How Strong Are Weak Patents? Joseph Farrell and Carl Shapro Supplementary Materal Lcensng Probablstc Patents to Cournot Olgopolsts * September 007 We study here the specal case n whch downstream competton takes the form of Cournot olgopoly wth lnear demand and constant margnal cost The frst (entrely standard) step s to construct the π ( ab, ) and x( ab, ) functons Let downstream demand be p A X, where X s total output If downstream frm produces wth constant margnal c, ts profts are π ( A X x c) x, where x s ts output and X s the combned output of all other frms Frm s best-response functon s A X c x 0 or A X c x 0 Note that p c x Addng up for all frms gves ( ) X NA ck and so ( ) p A+ ck We therefore have x ( A+ c Nc)/( ), where c cj Snce p c x, we get ( N 1) [ A c Nc ] + π + Wth c j a and cj b for all j, and ( A+ ( ) b Na) π ( ab, ) ( ) xab (, ) A + ( ) b Na * Ths document s avalable at Shapro s web ste, http://facultyhaasberkeleyedu/shapro A spreadsheet mplementng ths example s avalable upon request from Shapro, shapro@haasberkeleyedu Lcensng Probablstc Patents to Cournot Olgopolsts, Page 1
N ( ) We note for future reference that π 1( ab, ) xab (, ) and π ( ab, ) xab (, ) All of these formulas apply f and only f they predct xab (, ) 0, whch holds f and only f a ( A+ ( N 1) b)/ N In our proft and output notaton, we normalzed the cost levels so that the cost of producng usng the patented technology wthout any runnng royalty s zero, and the cost of producng usng the backstop technology s v If the cost of producng usng the patented technology s c > 0, c should be added to all cost expressons Ths would cause A to be replaced by A c n all of our expressons, so A should be nterpreted as the dfference between the ntercept on the lnear demand curve and the producton cost assocated wth the new technology Lkewse, A v should be nterpreted as the dfference between the ntercept on the lnear demand curve and the producton cost assocated wth the backstop technology In thnkng about whether v s small or large n ths context, the rato v/ A has meanng, not the absolute level of v The rato v/ A approxmates the fracton of the overall total welfare that can be generated by ths market that s attrbutable to the patented technology To see ths, observe that wthout the patented technology, maxmum total welfare equals the area under the lnear demand curve up to the pont where prce equals margnal cost under the backstop technology, v Ths area s ( A v) / Wth the patented technology, the maxmum total v v welfare s A / The fracton of ths total attrbutable to the patented technology, + ( ), s A A close to v/ A f v/ A s small A + ( N 1) b+ a A + Nr The downstream prce s pab (, ) Wth equal royalty rates, prr (,) A r (wth constant p '( r ) as noted n the text), each frm s margn s prr (,) r, ( A r) A r π (,) rr and xrr (,) In ths case, the patent holder s ncome from runnng ( ) royaltes s Nrx(,) r r, or N r ( A r ) The combned profts of the patent holder and the downstream frms, measured per downstream frm, are A r ( A r) ( A r)( A+ Nr) Tr () rxrr (,) + π (,) rr r +, or Tr () Dfferentatng, N + 1 ( N + 1) ( ) AN ( 1) rn T'( r), whch s decreasng n r ( ) Lcensng Probablstc Patents to Cournot Olgopolsts, Page
