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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?
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    $\begingroup$ 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. $\endgroup$
    – Giskard
    Oct 28, 2021 at 10:48
  • $\begingroup$ @Giskard Sure. Microeconomic Theory: Basic Principles and Extensions 12th, on Chap 12 section 12.9.1. Also see this lecture note on page 17: homepage.univie.ac.at/matan.tsur/courses/Lecture9.pdf $\endgroup$
    – user141240
    Oct 28, 2021 at 11:31
  • $\begingroup$ @Giskard They assume no cost of entry though. $\endgroup$
    – user141240
    Oct 28, 2021 at 11:34
  • $\begingroup$ 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. $\endgroup$
    – Bertrand
    Oct 28, 2021 at 11:52
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    $\begingroup$ @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. $\endgroup$
    – Bertrand
    Oct 28, 2021 at 19:02

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