Cournot Competition: Best-Response Dynamics and the Race to Equilibrium

Two firms will choose quantities. Each red or green curve will record one firm's best answer to the other firm's output. Their crossing is the destination of the lecture. At the yellow crossing, Firm 1 is already answering Firm 2 as well as it can, and Firm 2 is already answering Firm 1. Neither wants to change alone. That is the Cournot-Nash equilibrium. But I do not want that crossing to arrive as a mysterious pair of lines. We will build each line from profit, then let the firms chase one another toward the crossing. Begin with a deliberately simple market. The horizontal coordinate is total quantity Q. The vertical coordinate is the market price P. Demand is the blue line P equals one hundred minus Q. If total output is twenty, buyers pay eighty. If output rises, the price paid for every unit falls. Move total output from twenty to sixty. The market point slides down the demand curve, and price falls from eighty to forty. Quantity decisions therefore interact through one common price. Bring output back to forty. The corresponding price is sixty. We will later recognize this as the output and price a single profit-maximizing firm would choose. With two firms, total quantity is Firm 1's output plus Firm 2's output. The market does not care which firm supplied a unit. It sees only their sum. Both firms have constant marginal cost twenty and no fixed cost. The green line therefore stays flat at twenty, however much either firm produces. A firm's profit is its margin, price minus twenty, multiplied by its own quantity. The important word is own. Firm 1 earns the market margin on q one, not on its rival's production. Substitute the market price and total output. Profit becomes eighty minus q one minus q two, multiplied by the firm's own quantity. That expression contains the strategic tension. Producing another unit sells one more unit, but it also lowers the price received on every unit the firm already sells. Cournot competition asks each firm to choose its own quantity while treating the rival's quantity as given. We now solve that decision for Firm 1, first with a numerical guess and then for every possible guess.

Suppose Firm 1 believes Firm 2 will produce twenty units. Firm 1 now faces a one-variable decision: choose q one to maximize its own profit. The blue curve plots that profit against Firm 1's quantity. Profit is zero at quantity zero, rises, reaches a top, and eventually falls as the price effect overwhelms the extra sales. Try q one equal to ten. The market quantity is thirty, price is seventy, and Firm 1 earns a margin of fifty on ten units, or five hundred. Substituting the rival's twenty units leaves sixty minus q one, multiplied by q one. Expanding gives sixty q one minus q one squared. Move the trial quantity to thirty. The point climbs to the top of the profit curve. Here Firm 1 sells thirty units at a margin of thirty. Move farther to fifty. Profit falls back to five hundred. More output is not automatically better, because the common market price has fallen. Differentiate the quadratic. The marginal effect of q one is sixty minus two q one. Setting that equal to zero gives q one equal to thirty. So thirty is Firm 1's best response to a rival output of twenty. Now let the rival's quantity vary instead of fixing it at one number. Put Firm 2's output on the horizontal axis and Firm 1's output on the vertical axis. We want one point for every possible guess about q two. Firm 1's general profit is eighty minus q one minus q two, multiplied by q one. Expanding separates the own-output square from the interaction with the rival. Differentiate with respect to Firm 1's own quantity while holding q two fixed. The first-order condition is eighty minus two q one minus q two equal to zero. Solving gives q one equal to one half of eighty minus q two. This is not one answer. It is a rule assigning a profit-maximizing q one to every possible q two. Plot that rule in red. If Firm 2 produces zero, Firm 1 behaves like the market's sole producer and chooses forty. If Firm 2 produces twenty, Firm 1 chooses the thirty we just found. The red curve slopes downward because the quantities are strategic substitutes. A larger rival output depresses market price, so Firm 1's best reply is to produce less. Firm 2 solves the mirror-image problem. Its profit is the same market margin multiplied by q two. Differentiate with respect to q two. The first-order condition is eighty minus q one minus two q two equal to zero. Solving gives Firm 2's response: q two equals one half of eighty minus q one. The two firms have identical technologies and face the same demand. On axes ordered q two across and q one up, that response appears as the green curve. Every point on it is an output pair where Firm 2 is optimizing against Firm 1. The red curve records where Firm 1 is content. The green curve records where Firm 2 is content. Only their crossing makes both statements true at the same time. Because the firms are identical, the crossing lies on the dashed symmetry line q one equals q two. Call their common output q. Substitute q for the rival's output in either response rule. q equals one half of eighty minus q, so three q equals eighty. Each firm produces eighty thirds, about twenty-six point seven units. The yellow point is therefore the unique crossing of the two best-response curves. This algebra identifies the fixed point. It has not yet shown whether decentralized adjustment finds it. For that, we let the firms respond one after the other and watch the chase.