Two-Part Tarffs wth Downstream Competton We assume n the text that Tr ( ) strctly ncreases n the range 0 r v Snce T ' s decreasng AN ( 1) n r, ths wll be true f and only f T'( v) 0, whch holds f and only f v The ( ) prce/cost margn for a downstream monopolst usng the new technology wth no royalty burden s A /, so ths condton s satsfed f the sze of the patent s no greater than the prce/cost margn for a downstream monopolst tmes Of course, when N 1, due to the neffcency of double margnalzaton, Tr ( ) cannot ncrease wth r n the range 0 r v But AN ( 1) the condton v s very easly satsfed so long as there s some downstream ( ) A 1 competton Even wth N, the condton becomes v, whch ncludes moderately large 3 nnovatons responsble for roughly a thrd of total avalable market surplus; and the condton A 3 becomes even easer to satsfy as N becomes larger Wth N 4, t becomes v 5 We showed n the text that r( θ ) v f θ θ T'( v) / π( v, v) Substtutng, usng AN ( 1) vn ( 1) ( 1) T'( v) and (,) N (,) N A π vv xvv v, and smplfyng, we get ( ) θ 1 1 v/ A 1 v/ A If v/ A s small, then θ 1/, and all patents wth θ < 1/ are lcensed usng r( θ ) v As v/ Abecomes larger, θ falls When v/ A reaches ts upper lmt such that T'( v) 0, e, ( ) ( ), 1 θ For any gven level of / N + 3 v A, as N grows, θ rses, approachng 1 1 v/ A 1 v/ A As an example, wth N 5 and v/ A 1/10, θ 5/1 For θ > θ, the royalty rate r( θ ) s the soluton to Gr (, r θ ) T'() r θπ (,) v r 0subject to the ( 1) ( 1) ( 1) constrant 0 r v Combnng (,) N (,) N A + N vr xvr r π Nv wth AN ( 1) rn T'( r), we can solve G r (, r θ ) T'() r θπ (,) v r 0explctly for r( θ ), gvng: ( ) ( N 1)( A+ θ ( vn A)) r( θ ), ( N + θ ( ) ) Lcensng Probablstc Patents to Cournot Olgopolsts, Page 3
( )( vn A) so long ths expresson s non-negatve Snce r(1), so long as v A/N, we ( N + ) know that r(1) 0, whch mples that r( θ ) 0 for all θ If v< A/N, then r( θ ) 0 for suffcently strong patents However, the dervaton of ths formula for r( θ ) requred that π ( vr, ) 0, whch s true f and vn A only f A+ ( ) r Nv 0, e, f and only f r rˆ So we need to go back and ( N 1)( A+ θ ( vn A)) vn A check that r( θ ) rˆ Ths s equvalent to Cross-multplyng ( N + θ ( ) ) gves ( N 1) ( A+ θ( vn A)) ( N + θ( ) )( vn A) Smplfyng, ths becomes AN ( 1) NvN ( A) Expandng and smplfyng ths expresson gves ( A v) N + A 0 Snce v A/, ths nequalty holds so the analyss s consstent The patent holder never fnds t optmal to reduce r all the way down to ˆr ; whle settng r rˆ would make π (,) vr 0, at the margn, the lower value of r has no mpact on π (,) vr but does reduce Tr () 1 1 N + 1 v/ A Summarzng, we haveθ N 1 1 v/ A, r( θ ) v for θ θ, and ( N 1)( A+ θ ( vn A)) r( θ ) for θ θ We now compute the other relevant varables that ( N + θ ( ) ) are generated n ths lcensng equlbrum The fxed fee F( θ ) s gven by π ( r( θ), r( θ)) F( θ) θπ( v, r( θ)) + (1 θ) π(0,0) Usng the expressons already derved for the proft functons, we have ( A r( θ)) ( A+ ( ) r( θ) Nv) A F( θ) θ (1 θ) ( ) ( ) ( ) Next, we have P( θ ) Nr( θ) x( r( θ), r( θ)) + NF( θ) whch becomes ( A r( θ )) P( θ) N F( θ) + r( θ) ( ) Total welfare at prce p and correspondng output A p s equal to total profts, p( A p), plus 1 consumer surplus, ( ) A p, whch equals 1 ( )( ) A + p A p The prce resultng from royalty A+ Nr rate r s prr (,) Substtutng and smplfyng, the total welfare functon s N wr ( ) [( N+ ) A+ Nr]( A r) ( ) Lcensng Probablstc Patents to Cournot Olgopolsts, Page 4
