Industrial Organization and Econometrics Coming Together
Introduction
A common empirical task in economics is estimating demand curves, specifically, how much price changes in response to a shift in quantity demanded. This sounds deceptively simple: regress price on quantity, read off the coefficient. In practice, supply and demand are determined simultaneously in equilibrium. Any naive regression conflates movements along the demand curve with shifts of the curve itself, producing a biased estimate.
The standard remedy is instrumental variables (IV): find a variable that shifts the supply curve without being caused by demand shocks, use it to isolate exogenous variation in quantity, and identify the demand slope from that variation alone. This post focuses on a plausible but wrong instrument, market concentration: the number and size distribution of firms — is a tempting candidate. It contrasts it to wages as a valid instrument, and shows how the endogeneity of market structure leads to bias in the IV estimate.
We build the argument in two steps. First, we lay out the simultaneous-equations framework and the two conditions a valid instrument must satisfy, then check market concentration against both conditions informally. Second, we formalize the argument using a Cournot oligopoly model with free entry, which shows that the equilibrium number of firms is a direct function of the very demand shocks we need the instrument to be orthogonal to. The conclusion is a central maxim of empirical industrial organization: market structure is not an exogenous feature of a market; it is an equilibrium outcome co-determined by supply and demand.
Supply, Demand, and Equilibrium
We want to estimate the inverse demand equation — the causal effect of a change in quantity on price. Define a linear simultaneous system in equilibrium (\(Q_d = Q_s = Q\)):
Demand Equation: \[P = \beta_0 + \beta_1 Q + u_d\]
- \(\beta_1 < 0\) is the parameter of interest (the slope of the inverse demand curve).
- \(u_d\) is an unobserved demand shock (e.g., a sudden change in consumer preferences or aggregate income).
Supply Equation: \[P = \gamma_0 + \gamma_1 Q + \gamma_2 Z + u_s\]
- \(\gamma_1 > 0\) is the slope of the supply curve.
- \(Z\) is some supply-shifting variable (e.g., an input cost).
- \(u_s\) is an unobserved supply shock.
The Endogeneity Problem
Because we observe equilibrium outcomes, \(Q\) is determined by the intersection of supply and demand. Setting the two equations equal and solving for \(Q\) gives the reduced-form equation for quantity:
\[\beta_0 + \beta_1 Q + u_d = \gamma_0 + \gamma_1 Q + \gamma_2 Z + u_s\]
\[Q = \frac{(\gamma_0 - \beta_0) + \gamma_2 Z + (u_s - u_d)}{\beta_1 - \gamma_1}\]
The problem is immediate: \(Q\) is mathematically a function of \(u_d\), so \(\text{Cov}(Q, u_d) \neq 0\). Running OLS of \(P\) on \(Q\) yields a biased and inconsistent estimator of \(\beta_1\) because the regressor is correlated with the error term.
The IV Estimator
To consistently estimate \(\beta_1\), we need an instrument \(Z\) that is correlated with \(Q\) but uncorrelated with \(u_d\). The IV estimator is defined as:
\[\hat{\beta}_{1,\,\text{IV}} = \frac{\text{Cov}(Z,\, P)}{\text{Cov}(Z,\, Q)}\]
Substituting the true demand equation \(P = \beta_0 + \beta_1 Q + u_d\) into the numerator:
\[\text{Cov}(Z, P) = \beta_1\, \text{Cov}(Z, Q) + \text{Cov}(Z, u_d)\]
Dividing by \(\text{Cov}(Z, Q)\) gives the probability limit of the IV estimator:
\[\operatorname{plim}(\hat{\beta}_{1,\,\text{IV}}) = \beta_1 + \frac{\text{Cov}(Z,\, u_d)}{\text{Cov}(Z,\, Q)}\]
For this to converge to the true \(\beta_1\), the second term must vanish. This establishes two necessary conditions for a valid instrument:
- Relevance: \(\text{Cov}(Z, Q) \neq 0\). The instrument must be correlated with the endogenous regressor.
- Exogeneity (Exclusion Restriction): \(\text{Cov}(Z, u_d) = 0\). The instrument must be uncorrelated with the unobserved demand shock.
Why Market Concentration Is a Bad Instrument
Suppose we propose market concentration (denoted \(C\), measured by the Herfindahl–Hirschman Index or an \(N\)-firm concentration ratio) as our instrument. Let’s evaluate it against both conditions.
Condition 1 — Relevance
Does \(\text{Cov}(C, Q) \neq 0\)? Yes. Market concentration affects the markup that firms charge over marginal cost. High concentration leads firms to restrict quantity and raise prices, so \(C\) shifts the supply curve and is unambiguously correlated with equilibrium \(Q\).
