EconS 425 - Cournot Competition

Eric Dunaway

Washington State University [email protected]

Industrial Organization

Eric Dunaway (WSU) EconS 425 1 / 47 Introduction

Today, we’llsee what happens when we have two or more …rm competing with one anothey by setting their output levels. This is known as Cournot competition.

Eric Dunaway (WSU) EconS 425 Industrial Organization 2 / 47 Cournot Competition

Before we hop in to the Cournot model, however, let’sdo a quick review of . Remember that a monopolist faces an inverse demand function p(Q) and a cost function c(Q). The inverse demand function is decreasing dp(Q ) in quantity, i.e., dQ < 0 and the cost function is increasing in dc(Q ) quantity, i.e., dQ > 0. The monopolist’spro…t maximization problem is max p(Q)Q c(Q) Q with …rst-order condition, dπ dp(Q) dc(Q) = Q + p(Q) = 0 dQ dQ dQ MR MC The monopolist picks the| quantity{z that} maximizes| {z } their pro…ts by setting their marginal revenue equal to .

Eric Dunaway (WSU) EconS 425 Industrial Organization 3 / 47 Cournot Competition

dp Q + p MC = 0 dQ Rearranging this expression, we can derive the Lerner Index, which gives us the market price as a function of ε, the price elasticity of demand, 1 p MC = dQ 1 dp Q p MC 1 1 = = p dQ p dp Q ε We’ll need this later.

Eric Dunaway (WSU) EconS 425 Industrial Organization 4 / 47 Cournot Competition

Let’sadd some functional form. Suppose the inverse demand function for the monopolist is p(Q) = a bQ where a and b are positive constants. Marginal costs are constant at c. The monopolist’spro…t maximization problem becomes,

max (a bQ)Q cQ Q with …rst-order condition, dπ = a 2bQ c = 0 dQ Solving for Q, we have the monopolist’soptimal quantity, a c Q = 2b

Eric Dunaway (WSU) EconS 425 Industrial Organization 5 / 47 Cournot Competition

a c Q = 2b Next, we obtain the market price by plugging the market quantity back into the inverse demand function, a c a + c p = a bQ = a b = 2b 2   and lastly, our pro…ts are

a + c a c a c π = pQ cQ = c 2 2b 2b       (a c)2 = 4b

Eric Dunaway (WSU) EconS 425 Industrial Organization 6 / 47 Cournot Competition

In 1836, French economist Antoine Cournot published his work Researches into the Mathematical Principles of the Theory of Wealth. Up until this point, economists had a fairly robust model of monopoly, but models with more than one …rm were lacking. Cournot formalized a model of where …rms competed by selecting their output levels (quantities) independent of one another. Note that in the real world, …rms don’treally select quantities, but rather prices. We’llsee what happens when …rms select prices later on. This became a major source of criticism later on, by Cournot’scritics. Cournot’smodel is simply named after him: Cournot competition.

Eric Dunaway (WSU) EconS 425 Industrial Organization 7 / 47 Cournot Competition

Suppose now that we had two identical …rms, …rm 1 and …rm 2.

Firm 1 is able to set their own quantity, q1, while …rm 2 is able to independently set theirs, q2. Both …rms set their quantity at the same time. Think of …rms 1 and 2 as the players of a game. They have to strategically choose their output levels, taking into consideration the other …rm’soutput level. Essentially, we have a simultaneous move game

Eric Dunaway (WSU) EconS 425 Industrial Organization 8 / 47 Cournot Competition

Regardless of how many …rms are in the market, the market price for all of the …rms is the same. It is a function of the total output level,

p(Q) = p(q1 + q2)

The same relationship between price and total quantity holds as dp(Q ) before, dQ < 0, and since Q = q1 + q2, we have that both

∂p(q + q ) ∂p(q + q ) 1 2 < 0 and 1 2 < 0 ∂q1 ∂q2 i.e., the market price decreases if either …rm increases their output. Thus, neither …rm has complete control over the market price that they receive.

