Strategic Pricing of Technology License under Product Differentiation
|
|
- Barnaby Marshall Dorsey
- 6 years ago
- Views:
Transcription
1 Academc Forum Strategc Prcng of Technolog Lcense under Product Dfferentaton Abstract YoungJun Km, Ph.D. Assstant Professor of Economcs Ths paper studes factors that mght affect the prce (.e. a fed lcensng fee and the demand for technolog lcensng. The author fnds that the lcensor s lcensng fee s found to ncrease n the degree of product dfferentaton between technolog holders, the degree of knowledge napproprablt, and the level of transacton costs of technolog lcensng. Also, product dfferentaton between technolog holders rases the lcensees demand for technolog. 1. Introducton Technolog lcensng, where the appropraton and adaptaton of technologcal advances takes a central role, has become ncreasngl mportant for the compettve strateg of frms n hgh technolog ndustres. As Anand and Khanna (000 observe, t s one of onl a few sgnfcant methods of technolog transfer between frms, and one of the most commonl observed nter-frm contractual agreements these das. Reflectng an ncreasng mportance of lcensng actvt, t has been a popular subject of ndustral economcs. Snce Arrow has (196 acknowledged the profts a patent holder can obtan from lcensng, Kamen and Tauman (1986 and Kat and Shapro (1985, 1986 establsh a game theoretc framework for the analss of the technolog owner s optmal lcensng strateges. Issues related to technolog holders strategc ncentves to lcense are addressed b man other studes as well. In Galln (1984, lcensng can be used strategcall to lmt potental entr and reduce competton whle Shepard (1987 observes that lcensng can also be used to enhance demand b creatng a suppl. The relatonshp between sequental nnovatons and lcensng strateges are also eamned n Green and Scotchmer (1994. Volumnous pror studes focus on technolog holder s strategc lcensng behavor ncludng, for eample, the optmal number of lcenses to sell. The more realstc setup, however, would be that technolog holders compete wth prce rather than quantt of lcenses. In addton, lterature has mostl dealt wth the suppl sde of technolog lcensng (.e. lcensor, and the demand sde of technolog lcensng (.e. lcensee has been somewhat gnored n the model. Ths paper tres to fll ths vod and theoretcall eamnes factors that affect the lcensor s prce (.e. fed lcensng fee of lcense n the market where multple sellers and buers of technolog est. Technolog owners perform the prce competton based on a lcensee s demand for technolog assumng mperfect prce competton of duopol at the technolog market. We assume that multple technolog holders provde dfferentated products. Ths s a plausble assumpton snce companes ma tr hard to dfferentate themselves from each other to avod ferce competton at the product market. The net Secton develops the model and Secton 3 concludes. 11
2 Academc Forum The model Consder that there est two frms, frm and frm, that have ndependentl developed and patented propret technologes, and, for the producton of a good. Such a good can ether be perfectl homogeneous between the products produced wth the two technologes or dfferentated. Apart from two technolog holders, we assume that there are N ( = 1,, N heterogeneous frms that own b branches each who cannot nnovate, but can produce f the receve the rghts to use the technolog from one of ncumbent technolog holders. Assume that all potental entrant branches of frms are able to obtan technolog from ether one of technolog holders and technolog holders suppl ther technologes whenever there s a demand. It s assumed that ncumbent lcensor collects a fed lcensng fee, F, from each lcensee through the use of the lcensor s technolog. We also assume that technolog lcensng from the lcensor to the lcensee nvolves a fed transacton cost, T 0. These nclude costs of wrtng contracts, enforcng contracts, gatherng nformaton about lcensees and barganng wth them. The theoretcal framework of the model s a three-stage game. We use a backward nducton approach. Game Structure The Frst Stage: Technolog holders compete wth lcensng fee the charge to lcensees. The Second Stage: C.E.O.s of potental lcensee frms choose optmal technolog for ther branches of frms. The Thrd Stage: All frms (ncludng both lcensors and lcensees that have technolog produce products and compete at the product market. The Thrd Stage: Product Market Competton n output We assume Cournot competton n the product market. Inverse demand functons for product produced wth each technolog, and, are as follows: p = 1 ( + µ ( +, (1 where D D p = 1 ( + µ ( +, ( p, technolog, ( D and D p denote the prces, ( + D + D D s the quantt suppled b frms producng wth s the quantt suppled b frms endowed wth technolog, and D Z are total demand for technolog and respectvel. We assume that µ [ 0,1], wth products beng homogeneous for µ = 1 and completel dfferentated for µ = 0. Hgher values of µ represent more homogeneous products between ncumbent technolog holders. It s also assumed that, once the producton technologes have been acqured, the cost of producton s neglgble; these costs are set at ero. 1
