WEBVTT

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Two firms will choose quantities.

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Each red or green curve
will record one firm's best answer

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to the other firm's output.

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Their crossing
is the destination of the lecture.

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At the yellow crossing, Firm 1 is already
answering Firm 2 as well as it can,

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and Firm 2 is already answering Firm 1.

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Neither wants to change alone.

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That is the Cournot-Nash equilibrium.

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But I do not want that crossing
to arrive as a mysterious pair of lines.

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We will build each line from profit,

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then let the firms chase one
another toward the crossing.

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Begin with a deliberately simple market.

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The horizontal coordinate
is total quantity Q.

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The vertical coordinate
is the market price P.

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Demand is the blue line P
equals one hundred minus Q.

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If total output is twenty,
buyers pay eighty.

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If output rises, the price
paid for every unit falls.

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Move total output from twenty to sixty.

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The market point slides
down the demand curve,

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and price falls from eighty to forty.

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Quantity decisions therefore
interact through one common price.

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Bring output back to forty.

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The corresponding price is sixty.

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We will later recognize

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this as the output and price a single
profit-maximizing firm would choose.

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With two firms, total quantity
is Firm 1's output plus Firm 2's output.

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The market does not care which firm
supplied a unit. It sees only their sum.

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Both firms have constant marginal
cost twenty and no fixed cost.

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The green line therefore stays flat at
twenty, however much either firm produces.

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A firm's profit is its margin, price minus
twenty, multiplied by its own quantity.

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The important word is own.

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Firm 1 earns the market margin on q
one, not on its rival's production.

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Substitute the market
price and total output.

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Profit becomes eighty
minus q one minus q two,

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multiplied by the firm's own quantity.

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That expression contains
the strategic tension.

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Producing another unit
sells one more unit,

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but it also lowers the price received
on every unit the firm already sells.

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Cournot competition asks each firm

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to choose its own quantity while
treating the rival's quantity as given.

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We now solve that decision for Firm 1,

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first with a numerical guess
and then for every possible guess.

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Suppose Firm 1 believes Firm
2 will produce twenty units.

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Firm 1 now faces a one-variable decision:

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choose q one to maximize its own profit.

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The blue curve plots
that profit against Firm 1's quantity.

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Profit is zero at quantity zero, rises,

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reaches a top, and eventually falls as the
price effect overwhelms the extra sales.

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Try q one equal to ten.

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The market quantity is thirty, price
is seventy, and Firm 1 earns a margin

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of fifty on ten units, or five hundred.

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Substituting the rival's twenty
units leaves sixty minus q one,

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multiplied by q one.

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Expanding gives sixty q
one minus q one squared.

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Move the trial quantity to thirty.

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The point climbs
to the top of the profit curve.

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Here Firm 1 sells thirty
units at a margin of thirty.

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Move farther to fifty.

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Profit falls back to five hundred.

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More output is not automatically better,

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because the common
market price has fallen.

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Differentiate the quadratic.

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The marginal effect of q
one is sixty minus two q one.

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Setting that equal to zero
gives q one equal to thirty.

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So thirty is Firm 1's best
response to a rival output of twenty.

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Now let the rival's quantity vary
instead of fixing it at one number.

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Put Firm 2's output on the horizontal axis
and Firm 1's output on the vertical axis.

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We want one point for every
possible guess about q two.

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Firm 1's general profit is eighty minus
q one minus q two, multiplied by q one.

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Expanding separates the own-output square
from the interaction with the rival.

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Differentiate with respect to Firm 1's
own quantity while holding q two fixed.

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The first-order condition is eighty
minus two q one minus q two equal to zero.

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Solving gives q one equal
to one half of eighty minus q two.

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This is not one answer.

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It is a rule assigning a profit-maximizing
q one to every possible q two.

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Plot that rule in red.

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If Firm 2 produces zero,

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Firm 1 behaves like the market's
sole producer and chooses forty.

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If Firm 2 produces twenty, Firm 1
chooses the thirty we just found.

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The red curve slopes downward because
the quantities are strategic substitutes.

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A larger rival output
depresses market price,

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so Firm 1's best reply is to produce less.

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Firm 2 solves the mirror-image problem.

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Its profit is the same market
margin multiplied by q two.

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Differentiate with respect to q two.

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The first-order condition is eighty
minus q one minus two q two equal to zero.

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Solving gives Firm 2's response:

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q two equals one half
of eighty minus q one.

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The two firms have identical
technologies and face the same demand.

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On axes ordered q two across and q one up,
that response appears as the green curve.

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Every point on it is an output pair
where Firm 2 is optimizing against Firm 1.

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The red curve records
where Firm 1 is content.

