Short-run price competition with capacity contraints

Question

I am doing an undergrad course in Industrial Organization and at the end of the chapter on Short-run price competition (Bertrand and Cournot models mainly), there is a question with which I'm struggling. Here it is:

Consider a market with two price-setting firms producing a homogeneous product. The demand function is q = D(p) = 1 – p, which implies the inverse demand p = 1 – q1 – q2. The two firms have capacity constraints q1hat and q2hat , where q1hat + q2hat = 3/5. The marginal cost of production is zero for qi ≤ qihat and infinite for qi > qihat for any i = 1,2. Finally, assume that consumers are rationed according to the efficient rationing rule.

i. Show that if q1hat = q2hat , there is a unique Bertrand-Nash equilibrium where p1 = p2 = p* = 1 – q1hat – q2hat

ii. Show that when q1hat ≠ q2hat, the equilibrium under part (i) breaks down when the firms’ capacities are too dissimilar.

So for part i. I basically adapted part of the material from the chapter and said:

To show that this is a Nash equilibrium, we need to show that none of the firms has an incentive to unilaterally deviate from this equilibrium.

Is it profitable for firm 1 to set a price lower than p*, given that firm 2 sets price p*? The answer is no. By charging p* firm 1 sells exactly q1hat. Firm 1 cannot produce more than q1hat anyway, so by reducing its price below p* it would simply sell the same quantity at a lower price and would therefore make less profit.

Is it profitable for firm 1 to set a price higher than p*, given that firm 2 sets price p*? The answer is again no, but the argument now is more subtle. Suppose that firm 1 sets a price p ≥ p*. Then it has residual demand 1 – p – q2hat, because at price p total market demand is given by 1 – p and firm 2 sells q2hat. Firm 1 makes profit Π = p(1 – p – q2hat). Using the inverse demand function, the expression for profit can be written as (1 – q1 – q2hat)q1, where q1 is the quantity sold by firm 1 at price p. Note that the profit function Π = (1 – q1 – q2hat)q1 is exactly the same as the profit function of a firm that chooses output q given that the rival firm chooses output q2hat. This profit function is concave in q, that is Π''(q1) < 0. Also, ∂Π/∂q1 = 1 – 2q1 – q2hat. Evaluated at q1 = q1hat, this derivative is equal to 1 – 2q1hat – q2hat, which is positive because q1hat + q2hat = 3/5 and q1hat = q2hat so 2q1hat + q2hat = 9/10. In other words, if firm 1 starts from q1hat and marginally reduces its quantity, its profit will fall. This result and the concavity of the profit function ensure that any reduction of q1 below q1hat will reduce profit. Another way of saying this is that if firm 1 starts from p* and increases its price, its profit will fall.

So far so good, the concavity of the profit function implies that when we close in to the maximum marginal profit, the slope (or the derivative of profit with respect to q) will be closer and closer to 0.

Now part ii. gets me and I don't know how to attack it. I thought about making 2 cases, one where q1hat is close to 3/5 and one when it is close to 0. The profit function and its derivative are the same as for i. and the constraint q1hat + q2hat = 3/5 is still applicable from what I can understand, so demand can't be met. For me the part about setting a lower p is still the same as in part i. But the part about setting a p higher than p* is not as clear.

Can anybody help?

Answer 1

If Firm 1 assumes Firm 2 produces and sells the maximum it can produce, i.e. $\hat{q}_2$, then Firm 1 can aim to maximise its revenue and profit $$\Pi_1=q_1p = q_1(1-q_1-\hat{q}_2)$$ which is maximised at $\left(\dfrac{1-\hat{q}_2}{2}\right)^2$ with $q_1=\dfrac{1-\hat{q}_2}{2}$, at least when Firm 1's maximum production exceeds this quantity. If not, then Firm 1 should continue to produce as much as possible. So if Firm 1 is big enough then both firms producing as much as possible may not be an equilibrium

As an example, suppose $\hat{q}_1=0.5$ and $\hat{q}_2=0.1$, with a sum of $0.6=\frac35$ as required. If both firms produce at maximum capacity then the price will be $1-0.5-0.1=0.4$ making the Firm 1's revenue and profit $0.5\times 0.4=0.2$

But if Firm 2 produces at maximum capacity and Firm 1 were to produce only $0.45$, this would make the price $1-0.45-0.1=0.45$ and the Firm 1 's revenue and profit $0.45\times 0.45=0.2025$ which is higher than $0.2$, and the Firm 2 will also see profits rise due to the price increase, showing that both firms being at maximum production would not be an equilibrium in this case

With the overall maximum production being $\frac35=0.6$, the equilibrium will break down in this way if $\hat{q}_1\gt \frac25 =0.4$ and $\hat{q}_2 \lt \frac15 =0.2$