<p>I assume that the firm is a monopolist.</p>
<p>We know the inverse demand function, the fixed costs and the marginal costs:</p>
<ul>
<li><span class="math-container">$P(q) = 20 - q$</span></li>
<li><span class="math-container">$MC(q) = 12$</span></li>
<li><span class="math-container">$F = 16$</span></li>
</ul>
<p>As the marginal costs are constant, we can compute the total costs function as:</p>
<ul>
<li><span class="math-container">$TC(q) = MC\cdot q + F = 12 q + 16$</span></li>
</ul>
<p>The average total costs is obtained by dividing by <span class="math-container">$q$</span>:</p>
<ul>
<li><span class="math-container">$AC(q) = \frac{TC(q)}{q} = 12 + \frac{16}{q}$</span></li>
</ul>
<p>The demand curve is obtained by inverting the inverse demand function:</p>
<ul>
<li><span class="math-container">$q(P) = 20 - P$</span></li>
</ul>
<p>Total revenue is obtained by multiplying together price and quantity:</p>
<ul>
<li><span class="math-container">$R(q) = P(q)\cdot q = (20-q)\cdot q = 20 q - q^2$</span></li>
</ul>
<p>Marginal revenue is obtained by taking the derivative of revenue with respect to <span class="math-container">$q$</span></p>
<ul>
<li><span class="math-container">$MR(q) = \frac{\partial R(q)}{\partial q} = 20 - 2q$</span>.</li>
</ul>
<p>If the firm maximizes profits, then the marginal revenue equals the marginal costs so:</p>
<ul>
<li><span class="math-container">$12 = 20 - 2q \to q = 4$</span></li>
</ul>
<p>The optimal price can be found by substituting <span class="math-container">$q = 4$</span> into the inverse demand function:</p>
<ul>
<li><span class="math-container">$P = 20 - 4 = 16$</span></li>
</ul>
<p>Profits are determined as total revenue minus total costs:</p>
<ul>
<li><span class="math-container">$\pi = R - TC = P\cdot q - TC = 16 \cdot 4 - (12 \cdot 4 + 16) = 0$</span></li>
</ul>

CC BY-SA 4.0
2021-04-10T09:31:08.267Z
2021-04-10T09:31:08.267Z
tdm
<p>a)Determine marginal revenue curve if firm can only charge 1 price,List demand curve,marginal revenue curve,,marginal cost,average total cost equations.</p>
<p>b)What is optimal price and quantity?</p>
<p>c)If firm fixed cost are 16 dollars, what are the profits?</p>
<p>P=20-q</p>
<p>MC=12</p>
<p>Attempt:</p>
<p>I tried</p>
<p>mc=c/p</p>
<p>12*20-q=x</p>
<p>to find P*X=MR</p>
<p>b)</p>
<p>12=240-12q</p>
<p>q=-228/12</p>
<p>c)</p>
<p>MC=6</p>
<p>6=240-12q/20-q</p>
<p>q=24</p>
<p>The numerical answer is wrong so what mistake did I make? What's the correct equation to solve for test equation?</p>

<demand><marginal><cost-functions>
Determine marginal revenue given demand curve and marginal cost
JanusP
# Determine marginal revenue given demand curve and marginal cost

(Asked by: username_0 on 2021-04-08)

a)Determine marginal revenue curve if firm can only charge 1 price,List demand curve,marginal revenue curve,,marginal cost,average total cost equations.

b)What is optimal price and quantity?

c)If firm fixed cost are 16 dollars, what are the profits?

P=20-q

MC=12

Attempt:

I tried

mc=c/p

12\*20-q=x

to find P\*X=MR

b)

12=240-12q

q=-228/12

c)

MC=6

6=240-12q/20-q

q=24

The numerical answer is wrong so what mistake did I make? What's the correct equation to solve for test equation?

**Tags:** demand, marginal, cost-functions

---

## 1 answers

### Answer by username_1 on 2021-04-10. Score: 2

I assume that the firm is a monopolist.

We know the inverse demand function, the fixed costs and the marginal costs:

* $P(q) = 20 - q$
* $MC(q) = 12$
* $F = 16$

As the marginal costs are constant, we can compute the total costs function as:

* $TC(q) = MC\cdot q + F = 12 q + 16$

The average total costs is obtained by dividing by $q$:

* $AC(q) = \frac{TC(q)}{q} = 12 + \frac{16}{q}$

The demand curve is obtained by inverting the inverse demand function:

* $q(P) = 20 - P$

Total revenue is obtained by multiplying together price and quantity:

* $R(q) = P(q)\cdot q = (20-q)\cdot q = 20 q - q^2$

Marginal revenue is obtained by taking the derivative of revenue with respect to $q$

* $MR(q) = \frac{\partial R(q)}{\partial q} = 20 - 2q$.

If the firm maximizes profits, then the marginal revenue equals the marginal costs so:

* $12 = 20 - 2q \to q = 4$

The optimal price can be found by substituting $q = 4$ into the inverse demand function:

* $P = 20 - 4 = 16$

Profits are determined as total revenue minus total costs:

* $\pi = R - TC = P\cdot q - TC = 16 \cdot 4 - (12 \cdot 4 + 16) = 0$


---

