Changes
Jump to navigation
Jump to search
Shapiro (2001) - Navigating The Patent Thicket (view source)
Revision as of 22:48, 25 March 2013
, 22:48, 25 March 2013no edit summary
Each of these is discussed in turn.
===Model===
<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>.
Competition at the assembly level ensures that:
:<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>
Therefore the FOC is:
:<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.
===Hold-up===
