<p>The quote in the question isn't really rigorous about what a free market is, but it talks about monopolies and artificial scarcities, so I am interpreting the efficient outcome with price equal to marginal cost as being one necessary feature of what they understand as a free market.</p>

<hr>

<p>Let's look at the cournot model of competition. There are <span class="math-container">$n$</span> firms, each with idiosyncratic marginal cost <span class="math-container">$c_i$</span>. Each firm simultaneously chooses its output (supply) quantity, <span class="math-container">$q_i$</span>. Given consumers' demand, this results in some market price <span class="math-container">$P(Q)$</span>, where <span class="math-container">$Q=\sum_{i=1}^n q_i$</span> is the total industry output. We would expect the demand curve to be downward sloping: <span class="math-container">$P'(Q)&lt;0$</span>.</p>

<p>Thus, a firm's profits are
<span class="math-container">$$\pi_i(q_i,\mathbf{q}_{-i})=\left(P(Q)-c_i\right)q_i.$$</span>
The first-order condition for maximisation of profit is
<span class="math-container">$$\frac{\partial\pi_i}{\partial q_i}=P(Q)-c_i+\frac{\partial P(Q)}{\partial q_i}=0.$$</span>
Noting that <span class="math-container">$$\frac{\partial P(Q)}{\partial q_i}=\frac{\partial P(Q)}{\partial Q},$$</span>
the first-order condition can be re-written as
<span class="math-container">$$\frac{P(Q)-c_i}{c_i}=-\frac{\partial P(Q)}{\partial Q}\frac{Q}{P(Q)}\frac{q_i}{Q},$$</span>
<span class="math-container">$$\frac{P(Q)-c_i}{c_i}=-\frac{1}{\eta}m_i,$$</span>
where <span class="math-container">$\eta$</span> is the price elasticity of demand and <span class="math-container">$m_i=q_i/Q$</span> is firm <span class="math-container">$i$</span>'s market share.</p>

<p>The left-hand side measures a firm's price-cost margin, or its market power. We can compute the average market power in the industry by weighting each firm by its market share:</p>

<p><span class="math-container">$$\text{avg market power}=\sum_i m_i \frac{P(Q)-c_i}{c_i}=-\frac{\sum_i m_i^2}{\eta}.$$</span></p>

<p>Note that <span class="math-container">$\sum_i m_i^2$</span> is the HHI index for the market. So this equation tells us that the average market power in an industry will be positively related to the HHI. <strong>For any positive HHI, there will be a positive mark-up and the market will deviate from the perfectly competitive outcome in which artificial scarcities do not exist.</strong> Put differently, only if there is an infinite number of firms, each controling an infinitely small share of the industry do we obtain the textbook perfect competition outcome.</p>

CC BY-SA 4.0
2019-07-26T13:30:03.423Z
2019-07-26T13:30:03.423Z
Ubiquitous
<p>According to <a href="https://en.wikipedia.org/wiki/Free_market" rel="nofollow noreferrer">Free market</a></p>

<blockquote>
  <p>a free market is a system in which the prices for goods and services
  are determined by the open market and by consumers. In a free market,
  the laws and forces of supply and demand are free from any
  intervention by a government or other authority and from all forms of
  economic privilege, monopolies and artificial scarcities</p>
</blockquote>

<p>Being monopolies and oligopolies a force against free market, is there any study, theory or anything which shows how much of a market has to be in power of a few companies for them to be able to alter the free market laws/rules?</p>

CC BY-SA 4.0
2019-07-25T21:55:11.850Z
Are you aware of the [Herfindahl-Hirschman Index](https://www.investopedia.com/terms/h/hhi.asp)?
Herr K.
CC BY-SA 4.0
2019-07-26T13:06:32.427Z
@ Herr K. HHI is just an accounting mesure as far as market power is concerned it doesn't say much. That is why Competition Authorities only use it as a scrrening device. For example if a market is open to "hit-and-run" operations even a monopolist cannot charge much above marginal costs.
Grada Gukovic
<monopoly><oligopoly><free-market>
Is there any research, theory, or anything which shows how much of a market has to be in power of a few companies to be a force against free market?
Pablo
# Is there any research, theory, or anything which shows how much of a market has to be in power of a few companies to be a force against free market?

(Asked by: username_0 on 2019-07-25)

According to [Free market](https://en.wikipedia.org/wiki/Free_market)

> a free market is a system in which the prices for goods and services
> are determined by the open market and by consumers. In a free market,
> the laws and forces of supply and demand are free from any
> intervention by a government or other authority and from all forms of
> economic privilege, monopolies and artificial scarcities

Being monopolies and oligopolies a force against free market, is there any study, theory or anything which shows how much of a market has to be in power of a few companies for them to be able to alter the free market laws/rules?

**Tags:** monopoly, oligopoly, free-market
> **Comments:**
> **username_1**: Are you aware of the [Herfindahl-Hirschman Index](https://www.investopedia.com/terms/h/hhi.asp)?  
> **username_2**: @ Herr K. HHI is just an accounting mesure as far as market power is concerned it doesn't say much. That is why Competition Authorities only use it as a scrrening device. For example if a market is open to "hit-and-run" operations even a monopolist cannot charge much above marginal costs.  

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## 1 answers

### Answer by username_3 on 2019-07-26. Score: 5 | Accepted answer

The quote in the question isn't really rigorous about what a free market is, but it talks about monopolies and artificial scarcities, so I am interpreting the efficient outcome with price equal to marginal cost as being one necessary feature of what they understand as a free market.

---

Let's look at the cournot model of competition. There are $n$ firms, each with idiosyncratic marginal cost $c\_i$. Each firm simultaneously chooses its output (supply) quantity, $q\_i$. Given consumers' demand, this results in some market price $P(Q)$, where $Q=\sum\_{i=1}^n q\_i$ is the total industry output. We would expect the demand curve to be downward sloping: $P'(Q)<0$.

Thus, a firm's profits are
$$\pi\_i(q\_i,\mathbf{q}\_{-i})=\left(P(Q)-c\_i\right)q\_i.$$
The first-order condition for maximisation of profit is
$$\frac{\partial\pi\_i}{\partial q\_i}=P(Q)-c\_i+\frac{\partial P(Q)}{\partial q\_i}=0.$$
Noting that $$\frac{\partial P(Q)}{\partial q\_i}=\frac{\partial P(Q)}{\partial Q},$$
the first-order condition can be re-written as
$$\frac{P(Q)-c\_i}{c\_i}=-\frac{\partial P(Q)}{\partial Q}\frac{Q}{P(Q)}\frac{q\_i}{Q},$$
$$\frac{P(Q)-c\_i}{c\_i}=-\frac{1}{\eta}m\_i,$$
where $\eta$ is the price elasticity of demand and $m\_i=q\_i/Q$ is firm $i$'s market share.

The left-hand side measures a firm's price-cost margin, or its market power. We can compute the average market power in the industry by weighting each firm by its market share:

$$\text{avg market power}=\sum\_i m\_i \frac{P(Q)-c\_i}{c\_i}=-\frac{\sum\_i m\_i^2}{\eta}.$$

Note that $\sum\_i m\_i^2$ is the HHI index for the market. So this equation tells us that the average market power in an industry will be positively related to the HHI. **For any positive HHI, there will be a positive mark-up and the market will deviate from the perfectly competitive outcome in which artificial scarcities do not exist.** Put differently, only if there is an infinite number of firms, each controling an infinitely small share of the industry do we obtain the textbook perfect competition outcome.


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