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The green curve records
where Firm 2 is content.

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Only their crossing makes both
statements true at the same time.

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Because the firms are identical,

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the crossing lies on the dashed
symmetry line q one equals q two.

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Call their common output q.

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Substitute q for the rival's
output in either response rule.

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q equals one half of eighty
minus q, so three q equals eighty.

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Each firm produces eighty thirds,
about twenty-six point seven units.

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The yellow point is therefore the unique
crossing of the two best-response curves.

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This algebra identifies the fixed point.

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It has not yet shown whether
decentralized adjustment finds it.

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For that, we let the firms respond one
after the other and watch the chase.

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Return to the quantity plane.

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The red and green
response curves are fixed.

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The blue point will record
the firms' current output pair,

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with q two across and q one up.

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Start away from equilibrium at q two
equal to ten and q one equal to ten.

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The point is on neither response curve,
so both firms currently want to revise.

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Firm 1 moves first. It treats q two
equal to ten as fixed and chooses one

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half of seventy, so the point moves
vertically and lands at thirty-five.

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Now Firm 2 observes q
one equal to thirty-five.

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Its best response is twenty-two point
five, so the point moves horizontally

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and lands on the green curve.

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Firm 1 responds again.

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Against twenty-two point five,

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its maximizing quantity
is twenty-eight point seven five.

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The red vertical step is already
much shorter than the first.

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Firm 2 answers with twenty-five point six.

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The green horizontal
step is shorter as well.

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The pair has entered the narrow
region between the two curves.

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Another Firm 1 response gives
about twenty-seven point two.

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It still overshoots the crossing
slightly, but by less than before.

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Firm 2 then chooses about
twenty-six point four.

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The alternating steps produce a staircase:

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red movements are Firm 1,
and green movements are Firm 2.

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Firm 1 now moves to about
twenty-six point eight.

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The vertical correction
has become small enough

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that the displayed coordinates begin
to agree in their first two digits.

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Firm 2 replies with about
twenty-six point six.

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The horizontal
correction is smaller again.

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One more Firm 1 response gives
about twenty-six point seven.

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The point is now visually
almost at the crossing.

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Firm 2 answers at about
twenty-six point seven as well.

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The staircase has tightened
around the yellow Cournot-Nash point.

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The motion itself suggests stability.

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A displacement away
from the crossing generates

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a smaller response in the opposite
direction, then a still smaller reply.

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We can measure that shrinkage.

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Let e one and e two denote each firm's
output minus its equilibrium output.

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Firm 1's response curve
has slope minus one half.

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Therefore Firm 1's new error
is minus one half of Firm 2's old error.

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Firm 2's response also
has slope minus one half.

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Its new error is minus one
half of Firm 1's new error.

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Multiplying the two slopes
gives positive one quarter.

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After one complete
Firm 1 and Firm 2 round,

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the remaining error has only one
quarter of its previous magnitude.

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That is why the staircase contracts
rather than exploding outward.

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The negative slopes make
successive corrections alternate sides,

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while their product being smaller
than one makes the corrections shrink.

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The conclusion
is conditional, not universal.

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We assumed exact best responses,

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one firm moving at a time, and an
unchanged demand and cost environment.

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Different adjustment rules
can create slower motion,

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simultaneous jumps, or instability.

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Under this benchmark,
however, equilibrium is both

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a pair of mutual best responses and the
stable fixed point of the visible chase.

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We can now ask what market
price that fixed point produces.

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Use the same demand and cost curves
to compare three market organizations.

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The blue line is demand and the green
line is marginal cost twenty.

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Begin with monopoly.

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One owner controls
the entire market quantity,

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so its total revenue is price times Q:
one hundred minus Q, multiplied by Q.

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Differentiating total revenue gives
marginal revenue one hundred minus two Q.

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The red marginal-revenue curve
falls twice as quickly as demand.

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The monopolist chooses quantity where
marginal revenue equals marginal cost.

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One hundred minus two Q equals twenty.

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Solving gives monopoly quantity forty.

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Move up from quantity forty to demand,
and buyers pay the monopoly price sixty.

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Now return to Cournot.

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Each firm produces eighty
thirds, so together they

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supply one hundred sixty thirds,
about fifty-three point three units.

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Demand at that total quantity
gives price one hundred forty thirds,

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about forty-six point seven.

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The yellow Cournot point lies
down and to the right of monopoly.

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Why does Cournot
produce more than monopoly?

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Each firm considers how
its own output lowers price,

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but it does not fully internalize the
price loss imposed on the rival's units.

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For perfect competition,

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firms take price as given and expand
until price equals marginal cost.