Eric Dunaway (WSU) EconS 425 Industrial Organization 9 / 47 Cournot Competition

Setting up the pro…t maximization problem for …rm 1, we have

max p(Q)q1 c(q1) q1 and taking a …rst-order condition with respect to their quantitiy, we have, ∂π1 ∂p(Q) ∂Q ∂c(q1) = q1 + p(Q) = 0 ∂q1 ∂Q ∂q1 ∂q1 MR MC | {z } | {z }

Eric Dunaway (WSU) EconS 425 Industrial Organization 10 / 47 Cournot Competition

∂p(Q) ∂Q ∂c(q1) q1 + p(Q) = 0 ∂Q ∂q1 ∂q1

Since Q = q1 + q2, we have that ∂Q = 1 ∂q1 and we can rewrite our …rst-order condition as

∂p(Q) ∂c(q1) q1 + p(Q) = 0 ∂Q ∂q1

Notice that if q2 = 0, Q = q1 and we have the exact same …rst-order condition as the monopolist. Let’sderive the Lerner index again!

Eric Dunaway (WSU) EconS 425 Industrial Organization 11 / 47 Cournot Competition

∂p q1 + p MC = 0 ∂Q Rearranging terms, ∂p p MC = q1 ∂Q

This is actually a bit problematic. We have q1 on the right-hand side where we really need the total market quantity, Q. We can add it in. We just have to make sure we don’tchange the value of the right-hand side of the equation. Let’smultiply it by Q 1 = Q , which doesn’tchange the value.

∂p Q q1 1 p MC = q1 = ∂Q Q Q ∂Q 1 ∂p Q

Eric Dunaway (WSU) EconS 425 Industrial Organization 12 / 47 Cournot Competition

∂p Q q1 1 p MC = q1 = ∂Q Q Q ∂Q 1 ∂p Q

Lastly, dividing both sides by p, we get our Lerner index, which tells us by what proportion price is above marginal cost.

p MC q1 1 q1 1 = = p Q ∂Q p Q ε ∂p Q

Eric Dunaway (WSU) EconS 425 Industrial Organization 13 / 47 Cournot Competition

p MC q1 1 = p Q ε

Notice that the Lerner Index under this duopoly is almost the same as q1 under monopoly, but now it is multiplied by Q . Since Q = q1 + q2 and if both q1 and q2 are positive (as is required for duopoly), we q1 have that Q < 1. The price elasticity of demand didn’tchange between our monopoly and duopoly models. This means that if we have two …rms in the market, the Lerner index is smaller than it is under monopoly. q1 The term Q is the proportion of the total output belonging to …rm 1. We could perform a similar analysis for …rm 2 and obtain a symmetric expression. Intuitively, it means the market price under duopoly is lower.

Eric Dunaway (WSU) EconS 425 Industrial Organization 14 / 47 Cournot Competition

Let’ssee how our result changes with functional form now. We’ll use the same inverse demand function as before,

p(Q) = a bQ = a b(q1 + q2) with constant marginal cost c. Starting with …rm 1, their pro…t maximization problem it,

max (a b(q1 + q2))q1 cq1 q1 Taking a …rst-order condition with respect to their quantity,

∂π1 = a 2bq1 bq2 c = 0 ∂q1 Since this is a simultaneous move game, remember that we can derive an expression for …rm 1’sbest response to any chosen by …rm 2, i.e., their function.

Eric Dunaway (WSU) EconS 425 Industrial Organization 15 / 47 Cournot Competition

a 2bq1 bq2 c = 0

Solving for q1, we have our best response function,

a c q2 q1(q2) = 2b 2

Looking at this function, if q2 = 0, …rm 1’sbest response is to a c produce the monopoly quantity, 2b . As q2 increases, though, …rm 1’sbest response is to lower their output level as the market price decreases. a c If q2 b , it’sactually better to not produce at all, as it would yield negative pro…ts.

Eric Dunaway (WSU) EconS 425 Industrial Organization 16 / 47 Cournot Competition

q1

a • c 2b

q1(q2) q a • c 2 b

Eric Dunaway (WSU) EconS 425 Industrial Organization 17 / 47 Cournot Competition

We can perform the same analysis for …rm 2. Setting up their pro…t maximization problem,

max (a b(q1 + q2))q2 cq2 q2 and taking a …rst-order condition with respect to their quantity yields,

∂π2 = a bq1 2bq2 c = 0 ∂q2

Lastly, solving this expression for q2 gives us …rm 2’sbest response function to any output level chosen by …rm 1,

a c q1 q2(q1) = 2b 2

Eric Dunaway (WSU) EconS 425 Industrial Organization 18 / 47 Cournot Competition

q1

a • c q2(q1) b

q a • c 2 2b

Eric Dunaway (WSU) EconS 425 Industrial Organization 19 / 47 Cournot Competition

Remember from our lesson that we have a if neither player has an incentive to deviate from their given strategy. When your best response to my strategy is my best response to your strategy, and so on. In a continuous action space game, this occurs when the two best response functions intersect. Graphically, it’swhere the two lines cross. Mathematically, it’sjust a system of two equations and two unknowns.

