Why Demand Estimates Need Diagnosis
A demand model can converge numerically and still produce estimates that are fragile or economically implausible. A small change in the assumed market size may substantially change the outside-option share; a derivative calculated in the wrong units may imply enormous elasticities; and a small change in the instrument set may move estimated price sensitivity enough to alter markups and counterfactual prices. These are not peripheral issues: demand estimates often feed directly into supply-side calculations and policy or business counterfactuals.
Diagnosis is a collection of checks, not a single pass-or-fail test. Numerical stability, economic coherence, and sensitivity to defensible modeling choices answer different questions. A successful optimization or share inversion addresses only part of the evidence needed to judge an estimate.
Start With Shares and the Outside Option
For market \(t\), let \(s_{jt}\) be the observed share of inside product \(j\), and let \(s_{0t}\) be the share of consumers choosing the outside option. If the market contains a defined set of \(M_t\) potential consumers, inside shares should be consistent with the product counts and market-size definition, and the outside share is \(s_{0t}=1-\sum_j s_{jt}\). Shares must be nonnegative, and the inside shares must sum to no more than one. A zero or negative outside share is not a numerical nuisance to repair silently: it indicates that the market-size definition, counts, or choice-set construction needs review.
In the simple logit model, mean utility is \(\delta_{jt}=\log(s_{jt}/s_{0t})\). This identity makes the outside-option assumption consequential: changing the market size changes \(s_{0t}\), and therefore changes the inverted mean utilities. In a random-coefficient model, inversion is numerical rather than this direct log ratio, but the observed shares and the definition of the outside option remain essential inputs.
Worked Example: Market-Size Sensitivity in Share Inversion
Suppose a hypothetical market records 300 purchases of product 1 and 200 of product 2. First use a market size of \(M=1{,}000\) potential consumers, treating everyone who does not buy either inside product as choosing the outside option.
The observed shares are \(s_1=300/1{,}000=0.3000\), \(s_2=200/1{,}000=0.2000\), and \(s_0=1-0.3000-0.2000=0.5000\). Under simple logit, the inversions are
Now suppose \(M=1{,}200\) is also a defensible count of potential consumers, with the same purchase counts. Then \(s_1=0.2500\), \(s_2=1/6\), and \(s_0=700/1{,}200=7/12\). Using these unrounded shares gives
The market-size change shifts both inverted utilities even though the purchase counts are unchanged. This is a sensitivity result, not evidence that either market size is correct: the choice depends on the economically appropriate population at risk and the availability of the products to that population. In a random-coefficient application, repeat the inversion and estimation under each defensible definition, then compare the estimates and implications.
Numerical Stability Is Necessary, Not Sufficient
For a random-coefficient share inversion, a common numerical update compares observed shares with model-predicted shares at the current mean utilities. If the iteration converges, check more than whether the software reports success. Inspect the largest absolute share discrepancy at the solution, whether the stopping tolerance is tight enough for the intended use, and whether reasonable starting values or simulation draws lead to materially different results. A small share discrepancy establishes that the numerical inversion matches its target shares to that tolerance; it does not establish that the market definition, instruments, utility specification, or economic interpretation is sound.
Share discrepancies should be considered alongside their scale. A discrepancy of \(0.001\) is one tenth of a percentage point; whether that is acceptable depends on the application and the accuracy required for later calculations. If shares are extremely small, report relative discrepancies as well as absolute ones, since a small absolute error can be large relative to the target share. Products with zero observed shares need specific attention: a log-share inversion is undefined for zero, and adding an arbitrary small constant changes the data being fitted.
Check Elasticities Against Derivatives and Units
Let \(D_{jk,t}=\partial s_{jt}/\partial p_{kt}\) denote the derivative of product \(j\)'s share with respect to product \(k\)'s price in market \(t\). The price elasticity of \(s_{jt}\) with respect to \(p_{kt}\) is \(\varepsilon_{jk,t}=(p_{kt}/s_{jt})D_{jk,t}\). For an ordinary downward-sloping own-demand curve, the own-price derivative and own-price elasticity should be negative. Cross-price elasticities can have different signs depending on substitution patterns, so they should be interpreted in light of the model and products rather than assigned a universal sign.
