# How do we find a competitive equilibrium price when 2 firms operate with different marginal costs?

Consider a homogenous goods market with the demand function Q = 30 − P, where Q and P denote quantity and price respectively. There are two firms playing a price game in the following manner: firm 1 quotes a price and then firm 2 chooses a price. When they charge the same price they share the market equally and otherwise the market demand goes to the firm charging lower price. Firm 1 has a capacity constraint at the output level 5 units such that upto five units the marginal cost of production is Rs 3 per unit of output, however beyond 5 units it cannot produce any output. Firm 2 does not have any capacity constraint, it can produce any amount with the marginal cost Rs 6. What would be the equilibrium price in the market?

I'm confused, I think we have to put demand=supply firstly, then if the equilibrium price is 6 (the answer), firm 1 will produce 5 goods but firm 2 has no incentive to produce anything. Will the price not be a little greater than 6?

• The key for solving this is the concept of "residual demand." Once firm 1 makes its 5 units, firm 2 will face the demand curve of q2 = 25 - p2. However, I think there might be some issues with the way the problem is written. I don't believe 6 is the correct answer, and what does it mean to "share the market" given the capacity constraint? You might well be having trouble because the question doesn't make sense! I'd go back and read it again to make sure you're understanding the game correctly. – NickCHK Apr 16 '17 at 18:59
• Hey, thanks for the comment. The question is exactly this, doubled checked, and was a part of an entrance examination so I wouldn't really doubt the credibility. If we ignore the fact that the question may be flawed, what would be the approach to solve such a question? I understand equating demand and supply as a basic rule, but how do we arrive at the equilibrium behaviour of firms? – Arshdeep Singh Duggal Apr 16 '17 at 20:54
• Firm 1 wants to avoid setting a price which encourages Firm 2 to profitably undercut that price and take all the market, reducing Firm 1's profit to zero. Firm 2 then want to set a price which maximises profit taking account of its marginal cost and the residual demand. – Henry Apr 16 '17 at 21:03
• @NickCHK: It looks to me that if Firm 1 sets a price minimally below $18-\sqrt{53.75}$, let's say $p_1=10.66$, and Firm 2 then sets $p_2=15.5$, this leads to $q_1=5$ and $q_2=9.5$ and neither can make higher profits in this situation. (If I have not made a calculation error) It is curious that $p_2$ is so much higher than $p_1$ despite Firm 1 having lower marginal costs. – Henry Apr 16 '17 at 21:05
• @Henry I think that's probably about right. The only thing that makes me wonder if the problem is written incorrectly is that (1) as the question asker suspects, I don't think p2 = 6 is the correct answer, as seems to be coming from the answer key (?), and (2) it's not clear as to what "split the market" means in this context, since they can't actually share the market equally if Q>10. In any case, I'll write an answer below ignoring these issues. – NickCHK Apr 16 '17 at 21:54

Ignoring the previously mentioned possibility that the problem may be written incorrectly: The concept of "residual demand" will be helpful here.

As with any sequential game, it's good to work backwards. From the perspective of firm 2, firm 1 has quoted some price $p_1$, and firm 2 has an opportunity to respond.

Firm 2 knows that if it sets $p_2<p_1$, it will take the whole market, and will face the whole demand curve $Q = 30-p_2$. It can choose the profit-maximizing $p_2$, as long as $p_2<p_1$.

Similarly, if it sets $p_2=p_1$, it will share the market.

Finally, if it sets $p_2 > p_1$, it will face the residual demand left over after firm 1 makes its sales. Assuming that $p_1\leq 25$, firm 1 will sell its five units and then not be able to sell any more. The demand left over for firm 2 to sell to after firm 1 makes its sales will then be $q_2=30-p_2-5$, or $q_2=25-p_2$. Firm 2 can choose its profit-maximizing price given this demand curve.

