Supply-Side Estimation and Conduct Assumptions

From Demand Estimates to Supply-Side Information

Estimated demand describes how market shares respond to prices. By itself, it does not reveal firms’ marginal costs. A supply-side calculation connects the two: firms’ pricing conditions relate demand derivatives and market shares to the markups they choose over marginal cost. Because those pricing conditions depend on which products a firm controls, recovering costs requires a stated assumption about ownership and conduct.

The calculation has two stages. First, use demand derivatives, observed shares, and a conduct assumption to recover product-level markups and marginal costs. Second, relate those recovered costs to observed cost shifters to estimate parameters of a cost function. Both stages are conditional on the demand estimates and the assumed pricing model.

Definition: For product $j$, let $p_j$ be its price, $s_j$ its market share, and $c_j$ its marginal cost. Its markup is $m_j=p_j-c_j$. Write $D_{jk}=\partial s_j/\partial p_k$ for the effect of product $k$'s price on product $j$'s share. The ownership indicator $O_{kj}$ equals 1 when products $k$ and $j$ are controlled by the same firm and 0 otherwise.

Bertrand Pricing Conditions and Cost Recovery

Consider a market with a fixed set of differentiated products. Under differentiated-products Bertrand pricing, each firm chooses prices for its products, taking the other firms’ prices as given. Assume firms maximize the sum of profits from their own products, marginal costs do not vary with the firm’s own prices, shares are differentiable in prices, and the observed prices are an interior optimum. These assumptions give necessary first-order conditions; they do not, on their own, establish that the prices are a global or stable equilibrium.

For a firm owning a set of products $J_f$, write its profit as $\sum_{j\in J_f}(p_j-c_j)s_j$. Differentiating with respect to the price $p_k$ of one of its products gives

\frac{\partial \pi_f}{\partial p_k} =s_k+\sum_{j\in J_f}(p_j-c_j)\frac{\partial s_j}{\partial p_k}=0, \qquad k\in J_f.

The first term is the gain from raising product $k$'s price on its existing sales. The remaining terms capture the effect of that price change on the profits of the firm’s other products. A firm pricing products jointly accounts for these effects; a firm that does not own those other products does not.

Collect the first-order conditions across products. Let $m$ be the vector of markups and $s$ the vector of shares. Define the conduct matrix $\Delta$ by $\Delta_{kj}=-O_{kj}D_{jk}$. The negative sign is convenient because own-price share derivatives are usually negative. Under the assumptions above, the first-order conditions are $s=\Delta m$. If $\Delta$ is nonsingular, the markups and marginal costs implied by the model are as follows.

m=\Delta^{-1}s, \qquad c=p-m.

This is a recovery formula, not a demand estimation method: it takes shares and derivatives as given. It also makes the role of the ownership assumption explicit. Changing which products a firm controls changes entries of $\Delta$, and therefore can change every recovered markup and cost.

Worked Example: Recovering Costs When One Firm Owns Two Products

Suppose two products have observed shares $s_1=0.20$ and $s_2=0.15$, and prices $p_1=\$20$ and $p_2=\$18$. Assume one firm jointly prices both products. For this hypothetical market, let the estimated demand derivatives, in share units per dollar, be

D= \begin{pmatrix} -0.04 & 0.01\\ 0.01 & -0.03 \end{pmatrix}.

The diagonal entries are own-price effects, and the off-diagonal entries are cross-price effects. Under joint ownership, $O$ has all entries equal to 1, so

\Delta= \begin{pmatrix} 0.04 & -0.01\\ -0.01 & 0.03 \end{pmatrix}.

The determinant is $0.04(0.03)-(-0.01)(-0.01)=0.0011$, which is nonzero. Solving $\Delta m=s$ gives

m_1=\frac{0.03(0.20)+0.01(0.15)}{0.0011} =\frac{0.0075}{0.0011} \approx \$6.818, \qquad m_2=\frac{0.01(0.20)+0.04(0.15)}{0.0011} =\frac{0.0080}{0.0011} \approx \$7.273.

Subtracting each markup from its price gives $c_1=20-6.818\approx \$13.18$ and $c_2=18-7.273\approx \$10.73$. These are the marginal costs implied by the stated demand derivatives and joint-pricing assumption, not directly observed costs.

Conduct Assumptions Change the Inference

If the two products in the example are instead priced by separate firms, each firm’s first-order condition includes only its own product. The off-diagonal entries of the ownership matrix are then zero. The same observed shares and demand derivatives imply different markups, even though no demand estimate has changed.

Worked Example: Recalculating Costs Under Separate Ownership

Keep the hypothetical shares, prices, and demand derivatives from the previous calculation, but assume each product is owned and priced by a different firm. The conduct matrix is now diagonal:

\Delta= \begin{pmatrix} 0.04 & 0\\ 0 & 0.03 \end{pmatrix}.

The first-order conditions give $0.04m_1=0.20$ and $0.03m_2=0.15$. Thus

m_1=\frac{0.20}{0.04}=\$5, \qquad m_2=\frac{0.15}{0.03}=\$5.

The implied marginal costs are $c_1=\$20-\$5=\$15$ and $c_2=\$18-\$5=\$13$. Relative to joint ownership, separate ownership implies smaller markups and higher marginal costs here. Under joint pricing, a price change affects the other product’s share, and the firm accounts for that effect in its profit calculation. The comparison is conditional on the same demand inputs and the two alternative conduct assumptions.

