Market-Level Demand Data and Outside Options

Why Market Definitions Matter

Demand data are often organized as a set of products observed in several markets. A market might be a geographic area during a particular month, or a national market during a calendar year. Before comparing quantities or estimating demand, the analyst must decide what counts as a product, which purchases belong to each market, and how many potential purchase occasions the market contains. Those choices determine the market shares in the data.

An important feature of differentiated-product demand data is the outside option: the possibility that a consumer does not buy any of the products listed in the market. It can represent not purchasing during the period, buying an unlisted substitute, or both. The outside option is not merely a mathematical remainder. Its meaning depends on the market boundary and on the assumed population of potential purchasers.

Definition: A market is a defined set of potential purchase occasions, identified by a market index $t$. A product is one of the inside alternatives available in market $t$, indexed by $j=1,\ldots,J_t$. Let $Q_{jt}$ denote the observed quantity of product $j$ sold in market $t$, $p_{jt}$ its price per unit, and $M_t$ the market size: the number of potential purchase occasions over the same market and time period. When each occasion results in at most one choice among the inside products and the outside option, the product market share is $s_{jt}=Q_{jt}/M_t$.

Quantities, Prices, and Market Shares

The market index should capture all dimensions that define the relevant choice set and observation period. For example, if demand conditions differ across cities or months, a city-month can be a market. In that case, a product observed in two cities during one month has two market-level records, even if its name and physical characteristics are unchanged. The product-market pair, not the product name alone, is the unit of the demand data.

A quantity $Q_{jt}$ counts the units sold in the specified market and period. The price $p_{jt}$ should be measured for that same product, market, and period. It may be a posted price or an average transaction price, but the definition must be clear and applied consistently. A price per package cannot be compared directly with a price per ounce unless the unit conversion is made explicit. Likewise, sales measured over a month cannot be divided by a market size defined as annual purchase occasions without adjustment.

The market share is a ratio, not a raw sales count. A product that sells 10,000 units in a large market may have a smaller share than a product that sells 2,000 units in a small market. Shares are most directly interpretable when the numerator and denominator count the same kind of event, such as one purchase occasion or one consumer in the same period.

Definition: The outside-option quantity $Q_{0t}$ is the number of market-$t$ purchase occasions assigned to no listed inside product. The outside-option share is $s_{0t}=Q_{0t}/M_t$. If every potential occasion is assigned to exactly one inside product or the outside option, then $Q_{0t}=M_t-\sum_{j=1}^{J_t}Q_{jt}$, and $s_{0t}=1-\sum_{j=1}^{J_t}s_{jt}$.

These definitions rely on a specific accounting setup: the market size counts all relevant potential occasions, the inside products cover the listed alternatives, and each occasion is counted once. Under this setup, shares are nonnegative and add to one when the outside option is included. The identity is an accounting condition; it does not imply that consumers have identical preferences or that any particular behavioral model is correct.

Worked Example: Shares in One Market

Consider hypothetical data for one city during a month. The analyst defines the market as 100,000 potential purchase occasions, with at most one unit purchased per occasion. Three listed products have monthly sales of 18,000, 12,000, and 5,000 units. Their average prices are respectively $4.20, $3.80, and $5.10 per unit. Calculate the product and outside-option shares.

Check the measurement setup. The market size and quantities refer to the same city and month. The one-unit-per-occasion assumption means that sales units can be interpreted as occasions choosing the corresponding product. Suppose the three products plus the outside option exhaust the alternatives being counted.

For product 1, the share is $s_{1t}=18{,}000/100{,}000=0.1800$, or 18.00%. For product 2, $s_{2t}=12{,}000/100{,}000=0.1200$, or 12.00%. For product 3, $s_{3t}=5{,}000/100{,}000=0.0500$, or 5.000%. The prices describe the products in this market; they are not part of the share calculation.

