<p>OP cross posted to <a href="https://old.reddit.com/r/AskEconomics/comments/qhl23a/show_that_the_longrun_supply_curve_of_an/hidye4f" rel="nofollow noreferrer">r/AskEconomics</a>. I add MathJax.</p>
<blockquote>
<p>consider taking FOCs for x and n.</p>
<p>Firms take output price as given, and input cost is upward-sloping.</p>
<h3><span class="math-container">$dx/dp$</span> conditions:</h3>
<p>let's hold fixed the number of firms at n.</p>
<p>Each firm's profit is given by <span class="math-container">$p f(x) - w(x)$</span>, where f(x) is increasing and concave, and w(x) is increasing and not concave.</p>
<p>The FOC is <span class="math-container">$p f'(x) = w'(x)$</span>. As p increases, x must increase because f is concave.</p>
</blockquote>
<blockquote>
<h2>For n:</h2>
<p>if p increases but x is held constant, there is a profit and
new firms will enter.</p>
<p>Either way, increasing p will increase output <span class="math-container">$xn$</span>, so the supply curve slopes upward.</p>
</blockquote>

CC BY-SA 4.0
2024-05-21T21:40:29.010Z
2024-05-21T21:40:29.010Z
user48057
<p>Many intermediate microeconomics textbooks teach us that in perfect competition, the long-run supply curve of an increasing cost industry is upward-sloping. However, They usually give some hand-waving arguments such as &quot;entry of new firms shift the cost curves upward&quot;. How do I prove it analytically?</p>
<p><strong>My attempt:</strong></p>
<p>Suppose in a perfectly competitive market, there are <span class="math-container">$n$</span> firms with identical cost structures. Each firm takes inputs <span class="math-container">$ x = (x_1, \dots, x_m) \in \mathbb{R}^m_+ $</span> and produces <span class="math-container">$f(x) $</span> units of output according to the production function <span class="math-container">$ f: \mathbb{R}^m_+ \to \mathbb{R}_+ $</span>. Assume <span class="math-container">$f$</span> is continuously differentiable, strictly increasing and strictly concave.</p>
<p>For an increasing cost industry, the unit costs of inputs depend on the <strong>industry demands</strong> for inputs. Thus, for each <span class="math-container">$ 1 \leq i \leq m$</span>, let the unit cost of the <span class="math-container">$i$</span>-th input be <span class="math-container">$w_i(nx_i)$</span>, where <span class="math-container">$ w_i: \mathbb{R}_+ \to \mathbb{R}_+ $</span> is a continuously differentiable function. The industry demand for the <span class="math-container">$i$</span>-th input is <span class="math-container">$nx_i$</span> because the firms are identical. Increasing cost means that the derivatives <span class="math-container">$w_i' &gt; 0$</span> for all <span class="math-container">$i$</span>.</p>
<p>At output price <span class="math-container">$p &gt; 0$</span>, each firm earns the profit
<span class="math-container">$$ \pi(p; x, n) = p f(x) - \sum_{i=1}^m x_i w_i(nx_i) $$</span></p>
<p>In a perfectly competitive market, firms take output price as given and maximize their profit accordingly. Moreover, in the long-run, firms earn exactly zero profit because they are free to enter and exit. Thus, the long-run competitive equilibrium is a tuple <span class="math-container">$ (x_*(p), n_*(p)) $</span> such that</p>
<p><span class="math-container">$$ \pi(p; x_*(p), n_*(p)) = 0 \quad \text{and} \quad \nabla_x \pi(p; x_*(p), n_*(p)) = 0 \quad \forall p &gt; 0 $$</span></p>
<p>For each <span class="math-container">$p &gt; 0$</span>, we want to solve for <span class="math-container">$ x = (x_1, \dots, x_m) $</span> and <span class="math-container">$n$</span> in the following system of equations</p>
<p><span class="math-container">$$
\begin{align}
p f(x) &amp;= \sum_{i=1}^m x_i w_i(nx_i) \\
p \frac{\partial f}{\partial x_i}(x) &amp;= w_i(nx_i) + nx_i w_i'(nx_i)
\quad \text{for } i = 1, \dots, m
\end{align}
$$</span></p>
<p>There are <span class="math-container">$m + 1$</span> unknowns and <span class="math-container">$m + 1$</span> equations, which gives us hope that it is solvable. Suppose the system has a unique solution <span class="math-container">$ (x_*(p), n_*(p)) $</span> for each <span class="math-container">$p &gt; 0$</span>. Then the long-run market supply curve is</p>
<p><span class="math-container">$$ S(p) = n_*(p) f(x_*(p)) \quad \forall p &gt; 0 $$</span></p>
<p><strong>Question:</strong></p>
<ol>
<li>Did I model an increasing cost industry in perfect competition correctly?</li>
<li>How to prove that <span class="math-container">$S$</span> is strictly increasing?</li>
</ol>

