Market Shares, Inversion, and Demographic Micro Moments

Conceptual map

IV-03.01Individual-to-market aggregation
IV-03.02Share matching
IV-03.03Purchaser moments
IV-03.04Estimation objectives
IV-03.05Reconstruction boundaries

1. Market shares integrate individual probabilities

Aggregate shares and purchaser moments are different integrals of the same individual choice law. Their denominators and zero-share boundaries must remain visible.

Let consumer state $z_{i}$ have distribution $$F$$ , and let $P_{ij}(\delta,\theta)$ be the model probability of choosing product $$j$$ . Market share is $s_{j}(\delta,\theta)=\mathbb{E}_{F}[P_{ij}(\delta,\theta)]$ . For an integrable demographic characteristic $z_{i}$ ,

Proposition 1 · Purchaser conditional mean

\[\mathbb{E}[z_i\mid J_i=j]=\frac{\mathbb{E}[z_iP_{ij}]}{s_j},\qquad s_j>0.\](1)

Proof. Iterated expectation gives $\mathbb{E}[z_{i}1\{J_{i}=j\}]=\mathbb{E}[z_{i}P(J_{i}=j\mid z_{i})]$ . Divide by $P(J_{i}=j)=s_{j}$ . ∎

2. A purchaser moment reweights the population

Two equally common consumer types have $$z=0$$ and $$z=1$$ . Their purchase probabilities for product $$j$$ are 0.2 and 0.8. Aggregate share is 0.5. The numerator in (1) is $0.5\cdot0\cdot0.2+0.5\cdot1\cdot0.8=0.4$ , so the mean characteristic among purchasers is 0.8.

The population mean is 0.5. Conditioning on purchase shifts the type weights to 0.2 and 0.8. A simulation should reproduce both the aggregate denominator and the weighted numerator using the same draws.

3. Inversion is conditional on the demand model

In homogeneous logit with outside share $s_{0}>0$ and product shares $s_{j}>0$ , mean utility relative to the outside option is $\delta _{j}=\log s_{j}-\log s_{0}$ . Random-coefficient inversion instead solves a nonlinear fixed point for a fixed $\theta$ , heterogeneity law, simulation design, and choice set.

Matching shares determines mean utilities under that fixed model. A limited set of demographic moments adds selected restrictions on the joint law. It does not reconstruct every substitution pattern or latent utility component.

4. Zero shares and reused draws

Failure case · Micro moment evaluated at a zero predicted share

Equation (1) divides by $s_{j}$ . At zero share the conditional purchaser distribution is undefined, and a small denominator makes the moment numerically unstable.

A zero observed logit share implies $\delta _{j}=-\infty$ in the exact inversion limit. Sampling zeros, censored shares, and structural unavailability require separate treatment. Changing simulation draws while updating $\delta$ injects numerical noise into the inversion; fixed draws or controlled quasi-random sequences preserve a deterministic objective.

5. Implementation, exercises, and sources

Store market weights, demographic draws, choice sets, $\theta$ , and random seeds. At every inversion iterate, recompute shares and micro numerators from the same probabilities. Report share residuals, moment residuals, denominator floors, contraction or root tolerance, and the weighting matrix used to combine moments.

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Exercises

Reproduce the share and purchaser mean in Section 2.
Invert homogeneous-logit utilities for shares (0.5,0.3,0.2), including the outside option.
Construct two type-specific probability tables with the same aggregate share and purchaser mean but different probabilities for another product.

Partial solutions

1. The purchase-conditioned type-one weight is 0.4/0.5=0.8. 2. With outside share 0.5, utilities are $\operatorname{log}(0.3/0.5)$ and $\operatorname{log}(0.2/0.5)$ . 3. Preserve the two weighted restrictions for product $$j$$ while reallocating remaining probability across other products.

Steven Berry (1994), “Estimating Discrete-Choice Models of Product Differentiation,” RAND Journal of Economics 25, 242–262.Share inversion.
Steven Berry, James Levinsohn, and Ariel Pakes (1995), “Automobile Prices in Market Equilibrium,” Econometrica 63, 841–890.Aggregate random-coefficient demand.
Steven Berry, James Levinsohn, and Ariel Pakes (2004), “Differentiated Products Demand Systems from a Combination of Micro and Macro Data,” Journal of Political Economy 112, 68–105.Demographic micro moments.

6. Audit checkpoint

State the consumer law, market weights, choice sets, probability kernel, outside share, positive-share domain, inversion parameter vector, simulation draws, purchaser-moment numerator and denominator, weighting matrix, residuals, and denominator tolerance.

7. Scope boundary

The chapter covers share aggregation, elementary inversion, and purchaser moments. Full demand identification and GMM inference require specialized treatment.