Conceptual map
- IV-04.01Individual demand derivatives
- IV-04.02Aggregate share Jacobians
- IV-04.03Ownership matrices
- IV-04.04Markup equations
- IV-04.05Price counterfactuals
1. Ownership selects demand derivatives in the pricing system
Baseline shares enter pricing through their full price Jacobian. Matching shares while changing derivatives changes markups and counterfactual equilibrium.
Let \(s(p)\in \mathbb{R}^{J}\) be differentiable shares, \(c\) marginal costs, and \(\Omega\) the ownership matrix with \(\Omega _{jk}=1\) when products \(j\) and \(k\) share a profit-maximizing owner. Write \(\Delta _{jk}=\partial s_{j}/\partial p_{k}\).
At an interior Bertrand–Nash price vector with constant marginal costs,
Proof. Differentiate each firm's sum of product profits \((p_{j}-c_{j})s_{j}\) with respect to each price it controls. Ownership deletes derivatives of products outside that firm. Stack the conditions. ∎
If the transposed ownership-weighted Jacobian is nonsingular, markups equal \(-[(\Omega \odot \Delta )^{\top}]^{-1}s\). Computation should solve the linear system rather than form an inverse.
2. Two products show the ownership effect
Take simple logit with price coefficient \(\alpha=1\), outside share 0.5, and inside shares \(s_{1}=0.3\), \(s_{2}=0.2\). The price Jacobian is
Under separate ownership, cross derivatives are removed and markups are 0.3/0.21=10/\(7\approx1.42857\) and 0.2/0.16=1.25. Under joint ownership, solving the full system gives markup two for each product. The common joint markup equals 1/(\(\alpha s_{0}\)) in this homogeneous-logit example.
3. Counterfactual prices require a new equilibrium root
Changing ownership, costs, products, or demand parameters changes both shares and \(\Delta\) at the counterfactual price. Holding the baseline Jacobian fixed gives a local approximation. A structural price counterfactual solves (1) jointly with demand at the new price and checks the returned root, second-order incentives or global profit comparisons, and economic price bounds.
Multiple roots can arise in richer demand systems. Initialization and solver convergence select a computed root, while uniqueness needs a separate economic or mathematical condition.
4. Singular derivatives and corner products
Two demand systems can match \(s\) and have different \(\Delta\). Equation (1) then produces different markups under identical ownership and costs.
A zero-share product creates a boundary at which its percentage elasticities and markup equation can be unstable. A singular ownership-weighted Jacobian leaves the linear markup system unidentified or inconsistent. Capacity constraints, price caps, and product exit replace selected equalities in (1) with complementarity conditions.
5. Implementation, exercises, and sources
Calculate \(\Delta\) analytically and by finite differences at identical prices, verify its sign pattern and adding-up implications, construct \(\Omega\) from firm identifiers, solve for markups with a factorization, and report condition numbers and FOC residuals. Re-solve demand and supply for every counterfactual.
Download the volume verification script →Exercises
- Reproduce (2) and both markup vectors.
- Change outside share while preserving the ratio of inside shares and compute joint markups.
- Add a price cap and write complementarity conditions for the constrained product.
Partial solutions
1. Logit derivatives are \(-s_{j}(1-s_{j})\) on the diagonal and \(s_{j}s_{k}\) off diagonal. 2. Under joint ownership the common markup is the reciprocal of \(\alpha\) times the outside share. 3. Add a nonnegative multiplier to the FOC, cap slack, and their zero product.
- Steven Berry, James Levinsohn, and Ariel Pakes (1995), “Automobile Prices in Market Equilibrium,” Econometrica 63, 841–890.Demand derivatives and multi-product supply.
- Aviv Nevo (2000), “A Practitioner's Guide to Estimation of Random-Coefficients Logit Models of Demand,” Journal of Economics & Management Strategy 9, 513–548.Demand and markup implementation.
- Simon P. Anderson, André de Palma, and Jacques-François Thisse, Discrete Choice Theory of Product Differentiation.Pricing with discrete-choice demand.
6. Audit checkpoint
State demand, price units, derivative orientation, ownership, costs, interior or constrained pricing conditions, Jacobian rank and condition, baseline and counterfactual roots, initialization, FOC residual, and root multiplicity checks.
7. Scope boundary
The chapter covers static differentiated-product Bertrand pricing with constant marginal cost. Dynamic pricing, capacity games, and entry require separate models.