Units matter. If price is measured in dollars, a derivative is share change per dollar; if it is measured in cents, the numerical derivative changes by a factor of 100. The elasticity does not change when price and derivative are converted consistently. A derivative calculated per dollar but interpreted as per cent, or a derivative matrix with product indices reversed, can create apparently extreme elasticities.
Worked Example: Checking an Own-Price Elasticity
In a hypothetical market, a product has price \(p_j=10\) dollars and predicted share \(s_j=0.2000\). The model's predicted share is \(0.2008\) when its price is \(9.99\) dollars and \(0.1992\) when its price is \(10.01\) dollars, holding other inputs fixed.
A centered finite-difference check gives the approximate derivative
The implied own-price elasticity is
Thus, locally, a 1% increase in price corresponds to an approximately 4% decrease in share, with other prices held fixed. As a diagnostic, compare the derivative used in the elasticity calculation with finite differences at several step sizes. A result such as \(-40\) would not follow from the shares above and this price: it would signal a discrepancy to investigate, such as a price-unit error, an indexing error, or a numerical derivative problem. The finite-difference calculation is a check, not proof that the demand model is correctly specified.
Markups: Check the First-Order Conditions and Matrix Orientation
Markup calculations require a supply-side conduct assumption as well as demand derivatives. Under multiproduct Bertrand pricing, suppose firms choose interior prices, demand is differentiable, and marginal costs are represented by \(c_t\). Let \(\Omega_t\) be the ownership matrix, with \(\Omega_{jk,t}=1\) when products \(j\) and \(k\) have the same owner and \(0\) otherwise. With \(D_{jk,t}=\partial s_{jt}/\partial p_{kt}\), the first-order conditions in vector form are
Here \(\odot\) denotes element-by-element multiplication. If \((\Omega_t\odot D_t)^T\) is invertible, the implied markup vector is
The transpose follows from the price-\(j\) first-order condition: it uses the effects \(\partial s_k/\partial p_j=D_{kj,t}\) for products \(k\) owned by the same firm. A wrong orientation can produce a different markup vector whenever the derivative matrix is not symmetric.
Worked Example: A Two-Product Markup Orientation Check
Consider two hypothetical products owned by the same firm. At the estimated prices, let \(s=(0.2000,\,0.1000)^T\), and suppose the derivative matrix, with rows indexing shares and columns indexing prices, is
Because both products have the same owner, \(\Omega\) is a matrix of ones, so the first-order conditions use \(D^T\). Its determinant is \((-0.08)(-0.05)-(0.01)(0.02)=0.0038\), which is nonzero. Solving \(D^T m=-s\), where \(m=p-c\), gives
The implied markups are \(2.8947\) and \(3.1579\) in the price units used for the derivatives. Substitution checks the conditions: \(D^Tm=(-0.2000,\,-0.1000)^T=-s\), so \(s+D^Tm=0\), up to rounding. For prices \(p=(10,\,8)^T\) dollars, the implied costs are \(c=p-m=(7.1053,\,4.8421)^T\) dollars. If the matrix were not invertible or produced economically implausible costs or markups, that would call for investigation rather than automatic acceptance of the inversion.
For a single-product firm, the corresponding condition reduces to \(s_j+D_{jj}(p_j-c_j)=0\), so the markup is \(-s_j/D_{jj}\), provided \(D_{jj}\ne0\). This simpler calculation can help identify a unit or sign error before examining a multiproduct system. In either case, recovered markups are conditional on the demand derivatives and conduct assumptions; they are not direct observations of firms' costs.
Instrument and Specification Sensitivity
Compare demand estimates across instrument sets only when the alternatives are defensible and their assumptions are stated. For each set, assess relevance using scale-invariant first-stage evidence and its sampling uncertainty, and assess validity using the economic argument for instrument exogeneity and exclusion. A large difference between estimates is a warning about sensitivity; it does not, by itself, show which set is correct. Nor does a large first-stage statistic establish instrument validity.
