<p>The Bertrand case you mention is a little special because it induces a discontinuity in the demand function. Suppose that market demand is $D(P)=1-P$, zero marginal cost, and that all consumers buy from the low priced firm (and split equally in the event of a tie).</p>

<p>If the two firms try to collude around a price $p=0.5$ (with quantity $0.25$ each) then each firm's demand curve will look like this:
<a href="https://i.sstatic.net/eNnCq.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/eNnCq.png" alt="Residual demand under Bertrand competition."></a></p>

<p>Mathematically, a firm's demand will be: $Q_i(p_i,p_j)=0$ if $p_i&gt;p_j$; $Q_i(p_i,p_j)=D(p_i)/2$ if $p_i=p_j$; and $Q_i(p_i,p_j)=D(p_i)$ if $p_i&lt;p_j$.</p>

<hr>

<p>More generally, it <em>is</em> possible for the slope of a firm's residual demand curve to differ from that of the market demand curve. It all depends on how consumers are allocated across the two firms. The relevant material on rationing rules can be found in <a href="https://books.google.co.uk/books?id=HIjsF0XONF8C&amp;pg=PA213&amp;lpg=PA213&amp;dq=demand%20rationing%20rules&amp;source=bl&amp;ots=wspdQKfa7g&amp;sig=805ybvXIWBUzDbi_-_evPVpBscY&amp;hl=en&amp;sa=X&amp;ved=0CDwQ6AEwBGoVChMI4qWSpqnsyAIVg0oUCh1LnAIp#v=onepage&amp;q=demand%20rationing%20rules&amp;f=false" rel="nofollow noreferrer">Section 5.3 of Tirole's <em>The Theory of Industrial Organization</em>.</a></p>

CC BY-SA 3.0
2015-10-31T09:04:50.517Z
2015-10-31T09:11:02.133Z
2015-10-31T09:11:02.133Z
Ubiquitous
<p>For example, in tacit collusion for a Bertrand situation, demand would be split between the two colluding firms. How would the reflect upon the demand function?</p>

<p>For example D = 100 - 2Q, and for simplicity sake MC = 0.
I though to divide D by 2, D* = 50 - Q.</p>

<p>This does not make intuitive sense to me, as how does splitting the demand function change the slope?</p>

CC BY-SA 3.0
2015-10-31T09:52:35.923Z
I don't think $D = 100 - 2Q$ because usually demand is given as a function of price. In equilibrium $D(p) = Q$.
Giskard
<supply-and-demand><industrial-organisation><monopoly><oligopoly>
When a demand function is split, how do we algebraically change function
Dave
# When a demand function is split, how do we algebraically change function

(Asked by: username_0 on 2015-10-31)

For example, in tacit collusion for a Bertrand situation, demand would be split between the two colluding firms. How would the reflect upon the demand function?

For example D = 100 - 2Q, and for simplicity sake MC = 0.
I though to divide D by 2, D\* = 50 - Q.

This does not make intuitive sense to me, as how does splitting the demand function change the slope?

**Tags:** supply-and-demand, industrial-organisation, monopoly, oligopoly
> **Comments:**
> **username_1**: I don't think $D = 100 - 2Q$ because usually demand is given as a function of price. In equilibrium $D(p) = Q$.  

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## 1 answers

### Answer by username_2 on 2015-10-31. Score: 3

The Bertrand case you mention is a little special because it induces a discontinuity in the demand function. Suppose that market demand is $D(P)=1-P$, zero marginal cost, and that all consumers buy from the low priced firm (and split equally in the event of a tie).

If the two firms try to collude around a price $p=0.5$ (with quantity $0.25$ each) then each firm's demand curve will look like this:
[![Residual demand under Bertrand competition.](https://i.sstatic.net/eNnCq.png)](https://i.sstatic.net/eNnCq.png)

Mathematically, a firm's demand will be: $Q\_i(p\_i,p\_j)=0$ if $p\_i>p\_j$; $Q\_i(p\_i,p\_j)=D(p\_i)/2$ if $p\_i=p\_j$; and $Q\_i(p\_i,p\_j)=D(p\_i)$ if $p\_i

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More generally, it *is* possible for the slope of a firm's residual demand curve to differ from that of the market demand curve. It all depends on how consumers are allocated across the two firms. The relevant material on rationing rules can be found in [Section 5.3 of Tirole's *The Theory of Industrial Organization*.](https://books.google.co.uk/books?id=HIjsF0XONF8C&pg=PA213&lpg=PA213&dq=demand%20rationing%20rules&source=bl&ots=wspdQKfa7g&sig=805ybvXIWBUzDbi_-_evPVpBscY&hl=en&sa=X&ved=0CDwQ6AEwBGoVChMI4qWSpqnsyAIVg0oUCh1LnAIp#v=onepage&q=demand%20rationing%20rules&f=false)


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