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Ting (2009) - Organizational Capacity (view source)
Revision as of 21:13, 21 May 2010
, 21:13, 21 May 2010no edit summary
This is a principal-agent model where the agent can make an investment in either general capacity or capacity specific to a policy. The principal can choose policy and 'use' the capacity of the agent. The model is one of complete and perfect information, and has two stages. The solution concept is SPNE and the game is solved by backwards induction. In later section the choice of type of specialization is endogenized, as is whether the principal would rather delegate the choice of policy to the agent.
Key concepts:
*Fungible: Two investment are fungible if they can be mutually substituted. This leads to the notion of general capacity that can be used to support any policy.
There are two basic versions of the game: Generalized (GC) and Specific capacityCapacity (SC), then several variables are endogenized.
===Generalized Capacity(GC)===
This version of the model assumes that <math>z(c)\,</math> alone - that is <math>z(\cdot)\,</math> is independent of <math>x_t\,</math> and <math>y\,</math>. Thus the agent invests in generalized capacity that is not targeted at any specific purpose and the principal can use it in any fashion.
In both the GC and SC games:
:<math>c^o(x;x^A) = arg \max_c u^A(x, z(x,y,c); x^A) - k(c)\,</math>
In the GC game the unique SPNE is:
*<math>x_1^*=x_2^*=x^P\,</math>
*<math>c_1=c_2=c^0(x^p)\,</math>
*<math>c_1^*\,</math> and <math>z_1^*\,</math> are strictly decreasing in <math>x^A\,</math>
In the second period, the principal can choose any policy and so chooses <math>x^P\,</math> to maximise her utility. Further, utility is increasing in <math>z\,</math> and <math>z\,</math> is increasing in <math>c\,</math>, so <math>P\,</math> also wants to maximize <math>c_2\,</math> and does this by setting it to <math>c_1\,</math>.
In the first period, <math>P\,</math>'s best response to any <math>c\,</math> is <math>x^P\,</math>. <math>A\,</math> then maximizes utility by choosing <math>c_1\,</math> subject to this. However, as <math>z\,</math> is independent of <math>x\,</math> (and <math>y\,</math>), and as <math>A\,</math>'s utility is concave in <math>c\,</math>, but costs are convex, <math>A\,</math> chooses the interior maximum irrespective of <math>P\,</math>'s choice to maximize <math>z\,</math> and hence <math>u\,</math> (subject to the constraint that <math>x=x^p\,</math>). Any <math>y\,</math> can be choosen as it will have no effect.
===Specialized Capacity===
In this version of the model we assume that:
:<math>
z(x_t,y,c_t) =
\begin{cases}
z(c_t) & \mbox{ if } x_t = y
0 & \mbox ( otherwise}
\end{cases}
\,</math>
That is if the principal enacts policy <math>x_2 = y\,</math>, then the production function kicks in and the benefits to specialization are realized. Otherwise, there are no benefits to the agents investment in capacity.
Solving backwards we note that <math>P\,</math> can implement one of two policies:
*if <math>x_2 \ne y\,</math>, then <math>x_2 = x^P\,</math> is optimal, and <math>c_2=0\,</math> results
*otherwise <math>y\,</math> is optimal and <math>c_2 = c_1\,</math> is optimal
Supposing that <math>x_2 = x^P\,</math> is choosen, then <math>c_2 = 0\,</math> and <math>z=0\,</math> as <math>z(x_2,y,0)=0\,</math>. Otherwise if <math>x_2 = y\,</math> then <math>P\,</math> wants the full benefit of specialization and implements <math>c_2=c_1\,</math> as this maximises her utility.
The choice as to whether <math>x_2=y\,</math> is therefore:
:<math>u^P(x^P,0) \le u^P(y,z(c_1))\,</math>