Ths can be used to compute W( θ ) w( r( θ )), as well as W w() v Nvx(,) v v, gvng N( N + ) W ( A v ), ( ) whch can be used n turn to compute the patent holder contrbuton, K( θ) w( r( θ)) [ θw + (1 θ) w(0)] Negatve Fxed Fees Not Feasble, Downstream Competton We now calculate s( θ ) and P( θ ) usng π ( ss, ) θπ ( vs, ) + (1 θ ) π (0,0) A+ ( ) s vn A So long as xvs (,) 0, e, so long as v, or s rˆ, the formula N π (,) vs ( A vn+ ( N 1))/( s N+ 1) apples (Ths wll be true for all postve s f v< A/ N, n whch case r ˆ < 0 ) In the range, s > rˆ, the equaton defnng s( θ ) s ( A s) ( A vn ( N 1) s) (1 ) A explctly for s Wrtng the quadratc expresson n s n the form αs + βs+ γ 0, we have θ + + θ Ths expresson s quadratc n s, so we can solve α θ( ) 1, β A+ θ ( )( A vn) > 0, and γ θvn( vn A) < 0 All of these parameters are lnear n θ In ths case, for v< A/ N we get s'(0) / v N v/ A As v 0 ths gves s'(0) / v N, a specal case of the general fndng that for small values of v n Cournot olgopoly wth constant margnal costs s'(0) / v N However, f s > rˆ, then π ( vs, ) 0, and the equaton defnng s( θ ) becomes π (,) ss (1 θπ ) (0,0) Solvng ths gves s( θ ) A(1 1 θ ) If v> A/ N, defne ˆ θ by ˆ A(1 1 θ ) rˆ Solvng ths gves ˆ N θ 1 ( ) (1 v/ A) Puttng the peces together, we have s( θ ) A(1 1 θ ) for θ ˆ θ and s( θ ) as the soluton to ( A s) θ( A vn + ( N 1) s) + (1 θ) A for θ ˆ θ The resultng s( θ ) apples over the range of patent strengths for whch F( θ ) 0, or for all patent strengths f lcenses are constraned to be lnear If postve fxed fees are feasble, then f F( θ ) > 0 the per-unt royalty rate equals r( θ ), as already calculated n the secton wth two-part tarffs, not s( θ ) Once s( θ ) has been calculated, the patent holder s profts and the welfare and contrbuton functons are easy to derve usng s( θ ), exactly as n the case of two-part tarffs where we used r( θ ) Lcensng Probablstc Patents to Cournot Olgopolsts, Page 5
ertcal Integraton The patent holder s downstream dvson behaves just lke a frm wth zero cost, snce changes n ts output do not affect the output of the downstream frms and thus the patent holder s royalty ncome The formula for profts s ( ) π [ A+ c Nc] f there are N frms n total We have frms n total, so ths becomes ( N + ) π [ A+ c ( ) c ] Therefore, puttng + I π + + Snce c 1 0 for the patent holder, we get ( N ) ( a, b) [ A ( N 1) b ( N 1) a] I π I I I 1 π1 / π ( ) /( N 1), ths mples that ρ I I π + π 1 Lnear Lcenses to Downstream Frms That Do Not Compete Wth downstream frms that do not compete, we have a downstream monopolst Puttng N 1 nto the Cournot equatons above, and smplfyng the notaton, we get π () s ( A s)/4, xs () ( A s)/, and sx() s s( A s)/ The royalty rate s( θ ) solves ( A s) ( A s(1)) (1 ) A θ + θ The soluton to ths quadratc equaton s s( θ) A A θas(1) θs(1) s(1) +, an ncreasng convex functon of θ So long as v A/ v, and ths equaton becomes <, v s( θ) A A θav+ θv ), and s'(0) v(1 ) A Lcensng Probablstc Patents to Cournot Olgopolsts, Page 6