Recovering Markups from Demand Derivatives

From Demand Responses to Pricing Margins

Estimated demand describes how consumers’ choices respond to prices. To use that information to learn about firms’ costs, we also need a model of firms’ pricing decisions. Under multiproduct profit maximization, a firm accounts for how changing one of its prices affects demand for every product it owns. The resulting first-order conditions link demand derivatives to markups, the differences between prices and marginal costs.

The key calculation is a system of equations, not a separate single-product formula applied independently to every product. A price change can shift demand among a firm’s own products as well as toward products owned by other firms. Which demand changes enter a firm’s pricing condition depends on which products that firm owns.

The Multiproduct Pricing Conditions

Consider one market with $J$ products. Product $j$ has price $p_j$, quantity demanded $q_j(p)$, and marginal cost $c_j$, where $p=(p_1,\ldots,p_J)$. Let $J_f$ be the set of products owned by firm $f$. Assume the firm chooses the prices of its products to maximize the sum of their profits, taking other firms’ prices as given. Its profit is

\pi_f(p)=\sum_{k\in J_f}(p_k-c_k)q_k(p).

For the first-order condition for product $j\in J_f$, differentiate this profit with respect to $p_j$. The price change affects the margin on product $j$, and it can also affect the quantities sold of every product $k$ owned by the same firm.

\frac{\partial \pi_f}{\partial p_j} = q_j(p)+\sum_{k\in J_f}(p_k-c_k)\frac{\partial q_k(p)}{\partial p_j} =0.

This condition assumes an interior price choice and differentiable demand, with marginal costs held fixed when the firm varies its price. It is a necessary condition for a local profit maximum; by itself, it does not establish that a candidate price is a maximum.

Definition: The markup on product $j$ is $m_j=p_j-c_j$. Define the ownership indicator $\Omega_{jk}$ to be 1 if products $j$ and $k$ have the same owner, and 0 otherwise. Define the ownership-adjusted quantity-derivative matrix by $\Delta_{jk}=-\Omega_{jk}\,\partial q_k/\partial p_j$. The first-order conditions for all products can then be written $\Delta m=q$, where $m$ and $q$ are the vectors of markups and quantities.

The row index $j$ in $\Delta_{jk}$ identifies the price whose first-order condition is being written. The column index $k$ identifies a product whose margin contributes to that condition. This orientation follows directly from the derivative $\partial q_k/\partial p_j$: the response of quantity $k$ to a change in price $j$. Reversing these indices without also changing the equation can produce an incorrect system.

If the market has a fixed size $M$ and product shares $s_j=q_j/M$, then $q_j=M s_j$. The same conditions can be written using shares:

\Delta^s_{jk}=-\Omega_{jk}\frac{\partial s_k}{\partial p_j}, \qquad \Delta^s m=s.

The market-size factor cancels because both quantities and their price derivatives are multiplied by $M$. A share derivative has units of share per unit of price, so multiplying it by a markup in currency per unit gives a share. A quantity derivative has units of quantity per unit of price, so its product with a markup has units of quantity. Keeping the two versions distinct is an important check on the calculation.

Recovering Markups from the System

If the matrix $\Delta$ is nonsingular, the first-order conditions imply the unique algebraic solution $m=\Delta^{-1}q$. With share derivatives, the equivalent calculation is $m=(\Delta^s)^{-1}s$. After calculating each markup, recover marginal cost from $c_j=p_j-m_j$. Nonsingularity is a condition for a unique solution to this linear system; it does not, on its own, verify the profit-maximization assumptions or establish that the resulting prices are an equilibrium.

Worked Example: Two Products Owned by One Firm

Suppose a hypothetical market has two products, both owned by the same firm. Their shares are $s_1=0.20$ and $s_2=0.30$. At the observed prices, estimated share derivatives are $\partial s_1/\partial p_1=-0.080$, $\partial s_2/\partial p_1=0.030$, $\partial s_1/\partial p_2=0.030$, and $\partial s_2/\partial p_2=-0.105$, measured in share per dollar. The observed prices are $p_1=\$8$ and $p_2=\$10$. Assume the firm chooses both prices, demand is differentiable, and the observed prices satisfy interior profit-maximizing first-order conditions under fixed marginal costs.