Eric Dunaway (WSU) EconS 425 Industrial Organization 20 / 47 Cournot Competition

q1

a • c q2(q1) b

a • c 2b

q1(q2) q a • c a • c 2 2b b

Eric Dunaway (WSU) EconS 425 Industrial Organization 21 / 47 Cournot Competition

a c q2 q1 = 2b 2 a c q1 q2 = 2b 2 We can substitute the …rst equation into the second to obtain

a c 1 a c q2 q2 = 2b 2 2b 2   a c q2 = + 4b 4 Rearranging terms, and solving for q2, we have our equilibrium quantity,

3q2 a c = 4 4b a c q = 2 3b

Eric Dunaway (WSU) EconS 425 Industrial Organization 22 / 47 Cournot Competition

a c q = 2 3b Plugging this back into …rm 1’sbest response function, we have,

a c 1 a c a c q = = 1 2b 2 3b 3b   Thus, both …rms produce the exact same quantity, which is lower than a c the monopoly quantity, 2b . We can calculate the market price by plugging both of these values back into the inverse demand function,

a c a c p = a b(q + q) = a b + 1 2 3b 3b   a + 2c = 3

Eric Dunaway (WSU) EconS 425 Industrial Organization 23 / 47 Cournot Competition

Lastly, we can calculate our pro…t level. Note: Since both …rms produce the same quantity and face the same market price and costs, pro…ts are identical. The pro…t for …rm 1 is,

π = pq cq = (p c)q 1 1 1 1 a + 2c a c (a c)2 = c = 3 3b 9b     (a c)2 which is also below the monopoly pro…t level, 4b .

Eric Dunaway (WSU) EconS 425 Industrial Organization 24 / 47 Cournot Competition

Regarding welfare, since the price decreases from the monopoly level to a point closer to the perfectly competitive level, it’sfairly easy to show the following: Consumer surplus increases under duopoly, since the consumers pay a lower price. Producer surplus decreases under duopoly, since the pro…ts of the …rm decrease. Deadweight loss decreases under duopoly, since the price is lower and it allows more consumers to enter the market. To calculate these values, simple triangle formulas would su¢ ce.

Eric Dunaway (WSU) EconS 425 Industrial Organization 25 / 47 Cournot Competition

What if the …rms decided to work together and form a cartel? A cartel is an organization dedicated to …xing the price of a good or service, like OPEC. Both …rms could agree to supply half of the monopoly quantity to the market, Q a c q1 = q2 = = 2 4b and leave the market price unchanged from the monopoly price, a + c p = 2

Eric Dunaway (WSU) EconS 425 Industrial Organization 26 / 47 Cournot Competition

In this case, both …rms would receive a pro…t level of

π1 = pq1 cq1 = (p c)q1 a + c a c (a c)2 = c = 2 4b 8b     which is half the pro…ts under monopoly. More importantly, the pro…t under a cartel arrangement is strictly higher than that under Cournot competition. If both …rms were to simply restrict their output levels, they could both be better o¤. Then why don’twe see more of this?

Eric Dunaway (WSU) EconS 425 Industrial Organization 27 / 47 Cournot Competition

Just like in the prisoner’sdilemma, there are strong incentives to deviate from the cooperative solution. a c Suppose …rm 2 chose the cartel output level, 4b . Firm 1’sbest response to that output level is

a c q2 a c 1 a c 3(a c) q1 = = = 2b 2 2b 2 4b 8b   leading to a price level of

3(a c) a c 3a + 5c p = a b(q1 + q2) = a b + = 8b 4b 8  

Eric Dunaway (WSU) EconS 425 Industrial Organization 28 / 47 Cournot Competition

Lastly, for pro…ts (which will be di¤erent now since the …rms produce di¤erent quantity levels),

π1 = pq1 cq1 = (p c)q1 3a + 5c 3(a c) 9(a c)2 = c = 8 8b 64b     π2 = pq2 cq2 = (p c)q2 3a + 5c a c 3(a c)2 = c = 8 4b 32b     While the fractions make it a bit hard to see, the pro…ts for …rm 1 from cheating in the cartel agreement are higher than they are if they cooperated. Furthermore, the pro…ts for …rm 2 are quite terrible. Thus, …rm 1 (and …rm 2 for that matter) has strong incentives to deviate from any kind of cartel agreement to seize more pro…ts in the market. This is why cartels are naturally destructive.

