Ting (2009) - Organizational Capacity

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Reference(s)

Ting, Michael (2009), "Organizational Capacity", forthcoming JLEO, available on his website pdf (Class Slides: Set1 Set2, © Adrienne Hosek)


Abstract

Organizational capacity is critical to the effective implementation of policy. Consequently, strategic legislators and bureaucrats must take capacity into account in designing programs. This paper develops a theory of endogenous organizational capacity. Capacity is modeled as an investment that effects a policy's subsequent quality or implementation level. The agency has an advantage in providing capacity investments, and may therefore constrain the legislature's policy choices. A key variable is whether investments can be targeted" toward speciffc policies. If it cannot, then implementation levels decrease with the divergence in the players' ideal points, and policy-making authority may be delegated to encourage investment. If investment can be targeted, then implementation levels increase with the divergence of ideal points if the agency is suffciently professionalized, and no delegation occurs. In this case, the agency captures more benefits from its investment, and capacity is higher. The agency therefore prefers policy-specific technology.


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.


The Model

There is a:

  • Principal - [math]P\,[/math]
  • An Agent - [math]A\,[/math]


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
    • [math]c\,[/math] is assumed to be a vector, with [math]k(c)\,[/math] having weakly-negative cross-partials to allow for cost efficiencies, but this doesn't seem crucial to the model.
  • The implementation level, or production function, or capacity 'realization' function, [math]z(c,\cdots)\,[/math] is weakly concave and increasing in [math]c\,[/math].
    • [math]z\,[/math] is assumed to be [math]z(x,y,c)\,[/math], though it is hard to see why this full generality matters.
    • [math]z(c)\,[/math] is assumed to have positive cross-partials to allow for complementary investments, though this doesn't seem crucial to the model.
  • WLOG the model assumes that [math]x^P \lt x^A\,[/math]


The utility functions are:

[math]u_t^P = u^P(x_t,z; x^P)\,[/math]
[math]u_t^A = u^A(x_t,z; x^A) - k(c_t)\,[/math]


The utility functions are assumed to meet:

  • [math]\frac{\partial u^i}{\partial z} \gt 0\,[/math] : Utility is increasing in the effect of capacity
  • [math]\frac{\partial^2 u^i}{\partial z^2} \le 0\,[/math] : Utility is concave in the effect of capacity
  • [math]\frac{\partial^2 u^i}{\partial z \partial x} \gt 0 \mbox { for } x \lt x^i\,[/math]: Utility is increasing in policy if policy is below the ideal point, holding capacity fixed
  • [math]\frac{\partial^2 u^i}{\partial z \partial x} \lt 0 \mbox { for } x \gt x^i\,[/math]: Utility is decreasing in policy if policy is above the ideal point, holding capacity fixed
  • [math]\frac{\partial^2 u^A}{\partial z \partial x} = p \pi(x)\,[/math]: where [math]p\,[/math] represents the politization of the agent
    • For low politization the agent is largely indifferent to the policy
    • For high politization the agent wants lower implementation ([math]z\,[/math]) if the policy is far from [math]x^A\,[/math]
  • There are various assumptions on the utility and cost functions to avoid corner solutions
  • There exists some [math]z\,[/math] such that [math]u^P(x^P,0) = u^P(x,z)\,[/math], that is the principal can be made indifferent to her ideal point by some capacity implementation.
  • The total utility across both periods is additive with a discount factor: [math]u^i = u_1^i+\delta u_2^i\,[/math]


The sequence of the game is:

  1. [math]A\,[/math] chooses [math]c_1\,[/math] and [math]y\,[/math] (period 1a)
  2. [math]P\,[/math] chooses [math]x_1\,[/math] (period 1b)
  3. [math]P\,[/math] chooses [math]x_2\,[/math] and [math]c_2 \in \{c | 0 \le c \le c_1\}\,[/math] (period 2)


An equilibrium is characterized by: [math]y^*, x_t^*, c_t^*, z_t^*\,[/math]


There are two tie-break rules (it isn't clear when they are used):

  1. [math]P\,[/math] breaks ties in favour of lower levels of investment
  2. [math]A\,[/math] breaks ties by choosing the [math]x\,[/math] closest to [math]x_A\,[/math]


There are two basic versions of the game: Generalized (GC) and Specific Capacity (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) = \mbox{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 (SC)

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 agent's 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]


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 \gt x_c\,[/math]: then [math]z(c^0(x^A)) \lt \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^* \gt c_1^*\,[/math] and [math]\hat{z}_1^* \gt 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.
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