Product Entry and Exit in Demand Counterfactuals

Why Product Entry and Exit Require a New Counterfactual

When a product enters or exits a market, its own sales are only part of the demand response. Consumers who would have chosen an exiting product must reconsider the products that remain, while entry gives consumers another available option. The resulting shares depend on the entire available set, consumer preferences, and—if firms adjust prices—the prices chosen after the product-set change.

A useful counterfactual therefore separates two calculations. First, hold prices fixed and change which products are available to isolate the immediate demand reallocation. Second, if the question concerns a new market equilibrium, solve for prices under the counterfactual product set and specified firm conduct. Comparing these calculations helps distinguish a demand effect from a price-response effect.

Choice Probabilities Under a Changing Product Set

Let $J$ denote the set of products available in a market, with an outside option indexed by $0$. A consumer chooses one available alternative. For consumer $i$, write the utility from product $j$ as $V_{ij}+\epsilon_{ij}$, where $V_{ij}$ is systematic utility and $\epsilon_{ij}$ is a choice shock. A standard random-coefficient specification is $V_{ij}=\delta_j-\alpha_i p_j+\mu_{ij}$: $p_j$ is price, $\alpha_i>0$ is the consumer's price sensitivity, and $\delta_j$ and $\mu_{ij}$ capture mean and heterogeneous product utility. Normalize the outside option's systematic utility to zero.

With independent Type I extreme-value choice shocks, the choice probability conditional on consumer $i$'s preferences is the logit probability. In a random-coefficient model, the market share averages these probabilities over the distribution of consumer preferences. The same consumers and preference distribution are used before and after a product-set change; what changes is the available alternatives, and possibly their prices.

Definition: For product set $J$, conditional choice probabilities and market shares are $$ P_{ij}(J)=\frac{\exp(V_{ij})}{1+\sum_{k\in J}\exp(V_{ik})},\qquad s_j(J)=E_i[P_{ij}(J)]. $$ The outside-option probability is $P_{i0}(J)=1/(1+\sum_{k\in J}\exp(V_{ik}))$. In a simple logit model without consumer heterogeneity, the expectation is unnecessary. If market size is $M$, predicted sales of product $j$ are $M s_j(J)$.

The product set appears in the denominator of every available product's probability. Thus entry can reduce the probabilities of existing products even when their prices do not change. Exit removes one term from the denominator; the probabilities of all surviving alternatives, including the outside option, must then be recalculated. In a heterogeneous model, the change in an aggregate share also reflects which consumer types are more likely to choose each alternative.

Worked Example: Entry at Fixed Prices in a Simple Logit Market

Suppose a hypothetical market has an outside option and two products, A and B. At their current prices, their exponentiated systematic utilities are 2 for A and 1 for B; the outside option has weight 1. A new product C enters with weight 1. Prices are held fixed for this first calculation.

Before entry, the denominator is $1+2+1=4$. The probabilities are $P_A=2/4=0.5000$, $P_B=1/4=0.2500$, and $P_0=1/4=0.2500$. After entry, the denominator is $1+2+1+1=5$, so $P_A=2/5=0.4000$, $P_B=1/5=0.2000$, $P_C=1/5=0.2000$, and $P_0=1/5=0.2000$. The probabilities sum to 1 in each scenario.

If the market contains 10,000 consumers, predicted sales of A fall from $10{,}000(0.5000)=5{,}000$ to $10{,}000(0.4000)=4{,}000$, a decrease of 1,000. Product B falls from 2,500 to 2,000, while C attracts 2,000 consumers and the outside option falls by 500. These are fixed-price predictions: they do not yet account for firms changing prices after entry.

The example also illustrates why an entrant's sales cannot simply be added to the old allocation. Its probability comes from the same choice denominator as the incumbent products and the outside option.

Consumer Surplus for Entry and Exit

For a consumer with quasilinear utility and a constant price sensitivity $\alpha>0$, the logsum is an inclusive-value measure of the utility available across the alternatives. Under the logit assumptions, the expected maximum utility differs from the logsum by a constant that cancels when comparing two scenarios. Dividing the logsum difference by $\alpha$ expresses the change in money-metric consumer surplus in the same units as price.

With heterogeneous preferences, calculate the logsum change for each preference draw and average the resulting money-metric changes. This requires positive price sensitivity for the consumers included in the calculation. The formula describes a model-based consumer-surplus change under the specified choice and utility assumptions; it is not, by itself, a claim about producer profits or total welfare.

