RESEARCH ARTICLE HUMAN CAPITAL DEVELOPMENT FOR PROGRAMMERS USING OPEN SOURCE SOFTWARE Amit Mehra Indian School of Business, Hyderabad, INDIA {Amit_Mehra@isb.edu} Vijay Mookerjee School of Management, University of Texas at Dallas, Richardson, TX 75080 U.S.A. {vijaym@utdallas.edu} Appendix Proofs of Lemmas and Propositions Proof of Lemma 1 The maximum value of w needed to satisfy the IR constraint of the programmer (called w IR ) is obtained from w IR T = M 0 – ax(T)| v=0 . Here x(T)| v=0 represents the smallest possible skill level with which the programmer ends the contract. This skill level occurs when v = 0 is chosen throughout the contract duration. Next, note that the adjoint equation for the co-state variable μ(t) is = constant œ t.  () μ ∂ ∂ μ =- =  H y t 0 Thus μ must take the same value throughout the contract duration. If μ > 1, then > 0 implying that w must be the maximum possible and ∂ ∂ H w hence the optimal value of w > w IR , since the domain of the wage premium w is R + . However, such a choice of w results in a smaller net value for the firm than if the firm chose w = w IR since paying wages is a cost to the firm and has no impact on programmer productivity. This is a contradiction because a suboptimal value of w gives a better solution to the firm. Hence it must be that μ # 1. If μ < 1, then . ∂ ∂ H w w <  = 0 0 On the other hand, if μ = 1, then can take any possible positive value (singular solution). ∂ ∂ H w w =  0 Proof of Lemma 2 We represent by z. Thus, using Equation 4, we can write ∂ ∂ H v (6) λ α μ = + - z Kx a Next, taking the derivative of both sides of Equation 4 with respect to t, we have    (  ) (  ) z Kx x a x t x t x =- + + + + + λα λα α μα α MIS Quarterly Vol. 36 No. 1–Appendix/March 2012 A1