Total inside sales are $18{,}000+12{,}000+5{,}000=35{,}000$ units. Thus the outside-option quantity is $Q_{0t}=100{,}000-35{,}000=65{,}000$, and its share is $s_{0t}=65{,}000/100{,}000=0.6500$, or 65.00%. The shares add to $0.1800+0.1200+0.0500+0.6500=1.000$. Under the stated accounting interpretation, 65% of the potential monthly occasions resulted in no purchase of one of the three listed products.

The Market-Size Assumption

In many applications, the sales quantities are observed but the market size is not. The analyst must then justify a denominator. It might be a known number of eligible consumers, an estimate of purchase occasions, or a quantity based on population and an assumed purchase frequency. Each interpretation answers a different question. A count of households, for example, is not automatically a count of monthly purchase occasions: that conversion requires assumptions about who purchases and how often.

Changing $M_t$ while holding product sales fixed changes every inside share and the residual outside share. The denominator therefore has substantive consequences. A large assumed market implies that a smaller fraction of potential occasions chose the listed products; a smaller market implies larger inside shares. The analyst should report how market size is defined and, when it is uncertain, examine whether conclusions are sensitive to plausible alternatives.

Worked Example: How a Denominator Changes Shares

Suppose a hypothetical market has 40,000 observed sales of a listed product during one month. The analyst considers two possible market-size definitions: 100,000 potential monthly purchase occasions, based on eligible consumers, or 200,000 occasions, based on a broader population count. Assume that no other listed products are included and that each occasion can produce at most one purchase.

Using the first definition, the product share is $40{,}000/100{,}000=0.4000$, or 40.00%. The residual outside quantity is $100{,}000-40{,}000=60{,}000$, so the outside share is 60.00%.

Using the second definition, the product share is $40{,}000/200{,}000=0.2000$, or 20.00%. The residual outside quantity is $200{,}000-40{,}000=160{,}000$, so the outside share is 80.00%. The observed sales have not changed; the interpretation of those sales as a fraction of potential occasions has changed because the market-size assumption changed.

The second calculation is not automatically better because its denominator is larger. The analyst must decide which population actually represents potential purchase occasions in the defined month. If the 200,000 figure counts people who were ineligible or could not plausibly make the relevant purchase, it is not an appropriate market size for this question.

Deriving the Adding-Up Identity

The relationship between inside shares and the outside share follows directly from the partition of purchase occasions. Assume (1) there are $M_t$ relevant occasions, (2) each occasion is assigned to exactly one of the $J_t$ inside products or the outside option, and (3) quantities count those assignments in the same period and units. Then the inside quantities and outside quantity sum to the total number of occasions:

M_t = Q_{0t}+\sum_{j=1}^{J_t}Q_{jt}.

Divide both sides by the positive market size $M_t$. Since $s_{0t}=Q_{0t}/M_t$ and $s_{jt}=Q_{jt}/M_t$, this gives:

1=s_{0t}+\sum_{j=1}^{J_t}s_{jt}, \qquad s_{0t}=1-\sum_{j=1}^{J_t}s_{jt}.

This derivation also clarifies a useful diagnostic. If the computed inside shares sum to more than one, the residual outside share is negative, which is incompatible with the stated one-choice accounting setup. Possible explanations include an understated market size, quantities measured in units that do not correspond to purchase occasions, overlapping product categories, or consumers purchasing multiple units or products within an occasion. The arithmetic flags a mismatch; it does not by itself determine which data definition should be changed.

Markets Can Be Aggregated, but Shares Need Weights

When several markets are combined, quantities can be added if they use consistent product definitions, units, and time periods. Market shares cannot generally be averaged without weights. The pooled share for a product is total sales divided by total market size. Equivalently, it is the average of market-specific shares weighted by each market's size. An unweighted average gives a small market the same influence as a large one, which answers a different question.