More generally, the same framework can encode conduct assumptions other than the observed ownership structure, but those assumptions must be defined rather than inferred from the derivative matrix alone. For example, an analyst may specify weights on cross-product effects to represent a particular reduced-form conduct model. Such weights change the matrix used in the pricing equations; their economic interpretation depends on the model that justifies them. A weighted equation should not automatically be described as a firm’s standard profit-maximizing first-order condition.

Estimating Cost Parameters

Recovered marginal costs can be related to product characteristics or input prices. For a linear cost specification, let $w_j$ be a vector of observed cost shifters and suppose

c_j=w_j'\gamma+\xi_j,

where $\gamma$ is the vector of supply-side parameters and $\xi_j$ is an unobserved cost component. If the cost shifters are exogenous, meaning $E[w_j\xi_j]=0$, the parameters can be estimated by least squares using the recovered costs as the dependent variable. If some cost shifters are endogenous, use instruments $z_j$ satisfying the required exclusion and orthogonality conditions, in particular $E[z_j\xi_j]=0$. With more valid moments than parameters, generalized method of moments chooses $\gamma$ to make the sample moments $n^{-1}\sum_j z_j(\hat c_j-w_j'\gamma)$ small according to a specified weighting matrix.

The cost recovery and cost regression are linked. The recovered costs $\hat c_j$ depend on estimated demand derivatives, shares, and the conduct assumption. Consequently, an estimate of $\gamma$ is conditional on those inputs. In applications, uncertainty calculations should account for estimation in both stages, for example by resampling the full procedure rather than treating recovered costs as error-free observations.

Worked Example: Fitting a Linear Cost Function

Suppose four hypothetical recovered marginal costs, in dollars per unit, are $11, 13, 16,$ and $18$. Let $x_j$ be an observed, exogenous input-cost index, with values $0,1,2,3$, and fit $c_j=\gamma_0+\gamma_1x_j+\xi_j$ by least squares. The sample means are $\bar{x}=1.5$ and $\bar{c}=14.5$. The slope estimate is

\hat\gamma_1 = \frac{\sum_j(x_j-\bar{x})(c_j-\bar{c})} {\sum_j(x_j-\bar{x})^2} = \frac{12}{5} =2.4.

For the numerator, the four products of deviations are $5.25,0.75,0.75,$ and $5.25$, summing to 12. The denominator is $2.25+0.25+0.25+2.25=5$. The intercept is $\hat\gamma_0=\bar{c}-\hat\gamma_1\bar{x}=14.5-2.4(1.5)=10.9$. The fitted equation is therefore $\hat c_j=10.9+2.4x_j$, with costs in dollars per unit. Its fitted values are $10.9,13.3,15.7,$ and $18.1$, leaving residuals $0.1,-0.3,0.3,$ and $-0.1$ dollars per unit.

This calculation illustrates cost-parameter estimation conditional on the recovered costs. Its least-squares interpretation requires the input index to be exogenous in the stated cost equation. If that condition is not credible, least squares need not identify the cost parameters; valid instruments and the corresponding moment conditions are needed instead.

Interpretation, Limits, and a Common Pitfall

The pricing conditions are useful because they translate estimated substitution patterns into economically interpretable implications about costs. They also provide a way to compare supply-side implications across ownership or conduct scenarios. But a recovered cost is not an independent measurement. It is the value that makes the assumed pricing condition hold for the observed shares, prices, and estimated demand derivatives.

Several checks matter in practice. The conduct matrix must be nonsingular for the stated inversion to work. Recovered markups and costs should be assessed for economic plausibility, and the assumed demand derivatives should have sensible signs and magnitudes. A negative or implausibly large implied cost can signal a problem with the demand estimates, the conduct assumption, or the maintained pricing model; it is not automatically evidence that firms have unusual costs. Moreover, satisfying the first-order conditions is not sufficient to prove that observed prices maximize profit.

Pitfall: treating conduct as a harmless detail. The wrong reasoning is that demand estimates alone determine markups, so ownership can be added afterward without affecting the result. The correction is that ownership determines which cross-price effects enter the pricing equations, and therefore changes the inferred markups and costs.
Pitfall: calling recovered costs observed data. They are constructed from demand and conduct inputs. The correction is to describe them as model-implied costs and to account for their generated nature when estimating cost parameters and uncertainty.
Pitfall: interpreting a first-order condition as proof of equilibrium. The condition is necessary for an interior optimum under the stated assumptions, but does not establish a global optimum or rule out other equilibria. Those claims require additional analysis.

Key takeaway: Supply-side estimates combine observed prices and shares with estimated demand derivatives and an explicit pricing-conduct assumption. Marginal costs and cost-function parameters are conditional on all of these ingredients.

Check Your Understanding

Use the definitions and assumptions in this tutorial to answer the following.

Why does a firm that jointly prices two products include a cross-price derivative in each relevant first-order condition?
In the two-product example, what changed between the joint-ownership and separate-ownership calculations, and what stayed fixed?
What does nonsingularity of $\Delta$ permit the analyst to calculate?
State an exogeneity condition that would justify least-squares estimation of a linear marginal-cost equation.
Why should recovered marginal costs not be treated as directly observed, error-free data?