Return to the quantity plane. The red and green response curves are fixed. The blue point will record the firms' current output pair, with q two across and q one up. Start away from equilibrium at q two equal to ten and q one equal to ten. The point is on neither response curve, so both firms currently want to revise. Firm 1 moves first. It treats q two equal to ten as fixed and chooses one half of seventy, so the point moves vertically and lands at thirty-five. Now Firm 2 observes q one equal to thirty-five. Its best response is twenty-two point five, so the point moves horizontally and lands on the green curve. Firm 1 responds again. Against twenty-two point five, its maximizing quantity is twenty-eight point seven five. The red vertical step is already much shorter than the first. Firm 2 answers with twenty-five point six. The green horizontal step is shorter as well. The pair has entered the narrow region between the two curves. Another Firm 1 response gives about twenty-seven point two. It still overshoots the crossing slightly, but by less than before. Firm 2 then chooses about twenty-six point four. The alternating steps produce a staircase: red movements are Firm 1, and green movements are Firm 2. Firm 1 now moves to about twenty-six point eight. The vertical correction has become small enough that the displayed coordinates begin to agree in their first two digits. Firm 2 replies with about twenty-six point six. The horizontal correction is smaller again. One more Firm 1 response gives about twenty-six point seven. The point is now visually almost at the crossing. Firm 2 answers at about twenty-six point seven as well. The staircase has tightened around the yellow Cournot-Nash point. The motion itself suggests stability. A displacement away from the crossing generates a smaller response in the opposite direction, then a still smaller reply. We can measure that shrinkage. Let e one and e two denote each firm's output minus its equilibrium output. Firm 1's response curve has slope minus one half. Therefore Firm 1's new error is minus one half of Firm 2's old error. Firm 2's response also has slope minus one half. Its new error is minus one half of Firm 1's new error. Multiplying the two slopes gives positive one quarter. After one complete Firm 1 and Firm 2 round, the remaining error has only one quarter of its previous magnitude. That is why the staircase contracts rather than exploding outward. The negative slopes make successive corrections alternate sides, while their product being smaller than one makes the corrections shrink. The conclusion is conditional, not universal. We assumed exact best responses, one firm moving at a time, and an unchanged demand and cost environment. Different adjustment rules can create slower motion, simultaneous jumps, or instability. Under this benchmark, however, equilibrium is both a pair of mutual best responses and the stable fixed point of the visible chase. We can now ask what market price that fixed point produces.

Use the same demand and cost curves to compare three market organizations. The blue line is demand and the green line is marginal cost twenty. Begin with monopoly. One owner controls the entire market quantity, so its total revenue is price times Q: one hundred minus Q, multiplied by Q. Differentiating total revenue gives marginal revenue one hundred minus two Q. The red marginal-revenue curve falls twice as quickly as demand. The monopolist chooses quantity where marginal revenue equals marginal cost. One hundred minus two Q equals twenty. Solving gives monopoly quantity forty. Move up from quantity forty to demand, and buyers pay the monopoly price sixty. Now return to Cournot. Each firm produces eighty thirds, so together they supply one hundred sixty thirds, about fifty-three point three units. Demand at that total quantity gives price one hundred forty thirds, about forty-six point seven. The yellow Cournot point lies down and to the right of monopoly. Why does Cournot produce more than monopoly? Each firm considers how its own output lowers price, but it does not fully internalize the price loss imposed on the rival's units. For perfect competition, firms take price as given and expand until price equals marginal cost. Demand reaches price twenty at total quantity eighty. The ordering is now visible. Monopoly restricts output the most and has the highest price. Cournot lies between. Perfect competition has the largest output and the lowest price. Numerically, price falls from sixty under monopoly, to about forty-six point seven under Cournot, to twenty under perfect competition. The table collects the three outcomes. Read the headings first: market organization, total quantity, and market price. Monopoly produces forty and charges sixty. That row is the benchmark for coordinated quantity restriction. Cournot duopoly produces about fifty-three point three and charges about forty-six point seven. Strategic quantity competition moves the market toward competition, but not all the way. Perfect competition produces eighty and sets price equal to marginal cost, twenty. Relative to Cournot, another twenty-six point seven units are traded. The Cournot result is therefore intermediate because firms compete through quantities while retaining some control over market price. If firms compete directly through prices, the strategic logic changes sharply.

Cournot firms commit to quantities and let market demand determine one common price. Bertrand competition reverses the strategic choice: each firm posts a price. Keep the comparison disciplined. The product is identical, both firms have marginal cost twenty, either firm can serve the market, and buyers choose the lower price. Suppose Firm 1 posts fifty while Firm 2 posts forty-five. The two colored points display those prices on the same scale. The lower-price firm serves the market. Buyers do not pay fifty for what they can buy at forty-five, so Firm 2 sells and Firm 1 sells nothing. Firm 1 can respond by posting forty-four. It gives up one unit of margin relative to forty-five, but captures the market instead of selling zero. Firm 2 can answer with forty-three. Unlike the Cournot chase, an attractive response here is not a movement toward a smooth quantity curve. It is a price just below the rival. As long as the lower price remains above marginal cost, the higher-price firm can profitably undercut it. So no common price above twenty can be an equilibrium. Continue the undercutting pressure toward cost. Both price points move down the scale and arrive at twenty. At a common price of twenty, neither firm can gain by charging more, because it would lose its customers. Neither can gain by charging less, because price would fall below marginal cost. The Bertrand equilibrium is therefore price equal to marginal cost under these assumptions. Total demand at price twenty is eighty, the same total output as the perfectly competitive benchmark. This is the central contrast. Under Cournot, a firm choosing more output depresses the price on all units, so each firm restrains quantity. Under Bertrand, a slightly lower price can redirect the whole market. Quantity competition therefore leaves price above marginal cost in our two-firm example. Direct price competition drives the benchmark price down to marginal cost. Place the two games side by side. The table names the strategic choice, equilibrium price, and total output. In Cournot, firms choose quantities. Their best-response crossing gives price about forty-six point seven and total output about fifty-three point three. In the homogeneous-product Bertrand benchmark, firms choose prices. Undercutting gives price twenty and total output eighty. Do not treat that sharp result as a law for every price-setting market. If products differ, customers may stay with a higher-price seller. If capacity is limited, the cheaper firm may be unable to serve everyone. Different marginal costs, search costs, repeated interaction, and capacity constraints can also sustain prices above the simplest Bertrand level. The comparison works because we held the environment fixed and changed the strategic variable. The full lesson is now one connected argument. Demand turns two quantities into one price. Profit maximization turns each rival quantity into a best response. Alternating responses converge to the Cournot crossing. That crossing prices below monopoly but above competition, while direct price setting creates the undercutting race to marginal cost.