Worked Example: Pooling Two Markets

Consider two hypothetical monthly markets with the same three listed products. Market A has size 50,000 and product quantities 8,000, 7,000, and 5,000. Market B has size 30,000 and product quantities 3,000, 9,000, and 3,000. Assume the markets use the same one-purchase-occasion definition and that the listed products plus the outside option exhaust the alternatives in each market.

In Market A, the inside shares are $8{,}000/50{,}000=0.1600$, $7{,}000/50{,}000=0.1400$, and $5{,}000/50{,}000=0.1000$. Inside sales total 20,000, so $Q_{0A}=50{,}000-20{,}000=30{,}000$ and $s_{0A}=0.6000$.

In Market B, the inside shares are $3{,}000/30{,}000=0.1000$, $9{,}000/30{,}000=0.3000$, and $3{,}000/30{,}000=0.1000$. Inside sales total 15,000, so $Q_{0B}=30{,}000-15{,}000=15{,}000$ and $s_{0B}=0.5000$.

For the pooled data, total market size is $50{,}000+30{,}000=80{,}000$. Product quantities are 11,000, 16,000, and 8,000. Their pooled shares are therefore $11{,}000/80{,}000=0.1375$, $16{,}000/80{,}000=0.2000$, and $8{,}000/80{,}000=0.1000$. The pooled outside quantity is $30{,}000+15{,}000=45{,}000$, giving an outside share of $45{,}000/80{,}000=0.5625$. These four pooled shares sum to 1.

For product 1, the unweighted average of the two market shares would be $(0.1600+0.1000)/2=0.1300$, not the pooled share of 0.1375. The pooled share weights Market A by $50{,}000/80{,}000=0.6250$ and Market B by $30{,}000/80{,}000=0.3750$: $0.6250(0.1600)+0.3750(0.1000)=0.1375$. Thus the pooled statistic describes a randomly selected occasion from the combined markets, not a randomly selected market.

Practical Checks and Common Pitfalls

A well-constructed market-level data set should make its accounting choices visible. For each product-market row, record a consistent market identifier, product identifier, quantity, price, and the market size used to form shares. Document whether the outside option means no purchase, unlisted alternatives, or both. When the market definition changes across rows, make clear which characteristics are allowed to change and which are held fixed.

Using population as market size without justification. A geographic population is not necessarily the number of potential purchase occasions during the observation period. Specify eligibility and purchase frequency, or explain why the population count is an appropriate denominator.
Mixing time units. Monthly sales divided by an annual market size do not produce a monthly share unless the denominator has been converted to the same time basis.
Forgetting the outside option. Inside shares need not sum to one. Under an exhaustive one-choice setup, the remaining share belongs to the outside option; omitting it changes the interpretation of the data.
Treating all residual sales as nonpurchase. If unlisted brands or substitutes are omitted, the residual outside option includes them as well as no purchase. Its interpretation must match the alternatives excluded from the inside-product list.
Averaging shares across markets without considering size. An unweighted average describes an average market's share, while total sales divided by total market size describes the share across pooled purchase occasions.

Key takeaway: Market shares are defined by a numerator and a defensible denominator, both tied to the same product scope, market boundary, time period, and unit of observation. The outside-option share completes the accounting only when the inside alternatives and outside option form an exhaustive, non-overlapping partition of potential purchase occasions.

Check Your Understanding

Answer each question using the market definitions and assumptions stated.

A market has 80,000 potential monthly purchase occasions and two listed products sell 12,000 and 28,000 units. Under a one-unit-per-occasion, exhaustive-choice setup, calculate both inside shares and the outside share.
Explain why 5,000 units sold during a month cannot automatically be divided by the total population of a country to obtain a monthly market share.
What does the outside option include if the data list only three brands while consumers can also buy several unlisted brands?
Two markets have different sizes. Which calculation gives the pooled share of a product: the unweighted mean of its market shares or its total quantity divided by total market size? Explain.
Under the one-choice setup, what does it indicate if the sum of inside quantities exceeds the stated market size? Give two possible data or definition issues that could explain the discrepancy.