicted quantities Q₁ and Q₂ using the demand functions above.
Calculate own-price and cross-price elasticities using the demand functions above. Own-price elasticity measures how quantity responds to changes in its own price. Cross-price elasticity measures how quantity of one product responds to price changes of the other.
Four Pricing Scenarios
How Are Optimal Prices Calculated?
All four scenarios are solved analytically (closed-form solutions) — no numerical optimization or simulation is needed. Because the demand functions are linear, every first-order condition (FOC) produces a linear equation. The resulting systems are solved exactly using Cramer's rule.
The monopolist maximizes joint profit π = (P₁−C₁)Q₁ + (P₂−C₂)Q₂ over both prices simultaneously. The two FOCs (∂π/∂P₁ = 0, ∂π/∂P₂ = 0) form a 2×2 linear system, solved by Cramer's rule.
Solved by backward induction. Stage 2: the retailer maximizes its profit over retail prices P₁, P₂ given wholesale prices — yielding P(W) and Q(W) as linear functions of W. Stage 1: the manufacturer substitutes these into joint profit and optimizes over W₁, W₂. Both stages are 2×2 linear systems.
Each firm maximizes its own profit. The FOCs give best response functions: P₁ = f(P₂) and P₂ = g(P₁). Solving simultaneously yields the Bertrand-Nash equilibrium prices.
Backward induction again. The retailer's response P(W), Q(W) is derived first (same as S2). Then each manufacturer sets its own wholesale price to maximize its individual profit, anticipating the retailer's response — a Nash equilibrium in wholesale prices.
Scenario Comparison
Scenario 1: Single Owner, Direct Sales (Benchmark)
One manufacturer owns both products and sells directly to consumers
Scenario 2: Single Owner, Through Retailer
One manufacturer owns both products but sells through a single retailer who sets consumer prices
Scenario 3: Competing Brands, Direct Sales
Two separate manufacturers each sell their own product directly to consumers
Scenario 4: Competing Brands, Through Retailer
Two competing manufacturers both sell through the same single retailer
Key Insights
Comparison of profits across all four scenarios. Shows how channel structure and competition affect profit distribution.
Full comparison of outcomes across all scenarios.
| Metric | 1. Single Owner Direct | 2. Single Owner+Retailer | 3. Competitors Direct | 4. Competitors+Retailer |
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