Eric Dunaway (WSU) EconS 425 Industrial Organization 29 / 47 Cournot Competition

Let’slook at a couple of extensions of this model. Suppose now we had more than two …rms. Let n be the total number of identical …rms in the market. The inverse demand function becomes, n p = a b(q1 + q2 + ... + qn) = a b ∑ qi i=1 We can still derive our equilibrium quantities and price using the same method before. Setting up the pro…t maximization problem for …rm j, n max a b qi qi cqi q ∑ j i=1 ! we can calculate a …rst-order condition with respect to …rm j’s quantity, n ∂πj = a bqj b ∑ qi c = 0 ∂qj i=1

Eric Dunaway (WSU) EconS 425 Industrial Organization 30 / 47 Cournot Competition

n a bqj b ∑ qi c = 0 i=1

Unfortunately, we end up with n equations and n unknowns, which could be very problematic to solve. We can, however, use a math trick to help us. Since all n …rms are identical (meaning the face the same inverse demand function and have the exact same cost structure), we know that in equilibrium, all of their output levels will be the same, i.e.,

q1 = q2 = .. = qn = q

We can thus replace all of our individual values of qi with a general expression for quantity, q. This is known as invoking symmetry.

Eric Dunaway (WSU) EconS 425 Industrial Organization 31 / 47 Cournot Competition

A word of warning, though. You can only do this if both of the following conditions are met (and you better check for them on an exam!), All of the …rms are identical. You calculate your …rst-order condition …rst. If you invoke symmetry beforehand, you’llend up in the cartel solution. That being said, let’sinvoke symmetry here, let q = q1 = q2 = ... = qn

n a bqj b ∑ qi c = 0 i=1 n a bq b ∑ q c = 0 i=1 a bq nbq c = 0

Eric Dunaway (WSU) EconS 425 Industrial Organization 32 / 47 Cournot Competition

a bq nbq c = 0 From here, let’srearrange some terms,

(n + 1)bq = a c

and solving for q, we have our equilibrium quantity for all n …rms. a c q = q = i (n + 1)b

Note that if n = 1, we have our monopoly output level. Also, if n = 2, we have our duopoly output level.

Eric Dunaway (WSU) EconS 425 Industrial Organization 33 / 47 Cournot Competition

Next, let’scalculate our market price,

n n a c a + nc p = a b q = a b = ∑ i ∑ (n + 1)b n + 1 i=1 i=1   and lastly, our pro…t level,

π = pq cq = (p c)q i i i i a + nc a c (a c)2 = c = n + 1 (n + 1)b (n + 1)2b    

Eric Dunaway (WSU) EconS 425 Industrial Organization 34 / 47 Cournot Competition

a c a+nc (a c)2 qi = (n+1)b p = n+1 πi = (n+1)2b A couple of things to notice: As n increases, the individual …rm output level, market price, and individual …rm pro…t level all fall. In fact, as n approaches in…nity, look what happens, a c lim q = = 0 n ∞ i (n + 1)b ! a + nc lim p = = c n ∞ n + 1 ! (a c)2 lim π = = 0 n ∞ i (n + 1)2b ! As n becomes arbitrarily large, our market converges to the perfectly competitive solution, where the individual output level of each …rm is essentially 0, price is equal to marginal cost, and no economic pro…ts are made.

Eric Dunaway (WSU) EconS 425 Industrial Organization 35 / 47 Cournot Competition

Now, suppose we returned to our original two …rms, but now they have di¤erent marginal costs of production.

Suppose the constant marginal cost for …rm 1 was c1 while the constant marginal cost for …rm 2 was c2 and c1 < c2. This implies that …rm 1 has a cost advantage over …rm 2, as they can produce the good more cheaply. Also to note, the …rms in this case are not identical, so we can expect them to produce di¤erent quantities. This also means that we cannot invoke symmetry.