Result: At fixed prices, for the same consumer preferences before and after the change, the per-consumer surplus difference for preference draw $i$ is $$ \Delta CS_i=\frac{1}{\alpha_i}\left[ \log\left(1+\sum_{j\in J_{\mathrm{new}}}\exp(V_{ij}^{\mathrm{new}})\right) -\log\left(1+\sum_{j\in J_{\mathrm{old}}}\exp(V_{ij}^{\mathrm{old}})\right) \right]. $$ For a common $\alpha$ and a simple logit model, this reduces to the difference in log denominators divided by $\alpha$. With heterogeneous consumers, average $\Delta CS_i$ over the preference distribution; multiply by $M$ for aggregate consumer surplus in a market of size $M$.

Worked Example: Exit With Heterogeneous Consumers

Consider a hypothetical market with two equally common consumer types and products A, B, and C, plus the outside option. The exponentiated utilities for Type 1 are $1,3,1,1$ for the outside option, A, B, and C, respectively. For Type 2 they are $1,1,2,2$. Product C exits; prices and the utilities of the surviving options are held fixed. The common price sensitivity is $\alpha=0.5$ per dollar.

Before exit, each type's denominator is 6. Averaging the type-specific probabilities gives outside share $(1/6+1/6)/2=0.1667$, A share $(3/6+1/6)/2=0.3333$, B share $(1/6+2/6)/2=0.2500$, and C share $(1/6+2/6)/2=0.2500$.

After exit, Type 1's denominator is $1+3+1=5$, and Type 2's is $1+1+2=4$. The new aggregate outside share is $(1/5+1/4)/2=0.2250$; A's share is $(3/5+1/4)/2=0.4250$; and B's share is $(1/5+2/4)/2=0.3500$. The surviving shares sum with the outside share to 1. A's share rises by 0.09167 and B's by 0.1000, while the outside option rises by 0.05833. The different increases show that exit does not allocate C's former share in a fixed proportion across alternatives.

For Type 1, the per-consumer surplus change is $\log(5/6)/0.5=-0.3646$ dollars. For Type 2 it is $\log(4/6)/0.5=-0.8109$ dollars. Averaging equally gives $$ \Delta CS=\frac{1}{2}\left(\frac{\log(5/6)}{0.5}+\frac{\log(4/6)}{0.5}\right)=-0.5878\text{ dollars per consumer}. $$ The negative sign indicates a consumer-surplus loss from exit at the stated fixed prices. In a market of 10,000 consumers, the corresponding aggregate change is approximately $-\$5{,}878$.

Adding Price Responses: Re-solve the Market Equilibrium

If firms set prices strategically, a fixed-price share calculation is only the first stage of a counterfactual. The available products, firms' ownership, marginal costs, and assumed conduct determine the new pricing problem. Under static Bertrand competition with differentiated products, each firm chooses prices for its products to maximize its total operating profit, taking other firms' prices as given. For firm $f$, the first-order condition for the price of its product $j$ is $$ s_j+\sum_{k\in J_f}(p_k-c_k)\frac{\partial s_k}{\partial p_j}=0, $$ where $J_f$ is the set of products owned by firm $f$, and $c_k$ is product $k$'s marginal cost. The shares and derivatives must be evaluated at the counterfactual product set and prices. A product's exit changes the set of terms in these conditions; entry can also change the firm's ownership structure and its pricing incentives.

Define the scenario.
Specify which products are available, their characteristics and marginal costs, the outside option, ownership, and the market size.

Calculate fixed-price demand.
Hold the chosen prices fixed, update the available set, and recompute each consumer's choice probabilities and the integrated market shares.

Solve for counterfactual prices if required.
Use the specified conduct and ownership assumptions to solve the firms' pricing conditions, updating shares and derivatives at each candidate price vector.

Recompute outcomes and welfare.
At the solved prices, calculate shares, sales, and the logsum-based consumer-surplus change relative to the stated baseline.

Worked Example: Entry and Equilibrium Prices in a Symmetric Logit Market

Consider a hypothetical market with identical products owned by separate single-product firms. Each firm has marginal cost $c=1$ dollar, mean utility before the price term $\delta=3$, and all consumers have price sensitivity $\alpha=1$ per dollar. The outside option is available. Compare two-product and three-product Bertrand equilibria.

