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Ting (2009) - Organizational Capacity (view source)
Revision as of 19:14, 22 May 2010
, 19:14, 22 May 2010no edit summary
:<math>u^P(x^P,0) \le u^P(y,z(c_1))\,</math>
Define <math>\gamma(\cdot)\,</math> as the level of realized production needed to make <math>P\,</math> indifferent, as a function of a policy that <math>A\,</math> will choose. Therefore <math>\gamma(y)\,</math> makes the choice above hold with equality.
<math>A\,</math> has two variables to maximize over: the choice of policy <math>y\,</math> and the level of investment in capacity <math>c\,</math>. The second choice is constrained to be either the amount that maximizes the realized production <math>z(c^0(y;x^A))\,</math> or the amount that achieves <math>\gamma(y)\,</math>, which ever is lowest. The realized production is:
:<math>z_t^*=max\{\gamma(y), z(c^0(y,x^A))\} \mbox{ for some } y\,</math>
<math>A\,</math> doesn't want policy to move beyond <math>x_A\,</math>. In addition, <math>A\,</math> prefers policies closer to <math>x^A\,</math> than <math>x^P\,</math> and can prevent <math>P\,</math> from choosing <math>x^P\,</math> by investing in some <math>y closer to <math>x^A. Therefore, the equilibrium is:
:<math>x_1^* = x_2^* = y^* \mbox{ and } y^* \in \left ( x^P, x^A \right]\,</math>
A further refinement is possible. Suppose that there is some <math>x_c\,</math> that makes <math>P\,</math> exactly indifferent between choosing it (and it's investment outcome), and <math>x^A\,</math> and the optimal investment outcome that <math>A\,</math> would put into it <math>z(c^0(x^A))\,</math>:
:<math>\gamma(x_c) = z(c^0(x^A))\,</math>
This gives two cases for <math>A\,</math>:
*<math>x^A \le x_c\,</math>: then <math>z(c^0(x^A)) \ge \gamma(x_A)\,</math> and <math>A\,</math> can make his most preferred capacity investment in his ideal point.
*<math>x^A > x_c\,</math>: then <math>z(c^0(x^A)) <\gamma(x_A)\,</math> and <math>A\,</math> can not invest optimally in <math>x^A\,</math>, and must over invest to achieve it.
However, there must be some cutoff policy <math>y_c\,</math> below which <math>A\,</math> is no longer able to achieve indifference:
:<math>y_c = \max \{y | \gamma(y) = z(c^0(y;x^A)) \}\,</math>
Thus the solution must be:
:<math>y^* \in [y_c,x^A]\,</math>
===Comparing GC and SC===
The following points are important:
*In the SC game <math>c_1^*=c_2^* \ge c^0(x^P)\,</math> and <math>z_1^*\,</math> is strictly higher than in the GC game.
*<math>P\,</math> compromises on policy in the SC game in order to get the benefits of the investment in capacity
*For 'friendly' agents (whose ideal point is close to that of the principal, specifically <math>x^A \in [x^P,x_c]\,</math>) equilibrium policy is at <math>x^A\,</math>
*For 'unfriendly' agents, policy is a compromise: <math>y \in \left (x^P, x^A \right]\,</math>
*Specialized investment commits <math>P\,</math> not to unravel <math>A\,</math>'s investment, and makes <math>A\,</math>'s target policy at least as attractive as the principal's ideal point.
*Renegotiation doesn't happen in either game - in the GC game powerless, in the SC game <math>A\,</math> makes renegotiation prohibitively costly.
*<math>A\,</math>'s advantage in the SC game comes (at least partly) from moving first
*If <math>A\,</math> is unfriendly he must over-invest in capacity to get a policy closer to his ideal point.
*The model can also be interpretted in terms of politization of the agents.
**If <math>p\,</math> is low and the agent's utility is independent of policy, then implementation <math>z\,</math> strictly increases as with the distance between the agent's and the principal's ideal points.
**If <math>p\,</math> is high then an agent might be willing to shift the target policy away from <math>x^A\,</math>, and in doing so reduce the implementation needed to satisfy the principal.
