Supply Side and Markups

Summary

BLP can be paired with a supply side derived from firms’ Bertrand-Nash pricing first-order conditions. Given demand derivatives and an ownership matrix ${\mathcal{H}}_t$ (which products each firm owns), the model recovers markups ${\eta }_{jt}$ and hence marginal costs $c_{jt}=p_{jt}-{\eta }_{jt}$. Parameterizing marginal cost adds supply moments $E[{\omega }_{jt}{Z}_{jt}^S]=0$ that sharpen estimation. Solving the inverse problem — counterfactual equilibrium prices under a new ownership structure ${\mathcal{H}}_t^{\ast }$ (e.g. a merger) — uses the Morrow-Skerlos $\zeta$-markup fixed point, which is faster and more reliable than naive iteration or Newton’s method.

Overview

Including supply gives a model of marginal costs, enabling counterfactuals (merger price effects, pass-through) and adding identifying information. Its cost: the supply moments may be misspecified if the researcher does not know the functional form of marginal cost $f_{MC}(\cdot )$ or firm conduct ${\mathcal{H}}_t$. Conlon-Gortmaker find that a correctly specified supply side substantially improves finite-sample performance (bias all but eliminated with optimal instruments), but an incorrectly specified one is worse than no supply side because it biases $\alpha$. Validity is testable via overidentification (see GMM Estimation and Instruments for Price Endogeneity).

Main Content

Bertrand-Nash pricing FOCs and the markup ^foc

Multiproduct firm $f$ controlling products $J_{ft}$ maximizes ${\max }_{p_{jt}}{\sum }_{j\in J_{ft}}{s}_{jt}({\mathbf{p}}_t)(p_{jt}-c_{jt})$. The FOCs in matrix form for market $t$:

$${\mathbf{s}}_t({\mathbf{p}}_t)={\Delta }_t({\mathbf{p}}_t)\cdot ({\mathbf{p}}_t-{\mathbf{c}}_t),\underset{{\eta }_t({\mathbf{p}}_t,{\mathbf{s}}_t,{\theta }_2)}{\underbrace{{\Delta }_t({\mathbf{p}}_t{)}^{-1}{\mathbf{s}}_t({\mathbf{p}}_t)}}={\mathbf{p}}_t-{\mathbf{c}}_t.$$

The multiproduct Bertrand markup ${\eta }_t$ depends on the $J_t\times J_t$ intra-firm demand-derivative matrix

$${\Delta }_t({\mathbf{p}}_t)\equiv -{\mathcal{H}}_t\odot \frac{\partial {\mathbf{s}}_t}{\partial {\mathbf{p}}_t}({\mathbf{p}}_t),$$

the Hadamard (element-wise) product of the demand-derivative matrix ($\partial s_{jt}/\partial p_{kt}$) and the ownership matrix ${\mathcal{H}}_t$ (entry $(j,k)=1$ if the same firm produces $j$ and $k$, else 0). Alternative conduct (single-product, monopoly, Cournot, partial collusion) corresponds to different ${\mathcal{H}}_t$; Miller-Weinberg (2017) / Backus et al. (2020) even estimate a conduct parameter ${\mathcal{H}}_t(\kappa )$.

Marginal cost parameterization and supply moments ^supply-moments

Recover $c_{jt}=p_{jt}-{\eta }_{jt}({\theta }_2)$, then parameterize marginal cost:

$$f_{MC}(p_{jt}-{\eta }_{jt}({\theta }_2))=f_{MC}(c_{jt})=x_{jt}{\gamma }_1+w_{jt}{\gamma }_2+{\omega }_{jt},$$

with $f_{MC}(\cdot )$ commonly the identity (or $\log$ to keep costs positive). The cost depends on product characteristics $x_{jt}$ and cost shifters $w_{jt}$ excluded from demand. This yields the supply moment condition $E[{\omega }_{jt}{Z}_{jt}^S]=0$, stacked with the demand moments in the GMM objective.

Solving counterfactual pricing equilibria ^equilibrium

Counterfactuals (mergers, cost changes) require solving the $J_t\times J_t$ nonlinear system for new equilibrium prices, replacing ${\mathcal{H}}_t$ with a post-merger ${\mathcal{H}}_t^{\ast }$:

$${\mathbf{p}}_t={\mathbf{c}}_t+{\eta }_t({\mathbf{p}}_t,{\mathcal{H}}_t^{\ast }).$$

Naive iteration on this is not a contraction and can cycle (fails 1-5% of the time, Armstrong 2016). Newton’s method requires the demand Hessian and is costly. The preferred method (Morrow & Skerlos 2011) splits $\partial {\mathbf{s}}_t/\partial {\mathbf{p}}_t={\Lambda }_t-{\Gamma }_t$ (${\Lambda }_t$ diagonal, ${\Gamma }_t$ dense, ${\alpha }_i=\partial u_{ijt}/\partial p_{jt}$ the marginal disutility of price) and iterates the $\zeta$-markup fixed point

$${\mathbf{p}}_t\leftarrow {\mathbf{c}}_t+{\zeta }_t({\mathbf{p}}_t),{\zeta }_t({\mathbf{p}}_t)={\Lambda }_t^{-1}[{\mathcal{H}}_t^{\ast }\odot {\Gamma }_t]({\mathbf{p}}_t-{\mathbf{c}}_t)-{\Lambda }_t^{-1}{\mathbf{s}}_t({\mathbf{p}}_t),$$

which is 3-12x faster than Newton-type approaches and reliably finds an equilibrium. Fast, reliable equilibrium solving is also what makes the feasible optimal instruments computable.

Examples

Merger simulation (the canonical BLP use): estimate demand + supply on pre-merger data to recover $c_{jt}$ and demand derivatives. Construct ${\mathcal{H}}_t^{\ast }$ reflecting the merged firm now owning both parties’ products. Solve the $\zeta$-markup fixed point for post-merger prices ${\mathbf{p}}_t^{\ast }$; the difference ${\mathbf{p}}_t^{\ast }-{\mathbf{p}}_t$ is the predicted unilateral price effect. Because the merged firm internalizes substitution between formerly-rival products, markups on close substitutes rise.

Connections

• GMM Estimation and Instruments for Price Endogeneity — supply moments $E[{\omega }_{jt}{Z}_{jt}^S]=0$ and cross-equation restrictions enter the joint GMM objective.
• Random Coefficients Logit Model — demand derivatives $\partial s_{jt}/\partial p_{kt}$ that build ${\Delta }_t$ come from the RCL shares.
• The BLP Contraction Mapping — share inversion supplies the ${\mathbf{\delta }}_t$ used to evaluate demand derivatives.
• Numerical Integration and Optimization in PyBLP — equilibrium prices and optimal instruments reuse the numerical machinery.

See Also

• BLP Demand Estimation - Overview
• Market Share Models
• _Index