In each of these cases above, Firm 2 is selecting the profit-maximizing quantity by first setting up profit:

$$\Pi_2 = p_2q_2 - 6q_2 = (p_2-6)q_2$$

Plug in the appropriate $q_2$ using the demand functions determined above, then take the derivative with respect to $p_2$ and set the derivative equal to 0, solve for $p_2$. In each case you're constrained by what $p_2$ can be. If you're working with the $Q=30-p_2$ demand function (which you only get if $p_2<p_1$), then if profit-maximizing price is $p_2>p_1$, then you'll actually want to set $p_2$ just a hair underneath $p_1$ (you can just call it $p_2=p_1$ for simplicity).

Now you've got the optimal price for Firm 2 to pick given that it's decided to go for $p_2<p_1$, $p_2=p_1$, or $p_2>p_1$. To figure out which of those three it wants, you'll want to calculate Firm 2's profit under each of those scenarios. These profits should be dependent on $p_1$, since $p_1$ determines the boundaries of what $p_2$ can be while still satisfying $p_2<p_1$, $p_2=p_1$, or $p_2>p_1$.

So now Firm 1 knows how Firm 2 will respond to whatever $p_1$ is. And so Firm 1 will set $p_1$ in a way that maximizes Firm 1's profit. This step is a bit easier, since Firm 1 makes nothing under $p_2<p_1$, so it's unlikely that Firm 1 will want to set $p_1$ so that Firm 2 picks $p_2<p_1$.

To make this a bit more clear, let's do an example for Firm 2 making their decision.

Let's say Firm 1 had picked $p_1=10$.

If Firm 2 sets $p_2<p_1$, it is dealing with the demand function $$q_2 = 30-p_2$$. So we set up profit

$$\Pi_2 = (p_2-6)(30-p_2)$$ $$\frac{\partial \Pi_2}{\partial p_2} = 36 - 2p_2 = 0 \Rightarrow p_2=18$$

which is above $p_1=10$. Since profit is strictly increasing in $p_2$ for $p_2<10$, firm 2 will maximize its profit while holding $p_2<p_1$ by setting $p_2=10$ (a hair below 10, really, but let's just say 10). The profit from doing so is

$$\Pi_2 = (10-6)(30-10) = 80$$

And, of course, Firm 1 sells nothing and makes a profit of 0. Now if Firm 2 sets $p_2>p_1$, it's dealing with the demand function

$$q_2 = 25 - p_2$$

and wants to maximize profit

$$\Pi_2 = (p_2-6)(25-p_2)$$ $$\frac{\partial \Pi_2}{\partial p_2} = 31 - 2p_2 = 0 \Rightarrow p_2=15.5$$

which is above $p_1=10$, satisfying $p_2>p_1$. This price leads to the profit

$$\Pi_2 = (15.5-6)(25-15.5)=90.25$$

In this case, Firm 1 sells their five units and makes a profit of $(10-3)5=35$.

You will also want to check the $p_2=p_1$ case once it's clear what that actually leads to.

And so, if Firm 1 sets $p_1=10$, Firm 2 will want to set $p_2=15.5$, since a profit of $90.25$ is better than a profit of $80$.

This example is worked out with a specific $p_1$, and this particular $p_1$ makes Firm 2 not undercut Firm 1. The next step is to find out what the highest price that Firm 1 can set without getting Firm 2 to prefer undercutting. This will be $p_1$ in equilibrium, and Firm 2's best response to that $p_1$ will be $p_2$ in equilibrium.

• I think Firm 1 wants $(p_1-6)(20-p_1) \lt (15-6)(25-15)$ so Firm 2 does not decide to take all the market by charging minimally less than $p_1$ and instead takes all but $5$ of the quantity with the profit maximising price on the residual demand. This leads to $p_1 \lt 18-\sqrt{53.75} \approx 10.66856$ – Henry Apr 16 '17 at 23:50
• I believe the numerical answer you have is correct, but the inequality you've set up has some errors in it. Also, I was trying to get him started on the problem without finishing it. – NickCHK Apr 17 '17 at 2:36