Changes

<math>N\;</math> firms own patents on complementary inputs, with costs of producing a unit of <math>c_i\;</math>, charging a price <math>p_i\;</math>. The price of the product itself will be <math>p\;</math>, and assembling a unit will cost <math>\alpha\;</math>.

:<math>p=\alpha + \sum_i p_i\;</math>

Demand for the product is <math>D(p)\;</math>, and the price elasticity of demand is therefore:

:<math>\eta = - \frac{D'(p)\cdot p}{ D(p)}\;</math>

The <math>N\;</math> firms set their component prices independently and non-cooperatively. That is the model assumes that each firm is a monopolist so it sets price (quantity) to maximize profits.

:<math>\pi_i = D(p)(p_i-c)\;</math>

:<math>\frac{d \pi_i}{d p_i} = D(p) + D'(p)(p_i - c_i) = 0\;</math>

Summing across all <math>i\;</math>:

:<math>D(p)N + D'(p)\sum_i(p_i - c_i) = 0\;</math>

:<math>\therefore \frac{\sum_i(p_i - c_i)}{p} = \frac{D(p) N}{p D'(p)}\;</math>

Subbing in <math>sum_i p_i = p - \alpha\;</math>:

:<math>\frac{p - \underbrace{\alpha - \sum_i c_i}{c}}{p} = \frac{N}{\eta}\;</math>

With a single firm, <math>N=1\;</math>, the Lerner index is <math>\frac{1}{\eta}\;</math>, so with <math>N</math> firms the mark-up is <math>N\;</math> times the standard monopoly mark-up.