3 Academc Forum An frm (ether technolog holder or lcensee branches of frms producng wth technolog,, mames the followng profts at the product market, choosng the quantt of,, respectvel, ma π = p, (3 ma π = p, (4 The frst order condton of (3 s gven b: 1 µ ( + = 0. (5 D D Imposng smmetr above across frms usng same technolog, we obtan: 1 D µ µ D = 0, (6 from whch 1 µ (1 + D =. (7 + D Smlarl, for the frm wth technolog s: 1 µ (1 + D =. (8 + D From (7 and (8, we obtan the Nash equlbrum output b the frm wth technolog and technolog, respectvel: ( + D µ (1 + D =, (9 ( + D ( + D µ (1 + D (1 + D = ( + D ( + D ( + D µ (1 + D µ (1 + D (1 + D. (10 Substtutng (9 nto (1 and (3, (10 nto ( and (4 respectvel, we can compute the equlbrum prce and thus equlbrum proft at the product market for each frm endowed wth technolog as follows: ( + D µ (1 + D π = ( + D ( + D µ (1 + D (1 + D Smlarl, for the frm wth technolog s: π = ( + D ( + D ( + D µ (1 + D µ (1 + D (1 + D, (11. (1 Proposton 1. Each frm s proft n the product market s decreasng n µ. π π Proof. < 0 and < 0. µ µ The above proposton mples that each frm s proft decreases due to an ncreased competton at the product market when goods produced b frms are more homogeneous (.e. 13
4 Academc Forum the hgher µ between technolog holders. The more smlar goods produced b frms are, the fercer the competton the face at the goods market. Ths competton effect lowers frms profts accordngl. The Second Stage: Lcensee s demand for technolog Assume that each N ( = 1,,,N lcensee frm conssts of the same number of branches b that can adopt new technolog and produce outputs. Also, each branch s assumed to have a unque branch characterstc (.e. amount of know-how or tact knowledge, number of hghsklled engneers, commercalaton and marketng ablt, organatonal structure, management sklls, R&D ntenst, se. Thus lcensee frms are heterogeneous n a sense that the heterogenet of branch characterstcs among lcensee frms ma lead to heterogeneous behavor. It s assumed that the branch characterstc s unforml dstrbuted between 0 and 1, [ 0,1]. For nstance, 0 % chemcal engneer (100 % botechncans for = 0, whle 100% chemcal engneers (0 % botechncans for = 1. The C.E.O. of each potental lcensee frm wth branch characterstc s assumed to purchase ether varet or varet for hs branches from two ncumbent technolog holders, and. We assume that he consders both eplct proft and mplct (tact proft n decdng whch varet to choose. That s, the former s the proft lcensee branches can obtan at the product market eplctl, and the latter s the nherent proft the can generate from the more effcent producton, learnng, nventon, and the bgger n-house tact knowledge necessar for the obtaned technolog. Thus, the two avalable technologes are not the same to lcensee frms n terms of ther total potental profts (eplct and mplct that lcensee branches of frms can generate through technolog lcensng. For nstance, suppose varet s bo-related technolog. Then technolog s assumed to be best suted to branches wth branch characterstc = 0 (.e. 100 % botechncans. In ths case, technolog has the nherent advantage of mplct proft over technolog. In Fgure 1, as we move awa from branch characterstc 0 toward 1, ths nherent proft of technolog over technolog s assumed to decrease, reducng the per-unt total proft of technolog. On the other hand, technolog,.e. chemcal-related technolog, s best suted to branches wth branch characterstc = 1 (.e. 100 % chemcal engneers, and thus has the nherent mplct proft over technolog. As we move awa from the branch characterstc 1 toward 0, ths proft of technolog over technolog decreases. -- In Fgure 1, horontal as represents branch characterstcs, and lne (1 has a negatve slope whle ( has a postve slope. Before the ncrease of the lcensng fee, a potental lcensee frm wth *0 *0 less than equlbrum would adopt technolog, whle greater than equlbrum would purchase technolog. In Fgure, the demand for technolog s derved. Suppose a lcensng fee of technolog 0 1 ncreases from F to F, then per-unt total proft of technolog decreases and ths causes lne (1 to shft to lne (3 n Fgure 1. Accordngl, we can derve the demand for technolog b connectng those two ponts n Fgure. Hence, holdng others constant, the market share of technolog out of 14