Because both products have the same owner, every ownership indicator is 1. With row $j$ indexing the price condition and column $k$ indexing the affected product’s margin, the share-derivative matrix is

\Delta^s = \begin{pmatrix} -\partial s_1/\partial p_1 & -\partial s_2/\partial p_1\\ -\partial s_1/\partial p_2 & -\partial s_2/\partial p_2 \end{pmatrix} = \begin{pmatrix} 0.080 & -0.030\\ -0.030 & 0.105 \end{pmatrix}.

The negative off-diagonal entries arise because the cross-price derivatives are positive: raising one product’s price increases demand for the other product, which contributes to the firm’s pricing condition. Solve $\Delta^s m=s$. The determinant is $0.080(0.105)-(-0.030)(-0.030)=0.00750$, so the matrix is nonsingular. For a two-by-two system,

m_1=\frac{0.105(0.20)-(-0.030)(0.30)}{0.00750} =\frac{0.0300}{0.00750}=\$4.00, \qquad m_2=\frac{0.080(0.30)-(-0.030)(0.20)}{0.00750} =\frac{0.0300}{0.00750}=\$4.00.

The implied marginal costs are $c_1=\$8-\$4=\$4$ per unit and $c_2=\$10-\$4=\$6$ per unit. These are costs implied by the assumed pricing conditions and the supplied demand derivatives, not direct observations of production costs.

Worked Example: Ownership Changes the Markups

Keep the shares, prices, and derivative estimates from the preceding hypothetical example, but suppose each product is owned by a different firm. Each firm now chooses only its own product’s price. The ownership matrix is the identity matrix, so cross-product derivatives do not enter either firm’s first-order condition: neither firm receives the profit generated by the other product.

The ownership-adjusted share-derivative matrix therefore contains only the own-price terms:

\Delta^s= \begin{pmatrix} 0.080 & 0\\ 0 & 0.105 \end{pmatrix}, \qquad m_1=\frac{0.20}{0.080}=\$2.50, \qquad m_2=\frac{0.30}{0.105}\approx \$2.857.

The corresponding implied marginal costs are $c_1=\$8-\$2.50=\$5.50$ and $c_2=\$10-\$2.857\approx \$7.143$ per unit. The same demand shares and derivatives thus imply different markups under different ownership structures. This comparison isolates why the ownership indicators are part of the calculation rather than optional bookkeeping.

Using Quantity Derivatives and Checking Units

The share formulation is convenient when demand estimation produces shares. If the first-order conditions are instead formed in quantities, multiply both the shares and their derivatives by the market size. The resulting quantity system has the same markup solution. The following example verifies that equivalence and shows how to convert markups into costs.

Worked Example: Quantity-Derivative System and Implied Costs

Consider a hypothetical market with $M=200{,}000$ potential consumers and two products owned by one firm. The estimated shares are $s_1=0.30$ and $s_2=0.15$. The estimated share derivatives, in share per dollar, are $\partial s_1/\partial p_1=-0.10$, $\partial s_2/\partial p_1=0.04$, $\partial s_1/\partial p_2=0.02$, and $\partial s_2/\partial p_2=-0.08$. Prices are $p_1=\$12$ and $p_2=\$8$. Assume fixed market size, differentiable demand, common ownership, interior price choices, and that the observed prices satisfy the firm’s first-order conditions.

First convert shares into quantities: $q_1=200{,}000(0.30)=60{,}000$ and $q_2=200{,}000(0.15)=30{,}000$. Multiplying each share derivative by $M$ gives the quantity-derivative matrix, with rows indexed by quantity $k$ and columns by price $j$:

\begin{pmatrix} \partial q_1/\partial p_1 & \partial q_1/\partial p_2\\ \partial q_2/\partial p_1 & \partial q_2/\partial p_2 \end{pmatrix} = \begin{pmatrix} -20{,}000 & 4{,}000\\ 8{,}000 & -16{,}000 \end{pmatrix}.

