<p>Suppose that, in a perfectly competitive industry, the firms' technology have the following cost function: <span class="math-container">$C(x) = 100 + 3x + 0.04x^2$</span>. Assume the fixed costs are sunken.</p>
<p>a) If the demand for the product is given by <span class="math-container">$D(p) = 200(10-p)$</span>, what is the long-run equilibrium for this industry? (price, quantity demanded and consumed, number of firms).</p>
<p>In the long-run equilibrium, the price should equal the minimum average cost. That happens when <span class="math-container">$MC = AC$</span>, which means:
<span class="math-container">$$
3 + 0.08x = \frac{100}{x} + 3 + 0.04x \implies x = 50, x &gt; 0
$$</span></p>
<p>For this value of <span class="math-container">$x$</span>, we have, equating <span class="math-container">$p = CMg$</span>:
<span class="math-container">$$
p = 3 + 0.08 \cdot 50 = 7
$$</span>
and, therefore,
<span class="math-container">$$
D(7) = Q^D = 600
$$</span>
which means that the number of firms is <span class="math-container">$n = 600/50 = 12$</span>.</p>
<p>b) Under a negative and permanent demand shock, it dislocates to <span class="math-container">$D(p) = 200(8-p)$</span>. In the very-short run, the firms cannot change their production plans. What is the new equilibrium price? In the medium-run, however, the firms can adjust their production plans. Find the medium-run equilibrium (price, quantity produced and consumed, number of firms).</p>
<p>If in the very-short run the productions plans cannot be changed, let's assume the total market output is still 600. Using the new demand function:
<span class="math-container">$$
600 = 200(8-p) \implies p = 5
$$</span></p>
<p>In the medium-run, the firms can adjust their production plans and therefore we should find the value of <span class="math-container">$q$</span> that satisfies <span class="math-container">$MC = AVC$</span>:
<span class="math-container">$$
3 + 0.08x = 3 + 0.04x \implies x = 0
$$</span>
And so:
<span class="math-container">$$
p = 3 + 0.08 \cdot 0 \implies p = 3
$$</span>
and therefore
<span class="math-container">$$
D(3) = Q^D = 1000
$$</span>
However, since the best option for the firms is to not produce, we can (can we?) suppose that all of them leave the market, resulting in <span class="math-container">$n = 0$</span>.</p>
<p>c) Find the new long-run equilibrium (price, quantity produced and consumed, number of firms). Compare your results with the medium-run equilibrium and interpret.</p>
<p>In the new long-run, we will again have <span class="math-container">$MC = AC$</span>, resulting in a price of <span class="math-container">$p = 7$</span> and <span class="math-container">$x = 50$</span> again. However, given the change in the demand function, the quantity demanded now will be
<span class="math-container">$$
D(7) = 200(8-7) = 200
$$</span>
which means that the new number of firms is <span class="math-container">$n = 200/50 = 4$</span>, an increase for the medium-run, due to the absence of fixed-costs in the long-run.</p>
<p>Are my answers correct? I had never encountered a question asking about medium-run before. Also, is the profit maximization level of production different from the equilibrium level of production?</p>
<p>Thanks!</p>

CC BY-SA 4.0
2021-03-14T21:29:30.923Z
b) looks wrong to me. The way I read the assignment firms are in the medium term able to adjust quantity but firms are not able to enter or exit market. So p=MC and Supply(p) = Demand(p), with supply being 12*(p-3)/0.08 = 12x ... 12 firms all same production technology/cost function.
Jesper Hybel
CC BY-SA 4.0
2021-03-14T23:49:01.713Z
That was very helpful. Following that path, I found x = 70, p' = 3.8 and q' = 840.  How to interpret the change in the number of firms? Some firms decided to leave the market in the long-run after the reduction in demand? Thanks!
Pedro Cunha
<microeconomics><profit-maximization><competitive-equilibrium><nonexpert>
Short, Medium and Long-Run Profit Maximization
Pedro Cunha
# Short, Medium and Long-Run Profit Maximization

(Asked by: username_0 on 2021-03-14)

Suppose that, in a perfectly competitive industry, the firms' technology have the following cost function: $C(x) = 100 + 3x + 0.04x^2$. Assume the fixed costs are sunken.

a) If the demand for the product is given by $D(p) = 200(10-p)$, what is the long-run equilibrium for this industry? (price, quantity demanded and consumed, number of firms).

In the long-run equilibrium, the price should equal the minimum average cost. That happens when $MC = AC$, which means:
$$
3 + 0.08x = \frac{100}{x} + 3 + 0.04x \implies x = 50, x > 0
$$

For this value of $x$, we have, equating $p = CMg$:
$$
p = 3 + 0.08 \cdot 50 = 7
$$
and, therefore,
$$
D(7) = Q^D = 600
$$
which means that the number of firms is $n = 600/50 = 12$.

b) Under a negative and permanent demand shock, it dislocates to $D(p) = 200(8-p)$. In the very-short run, the firms cannot change their production plans. What is the new equilibrium price? In the medium-run, however, the firms can adjust their production plans. Find the medium-run equilibrium (price, quantity produced and consumed, number of firms).

If in the very-short run the productions plans cannot be changed, let's assume the total market output is still 600. Using the new demand function:
$$
600 = 200(8-p) \implies p = 5
$$

In the medium-run, the firms can adjust their production plans and therefore we should find the value of $q$ that satisfies $MC = AVC$:
$$
3 + 0.08x = 3 + 0.04x \implies x = 0
$$
And so:
$$
p = 3 + 0.08 \cdot 0 \implies p = 3
$$
and therefore
$$
D(3) = Q^D = 1000
$$
However, since the best option for the firms is to not produce, we can (can we?) suppose that all of them leave the market, resulting in $n = 0$.

c) Find the new long-run equilibrium (price, quantity produced and consumed, number of firms). Compare your results with the medium-run equilibrium and interpret.

In the new long-run, we will again have $MC = AC$, resulting in a price of $p = 7$ and $x = 50$ again. However, given the change in the demand function, the quantity demanded now will be
$$
D(7) = 200(8-7) = 200
$$
which means that the new number of firms is $n = 200/50 = 4$, an increase for the medium-run, due to the absence of fixed-costs in the long-run.

Are my answers correct? I had never encountered a question asking about medium-run before. Also, is the profit maximization level of production different from the equilibrium level of production?

Thanks!

**Tags:** microeconomics, profit-maximization, competitive-equilibrium, nonexpert
> **Comments:**
> **username_1**: b) looks wrong to me. The way I read the assignment firms are in the medium term able to adjust quantity but firms are not able to enter or exit market. So p=MC and Supply(p) = Demand(p), with supply being 12*(p-3)/0.08 = 12x ... 12 firms all same production technology/cost function.  
> **username_0**: That was very helpful. Following that path, I found x = 70, p' = 3.8 and q' = 840.  How to interpret the change in the number of firms? Some firms decided to leave the market in the long-run after the reduction in demand? Thanks!  

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## 0 answers