5 Academc Forum *0 *1 total technolog market decreases from to (note that snce s unforml dstrbuted between 0 and 1, we can nterpret as the market share. Smlarl, we can derve the demand for technolog. -- <Fgure.1> The per-unt total proft as a functon of $ (1 ( (1: per-unt total proft of technolog (before (3 the change of a lcensng fee; 0 [ π + ( 1 µ π (1 ] F (: per-unt total proft of technolog 0 [ π + ( 1 µ π ] F (3: per-unt total proft of technolog (after the ncrease of a lcensng fee; *1 * [ π + ( 1 µ π (1 ] F <Fgure.> Demand for technolog $ (4: Demand for technolog ; D = N b = F 1 0 Nb[ π π + (1 µ π F + F ] F (4 * (1 µ ( π + π *1 *0 Nb Therefore, the C.E.O of each potental lcensee frm mames the followng total profts (both eplct and mplct from lcensng b selectng the proporton ( φ of ts branches endowed wth technolog, subject to nequalt constrant, 0 φ 1: ma v = [ π + 1 µ π (1 ] bφ F bφ φ ( +[ π 1 µ π ] b(1 φ F b(1 φ Lagrangan s as follows: π + 1 µ π (1 bφ F +, (13 ( L = [ ( ] bφ + [ π ( 1 µ π ] b(1 φ F b(1 φ +, + λ 1 φ + λ ( φ, (14 ( where φ = the proporton of branches endowed wth technolog b lcensee frm. ( 1 φ = the proporton of branches endowed wth technolog b lcensee frm. b = the number of branches of each lcensee frm. F, F = fed lcensng fee pad to lcensor frm, and frm, respectvel. 15
6 Academc Forum π, π = eplct proft assocated wth technolog and technolog n the product market, respectvel. ( 1 µπ = dfference of mplct proft between technologes. Note that µ stands for the degree of product dfferentaton across varetes. ( 1 µ π (1 =mplct proft of varet for the branch wth branch characterstc. ( 1 µπ = mplct proft of varet for the branch wth branch characterstc. L φ The frst order condton of the Lagrangan wth respect to φ s gven b: = [ π + ( 1 µ π (1 ] b F b [ π + (1 µ π ] b + F b The Kuhn Tucker condtons are: λ 0, λ 0, ( 1 φ 0, φ 0, λ + λ = 0. (15 λ ( 1 φ = 0, λ φ = 0, Frst, n case where φ = 1, and λ 0, λ = 0, then: [ π + ( 1 µ π (1 ] b F b [ π + ( 1 µ π ] b + F b = λ. (16 That s, total value of technolog mnus total value of technolog s greater than or equal to ero snce λ 0. Ths means that technolog s more valuable than, and thus lcensees would prefer to lcense technolog. Second, n the case where φ = 0, and π + 1 µ π b F + 1 λ = 0, λ 0, then: [ ( ] b [ ( µ π (1 ] b + F b π = λ. (17 Ths means that technolog s more valuable than, and thus lcensees would prefer to lcense technolog. Fnall, n the case of nteror soluton where 0 < φ < 1, and λ = 0, λ = 0, then: [ π + ( 1 µ π (1 ] b F b [ π + 1 µ π ] b + F b ( = 0. (18 Ths means that technolog and are equall valuable and lcensee s ndfferent between the two technologes. From (18 * π π + (1 µ π F + F =, (19 (1 µ ( π + π * where denotes the branch characterstc for the potental lcensee that s ndfferent between * technolog and. Thus all branches wth the branch characterstc less than obtan * technolog, whle all branches of the branch characterstc greater than purchase technolog. Snce we assume that the branch characterstc s unforml dstrbuted between ero and one, s equal to the market share of technolog and ( 1 s the market share of technolog out of total technolog market. Therefore, as Fgure llustrates, the total demand 16
7 Academc Forum for technolog ( D s equal to the product of the number of potental lcensee frms (N, the number of branches each lcensee has (b, and the market share of technolog ( : Nb[ π π + (1 µ π F + F ] = D = N b *. (0 (1 µ ( π + π Smlarl, we can derve the total demand for technolog : * Nb[ π π + (1 µ π F + F ] D = N b (1 =. (1 (1 µ ( π + π Proposton. The total demand for each technolog s decreasng n µ. D D Proof. < 0; < 0. µ µ The above proposton mples that the total demand for each technolog decrease wth the homogenet of products between technolog holders. Consderng that each frm s proft decreases due to an ncreased competton at the product market when goods are more homogeneous (.e. the hgher µ between technologes (Proposton 1, product homogenet gves potental lcensees less ncentve to adopt technolog. Proposton 3. The total demand for each technolog s ncreasng n terms of the lcensng fee of the other technolog, and decreasng n terms of ts own lcensng fee. D D D D Proof. < 0 and > 0, < 0 and > 0. The proposton 3 s ver ntutve. Frms tend to demand the cheaper technolog. The Frst Stage: Market for Technolog Competton n a fed lcensng fee Gven the proft at the product market and total demand for each technolog, each ncumbent technolog owner sets ts prce of technolog, a fed lcensng fee, to mame proft at the technolog market (.e. assumng mperfect prce competton of duopol, no frm can acheve the hgher proft b changng the lcensng fee charged for ts technolog. The technolog holder solves the followng: ma V = π + θf D TD, ( F where θ denotes the degree of knowledge approprablt (.e. the level of patent protecton enforcement. We assume that θ [ 0,1], wth perfect knowledge approprablt for θ = 1 and no knowledge approprablt for θ = 0. For nstance, lower value of θ represents the weaker patent protecton enforcement. The low degree of knowledge approprablt (.e. strong patent protecton enforcement ncreases the danger of lcensors patents beng nfrnged and lcensee frms can freel cop the lcensor s patented technolog and use t for generatng etra proft wthout pang for t. Further, lcensees ma have an ncentve to shrk a contract fee under the weak patent protecton regme. * 17
