Estimating Total Demand with a Price-Based Logit Share Model

Summary

The document asks how to estimate combined demand when two firms charge fixed prices. It starts from a linear demand expression for the overall market and allocates that demand between firms using a price-sensitive logit share. The central modeling issue is that the total market demand and the allocation of that demand are distinct: the logit rule specifies each firm’s share, but does not by itself determine total demand.

The question proposes a logarithmic equation as a possible solution to the resulting implicit problem, but supplies no derivation, validation, or answer confirming that equation. Its parameters and notation also appear inconsistently in the proposed expression, so readers should not treat it as an established estimator. The material is useful as a prompt about combining aggregate demand with discrete-choice shares, while leaving the correct derivation and assumptions unresolved.

Key ideas

• A price-based logit rule allocates market demand across competing firms according to their relative prices.
• The share formula alone does not specify the level of total demand.
• The document proposes an implicit logarithmic equation but does not establish its correctness.
• Any estimator depends on clearly defining how aggregate demand interacts with firm-level shares.

Tags

# Total demand under logit model


# Total demand under logit model












The setting is simple, i.e. formula for demand of service/product is linear $$ d = \alpha - \beta p $$ where $ \alpha $ is maximum demand, $ \beta $ is some coefficient, and $ p $ is price. There are two firms, with different prices $p_1$ and $ p_2 $, and the prices are fixed, so there is no competition. The fraction of total demand is captured by the logit rule (price-oriented) $$ f = \frac{e^{-\gamma p_1}}{e^{-\gamma p_1} + e^{-\gamma p_2}} $$ where $ \gamma $ is some sensitivity coefficient.

Question: how can I estimate total demand in this setting?

EDIT: I found an answer by my own, but I'm not sure if it is OK. The answer is the solution of this, I belive, transcedent equation $$ d = \beta - \beta \ln q + \beta \ln (\alpha (e^{-p_1} + e^{-p_2})) $$

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