Changes

#A firm can then assign workers to assets in 'richer and more varied' manner. This makes the firm more responsive to chamnges changes that it could not anticipate ex-ante.

"My argument is that it allows the firm in internalize many of the externalities

to exploit these constituents."  ==Regulating Trade Within the Firm== This section basis much of it's analysis on the implications of [Holmstrom Milgrom (1991) - Multi Task Principal Agent Analyses| Holmstrom and Milgrom (1991)] (the multitask principal agent problem). With that in mind it sets up three moral hazard models. Moral hazard stems from imperfect performance measurement. Three ways that measurement can be imperfect are:#<math>x=f(e,\theta)\;</math> where <math>\theta\;</math> is realized by nature, and the agent is risk averse (or has limited wealth)#<math>x=\theta e\;</math> where <math>\theta\;</math> is observed by the agent prior to exerting effort, <math>c(e)\;</math> is strictly convex and <math>\mathbb{E}(\theta) =1 \;</math>#<math>x = e + m\;</math>, where <math>m\;</math> is the degree to which the agent manipulates <math>x\;</math>, and is privately costly (this can be interpretted as shading on quality)  ===A simple model=== Using the shading model (<math>x = e + m\;</math>), with private cost: :<math>cost = c(e) + d(m) = \frac{1}{2}e^2 + \frac{1}{2}\lambda m^2\;</math>  The value of output is: :<math>y = pe\;</math>  where <math>p\;</math> is the value of the agents input(s).  Using linear sharing rules of the form: :<math>s(x) = \alpha x + \beta\;</math>  The agent's FOCs from <math>s(x) - (c(e) + d(m))\;</math> are: :<math>\alpha = e \quad \mbox{and} \quad \alpha = m\cdot \lambda \quad \therefore \frac{e}{m} = \lambda\;</math>  So the ratio above will be the same irrespective of the sharing rule (incentive scheme) that is used (even if it is non-linear).  The total surplus is given by: :<math>S = y - (c(e) + d(m)) = pe - \left( \frac{1}{2}e^2 + \frac{1}{2}\lambda m^2 \right)\;</math>  The FOCs are: :<math>p = e \quad \mbox{and} \quad \lambda m = 0\;</math>  As <math>\alpha = e\;</math> from the agent's FOC, the first best contract is <math>\alpha = p\;</math>. However, subbing in the agent's solutions for <math>e\;</math> and <math>m\;</math> into <math>S\;</math> gives: :<math>S = p\alpha - \left( \frac{1}{2} \alpha ^2 + \frac{1}{2}\lambda \frac{\alpha}{\lambda}^2 \right)\;</math>  Then the best choice is to set: :<math>p - \frac{1}{2} \left (2 \alpha + 2\frac{\alpha}{\lambda} \right ) = 0 \;\therefore\; \alpha = \frac{\lambda p}{1+\lambda}\;</math>  ===Introducing Multitasking=== Now we allow for two tasks, and two performance measures, by changing the cost to: :<math>cost = \frac{1}{2}(e_1 + e_2)^2 + \frac{1}{2}\lambda m^2\;</math>  and adding the two performance measures: :<math>x_1 = R(e_1) \;\mbox{and}\; x_2 = e_2+m\;</math>  and changing the Principal's return function to be: :<math>y = p_1 R(e_1)+p_2 e_2\;</math>  where <math>R(\cdot)\;</math> is strictly concave and increasing to prevent corners.  Now the agent maximizes: :<math>\alpha_1 R(e_1) + \alpha_2 (e_2 +m) - \frac{1}{2}(e_1 + e_2)^2 + \frac{1}{2}\lambda m^2\;</math>  The three FOCs wrt to <math>e_1,e_2,m\;</math> are: :<math>\alpha_1 R'(e) = \alpha_2\;</math> :<math>e_2 = \alpha_2 -e_1\;</math> :<math>m = \frac{\alpha_2}{\lambda}\;</math>  The deadweight loss is immediately obvious from the equations above - the first best would have no shading so <math>\frac{1}{2}\lambda m^2\;</math> is the loss, and from the last FOC we have: <math>\frac{1}{2}\lambda m^2 = \frac{\alpha_2^2}{2\lambda}\;</math> Without specifying <math>R(\cdot)\;</math> above, there is not simple analytical solution to either the first best or the second best. However, as manipulation becomes infinitely costly (i.e. <math>\lambda \to \infty\;</math>) the principal sets: :<math>\alpha_1 - p_1 and \alpha_2 = p_2\;</math>  When manipulation is possible, the second best solution sets low powered incentives on <math>e_2\;</math>, to reduce wasteful manipulation. This makes is it optimal to set lower powered incentives in <math>e_1\;</math> too, as otherwise too much attention will be diverted towards it. As <math>\lambda\;</math> decreases (manipulation becomes less costly) then the incentives will become even weaker. Holmstrom says:  "...an optimal incentive design should consider not only rewards but also instruments for influencing the agent's opportunity cost..." ==Firm Boundaries== In this section the paper provides two variants of former models to illustrate some points. We quickly summarize their conclusions below. ===Variant 1=== This is a variant on the above model of multitasking but using asset ownership instead. In this variant there is one asset that is used by employee 1 in production of just one factor, but not the other factor, and not by employee 2. It turns out that it is optimal '''not''' to have employee 1 own the asset. Here balanced incentives, even when low powered, turn out to be better than imbalanced incentives (for all parameter values). Holmstrom says that here:  "...the logic of integration is not one of asset complementarities as defined in Hart and Moore (1990), but rather one of incentive complementarities caused by contractual externalitities as in Holmstrom and Milgrom (1994)." ===Variant 2=== A worker works for firm <math>A\;</math> which has the multitasking model above, and a key asset that enables it and it alone to produce output <math>y_2\;</math>. Firm <math>B\;</math> wants to buy just output <math>y_2\;</math>, but can only contract on measure <math>x_2\;</math> (not on <math>x_1\;</math>). Only firm <math>A\;</math> can contract with its worker, and firm <math>B\;</math> can not observe this contract. With cheap enough manipulation (<math>\lambda\;</math> low enough), firm <math>B\;</math> will stop contracting for the output <math>y_2\;</math> altogether, because the worker will massively mis-allocate his effort. Firm <math>B\;</math> can then decide if it wants to live without <math>y_2\;</math> or try to buy the key asset. The problem is the the output prices have measurement costs in them. A price contract does not allow <math>B\;</math> to specify what it wants from <math>A\;</math> precisely enough. This contrasts with Bernheim and Whinston (1986) (see [Baron 2001 - Theories of Strategic Nonmarket Participation| Baron (2001) for a write up and references), where there are two principals and one agent (i.e. a common agency problem).