8 Academc Forum Thus, ncumbent technolog holder s total proft s the sum of the profts from ts own producton (= π and total fed fee paments from D lcensees (= θ F D mnus transacton costs of lcensng (=T D. Smlarl, the problem for technolog holder s: ma V = π + θf D TD, (3 F From the sstem of the two frst order condtons obtaned from ( and (3, we can obtan, b mposng smmetr, the equlbrum lcensng fee: θ (1 µ π + T F =. (4 θ Proposton 4. The Equlbrum lcensng fee F s decreasng n µ. Proof. = ( 1 µ π µ π < 0 snce π µ < 0. µ The proposton 4 mples that the ncumbent technolog holder frm can charge a hgher lcensng fee for ts technolog when goods between technolog holders are more dfferentated (.e. the lower µ. The ntuton s that when the good s hghl dfferentated, each ncumbent technolog holder has ts own market nche. Hence product and technolog dfferentaton gve each technolog holder frm more market power. Consderng that market power s postvel related wth the lcensng fee technolog holders can charge, a product dfferentaton between technolog holders leads to the hgher lcensng fee. Proposton 5. The Equlbrum lcensng fee F s ncreasng n T and decreasng n θ Proof. > 0, < 0. T θ The above proposton shows that equlbrum lcensng fee ncreases wth transacton costs of lcensng whle the hgh degree of knowledge approprablt (.e. strong patent protecton enforcement nduces frms to charge less for ther technolog. In the presence of low transacton costs of lcensng, weaker pressure ma be brought to bear on technolog holders to charge the hgher prce for ther technolog due to an effcent transacton of technolog at the market. In addton, gven that strength of patent protecton s negatvel correlated to transacton costs of technolog lcensng (Arrow, 196; Merges, 1998, a strong patent protecton leads to fewer ncentves for technolog holders to set the hgh prce for ther lcenses due to the smlar arguments. 3. Concluson Ths paper studes lcensors strategc prcng of lcense and lcensees optmal demand for technolog when a product s dfferentated. The man aspect of the model s the endogenet of the degree of product dfferentaton n the olgopol market, endogenet that s a functon of the relatve proportons of frms adoptng each technolog. Snce olgopol profts are a functon of the degree of product dfferentaton, technolog holders wll nternale ths effect when the set ther lcense prce. 18
9 Academc Forum We fnd that the lcensor s lcensng fee s found to ncrease n the degree of product dfferentaton between technolog holders, the degree of knowledge napproprablt, and the level of transacton costs of technolog lcensng. Also, product dfferentaton between technolog holders rases the lcensee s demand for technolog. References Anand, B.N., and T. Khanna, The structure of lcensng contracts, Journal of ndustral Economcs 48(1, 000, Arrow, K.J., Economc welfare and the allocaton of resources for nventon, In The Rate and Drecton of Inventve Actvt: Economc and Socal Factors, ed. R.R. Nelson. Prnceton Unverst Press Galln, N.T., Deterrence through Market Sharng: A Strategc Incentve for Lcensng, Amercan Economc Revew 74, 1984, Galln, N.T., and B.D. Wrght, Technolog Transfer under asmmetrc nformaton, Rand Journal of Economcs 1, 1984, Green, J. and Scotchmer, S., On the dvson of proft n sequental nnovaton, Rand Journal of Economcs, 6, 1995, Kamen M.I., Y. Tauman, Fees versus roaltes and the prvate value of a patent, Quarterl Journal of Economcs 101, 1986, Kat M.L. and C. Shapro, On the lcensng of nnovaton, Rand Journal of Economcs 16, 1985, Kat M.L. and C. Shapro, How to lcense ntangble propert, Quarterl Journal of Economcs 101, 1986, Merges, R., Propert rghts, transactons, and the value of ntangble assets. Mmeo, Boalt School of Law, Unverst of Calforna, Berkele,CA, Shepard, A., Lcensng to Enhance Demand for New Technolog, Rand Journal of Economcs 18, 1987,
10 Academc Forum Bograph Dr. YoungJun Km has been an assstant professor of economcs at Henderson State Unverst snce August 005. Before jonng to Henderson State, he taught at The George Washngton Unverst n Washngton D.C. as an adjunct professor ( Dr. Km's research nterests nclude technologcal strategc allances, manageral economcs, economcs of technolog and nnovaton, ndustral organaton, econometrcs, and development, and he has publshed hs papers n such journals as Manageral and Decson Economcs, Journal of Economcs and Busness, S.A.M. Advanced Management Journal, Appled Economcs Letters, The Journal of Amercan Academ of Busness, Cambrdge, Journal of Internatonal Busness and Economcs. Dr. Km's paper has been selected as one of the Best Papers from the 64th Academ of Management Annual Meetngs. 0
How Strong Are Weak Patents? Joseph Farrell and Carl Shapiro. Supplementary Material Licensing Probabilistic Patents to Cournot Oligopolists *
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
More informationMarket structure and Innovation
Market structure and Innovaton Ths presentaton s based on the paper Market structure and Innovaton authored by Glenn C. Loury, publshed n The Quarterly Journal of Economcs, Vol. 93, No.3 ( Aug 1979) I.