CC BY-SA 4.0
2021-10-28T10:48:34.957Z
Hi! Do you have a reference for "*Many intermediate microeconomics textbooks teach us that in perfectly competitive market, the long-run supply curve of an increasing cost industry is upward-sloping.*"? I don't think this is true in the long-run if you assume constant cost of entry, so please provide a reference to back up your claim.
Giskard
CC BY-SA 4.0
2021-10-28T11:31:58.013Z
@Giskard Sure. Microeconomic Theory: Basic Principles and Extensions 12th, on Chap 12 section 12.9.1. Also see this lecture note on page 17: https://homepage.univie.ac.at/matan.tsur/courses/Lecture9.pdf
user141240
CC BY-SA 4.0
2021-10-28T11:34:11.653Z
@Giskard They assume no cost of entry though.
user141240
CC BY-SA 4.0
2021-10-28T11:52:49.460Z
In perfect competition, it is not clear to me either, how "entry of new firms shift the cost curves upward". The cost curve of firm $i$ depends upon the input prices (given) and its production level only.
Bertrand
CC BY-SA 4.0
2021-10-28T13:43:38.920Z
@Bertrand It says that in an increasing cost industry, entry of new firms bid up the input price, thus increasing the cost for every firm.
user141240
CC BY-SA 4.0
2021-10-28T19:02:09.587Z
@user141240 OK, this is plausible and close to what Silberberg (1974) did in a similar context. However, if competition is perfect, firms have an infinitesimal weight, or are not aware that their actions change $w$. I would set $w'=0$ in your marginal revenue = marginal cost equations.
Bertrand
<microeconomics><mathematical-economics><supply-and-demand><competitive-equilibrium>
Show that the long-run supply curve of an increasing cost industry in perfect competition is upward sloping
user141240
# Show that the long-run supply curve of an increasing cost industry in perfect competition is upward sloping

(Asked by: username_0 on 2021-10-28)

Many intermediate microeconomics textbooks teach us that in perfect competition, the long-run supply curve of an increasing cost industry is upward-sloping. However, They usually give some hand-waving arguments such as "entry of new firms shift the cost curves upward". How do I prove it analytically?

**My attempt:**

Suppose in a perfectly competitive market, there are $n$ firms with identical cost structures. Each firm takes inputs $ x = (x\_1, \dots, x\_m) \in \mathbb{R}^m\_+ $ and produces $f(x) $ units of output according to the production function $ f: \mathbb{R}^m\_+ \to \mathbb{R}\_+ $. Assume $f$ is continuously differentiable, strictly increasing and strictly concave.

For an increasing cost industry, the unit costs of inputs depend on the **industry demands** for inputs. Thus, for each $ 1 \leq i \leq m$, let the unit cost of the $i$-th input be $w\_i(nx\_i)$, where $ w\_i: \mathbb{R}\_+ \to \mathbb{R}\_+ $ is a continuously differentiable function. The industry demand for the $i$-th input is $nx\_i$ because the firms are identical. Increasing cost means that the derivatives $w\_i' > 0$ for all $i$.

