Why Model Selection Matters
An oligopoly model is a structured account of how a small number of firms make decisions while taking account of their rivals. The same market could look very different depending on whether firms choose quantities or prices, whether customers regard products as interchangeable, and whether a firm can satisfy the demand it attracts. Selecting a model is therefore not a matter of attaching a label to an industry. It means identifying the decision variable and checking whether the model’s assumptions fit the evidence.
This tutorial is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It focuses on choosing among Cournot, homogeneous Bertrand, differentiated Bertrand, and kinked-demand reasoning. The goal is not to infer a unique model from one observation: more than one model may fit incomplete evidence, and an observed price or quantity by itself does not reveal the firms’ decision process.
A Practical Selection Framework
Begin with evidence about the firms’ strategic variable. If firms commit to production or capacity quantities and price subsequently responds to aggregate output, Cournot is the natural starting point. If firms post prices and buyers respond to the prices, a Bertrand model is more appropriate. A firm’s capacity choice does not, by itself, establish Cournot behavior: capacity may constrain a firm that competes by setting prices.
Next ask how readily buyers switch between the products. If buyers view the products as identical and purchase from the lowest-priced seller, homogeneous Bertrand is a candidate. If brand, location, service, or other characteristics give customers a reason to remain with a higher-priced seller, differentiated Bertrand is more plausible. In a differentiated-price model, a firm’s sales generally depend on both its own price and its rivals’ prices.
Then check whether capacity changes the implications of price competition. The standard homogeneous Bertrand result depends on firms being able to serve the demand they attract. If a low-priced firm cannot meet that demand, some customers may remain with higher-priced firms, so the unrestricted-capacity version does not describe the market well. Evidence of binding capacity calls for a model that explicitly accounts for that constraint.
Finally, consider rivals’ expected responses to price changes. Kinked-demand reasoning is relevant when a firm expects rivals to match price cuts but not price increases. That asymmetry can make demand less responsive to a price cut and more responsive to a price increase around the current price. A stable observed price alone is not enough to establish this model; the asymmetric response expectation is the key assumption.
| Evidence to assess | Model that may fit | Important qualification |
|---|---|---|
| Firms select quantities; price responds to total output | Cournot | Confirm that quantity is the strategic choice, not merely a capacity limit |
| Firms select prices; products are identical; capacity is adequate | Homogeneous Bertrand | Binding capacity can invalidate the unrestricted-capacity setup |
| Firms select prices; products are substitutes but not identical | Differentiated Bertrand | Each firm’s demand should depend on its own and rivals’ prices |
| Rivals are expected to match cuts but not increases | Kinked-demand reasoning | Price rigidity does not establish the explanation without this response pattern |
Distinguish a chosen output from a chosen posted price. Do not treat capacity as equivalent to current production or as proof that quantity is the strategic variable.
Ask whether customers switch entirely to the lower-priced firm or whether some retain a preference for a particular seller.
Test whether a low-priced firm can serve the demand it attracts, and whether evidence supports the particular way rivals are expected to react.
Name the model that best fits the stated evidence and identify a missing assumption or alternative if the evidence is incomplete.
How the Strategic Variable Shows Up in a Model
A short derivation illustrates why the choice variable matters. Suppose two firms sell differentiated substitutes and firm 1’s demand is \(q_1=a-bp_1+dp_2\), where \(a>0\), \(b>0\), and \(d>0\). The negative own-price term means a higher \(p_1\) reduces firm 1’s sales; the positive cross-price term means a higher rival price \(p_2\) shifts some demand to firm 1. Let firm 1’s constant marginal cost be \(c_1\), with no fixed cost in this calculation. Its profit is \(\pi_1=(p_1-c_1)q_1\).
Holding the rival’s price fixed, differentiating with respect to firm 1’s chosen price gives:
The first term is the extra contribution from selling the units already demanded at the new price; the second reflects the loss of sales when the price rises. An interior profit maximum satisfies \(q_1=b(p_1-c_1)\), provided the resulting quantity is feasible and the second-order condition is satisfied. This condition follows from a price choice and demand responding to prices. If firms instead select quantities and price is determined by total output, the relevant optimization is over quantity, as in Cournot. The derivation does not establish which model describes a real market; the observed decision process and the model assumptions must still be checked.