The size of a raw covariance cannot diagnose instrument strength. In a one-regressor IV calculation after residualizing other controls, the ratio \(\operatorname{Cov}(z,y)/\operatorname{Cov}(z,x)\) is unchanged if \(z\) is multiplied by any nonzero constant \(a\), because both covariances are multiplied by \(a\). Therefore, a small raw denominator can reflect the instrument's units or scale, not weak relevance. Use scale-invariant diagnostics and their uncertainty instead.
Worked Example: Interpreting Instrument Sensitivity
Suppose a hypothetical random-coefficient demand analysis estimates a price coefficient of \(-0.40\) using one defensible instrument set and \(-0.12\) using another. In addition, scale-invariant first-stage diagnostics show substantially less predictive content for the second set, with greater uncertainty in its first-stage relationship. The numerical difference is consequential: the implied price response, elasticities, and subsequent markups may change.
The appropriate conclusion is that the demand estimate is sensitive to instrument choice and that the second set provides weaker relevance evidence in this application. It is not correct to conclude that the second instrument is weak merely because its raw covariance is smaller, or that the first estimate is valid merely because it is more stable. Review the instrument construction and exclusion argument, report the first-stage evidence and uncertainty, and show how central demand and counterfactual results change across credible specifications.
A Practical Diagnostic Sequence
A useful audit keeps numerical, data-definition, and economic checks distinct. The following sequence makes it easier to locate the source of a surprising result without treating one favorable diagnostic as a general endorsement.
Verify product counts, availability, potential consumers, and the outside share. Recalculate shares from the stated market size and confirm that they sum consistently.
Record share-fit tolerances and test numerical stability. Check selected demand derivatives with finite differences and confirm the price units and product-index convention.
Review own-price signs, elasticity magnitudes, recovered markups, implied costs, and the conduct assumptions used to obtain them. Trace an implausible quantity back to the data and calculations that determine it.
Repeat key calculations under defensible alternatives for market size, outside-option definition, instrument set, and numerical choices. Report which conclusions persist and which depend on a particular choice.
Common Pitfalls and Why the Checks Matter
- “The inversion converged, so the demand estimates are credible.” Convergence only says that an iterative calculation met its numerical criterion. It does not validate the market definition, instruments, model, or economic implications.
- “A very large own-price elasticity must be wrong.” It may be an error, but it could also reflect the estimated model and data. Verify units, derivative calculations, share magnitudes, and local finite differences before judging plausibility.
- “The raw covariance is smaller, so that instrument is weaker.” Covariance depends on scale. Examine scale-invariant relevance evidence and sampling uncertainty; evaluate validity separately.
- “The markup formula is just a matrix inverse.” Its orientation follows from the stated derivative indices and the first-order conditions. With \(D_{jk}=\partial s_j/\partial p_k\), the vector condition contains \(D^T\), not \(D\), for a single owner of all included products.
- “An alternative market size is just a harmless rescaling.” It changes the outside share and therefore the inversion. Treat a different market-size definition as a substantive specification choice and explain why it is defensible.
These checks matter because a fragile demand estimate can propagate into conclusions about substitution, recovered costs, consumer surplus, and equilibrium prices. Good practice is to document the source of each important input, show the checks most relevant to the application, and distinguish a numerical failure from an economic or identification concern. Robustness does not guarantee correctness, but unexplained instability is evidence that the conclusions require qualification.
Check Your Understanding
Answer each question by stating the relevant calculation or diagnostic and what it can—and cannot—establish.
- Why does changing a defensible market-size assumption change simple-logit mean-utility inversion when purchase counts stay fixed?
- If a random-coefficient share inversion converges to a small share discrepancy, which important economic questions remain unanswered?
- A price derivative is reported per cent, but prices are recorded in dollars. What should you verify before interpreting the resulting elasticity?
- With \(D_{jk}=\partial s_j/\partial p_k\), why does the multiproduct Bertrand first-order condition use \((\Omega\odot D)^T\)?
- Why does a smaller raw covariance between an instrument and price not, by itself, establish weaker instrument relevance?