More informationOnline Appendix. t=1 (p t w)q t. Then the first order condition shows that
Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate
More informationWelfare Properties of General Equilibrium. What can be said about optimality properties of resource allocation implied by general equilibrium?
APPLIED WELFARE ECONOMICS AND POLICY ANALYSIS Welfare Propertes of General Equlbrum What can be sad about optmalty propertes of resource allocaton mpled by general equlbrum? Any crteron used to compare
More informationPrice competition with capacity constraints. Consumers are rationed at the low-price firm. But who are the rationed ones?
Prce competton wth capacty constrants Consumers are ratoned at the low-prce frm. But who are the ratoned ones? As before: two frms; homogeneous goods. Effcent ratonng If p < p and q < D(p ), then the resdual
More informationHila Etzion. Min-Seok Pang
RESERCH RTICLE COPLEENTRY ONLINE SERVICES IN COPETITIVE RKETS: INTINING PROFITILITY IN THE PRESENCE OF NETWORK EFFECTS Hla Etzon Department of Technology and Operatons, Stephen. Ross School of usness,
More information(1 ) (1 ) 0 (1 ) (1 ) 0
Appendx A Appendx A contans proofs for resubmsson "Contractng Informaton Securty n the Presence of Double oral Hazard" Proof of Lemma 1: Assume that, to the contrary, BS efforts are achevable under a blateral
More informationQuantity Precommitment and Cournot and Bertrand Models with Complementary Goods
Quantty Precommtment and Cournot and Bertrand Models wth Complementary Goods Kazuhro Ohnsh 1 Insttute for Basc Economc Scence, Osaka, Japan Abstract Ths paper nestgates Cournot and Bertrand duopoly models
More information3.2. Cournot Model Cournot Model
Matlde Machado Assumptons: All frms produce an homogenous product The market prce s therefore the result of the total supply (same prce for all frms) Frms decde smultaneously how much to produce Quantty
More informationSupporting Materials for: Two Monetary Models with Alternating Markets
Supportng Materals for: Two Monetary Models wth Alternatng Markets Gabrele Camera Chapman Unversty Unversty of Basel YL Chen Federal Reserve Bank of St. Lous 1 Optmal choces n the CIA model On date t,
More informationEconomics 101. Lecture 4 - Equilibrium and Efficiency
Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of
More informationThe oligopolistic markets
ernando Branco 006-007 all Quarter Sesson 5 Part II The olgopolstc markets There are a few supplers. Outputs are homogenous or dfferentated. Strategc nteractons are very mportant: Supplers react to each
More informationCopyright (C) 2008 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of the Creative
Copyrght (C) 008 Davd K. Levne Ths document s an open textbook; you can redstrbute t and/or modfy t under the terms of the Creatve Commons Attrbuton Lcense. Compettve Equlbrum wth Pure Exchange n traders
More informationIn the figure below, the point d indicates the location of the consumer that is under competition. Transportation costs are given by td.