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Demand reaches price twenty
at total quantity eighty.

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The ordering is now visible.

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Monopoly restricts output
the most and has the highest price.

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Cournot lies between.

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Perfect competition has the largest
output and the lowest price.

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Numerically, price falls

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from sixty under monopoly, to about
forty-six point seven under Cournot,

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to twenty under perfect competition.

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The table collects the three outcomes.

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Read the headings first:

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market organization, total
quantity, and market price.

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Monopoly produces forty and charges sixty.

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That row is the benchmark
for coordinated quantity restriction.

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Cournot duopoly produces
about fifty-three point three

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and charges about forty-six point seven.

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Strategic quantity competition
moves the market toward competition,

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but not all the way.

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Perfect competition produces eighty and
sets price equal to marginal cost, twenty.

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Relative to Cournot, another
twenty-six point seven units are traded.

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The Cournot result is therefore
intermediate because firms compete

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through quantities while retaining
some control over market price.

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If firms compete directly through prices,
the strategic logic changes sharply.

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Cournot firms commit to quantities and let
market demand determine one common price.

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Bertrand competition
reverses the strategic choice:

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each firm posts a price.

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Keep the comparison disciplined.

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The product is identical,
both firms have marginal cost twenty,

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either firm can serve the market,
and buyers choose the lower price.

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Suppose Firm 1 posts fifty
while Firm 2 posts forty-five.

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The two colored points display
those prices on the same scale.

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The lower-price firm serves the market.

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Buyers do not pay fifty
for what they can buy at forty-five,

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so Firm 2 sells and Firm 1 sells nothing.

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Firm 1 can respond by posting forty-four.

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It gives up one unit
of margin relative to forty-five,

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but captures the market
instead of selling zero.

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Firm 2 can answer with forty-three.

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Unlike the Cournot chase,

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an attractive response here is not
a movement toward a smooth quantity curve.

00:14:24.983 --> 00:14:27.689
It is a price just below the rival.

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As long as the lower price
remains above marginal cost,

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the higher-price firm
can profitably undercut it.

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So no common price above
twenty can be an equilibrium.

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Continue the undercutting
pressure toward cost.

00:14:43.435 --> 00:14:47.221
Both price points move down
the scale and arrive at twenty.

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At a common price of twenty,

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neither firm can gain by charging more,
because it would lose its customers.

00:14:55.610 --> 00:15:00.998
Neither can gain by charging less, because
price would fall below marginal cost.

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The Bertrand equilibrium
is therefore price equal

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to marginal cost under these assumptions.

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Total demand at price twenty is eighty,

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the same total output
as the perfectly competitive benchmark.

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This is the central contrast.

00:15:19.101 --> 00:15:24.184
Under Cournot, a firm choosing more
output depresses the price on all units,

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so each firm restrains quantity.

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Under Bertrand, a slightly lower
price can redirect the whole market.

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Quantity competition therefore
leaves price above marginal cost

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in our two-firm example.

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Direct price competition drives
the benchmark price down to marginal cost.

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Place the two games side by side.

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The table names the strategic choice,
equilibrium price, and total output.

00:15:53.057 --> 00:15:56.517
In Cournot, firms choose quantities.

00:15:56.969 --> 00:16:00.658
Their best-response crossing gives
price about forty-six point seven

00:16:00.742 --> 00:16:04.017
and total output about
fifty-three point three.

00:16:05.232 --> 00:16:10.294
In the homogeneous-product Bertrand
benchmark, firms choose prices.

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Undercutting gives price
twenty and total output eighty.

00:16:15.015 --> 00:16:19.068
Do not treat that sharp result
as a law for every price-setting market.

00:16:19.172 --> 00:16:23.573
If products differ, customers
may stay with a higher-price seller.

00:16:23.769 --> 00:16:28.925
If capacity is limited, the cheaper
firm may be unable to serve everyone.

00:16:29.721 --> 00:16:34.049
Different marginal costs,
search costs, repeated interaction,

00:16:34.133 --> 00:16:39.707
and capacity constraints can also sustain
prices above the simplest Bertrand level.

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The comparison works

00:16:41.320 --> 00:16:46.278
because we held the environment fixed
and changed the strategic variable.

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The full lesson is now
one connected argument.

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Demand turns two
quantities into one price.

00:16:54.203 --> 00:16:59.335
Profit maximization turns each
rival quantity into a best response.

00:16:59.787 --> 00:17:03.445
Alternating responses converge
to the Cournot crossing.

00:17:03.642 --> 00:17:07.645
That crossing prices below
monopoly but above competition,

00:17:07.729 --> 00:17:12.153
while direct price setting creates
the undercutting race to marginal cost.