At output price $p > 0$, each firm earns the profit
$$ \pi(p; x, n) = p f(x) - \sum\_{i=1}^m x\_i w\_i(nx\_i) $$

In a perfectly competitive market, firms take output price as given and maximize their profit accordingly. Moreover, in the long-run, firms earn exactly zero profit because they are free to enter and exit. Thus, the long-run competitive equilibrium is a tuple $ (x\_\*(p), n\_\*(p)) $ such that

$$ \pi(p; x\_\*(p), n\_\*(p)) = 0 \quad \text{and} \quad \nabla\_x \pi(p; x\_\*(p), n\_\*(p)) = 0 \quad \forall p > 0 $$

For each $p > 0$, we want to solve for $ x = (x\_1, \dots, x\_m) $ and $n$ in the following system of equations

$$
\begin{align}
p f(x) &= \sum\_{i=1}^m x\_i w\_i(nx\_i) \\
p \frac{\partial f}{\partial x\_i}(x) &= w\_i(nx\_i) + nx\_i w\_i'(nx\_i)
\quad \text{for } i = 1, \dots, m
\end{align}
$$

There are $m + 1$ unknowns and $m + 1$ equations, which gives us hope that it is solvable. Suppose the system has a unique solution $ (x\_\*(p), n\_\*(p)) $ for each $p > 0$. Then the long-run market supply curve is

$$ S(p) = n\_\*(p) f(x\_\*(p)) \quad \forall p > 0 $$

**Question:**

1. Did I model an increasing cost industry in perfect competition correctly?
2. How to prove that $S$ is strictly increasing?

**Tags:** microeconomics, mathematical-economics, supply-and-demand, competitive-equilibrium
> **Comments:**
> **username_1**: Hi! Do you have a reference for "*Many intermediate microeconomics textbooks teach us that in perfectly competitive market, the long-run supply curve of an increasing cost industry is upward-sloping.*"? I don't think this is true in the long-run if you assume constant cost of entry, so please provide a reference to back up your claim.  
> **username_0**: @Giskard Sure. Microeconomic Theory: Basic Principles and Extensions 12th, on Chap 12 section 12.9.1. Also see this lecture note on page 17: https://homepage.univie.ac.at/matan.tsur/courses/Lecture9.pdf  
> **username_0**: @Giskard They assume no cost of entry though.  
> **username_2**: In perfect competition, it is not clear to me either, how "entry of new firms shift the cost curves upward". The cost curve of firm $i$ depends upon the input prices (given) and its production level only.  
> **username_0**: @Bertrand It says that in an increasing cost industry, entry of new firms bid up the input price, thus increasing the cost for every firm.  
> **username_2**: @user141240 OK, this is plausible and close to what Silberberg (1974) did in a similar context. However, if competition is perfect, firms have an infinitesimal weight, or are not aware that their actions change $w$. I would set $w'=0$ in your marginal revenue = marginal cost equations.  

---

## 1 answers

### Answer by username_3 on 2024-05-21. Score: 0

OP cross posted to [r/AskEconomics](https://old.reddit.com/r/AskEconomics/comments/qhl23a/show_that_the_longrun_supply_curve_of_an/hidye4f). I add MathJax.

> consider taking FOCs for x and n.
>
> Firms take output price as given, and input cost is upward-sloping.
>
> ### $dx/dp$ conditions:
>
> let's hold fixed the number of firms at n.
>
> Each firm's profit is given by $p f(x) - w(x)$, where f(x) is increasing and concave, and w(x) is increasing and not concave.
>
> The FOC is $p f'(x) = w'(x)$. As p increases, x must increase because f is concave.

> ## For n:
>
> if p increases but x is held constant, there is a profit and
> new firms will enter.
>
> Either way, increasing p will increase output $xn$, so the supply curve slopes upward.


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