Estimating Demand for a New Product

Abstract: This paper develops an approach for estimating demand for a new product. Taking willingness to pay (WTP) as primitive, it establishes a general and yet analytically simple demand function, and proposes an estimation procedure that consistently recovers the underlying demand function from the WTP data. Monte Carlo simulations find the estimation procedure works well in identifying the demand function. This approach complements existing methods of demand estimation, and can be applied both within and outside academia, for example in teaching economics, for a business to launch new products, and for policymakers to conduct non-market valuation.

• The paper introduces a demand estimation methodology based on WTP, inferring demand functions directly from consumer data rather than relying on market equilibrium.
• It employs nonparametric techniques and simulation studies across various WTP distributions to validate estimator consistency and robustness to misreporting.
• The empirical illustration on coffee demand demonstrates the method's flexibility and accuracy, with a semi-log model achieving an R² of 0.812.

Estimating Demand for a New Product: An Expert Analysis

Theoretical Framework

The paper "Estimating Demand for a New Product" (2606.15748) advances a demand estimation methodology where consumer willingness to pay (WTP) is treated as the primitive. Rather than assuming a specific functional form for demand or relying on market equilibrium outcomes (which are unobservable for genuinely new products), the approach infers demand functions directly from WTP distributions under minimal behavioral assumptions:

1. Consumers purchase a single unit (relaxable).
2. Heterogeneous WTP is distributed as $F_{W|Z}$, with $Z$ capturing relevant covariates (e.g., income, substitute prices).
3. Purchase occurs if and only if $p < W$.

The core theoretical contribution is a general demand function,

$q(p, z) = [1 - F_{W|Z=z}(p)] \Pr(Z = z) L$

(discrete $Z$), or, for continuous $Z$,

$q(p, z) = [1 - F_{W|Z}(p)] f_Z(z) L,$

where $L$ is market size. The framework naturally encompasses canonical demand forms as special cases (e.g., linear demand via uniformly distributed WTP, log-linear under Pareto, semi-log via exponential).

Importantly, when the WTP and $Z$ distributions are unknown, a Taylor expansion about a reference point provides a principled route to parametric approximations, supporting classical and flexible functional forms for empirical implementation. The method generically enforces the law of demand regardless of distributional choices.

Estimation Procedure

The empirical procedure follows five main steps:

1. Collect WTP and $Z$ data: Implementable via stated or revealed preference mechanisms. The paper emphasizes practical survey-based mechanisms, such as Vickrey auctions (truth-inducing) or Contingent Valuation Method (CVM), with explicit guidance on controlling for biases (e.g., anchor effects).
2. Estimate empirical CDF of WTP: Nonparametric estimator, convergent by Glivenko-Cantelli.
3. Estimate PDF or mass function of $Z$0: Standard nonparametric estimators.
4. Calculate demand at each observed price: $Z$1.
5. Estimate demand function parameters: Use OLS (or GMM if endogenous covariates suspected), leveraging the asymptotic orthogonality between regressors and errors inherent to the non-market nature of the WTP data.

The estimator is consistent; the construction ensures that potential endogeneity that plagues market data is neutralized.

Robustness to Misreporting

The estimator’s robustness to misreported WTP is rigorously established. When WTP errors are uniform and independent of price, the estimator recovers an accurate coefficient for price, even under arbitrary levels of misreporting—critical in survey settings where hypothetical bias and information effects persist. For non-uniform errors, robustness holds when WTP's density is approximately flat near the relevant price or misreporting is minor. The central theoretical result is that the impact of reporting errors on price coefficients is second-order in the error magnitude for sufficiently regular WTP distributions—in practice, bias tends to be negligible for "well-behaved" reporting scenarios.

Simulation Studies

Monte Carlo simulations systematically assess the estimator under diverse WTP distributions (uniform, exponential, Pareto, log-normal) with and without added small errors. The estimator accurately recovers true demand parameters, including the price coefficient, in all but the most irregular (tailed, skewed) distributions—even in the presence of realistic reporting errors. Notably, for the uniform and exponential WTP distributions, the estimation error for the price coefficient remains below 6.2% and is essentially unchanged under minor misreporting. However, with Pareto or log-normal WTP, larger deviations appear as anticipated due to the non-flatness, confirming the theoretical results on sensitivity to WTP distributional shape.

Empirical Illustration

A real-world application involving coffee demand among university staff and students employs the procedure with CVM-based WTP collection. The survey design controls for price anchoring by randomizing starting price offers. Empirical CDFs of WTP are highly non-normal, centering most mass between \$Z$26 for a medium long black.

Three regression specifications (linear, semi-log, log-linear) were estimated, with the semi-logarithmic model providing the best fit ($Z$3), and a price coefficient of $Z$4, indicating substantial price sensitivity. These results illustrate the method’s applicability even for established goods and underscore its flexibility in model specification and functional form recovery.

Implications and Future Directions

The paper provides a practical and theoretically sound tool for estimating demand curves when traditional market data are unavailable or confounded by endogeneity. The generality of the approach, accommodating a rich set of distributions and supporting nonparametric or flexible parametric approximations, enables both pedagogical and policy applications (e.g., rapid classroom demonstrations, government non-market valuation, private sector market-entry forecasts).

A methodological limitation—also highlighted—lies in the static nature of the model, as forward-looking or durable-goods settings (dynamic demand) are not modeled. Extending this framework to nested or dynamic choice settings, with inter-temporal WTP and experience effects, represents a fertile avenue for future research.

Conclusion

The proposed WTP-based demand estimation framework offers analytical simplicity, broad applicability, and strong theoretical soundness, substantially lowering technical entry barriers for non-market demand estimation. The estimator is both statistically robust and practically effective across an array of WTP environments. The methodology complements and, in non-market circumstances, supersedes classic market equilibrium-based identification strategies. Future extension to dynamic settings could unify static and dynamic demand estimation under the same robust WTP-centric methodology.