<p><strong>Hint:</strong> To find equilibrium price and quantity, you want to find when the system is in equilibrium. In equilibrium, quantity demanded should equal quantity supplied.</p>

<p>$$Q_d = Q_s \implies -300,000P + 6,000,000 = 200,000 \sum_{i=1}^n (c_i'(q_i))^2 - 200,000$$</p>

<p>Where $c_i(q_i)$ is the cost function for firm $i$ with respect to $q_i$, firm $i$'s production. I'm assuming when you wrote $MC$ you meant marginal cost.</p>

<p>$$\implies 6,200,000 - 200,000\sum_{i=1}^n (c_i'(q_i))^2 = 300,000P$$</p>

<p>$$\implies P = \frac{62}{3} - \frac{2}{3} \sum_{i=1}^n (c_i'(q_i))^2$$</p>

<p>Now that we have an inverse demand function with respect to $q$, we have to solve the firms' maximization problem. Since there's no information you've given to us directly about that, let's assume technology is homogeneous and that firms compete via Cournot competition.</p>

<p>$$\Pi_j = q_j(P(q)) - c(q_j)$$</p>

<p>$$\Pi_j =  q_j(\frac{62}{3} - \frac{2}{3} \sum_{i=1}^n (c'(q_i))^2) - c(q_j)$$</p>

<p>Now take the derivative w.r.t $q_j$ and set to zero to find the maximum.</p>

<p>$$\Pi_j' =  (\frac{62}{3} - \frac{2}{3} \sum_{i=1, i \neq j}^n (c'(q_i))^2) + \frac{2}{3} \frac{d}{dq_j} (c'(q_j)^2 \cdot q_j) - c'(q_j) = 0$$</p>

<p>Where that derivative is evaluated with the product and chain rule.</p>

<p>From here, you solve for $q_j$ as an expression of all $q_{i, i \neq j}$. Since the cost functions are homogeneous, it shouldn't be that bad to do. Also remember that the number of suppliers is 1000, so the problem can be further simplified:</p>

<p>$$\Pi_j' =  (\frac{62}{3} - \frac{2}{3} \cdot 999(c'(q_i))^2) + \frac{2}{3} \frac{d}{dq_j} (c'(q_j)^2 \cdot q_j) - c'(q_j) = 0$$</p>

<p>$$\implies \Pi_j' =  (\frac{62}{3} - 666(c'(q_i))^2) + \frac{2}{3} (2c'(q_j) \cdot c''(q_j) \cdot q_j + c'(q_j)^2) - c'(q_j) = 0$$</p>

<p>and then you solve for $q_j$ in terms of $q_{i, i \neq j}$ and do the rest of the Cournot derivation.</p>

CC BY-SA 3.0
2015-10-20T00:59:05.780Z
The crucial hint that can be exploited is the number of suppliers.
Alecos Papadopoulos
CC BY-SA 3.0
2015-10-20T01:03:05.533Z
Aha! I knew I was forgetting something.
Kitsune Cavalry
CC BY-SA 3.0
2015-10-20T00:56:55.897Z
2015-10-20T01:17:31.477Z
2015-10-20T01:17:31.477Z
Kitsune Cavalry
<p>I have a market with 1000 suppliers.</p>

<p>The total demand can be described by</p>

<p>$$
Q=-300000 P + 6000000
$$</p>

<p>while the total supply can be described by</p>

<p>$$
Q=200000 \sum(MC^2)-200000
$$</p>

<p>What is the price and quantity in equilibrium?</p>

<p>I have no idea what to do with $\sum(MC^2)$ in the supply.</p>

<p>I guess I should isolate $P$ in the first equation:
$$
P(Q) = 20-\frac{1}{300000} Q
$$
but I don't know what to do now.</p>

CC BY-SA 3.0
2015-10-20T01:15:14.340Z
I guess you are allowed to assume that this is a perfectly competitive market (which among other things implies identical suppliers). Then a) remember what is the relation between price, marginal cost and average cost in such a market in long run equilibrium, and b) if something is a derivative of something else, then integration can lead you to this something else.
Alecos Papadopoulos
CC BY-SA 3.0
2015-11-19T03:47:11.580Z
Please see http://meta.economics.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions?cb=1 for the policy on homework questions.
HRSE
<supply-and-demand>
Price and quantity in equilibrium
Jamgreen
# Price and quantity in equilibrium

(Asked by: username_0 on 2015-10-19)

I have a market with 1000 suppliers.