Worked Example: Testing the Homogeneous Bertrand Conditions
Two hypothetical sellers offer the same unbranded product. Each posts a price. At the prices under consideration, market demand at the lower price is 100 units. Seller A posts $18, seller B posts $19, each has capacity for 120 units, and each has constant marginal cost of $12. Which model is the best fit, and what happens to sales at these posted prices?
Method. Check the decision variable, product similarity, and whether the lower-priced firm can serve the demand it attracts. These are the main conditions for applying the standard homogeneous Bertrand setup.
Both sellers choose prices, and buyers regard the products as identical. Seller A’s price is lower because \(18<19\), so it attracts the 100 units of demand. Its capacity is 120 units, which is sufficient because \(120\geq100\). Seller A’s sales revenue is \(18\times100=\$1{,}800\), and its variable cost is \(12\times100=\$1{,}200\). Its contribution after variable cost is therefore \(\$1{,}800-\$1{,}200=\$600\). Seller B sells zero units at these prices and earns zero variable contribution from sales.
Conclusion. Homogeneous Bertrand is a suitable model for analyzing this price competition: the products are identical, price is the decision variable, and the lower-priced firm can meet demand. This calculation describes the sales allocation at the stated prices; it does not show that those prices are an equilibrium. The tempting mistake is to apply the standard model without checking capacity. Here capacity is adequate, but if Seller A could supply only 60 units, the unrestricted-capacity assumption would fail.
Worked Example: Recognizing Quantity Competition
Two hypothetical producers supply an identical input. Each chooses its production quantity before a market price is established. Inverse market demand is \(P=90-Q\), where \(Q=q_1+q_2\). The firms produce \(q_1=20\) and \(q_2=15\) units, and each has constant marginal cost of $10. Identify the appropriate model and calculate the price and each firm’s contribution after variable cost.
Method. Since each firm chooses output and the market price depends on total output, use the Cournot description. First add the chosen quantities: \(Q=20+15=35\). Substitute total output into inverse demand: \(P=90-35=\$55\).
Firm 1’s contribution is \((P-10)q_1=(55-10)\times20=45\times20=\$900\). Firm 2’s contribution is \((P-10)q_2=(55-10)\times15=45\times15=\$675\). The total contribution is \(\$900+\$675=\$1{,}575\), which also equals \((55-10)\times35=45\times35=\$1{,}575\).
Conclusion. Cournot is the appropriate model because the strategic choices are quantities and price clears the market given their sum. The arithmetic checks both firm contributions against total contribution. A common mistake is to choose Bertrand merely because the firms sell the same product; product identity is relevant, but the strategic variable is decisive here. These calculations do not claim that the stated quantities are equilibrium quantities.
Worked Example: Distinguishing Differentiated Bertrand Competition
Two hypothetical firms set prices for related products. Firm 1’s demand is \(q_1=100-2p_1+0.5p_2\), and its marginal cost is $10. Suppose firm 2’s price is fixed at \(p_2=\$20\) while firm 1 considers its own price. Identify the model and calculate firm 1’s profit-maximizing price conditional on \(p_2\), assuming a nonnegative quantity and an interior maximum.
Method. The firms choose prices, while the positive cross-price term means firm 1’s demand responds to firm 2’s price. The demand specification therefore fits differentiated Bertrand competition, not the identical-product version of Bertrand. Substitute \(p_2=20\) to get \(q_1=100-2p_1+0.5(20)=110-2p_1\).
Firm 1’s profit is \(\pi_1=(p_1-10)(110-2p_1)\). Differentiating with respect to \(p_1\) gives:
Set the derivative equal to zero: \(130-4p_1=0\), so \(p_1=\$32.50\). The second derivative is \(-4<0\), confirming an interior maximum. At this price, \(q_1=110-2(32.50)=110-65=45\) units. Profit is \((32.50-10)\times45=22.50\times45=\$1{,}012.50\).
Conclusion. The firm’s demand depends on its own price and its rival’s price, so differentiated Bertrand is the relevant model. The calculation is a best response conditional on the rival price; it does not by itself calculate a market equilibrium. The common mistake is to read “Bertrand” as necessarily meaning identical products. Bertrand identifies price competition; the demand relationship determines whether the products are modeled as homogeneous or differentiated.