With $J$ identical products at common price $p$, each product's share is $s=x/(1+Jx)$, where $x=\exp(3-p)$. Its own-price derivative is $\partial s/\partial p=-s(1-s)$, since $\alpha=1$. A single-product firm's price condition is $s+(p-c)\partial s/\partial p=0$; substituting the derivative yields $p-c=1/(1-s)$. Combining this with the share expression gives $$ p=1+\frac{1+Jx}{1+(J-1)x},\qquad x=\exp(3-p). $$ The equality follows because $1-s=[1+(J-1)x]/(1+Jx)$. Thus prices must satisfy the pricing condition and the demand equation simultaneously.

For $J=2$, solving these equations gives approximately $p=2.599$, $x=\exp(3-2.599)=1.493$, and denominator $1+2x=3.987$. Each product's share is $1.493/3.987=0.3745$, and the outside share is $1/3.987=0.2508$. The implied markup is $p-c=1.599$, matching $1/(1-0.3745)$ up to rounding.

For $J=3$, the solution is approximately $p=2.393$, $x=\exp(3-2.393)=1.835$, and denominator $1+3x=6.505$. Each product's share is $1.835/6.505=0.2821$, and the outside share is $1/6.505=0.1537$. The markup is $1.393$, matching $1/(1-0.2821)$ up to rounding. Entry therefore lowers the equilibrium price of each product by about $2.599-2.393=0.206$ dollars in this example; it also lowers each incumbent's share.

At equilibrium, per-consumer consumer surplus is the log denominator divided by $\alpha=1$. It rises from $\log(3.987)=1.383$ dollars in the two-product market to $\log(6.505)=1.873$ dollars in the three-product market, an increase of about $0.490$ dollars per consumer. This comparison includes both the additional option and the equilibrium price response. It relies on the stated logit demand, costs, ownership, and Bertrand conduct assumptions.

Assumptions, Interpretation, and a Common Pitfall

A product-entry or exit result is conditional on how the counterfactual is defined. An entrant needs specified characteristics, price or cost information, and ownership; an exiting product must be removed from the choice set. The consumer population and preference distribution are typically held fixed unless the question explicitly changes them. If preferences are integrated by simulation, using the same preference draws in the baseline and counterfactual makes the comparison less affected by simulation noise. These choices do not establish that the simulated scenario will occur in an observed market; they make clear what the model predicts under stated assumptions.

For equilibrium outcomes, additional assumptions about firms' conduct and costs are necessary. A fixed-price comparison answers how choice probabilities change if prices do not respond. It does not answer what equilibrium prices will be. Conversely, a solved equilibrium comparison includes price changes and should not be described as the direct demand effect alone. The two comparisons can be informative precisely because they answer different questions.

Pitfall: removing the exiting product's share and leaving all other shares unchanged. Shares must sum to one, and consumers who had chosen the exiting product must choose among the alternatives that remain, including the outside option. Recalculate the probabilities using the new choice denominator.
Pitfall: treating entry as a new source of demand without reducing any existing probability. With a fixed consumer population and one choice per consumer, the entrant's probability comes from the choice allocation. Recompute the probabilities of incumbents and the outside option as well.
Pitfall: calling fixed-price predictions an equilibrium counterfactual. These predictions hold prices fixed by construction. If firms reprice, solve the relevant pricing conditions and then recompute demand and consumer surplus at the resulting prices.
Pitfall: interpreting a share change as a causal empirical estimate. A model counterfactual is a prediction conditional on estimated preferences and specified assumptions. Its interpretation depends on those inputs; a predicted reallocation is not, by itself, evidence that an observed entry or exit caused the same change.

Key takeaway: Product entry and exit change the set of alternatives in every consumer's choice problem. Recalculate probabilities and integrate over consumers to measure fixed-price demand reallocation; when prices respond, solve the counterfactual equilibrium before reporting shares or consumer surplus.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

Why does adding an alternative to the choice set generally require recalculating the outside-option probability as well as incumbent product probabilities?
In a market with consumer heterogeneity, why can the aggregate share of a surviving product change by a different amount from the share of another surviving product after an exit?
What additional assumptions are needed to turn a fixed-price entry calculation into a counterfactual equilibrium price prediction?
How should a logsum difference be converted to money-metric consumer surplus when consumers have a common price sensitivity $\alpha$?
Explain why predicted sales of an entrant cannot simply be added to baseline sales while leaving all other choices unchanged.