# When a demand function is split, how do we algebraically change function

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?

• I don't think $D = 100 - 2Q$ because usually demand is given as a function of price. In equilibrium $D(p) = Q$. – Giskard Oct 31 '15 at 9:52

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:
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 $p_i<p_j$.