UC Berkeley Economcs 11 Sprng 006 Prof. Joseph Farrell / GSI: Jenny Shanefelter Problem Set # - Suggested Solutons. 1.. In ths problem, we are extendng the usual Hotellng lne so that now t runs from [-a,
More informationPROBLEM SET 7 GENERAL EQUILIBRIUM
PROBLEM SET 7 GENERAL EQUILIBRIUM Queston a Defnton: An Arrow-Debreu Compettve Equlbrum s a vector of prces {p t } and allocatons {c t, c 2 t } whch satsfes ( Gven {p t }, c t maxmzes βt ln c t subject
More informationEnvironmental taxation: Privatization with Different Public Firm s Objective Functions
Appl. Math. Inf. Sc. 0 No. 5 657-66 (06) 657 Appled Mathematcs & Informaton Scences An Internatonal Journal http://dx.do.org/0.8576/ams/00503 Envronmental taxaton: Prvatzaton wth Dfferent Publc Frm s Objectve
More informationEndogenous timing in a mixed oligopoly consisting of a single public firm and foreign competitors. Abstract
Endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors Yuanzhu Lu Chna Economcs and Management Academy, Central Unversty o Fnance and Economcs Abstract We nvestgate endogenous
More informationSupporting Information for: Two Monetary Models with Alternating Markets
Supportng Informaton for: Two Monetary Models wth Alternatng Markets Gabrele Camera Chapman Unversty & Unversty of Basel YL Chen St. Lous Fed November 2015 1 Optmal choces n the CIA model On date t, gven
More informationPricing and Resource Allocation Game Theoretic Models
Prcng and Resource Allocaton Game Theoretc Models Zhy Huang Changbn Lu Q Zhang Computer and Informaton Scence December 8, 2009 Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009
More informationPerfect Competition and the Nash Bargaining Solution
Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange
More informationCan the Market be Fixed? (II) Does the Market Need Fixing? The Beneficent Effects of Assigning Property Rights. The Coase Theorem
Eternaltes An eternalt occurs whenever the actvtes of one economc agent affect the actvtes of another agent n was that are not taken nto account b the operaton of the market. Eamples: One frm produces
More informationStrategy and Cost in Technology Licensing
Strategy and Cost n Technology Lcensng YoungJun Km* Department of Economcs The George Washngton Unversty Ncholas S. Vonortas** Department of Economcs & Center for Internatonal Scence and Technology Polcy
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationIntroduction. 1. The Model
H23, Q5 Introducton In the feld of polluton regulaton the problems stemmng from the asymmetry of nformaton between the regulator and the pollutng frms have been thoroughly studed. The semnal works by Wetzman
More informationInvestment Secrecy and Competitive R&D
BE J. Econ. nal. Polcy 2016; aop Letter dt Sengupta* Investment Secrecy and Compettve R&D DOI 10.1515/beeap-2016-0047 bstract: Secrecy about nvestment n research and development (R&D) can promote greater
More informationConjectures in Cournot Duopoly under Cost Uncertainty
Conjectures n Cournot Duopoly under Cost Uncertanty Suyeol Ryu and Iltae Km * Ths paper presents a Cournot duopoly model based on a condton when frms are facng cost uncertanty under rsk neutralty and rsk
More informationCHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION
CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr
More informationCredit Card Pricing and Impact of Adverse Selection
Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n
More informationWelfare Analysis of Cournot and Bertrand Competition With(out) Investment in R & D
MPRA Munch Personal RePEc Archve Welfare Analyss of Cournot and Bertrand Competton Wth(out) Investment n R & D Jean-Baptste Tondj Unversty of Ottawa 25 March 2016 Onlne at https://mpra.ub.un-muenchen.de/75806/
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationExponential Type Product Estimator for Finite Population Mean with Information on Auxiliary Attribute
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 193-9466 Vol. 10, Issue 1 (June 015), pp. 106-113 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) Exponental Tpe Product Estmator
More informationLecture Notes, January 11, 2010
Economcs 200B UCSD Wnter 2010 Lecture otes, January 11, 2010 Partal equlbrum comparatve statcs Partal equlbrum: Market for one good only wth supply and demand as a functon of prce. Prce s defned as the
More informationGoldilocks and the licensing firm:
Goldlocks and the lcensng frm: Choosng a partner when rvals are heterogeneous * Anthony Creane and Hdeo onsh We consder how the amount of the technology transferred and the characterstcs of the partner
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationOne Dimensional Axial Deformations
One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the
More informationCS286r Assign One. Answer Key
CS286r Assgn One Answer Key 1 Game theory 1.1 1.1.1 Let off-equlbrum strateges also be that people contnue to play n Nash equlbrum. Devatng from any Nash equlbrum s a weakly domnated strategy. That s,
More informationUpstream competition. Stéphane Caprice Toulouse School of Economics
Upstream competton and buyer mergers Stéphane Caprce Toulouse School of Economcs GREMAQ-IRA 1 Buyer frms often merge n order to obtan lower prces from supplers What s the contrbuton of ths paper? Explan
More informationOn endogenous Stackelberg leadership: The case of horizontally differentiated duopoly and asymmetric net work compatibility effects
On endogenous Stackelberg leadershp: The case of horzontally dfferentated duopoly and asymmetrc net work compatblty effects Tsuyosh TOSHIMITSU School of Economcs,Kwanse Gakun Unversty Abstract Introducng
More informationf(x,y) = (4(x 2 4)x,2y) = 0 H(x,y) =
Problem Set 3: Unconstraned mzaton n R N. () Fnd all crtcal ponts of f(x,y) (x 4) +y and show whch are ma and whch are mnma. () Fnd all crtcal ponts of f(x,y) (y x ) x and show whch are ma and whch are
More informationAllocative Efficiency Measurement with Endogenous Prices
Allocatve Effcency Measurement wth Endogenous Prces Andrew L. Johnson Texas A&M Unversty John Ruggero Unversty of Dayton December 29, 200 Abstract In the nonparametrc measurement of allocatve effcency,
More informationResource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis
Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques
More informationEconomics 8105 Macroeconomic Theory Recitation 1
Economcs 8105 Macroeconomc Theory Rectaton 1 Outlne: Conor Ryan September 6th, 2016 Adapted From Anh Thu (Monca) Tran Xuan s Notes Last Updated September 20th, 2016 Dynamc Economc Envronment Arrow-Debreu
More informationWelfare Comparisons with a Consumer-Friendly Upstream Firm: Centralized vs. Decentralized Bargaining
Open Journal of Socal Scences 07 5 8-97 http://www.scrp.org/ournal/ss ISSN Onlne: 37-5960 ISSN Prnt: 37-595 Welfare Comparsons wth a Consumer-Frendly Upstream Frm: Centralzed vs. Decentralzed Barganng
More informationSCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.
SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationComparative Advantage and Optimal Trade Taxes
Comparatve Advantage and Optmal Trade Taxes Arnaud Costnot (MIT), Dave Donaldson (MIT), Jonathan Vogel (Columba) and Iván Wernng (MIT) June 2014 Motvaton Two central questons... 1. Why do natons trade?
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationAvailable online Journal of Chemical and Pharmaceutical Research, 2014, 6(5): Research Article
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research 4 6(5):7-76 Research Artcle ISSN : 975-7384 CODEN(USA) : JCPRC5 Stud on relatonshp between nvestment n scence and technolog and
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationConstant Best-Response Functions: Interpreting Cournot
Internatonal Journal of Busness and Economcs, 009, Vol. 8, No., -6 Constant Best-Response Functons: Interpretng Cournot Zvan Forshner Department of Economcs, Unversty of Hafa, Israel Oz Shy * Research
More informationMergers among leaders and mergers among followers. Abstract
Mergers among leaders and mergers among followers John S. Heywood Unversty of Wsconsn - Mlwaukee Matthew McGnty Unversty of Wsconsn-Mlwaukee Abstract We are the frst to confrm that suffcent cost convexty
More informationIntensity of competition and the choice between product and process innovation
Internatonal Journal of Industral Organzaton 16 (1998) 495 510 Intensty of competton and the choce between product and process nnovaton Gacomo onanno *, arry Haworth a, b a Department of Economcs, Unversty
More informationACTM State Calculus Competition Saturday April 30, 2011
ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward
More informationA Cournot-Stackelberg Advertising Duopoly Derived From A Cobb-Douglas Utility Function
MDEF Workshop 01, Urbno, 0- September A Cournot-Stackelberg Advertsng Duopoly Derved From A Cobb-Douglas Utlty Functon Alna Ghrvu * and Tönu Puu** *Faculty of Economc Studes and Busness Admnstraton, Babeş-
More informationPrice Discrimination of Digital Content
Prce Dscrmnaton of Dgtal Content Prce Dscrmnaton of Dgtal Content Koj Domon Faculty of Socal Scences, Waseda Unversty -6- Nshwaseda, Shnjuku-ku, Tokyo 69-8050, Japan Tel/Fax: +8 3 5286-45, E-mal: domon@waseda.jp
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationProblem Set 4: Sketch of Solutions
Problem Set 4: Sketc of Solutons Informaton Economcs (Ec 55) George Georgads Due n class or by e-mal to quel@bu.edu at :30, Monday, December 8 Problem. Screenng A monopolst can produce a good n dfferent
More informationManaging Capacity Through Reward Programs. on-line companion page. Byung-Do Kim Seoul National University College of Business Administration
Managng Caacty Through eward Programs on-lne comanon age Byung-Do Km Seoul Natonal Unversty College of Busness Admnstraton Mengze Sh Unversty of Toronto otman School of Management Toronto ON M5S E6 Canada
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationSolutions to Homework 7, Mathematics 1. 1 x. (arccos x) (arccos x) 1
Solutons to Homework 7, Mathematcs 1 Problem 1: a Prove that arccos 1 1 for 1, 1. b* Startng from the defnton of the dervatve, prove that arccos + 1, arccos 1. Hnt: For arccos arccos π + 1, the defnton
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationJanuary Examinations 2015
24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)
More informationLet p z be the price of z and p 1 and p 2 be the prices of the goods making up y. In general there is no problem in grouping goods.
