<p>In dealing with problems like this, it's helpful to do it in stages. You know that the two dealers take the wholesale price <span class="math-container">$r$</span> as a given (they can't change that), and car dealer <span class="math-container">$i$</span> choose retail price <span class="math-container">$q_i$</span> to try to maximize their profits <span class="math-container">$q_i \cdot (P - r)$</span>.</p>

<p>You also know that these car dealers are exactly the same... they have the same cost, set some price, face the same demand (we say the two firms are "symmetric"). So in the end you know that whatever is the best quantity <span class="math-container">$q_i$</span> for the first car dealer would be the best choice for the second dealer as well, so we know <span class="math-container">$q_1 = q_2 = Q/2$</span>.</p>

<p>With this, the dealers are choosing <span class="math-container">$Q$</span> to maximize <span class="math-container">$\pi(Q, r) = \frac Q 2 \cdot (P - r)$</span>, with <span class="math-container">$P$</span> given by the demand function (solve the inverse total demand for <span class="math-container">$P$</span>). Let's call this profit-maximizing quantity <span class="math-container">$Q^* = 2q_1^* = 2q_2^*$</span> and the resulting price <span class="math-container">$P^*$</span>.</p>

<p>Now that we're done with the car dealer, let's step in Toyota's shoes. You want to make as much money as you could, so the higher <span class="math-container">$r$</span> is the better off you are. But you also know that if you set the retail price too high, the dealers will lose money with each car they sell and will likely stop ordering from you (maybe switch to selling Hondas instead). What's the maximum you could charge while still keep the dealers not losing money?</p>

CC BY-SA 4.0
2019-09-24T02:02:43.247Z
2019-09-24T02:02:43.247Z
Art
<p>So I´m a bit stuck on this one. There are two firms in the country that sell cars. Both sell Toyotas and they buy their cars directly from the manufacturer, which is Toyota. We can assume that quantity is the strategic variable, and that the two car dealers have no other marginal costs than the wholesale price "r".</p>

<p>The inverse total demand:
P = 1500 - 5(q1 + q2)</p>

<p>MC = 30 </p>

<p>And I got:</p>

<p>q1 + q2 = Q</p>

<p>Q = 2(1500-r)/(2+1)5 = (3000-2r)/15</p>

<p>Which <strong>wholesale price, r</strong>, will the Toyota set? What is the <strong>equilibrium retail price, p,</strong> and the <strong>number of units sold?</strong></p>

<p>When I solve for "r" I get:</p>

<p>r = 1500 - 7.5Q </p>

<p>But after this I am stuck and I can´t figure out what to do next. Anyone with any suggestions on how I should proceed or point me in the right direction?</p>

<microeconomics><financial-economics>
Two firms competing ala Cournot
Batman Team
# Two firms competing ala Cournot

(Asked by: username_0 on 2019-09-24)

So I´m a bit stuck on this one. There are two firms in the country that sell cars. Both sell Toyotas and they buy their cars directly from the manufacturer, which is Toyota. We can assume that quantity is the strategic variable, and that the two car dealers have no other marginal costs than the wholesale price "r".

The inverse total demand:
P = 1500 - 5(q1 + q2)

MC = 30

And I got:

q1 + q2 = Q

Q = 2(1500-r)/(2+1)5 = (3000-2r)/15

Which **wholesale price, r**, will the Toyota set? What is the **equilibrium retail price, p,** and the **number of units sold?**

When I solve for "r" I get:

r = 1500 - 7.5Q

But after this I am stuck and I can´t figure out what to do next. Anyone with any suggestions on how I should proceed or point me in the right direction?

**Tags:** microeconomics, financial-economics

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## 1 answers

### Answer by username_1 on 2019-09-24. Score: 1

In dealing with problems like this, it's helpful to do it in stages. You know that the two dealers take the wholesale price $r$ as a given (they can't change that), and car dealer $i$ choose retail price $q\_i$ to try to maximize their profits $q\_i \cdot (P - r)$.

You also know that these car dealers are exactly the same... they have the same cost, set some price, face the same demand (we say the two firms are "symmetric"). So in the end you know that whatever is the best quantity $q\_i$ for the first car dealer would be the best choice for the second dealer as well, so we know $q\_1 = q\_2 = Q/2$.

With this, the dealers are choosing $Q$ to maximize $\pi(Q, r) = \frac Q 2 \cdot (P - r)$, with $P$ given by the demand function (solve the inverse total demand for $P$). Let's call this profit-maximizing quantity $Q^\* = 2q\_1^\* = 2q\_2^\*$ and the resulting price $P^\*$.

Now that we're done with the car dealer, let's step in Toyota's shoes. You want to make as much money as you could, so the higher $r$ is the better off you are. But you also know that if you set the retail price too high, the dealers will lose money with each car they sell and will likely stop ordering from you (maybe switch to selling Hondas instead). What's the maximum you could charge while still keep the dealers not losing money?


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