Eric Dunaway (WSU) EconS 425 Industrial Organization 36 / 47 Cournot Competition

Firm 1’spro…t maximization problem is

max (a b(q1 + q2))q1 c1q1 q1 with …rst-order condition with respect to their quantity,

∂π1 = a 2bq1 bq2 c1 = 0 ∂q1

Like before, if we solve this expression for q1, we have …rm 1’s optimal quantity given any output level from …rm 2, i.e., their best response function, a c1 q2 q1(q2) = 2b 2

Eric Dunaway (WSU) EconS 425 Industrial Organization 37 / 47 Cournot Competition

q1

a • c1 2b

q1(q2)

q2

Eric Dunaway (WSU) EconS 425 Industrial Organization 38 / 47 Cournot Competition

Likewise, performing the same analysis for …rm 2, their pro…t maximization problem is

max (a b(q1 + q2))q2 c2q2 q2 with …rst-order condition with respect to their quantity,

∂π2 = a bq1 2bq2 c2 = 0 ∂q2

And again, if we solve this expression for q2, we have …rm 2’sbest response function, a c2 q1 q2(q1) = 2b 2

Eric Dunaway (WSU) EconS 425 Industrial Organization 39 / 47 Cournot Competition

q1

q2(q1)

q2 a • c2 2b

Eric Dunaway (WSU) EconS 425 Industrial Organization 40 / 47 Cournot Competition

q1

q2(q1)

a • c1 2b

q1(q2)

q2 a • c2 2b

Eric Dunaway (WSU) EconS 425 Industrial Organization 41 / 47 Cournot Competition

a c1 q2 q1 = 2b 2 a c2 q1 q2 = 2b 2 Once again, we can solve this system of two equations and two unknowns. Substituting the top equation into the bottom,

a c2 1 a c1 q2 q2 = 2b 2 2b 2   a + c1 2c2 q2 = + 4b 4 and rearranging terms,

3q2 a + c1 2c2 = 4 4b a + c1 2c2 q = 2 3b

Eric Dunaway (WSU) EconS 425 Industrial Organization 42 / 47 Cournot Competition

a + c1 2c2 q = 2 3b Plugging this value back into …rm 1’sbest response function,

a c1 1 a + c1 2c2 a 2c1 + c2 q = = 1 2b 2 3b 3b   Since we assumed that c1 < c2, we have that q1 > q2. This should make sense. Since …rm 1 has a cost advantage over …rm 2, they can supply a higher quantity to the market and enjoy higher pro…ts even with a lower pro…t margin. Our market price is,

a 2c1 + c2 a + c1 2c2 p = a b(q + q) = a b + 1 2 3b 3b   a + c + c = 1 2 3

Eric Dunaway (WSU) EconS 425 Industrial Organization 43 / 47 Cournot Competition

Lastly, for pro…ts, since the …rms are asymmetric, we’ll calculate them separately,

π1 = pq1 c1q1 = (p c1)q1 2 a + c1 + c2 a 2c1 + c2 (a 2c1 + c2) = c1 = 3 3b 9b     π2 = pq2 c2q2 = (p c2)q2 2 a + c1 + c2 a + c1 2c2 (a + c1 2c2) = c2 = 3 3b 9b    

Again, it should be fairly clear that since c1 < c2, the pro…ts for …rm 1 are higher than those for …rm 2.

Note: If we set c1 = c2, all of our results collapse back into the original Cournot model.

Eric Dunaway (WSU) EconS 425 Industrial Organization 44 / 47 Summary

Cournot competition allows for …rms to independently set their output levels and compete in a market. As more …rms enter the market, the price gradually falls from the monopoly price to the perfectly competitive price.

Eric Dunaway (WSU) EconS 425 Industrial Organization 45 / 47 Next Time

Bertrand Competition. What happens when two or more …rms compete in prices? Comparing the di¤erences between price and quantity competition. Also, some useful applications. Reading: 7.4-7.5 Note: No class Monday due to Holiday

Eric Dunaway (WSU) EconS 425 Industrial Organization 46 / 47 Practice Problem

Return to our Cournot competition model with two …rms. Suppose now that the products aren’tquite identical. In fact, both …rms face di¤erent demand functions,

p1 = a b(q1 + dq2) p2 = a b(dq1 + q2) where d is some number between 0 and 1 that speci…es how similar the products are. Everything else about the model remains the same. 1. Calculate the best response functions for each …rm. How does the slope of those best responses change with d? What happens when d = 0 or d = 1? 2. Calculate the equilibrium prices and quantities in this market. (Note: the …rms are not identical).

Eric Dunaway (WSU) EconS 425 Industrial Organization 47 / 47