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Ting (2009) - Organizational Capacity (view source)
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|Has article title=Organizational Capacity
|Has author=Ting
|Has year=2009
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*This page is referenced in [[BPP Field Exam Papers]]
The variables are:
*Policy set by the principal is <math>x_t \in X\,</math>, for <math>t=\{1,2\}\,</math>.
*Agents have ideal points of <math>x^A\,</math> and <math>x^P\,</math>.
*Agents can invest in capacity <math>c\,</math>, at a cost <math>k(c)\,</math> which is continuous, increasing and convex
In both the GC and SC games:
:<math>c^o(x;x^A) = \mbox{arg } \max_c u^A(x, z(x,y,c); x^A) - k(c)\,</math>
:<math>
z(x_t,y,c_t) =
\begin{casecases}
z(c_t) & \mbox{ if } x_t = y \\
0 & \mbox ( { otherwise}\end{casecases}
\,</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 agent's investment in capacity.
:<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\,</math> closer to <math>x^A\,</math>. 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.
===Endogenous Specialization===
Whether specialization would occur if it were endogenously choosen depends on both who is choosing, and if the prinicpal is choosing where the agent's ideal point is:
*The utility of the agent is strictly higher in SC than in GC
*The utility of the principal is strictly lower in SC than in GC if <math>x^A \ge x_c\,</math>.
===Delegation===
In this section the paper first considers what would happen if the principal gave the agent the authority to choose policy in the first period, and then considers whether the principal would endogenously choose to do so.
In the GC game:
*The principal's choices in period 2 remain unchanged.
*However, the agent now chooses his ideal point as the policy in period 1.
*Recall that the total utility is the sum over the two periods with the second period discounted.
**Therefore the agent invests more heavily in capacity and realized implementation is higher, as compared with the standard GC model, particularly if the discount factor is low so the second period gets little weighting.
*<math>\hat{c}_1^* > c_1^*\,</math> and <math>\hat{z}_1^* > z_1^*\,</math>.
In the SC game:
*There are three possibilities for the agent:
**Choose <math>x_1 = y\,</math> and the game is as before
**Choose <math>x_1 = x^A\,</math> and pay twice for the capacity investment that is only realized once (in period 2), but get some benefits from <math>x^A\,</math> in the first period.
**Choose <math>x_1 = x^A\,</math> and invest nothing, taking just the benefits from the first period - this can be particularly ideal if <math>\delta\,</math> is very low.
In choosing delegation endogenously:
*The payoffs from the two GC games (to the principal) are compared. Under "many functional assumptions" <math>P\,</math> is less likely to delegate as the distance between <math>x^A\,</math> and <math>x^P\,</math> diverge.
*In the SC game, delegating authority to the agent is weakly dominated.
