Take the M R = M C condition for a monopolist.
p(y) + yp0 (y) = c0 (y)
Simplify the notation slightly
p + yp0 (y) = M C
(⇤)
If we take the demand curve y(p) the elasticity of demand is
" = y 0 (p)
p
y
The Inverse Function Theorem says that, for a differentiable, monotonic function of a single variable y = f (x) and
its inverse x = g(y), it is the case that
1
f 0 (x) = 0
g (y)
Applying this to the market demand curve and inverse demand curve we have
y 0 (p) =
so the elasticity of demand can be written
"=
1
p0 (y)
1 p
p0 (y) y
Rearrange for
p0 (y) =
Substitute back into (⇤) to get
p+
1p
y"
p
= MC
"
Then rearrange for the mark-up
p
p
p
p
"
1
"
MC =
MC
=
p
MC
1
=
p
|"|
[since " < 0]
For an oligopolist (Firm i) the equivalent of (⇤) is
p + yi p0 (y) = M Ci
(†)
As before we have the rearrangement of the elasticity
p0 (y) =
Substituting into (†) gives
p + yi
Let si =
yi
y
denote the market share. Then

1p
= M Ci
y"
p
= M Ci
"
p
M Ci = si
"
M Ci
si
=
p
|"|
p + si
p
p
1p
y"
1