Economcs 90 Prce Theory ON THE QUESTION OF SEPARABILITY What we would lke to be able to do s estmate demand curves by segmentng consumers purchases nto groups. In one applcaton, we aggregate purchases
More informationElshaboury SM et al.; Sch. J. Phys. Math. Stat., 2015; Vol-2; Issue-2B (Mar-May); pp
Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 Scholars Journal of Phscs Mathematcs Statstcs Sch. J. Phs. Math. Stat. 5; (B):69-75 Scholars Academc Scentfc Publshers
More informationTemperature. Chapter Heat Engine
Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationModule 17: Mechanism Design & Optimal Auctions
Module 7: Mechansm Desgn & Optmal Auctons Informaton Economcs (Ec 55) George Georgads Examples: Auctons Blateral trade Producton and dstrbuton n socety General Setup N agents Each agent has prvate nformaton
More informationThe Value of Demand Postponement under Demand Uncertainty
Recent Researches n Appled Mathematcs, Smulaton and Modellng The Value of emand Postponement under emand Uncertanty Rawee Suwandechocha Abstract Resource or capacty nvestment has a hgh mpact on the frm
More informationECE559VV Project Report
ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUM-RATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sum-rate
More informationElectrical double layer: revisit based on boundary conditions
Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843-318, USA Abstract The electrcal double layer
More informationOpen Source Software, Competition and Potential Entry
Open Source Software, Competton and Potental Entry Po Baake fo Insttut for Economc Research Thorsten Wchmann Berlecon Research GmbH May 2003 Prelmnary Verson Abstract We analyze a model wth two software
More informationfind (x): given element x, return the canonical element of the set containing x;
COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:
More informationA NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegian Business School 2011
A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegan Busness School 2011 Functons featurng constant elastcty of substtuton CES are wdely used n appled economcs and fnance. In ths note, I do two thngs. Frst,
More informationOn the correction of the h-index for career length
1 On the correcton of the h-ndex for career length by L. Egghe Unverstet Hasselt (UHasselt), Campus Depenbeek, Agoralaan, B-3590 Depenbeek, Belgum 1 and Unverstet Antwerpen (UA), IBW, Stadscampus, Venusstraat
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationMixed Taxation and Production Efficiency
Floran Scheuer 2/23/2016 Mxed Taxaton and Producton Effcency 1 Overvew 1. Unform commodty taxaton under non-lnear ncome taxaton Atknson-Stgltz (JPubE 1976) Theorem Applcaton to captal taxaton 2. Unform
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationBilateral Trade Flows and Nontraded Goods
The Emprcal Economcs Letters, 7(5): (May 008) ISSN 1681 8997 Blateral Trade Flows and Nontraded Goods Yh-mng Ln Department of Appled Economcs, Natonal Chay Unversty. 580 Snmn Road, Chay, 600, Tawan Emal:
More informationMarkov Chain Monte Carlo Lecture 6
where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways
More informationDummy variables in multiple variable regression model
WESS Econometrcs (Handout ) Dummy varables n multple varable regresson model. Addtve dummy varables In the prevous handout we consdered the followng regresson model: y x 2x2 k xk,, 2,, n and we nterpreted
More informationEquilibrium with Complete Markets. Instructor: Dmytro Hryshko
Equlbrum wth Complete Markets Instructor: Dmytro Hryshko 1 / 33 Readngs Ljungqvst and Sargent. Recursve Macroeconomc Theory. MIT Press. Chapter 8. 2 / 33 Equlbrum n pure exchange, nfnte horzon economes,
More informationNorm Bounds for a Transformed Activity Level. Vector in Sraffian Systems: A Dual Exercise
ppled Mathematcal Scences, Vol. 4, 200, no. 60, 2955-296 Norm Bounds for a ransformed ctvty Level Vector n Sraffan Systems: Dual Exercse Nkolaos Rodousaks Department of Publc dmnstraton, Panteon Unversty
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationEconomies of Scale: Replicating Christensen and Greene (1976) by Arianto A. Patunru Department of Economics, University of Indonesia 2004
Economes of Scale: Replcatng Chrstensen and Greene (976) b Aranto A. Patunru Department of Economcs, Unverst of Indonesa 4. Introducton Ths exercse s based on a class assgnment n Unverst of Illnos, nstructed
More informationMA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials
MA 323 Geometrc Modellng Course Notes: Day 13 Bezer Curves & Bernsten Polynomals Davd L. Fnn Over the past few days, we have looked at de Casteljau s algorthm for generatng a polynomal curve, and we have
More informationPART 8. Partial Differential Equations PDEs
he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef
More informationOPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
More informationR&D investment, asymmetric costs, and research joint ventures
R&D nvestment, asymmetrc costs, and research jont ventures Alejandro Montecnos Thomas Gresk July 6, 207 Abstract Ths paper nvestgates how an ntal asymmetry n producton costs affects the welfare dfferences
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationAmusing Properties of Odd Numbers Derived From Valuated Binary Tree
IOSR Journal of Mathematcs (IOSR-JM) e-iss: 78-578, p-iss: 19-765X. Volume 1, Issue 6 Ver. V (ov. - Dec.016), PP 5-57 www.osrjournals.org Amusng Propertes of Odd umbers Derved From Valuated Bnary Tree
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationGravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)
Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationVolume 29, Issue 4. Incomplete third-degree price discrimination, and market partition problem. Yann Braouezec ESILV
Volume 29, Issue 4 Incomplete thrd-degree prce dscrmnaton, and market partton problem Yann Braouezec ESILV Abstract We ntroduce n ths paper the "ncomplete" thrd-degree prce dscrmnaton, whch s the stuaton
More informationSubjective Uncertainty Over Behavior Strategies: A Correction
Subjectve Uncertanty Over Behavor Strateges: A Correcton The Harvard communty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton Publshed Verson Accessed
More information