Applying $\Delta_{jk}=-\partial q_k/\partial p_j$ gives

\Delta= \begin{pmatrix} 20{,}000 & -8{,}000\\ -4{,}000 & 16{,}000 \end{pmatrix}, \qquad \Delta m= \begin{pmatrix} 60{,}000\\ 30{,}000 \end{pmatrix}.

The determinant is $20{,}000(16{,}000)-(-8{,}000)(-4{,}000)=288{,}000{,}000$. Thus,

m_1= \frac{16{,}000(60{,}000)-(-8{,}000)(30{,}000)} {288{,}000{,}000} \approx \$4.167, \qquad m_2= \frac{20{,}000(30{,}000)-(-4{,}000)(60{,}000)} {288{,}000{,}000} \approx \$2.917.

Each markup is in dollars per unit: quantity on the right-hand side divided by a quantity-per-dollar derivative matrix gives dollars. The implied marginal costs are $c_1=\$12-\$4.167\approx \$7.833$ and $c_2=\$8-\$2.917\approx \$5.083$ per unit. Solving with the share system would give the same markups because the entire quantity system is the share system multiplied by $200{,}000$.

Derivation and Interpretation

The matrix equation is simply a compact arrangement of the individual first-order conditions. For product $j$, the condition is $q_j+\sum_{k\in J_f}m_k\,\partial q_k/\partial p_j=0$. Moving the derivative terms to the other side gives $q_j=\sum_k-\Omega_{jk}(\partial q_k/\partial p_j)m_k$. By the definition of $\Delta_{jk}$, the right-hand side is $\sum_k\Delta_{jk}m_k$. Writing this equation for each product yields $\Delta m=q$. The same argument with $q_j=M s_j$ yields $\Delta^s m=s$.

This derivation also clarifies the economic role of cross-price responses. When products $j$ and $k$ have the same owner, the effect of price $p_j$ on demand for product $k$ affects the firm’s total profit and appears in the condition for $p_j$. When they have different owners, that effect does not enter firm $j$’s profit derivative. A positive cross-price response therefore appears as a negative off-diagonal element in the corresponding same-owner row of the markup matrix.

Common Mistake: Ignoring Ownership or Reversing Derivatives

A common mistake is to calculate each markup from only the product’s own-price derivative, even when one firm owns several products. That reasoning treats the firm as if it ignored changes in profit on its other products. The correction is to include every product owned by the firm in the relevant first-order condition, using the derivative of each product’s quantity with respect to the price being changed.

A related error is to place $\partial q_j/\partial p_k$ in row $j$, column $k$ while retaining the definition $\Delta_{jk}=-\Omega_{jk}\partial q_k/\partial p_j$. The indices are not interchangeable in general: demand derivative matrices need not be symmetric. Write the first-order condition first, identify which price is being differentiated, and then fill in the matrix. Finally, check that the solved markup has units of currency per unit and that $c_j=p_j-m_j$ is interpreted as an implied marginal cost under the maintained assumptions.

If the derivative matrix is singular or nearly singular, a unique or numerically stable markup calculation may not be available. Even when a unique solution exists, estimated derivatives can be imprecise, and small changes in them may materially change the recovered costs. The matrix calculation translates demand estimates into supply-side implications; its output should not be mistaken for an assumption-free measurement of costs.

Check Your Understanding

Use the definitions and first-order conditions above to answer the following questions.

In the condition for the price of product $j$, why can the derivative of another product’s quantity appear?
For a same-owner pair, which derivative belongs in matrix entry $\Delta_{jk}$: $\partial q_j/\partial p_k$ or $\partial q_k/\partial p_j$?
If market size is fixed, explain why solving with shares and share derivatives gives the same markups as solving with quantities and quantity derivatives.
Suppose two products have separate owners. Which entries of the ownership-adjusted matrix are zero, and why?
What assumption is needed to interpret $p_j-m_j$ as an implied marginal cost under this calculation?