Condition 2 — Exogeneity
Does \(\text{Cov}(C, u_d) = 0\)? No. It is a foundational principle of IO that market structure is endogenous. Suppose there is a positive demand shock (\(u_d > 0\)) — a sudden fad for the product. Prices and profits rise, attracting new entrants. As firms enter, concentration falls. The demand shock therefore directly causes changes in \(C\), so \(\text{Cov}(C, u_d) \neq 0\).
With \(\text{Cov}(C, u_d) \neq 0\), the bias term in the IV formula does not vanish. Market concentration is a structurally invalid instrument for estimating the demand slope.
What Would Be a Good Instrument?
A valid instrument must shift the supply curve without being driven by demand shocks. The canonical examples are exogenous input costs — for instance, a regional wage rate or the price of a raw material. A region-wide wage increase raises firms’ marginal costs and shifts the supply curve up, tracing out the demand curve, while remaining plausibly uncorrelated with idiosyncratic demand shocks for a particular product.
Formalizing the Argument
The intuitive argument above becomes a formal proof once we embed it in a standard IO model. We use a Cournot oligopoly with free entry — the workhorse model for analyzing the endogeneity of market structure.
Short-Run Equilibrium (Fixed \(N\))
Assume \(N\) identical firms producing a homogeneous good.
- Inverse Demand: \(P = \alpha - bQ + u_d\) (we want to estimate \(-b\), where \(b > 0\)).
- Cost Structure: Each firm has constant marginal cost \(c\) and a fixed entry cost \(F\).
- Total Quantity: \(Q = \sum_{i=1}^N q_i\).
Each firm chooses \(q_i\) to maximize profit:
\[\pi_i = (P - c)\,q_i - F = (\alpha - bQ + u_d - c)\,q_i - F\]
The first-order condition is:
\[\frac{\partial \pi_i}{\partial q_i} = \alpha - bQ + u_d - c - b\,q_i = 0\]
In a symmetric equilibrium \(q_i = q\) for all \(i\), so \(Q = Nq\) and \(q_i = Q/N\). Substituting into the FOC and solving:
\[Q^* = \frac{N(\alpha - c + u_d)}{b(N+1)}, \qquad P^* = \frac{\alpha + u_d + Nc}{N+1}\]
The equilibrium per-firm operating profit (before fixed costs) is:
\[\pi_i^* = \frac{(\alpha - c + u_d)^2}{b(N+1)^2}\]
Long-Run Equilibrium: Endogenizing \(N\)
Under free entry, firms enter until profits are driven to zero. Setting \(\pi_i^* = F\):
\[\frac{(\alpha - c + u_d)^2}{b(N^*+1)^2} = F\]
Solving for the equilibrium number of firms:
\[N^* = \frac{\alpha - c + u_d}{\sqrt{bF}} - 1\]
The Formal Proof of Endogeneity
Now we can evaluate \(N^*\) (or equivalently \(1/N^*\), which maps to the HHI) as a proposed instrument for \(Q\).
Condition 1 — Relevance
From the expression for \(Q^*\), a larger \(N\) pushes output toward the competitive level. \(\text{Cov}(N^*, Q) \neq 0\).
Condition 2 — Exogeneity
Differentiating the equilibrium \(N^*\) with respect to the demand shock \(u_d\):
\[\frac{\partial N^*}{\partial u_d} = \frac{1}{\sqrt{bF}} > 0\]
A positive demand shock strictly increases the equilibrium number of firms, so \(\text{Cov}(N^*, u_d) > 0\). Plugging into the IV probability limit:
\[\operatorname{plim}(\hat{\beta}_{1,\,\text{IV}}) = -b + \underbrace{\frac{\text{Cov}(N^*,\, u_d)}{\text{Cov}(N^*,\, Q)}}_{> \,0}\]
The bias is positive, which means the IV estimate is biased toward zero — it systematically underestimates the steepness of the demand curve.
Conclusion
Market structure is not an exogenous feature of a market; it is an equilibrium outcome co-determined by supply and demand. Using market concentration as an instrument for quantity in demand estimation violates the exclusion restriction, producing a biased estimate.
The Cournot model makes this concrete: the same demand shocks that contaminate OLS also shift the number of firms, ruling out concentration as a valid instrument.
The preferred alternative — an exogenous cost shifter such as a regional wage \(w\) — passes both tests. If \(c = w\), then \(w\) appears in the numerator of \(Q^*\) (relevance) but, under competitive labor markets, is orthogonal to product-specific demand shocks (exogeneity). Instrumenting for \(Q\) with \(w\) isolates supply-side variation and correctly traces out the demand curve slope \(-b\).