The total demand can be described by

$$
Q=-300000 P + 6000000
$$

while the total supply can be described by

$$
Q=200000 \sum(MC^2)-200000
$$

What is the price and quantity in equilibrium?

I have no idea what to do with $\sum(MC^2)$ in the supply.

I guess I should isolate $P$ in the first equation:
$$
P(Q) = 20-\frac{1}{300000} Q
$$
but I don't know what to do now.

**Tags:** supply-and-demand

> **Comments:**
> **username_1**: I guess you are allowed to assume that this is a perfectly competitive market (which among other things implies identical suppliers). Then a) remember what is the relation between price, marginal cost and average cost in such a market in long run equilibrium, and b) if something is a derivative of something else, then integration can lead you to this something else.  
> **username_2**: Please see http://meta.economics.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions?cb=1 for the policy on homework questions.  

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## 1 answers

### Answer by username_3 on 2015-10-20. Score: 0

**Hint:** To find equilibrium price and quantity, you want to find when the system is in equilibrium. In equilibrium, quantity demanded should equal quantity supplied.

$$Q\_d = Q\_s \implies -300,000P + 6,000,000 = 200,000 \sum\_{i=1}^n (c\_i'(q\_i))^2 - 200,000$$

Where $c\_i(q\_i)$ is the cost function for firm $i$ with respect to $q\_i$, firm $i$'s production. I'm assuming when you wrote $MC$ you meant marginal cost.

$$\implies 6,200,000 - 200,000\sum\_{i=1}^n (c\_i'(q\_i))^2 = 300,000P$$

$$\implies P = \frac{62}{3} - \frac{2}{3} \sum\_{i=1}^n (c\_i'(q\_i))^2$$

Now that we have an inverse demand function with respect to $q$, we have to solve the firms' maximization problem. Since there's no information you've given to us directly about that, let's assume technology is homogeneous and that firms compete via Cournot competition.

$$\Pi\_j = q\_j(P(q)) - c(q\_j)$$

$$\Pi\_j = q\_j(\frac{62}{3} - \frac{2}{3} \sum\_{i=1}^n (c'(q\_i))^2) - c(q\_j)$$

Now take the derivative w.r.t $q\_j$ and set to zero to find the maximum.

$$\Pi\_j' = (\frac{62}{3} - \frac{2}{3} \sum\_{i=1, i \neq j}^n (c'(q\_i))^2) + \frac{2}{3} \frac{d}{dq\_j} (c'(q\_j)^2 \cdot q\_j) - c'(q\_j) = 0$$

Where that derivative is evaluated with the product and chain rule.

From here, you solve for $q\_j$ as an expression of all $q\_{i, i \neq j}$. Since the cost functions are homogeneous, it shouldn't be that bad to do. Also remember that the number of suppliers is 1000, so the problem can be further simplified:

$$\Pi\_j' = (\frac{62}{3} - \frac{2}{3} \cdot 999(c'(q\_i))^2) + \frac{2}{3} \frac{d}{dq\_j} (c'(q\_j)^2 \cdot q\_j) - c'(q\_j) = 0$$

$$\implies \Pi\_j' = (\frac{62}{3} - 666(c'(q\_i))^2) + \frac{2}{3} (2c'(q\_j) \cdot c''(q\_j) \cdot q\_j + c'(q\_j)^2) - c'(q\_j) = 0$$

and then you solve for $q\_j$ in terms of $q\_{i, i \neq j}$ and do the rest of the Cournot derivation.

> **Comments:**
> **username_1**: The crucial hint that can be exploited is the number of suppliers.  
> **username_3**: Aha! I knew I was forgetting something.  

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