Worked Example: When Kinked-Demand Reasoning Fits
A hypothetical firm faces a current price of $30 and sells 100 units. It expects rivals to match a price cut, so a $1 price decrease would raise its sales by only 2 units. It expects rivals not to match a price increase, so a $1 increase would reduce its sales by 8 units. Using the local demand slopes, calculate marginal revenue on each side of the kink and assess whether marginal cost of $10 is consistent with a price held at the kink.
Method. The stated asymmetric rival response is the defining evidence for kinked-demand reasoning. At the kink, the quantity-price slope below the current price is \(\Delta Q/\Delta P=2/(-1)=-2\). Above the current price it is \(\Delta Q/\Delta P=(-8)/(1)=-8\).
For a differentiable demand segment, marginal revenue can be written \(MR=P+Q/(dQ/dP)\). Applying the lower-price slope at the kink gives \(MR=30+100/(-2)=30-50=-\$20\). Applying the higher-price slope gives \(MR=30+100/(-8)=30-12.50=\$17.50\). The marginal-revenue gap therefore extends from \(-\$20\) on the lower-price side to \(\$17.50\) on the higher-price side.
Marginal cost of $10 lies inside that gap because \(-20<10<17.50\). In the kinked-demand model, this is consistent with the firm retaining the current price as marginal cost changes within the gap. The arithmetic is a local implication of the stated slopes, not evidence that all firms in a market respond this way. A stable price without the asymmetric rival-response assumption is not enough to justify this model.
Common Selection Errors
- Classifying by product alone. Identical products do not imply homogeneous Bertrand if firms choose quantities. Identify the strategic variable first.
- Treating capacity as the strategic choice. Capacity may limit output under price competition; it is not automatically the quantity firms choose in a Cournot model.
- Ignoring the capacity condition. In homogeneous Bertrand, the standard undercutting logic assumes the lower-priced firm can serve the demand it attracts. When it cannot, the standard model’s assumptions do not hold.
- Calling all price competition homogeneous Bertrand. If customers retain preferences for particular products and demand responds to both firms’ prices, differentiated Bertrand is more appropriate.
- Inferring a kink from price stability alone. Kinked-demand reasoning requires an asymmetric expectation about rival responses to price increases and decreases. A constant observed price does not reveal that expectation by itself.
- Claiming too much from one calculation. A model may fit stated conditions without proving that its predicted equilibrium occurs. Separate the model’s assumptions from evidence about actual choices and outcomes.
Check Your Understanding
For each situation, identify the best-fitting model and complete the requested calculation or qualification.
- Two identical suppliers choose quantities of 18 and 12 units. Market demand is \(P=80-Q\), and both firms have marginal cost of $10. Name the model, calculate the market price, and find firm 1’s contribution after variable cost. (Answer: Cournot; \(Q=30\), \(P=\$50\), firm 1 contribution \(=\$720\).)
- Two sellers offer identical goods and set prices of $25 and $27. Demand at the lower price is 80 units; each seller can supply 100 units, and marginal cost is $14. Identify the model and calculate the lower-priced seller’s contribution after variable cost. (Answer: Homogeneous Bertrand; the lower-priced seller serves 80 units and earns \((25-14)\times80=\$880\).)
- Firm A sets a price, has marginal cost of $9, and faces demand \(q_A=90-3p_A+p_B\). If \(p_B=15\), calculate A’s conditional profit-maximizing price and quantity, assuming an interior maximum. Which Bertrand model fits? (Answer: Differentiated Bertrand; \(q_A=105-3p_A\), \(p_A=\$22\), \(q_A=39\).)
- At a current price of $40 and quantity 60, perceived demand slopes are \(-3\) units per dollar below the price and \(-6\) above it. Calculate the marginal revenue on each side and assess marginal cost of $25. (Answer: Kinked-demand reasoning; marginal revenues are $20 below and $30 above, so marginal cost lies in the gap and is consistent with price rigidity in the model.)
- Two firms sell identical products and set prices, but each can supply only 30 units while demand at the lower price is 70 units. Is the standard unrestricted-capacity homogeneous Bertrand model a good fit? (Answer: No. Price competition is present, but the lower-priced firm cannot serve all 70 units; a model incorporating capacity constraints is needed.)