## Tuesday, 11 August 2009

### Just for fun: the theory of the firm 5

The incentive-system theory. This approach to the theory of the firm was developed by Holmstrom and Milgrom (1991, 1994); Holmstrom and Tirole (1991) and Holmstrom (1999) and has been described by Gibbons (2005: 206) as an "accidental theory of the firm". The reason for Gibbons's description is that the focus of these papers was not on the make-or-buy problem of the transaction cost or Grossman/Hart/Moore approaches but rather on a multi-task, multi-instrument principal-agent problem and its application to the firm was an "accidental" outcome of this endeavour.

To analyse this approach to the firm, we will take advantage of Gibbons (2005: 210-2) "stick-figure rendition" of the theory. In the simple Gibbons model, there is a technology of production which is a linear combination of the agent's actions: $y=f_1a_1+f_2a_2+\varepsilon$ where the $a_1$ and $a_2$ are actions chosen by the agent and $\varepsilon$ is a noise term. Evaluation of performance by the agent is based on an indicator $p$ which is a different linear combination of the agent's actions: $p=g_1a_1+g_2a_2+\phi$, where $\phi$ is another noise term. Gibbons assumes that both parties are risk-neutral, $\omega$ is the total compensation paid by the principal to the agent and $c(a_1,a_2)$ represents the agent's cost function. Gibbons makes the assumption that:
$$c(a_1,a_2)=\frac{1}{2}a^2_1+\frac{1}{2}a^2_2.\nonumber$$
In addition, Gibbons assumes that the principal and the agent sign a linear contract, $\omega=s+bp$, based on the performance indicator $p$.

To provide a theory of the firm, this model has to be extended to include physical capital, a machine, which is used by the agent during the production of $y$. Post production this capital has a value determined by a third linear combination of the agent's actions: $v=h_1a_1+h_2a_2+\xi$ where $\xi$ is a third noise term. The choice variables in the model are therefore the agent's actions $a_i, i=1,2$ and $b$ the slope of the optimal contract. As a point of comparison, note that the first-best actions of the agent are those which maximise the expected total surplus, that is, they will maximise the expected value of the sum of the principal's payoff, $y-\omega$, the agent's payoff, $\omega-c(a_1,a_2)$, and the value of the physical asset, $v$.
\begin{eqnarray}TS^{FB}&=&E(y-\omega+\omega-c(a_1,a_2)+v)\nonumber\\&=&E(y+v)-c(a_1,a_2)\nonumber\\&=&E(f_1a_1+f_2a_2+\varepsilon+h_1a_1+h_2a_2+\xi)-c(a_1,a_2)\nonumber\\&=&f_1a_1+f_2a_2+h_1a_1+h_2a_2-c(a_1,a_2)\nonumber~~\textrm{assuming}~E(\varepsilon)=E(\xi)=0\\&=&f_1a_1+f_2a_2+h_1a_1+h_2a_2-\left(\frac{1}{2}a^2_1+\frac{1}{2}a^2_2\right)\nonumber\end{eqnarray}
and therefore $a^{FB}_1=f_1+h_1$ and $a^{FB}_2=f_2+h_2$. $TS^{FB}$ is independent of the value of $b$.

If the principal owns the machine, then the agent is an employee of his firm and the principal's payoff is $y+v-\omega$, while the agent's payoff is $\omega-c$. In this case, the agent's optimal actions maximise
\begin{eqnarray*}E(\omega)-c(a_1,a_2)&=&E(s+bp)-\left(\frac{1}{2}a^2_1+\frac{1}{2}a^2_2\right)\\&=&E(s+b(g_1a_1+g_2a_2+\phi))-\left(\frac{1}{2}a^2_1+\frac{1}{2}a^2_2\right)\\&=&s+bg_1a_1+bg_2a_2-\left(\frac{1}{2}a^2_1+\frac{1}{2}a^2_2\right)~~\textrm{assuming}~E(\phi)=0.\end{eqnarray*}
The optimal actions are therefore, $a^\star_{1E}(b)=bg_1$ and $a^\star_{2E}(b)=bg_2$. The efficient contract slope, $b^\star_E$, maximises the expected total surplus, $E(y+v)-c(a_1,a_2)$ or
$$TS_E(b)=(f_1+h_1)a^\star_{1E}(b)+(f_2+h_2)a^\star_{2E}(b)-\left(\frac{1}{2}a^\star_{1E}(b)^2+\frac{1}{2}a^\star_{2E}(b)^2\right).\nonumber$$

Alternatively, the machine can be owned by the agent. Gibbons interprets this case as the agent being an independent contractor. In this situation, the payoffs for the principal will be $y-w$ and for the agent they are $w+v-c$. The optimal actions for the agent will therefore be, $a^\star_{1C}(b)=g_1b+h_1$ and $a^\star_{2C}(b)=g_2b+h_2$. For this case, the efficient slope, $b^\star_C$, will maximise the expected total surplus of
$$TS_C(b)=(f_1+h_1)a^\star_{1C}(b)+(f_2+h_2)a^\star_{2C}(b)-\left( \frac{1}{2}a^\star_{1C}(b)^2+\frac{1}{2}a^\star_{2C}(b)^2\right).\nonumber$$
Gibbons (2005: 211) summaries the analysis so far as:
"[...] having the agent own the asset causes the agent to respond to a given contract slope $(b)$ differently than when the agent does not own the asset, so the make-or-buy problem amounts to determining which of the agent's best-response functions $-$ that of the employee, $(a^\star_{1E}(b), a^\star_{2E}(b))$, or that of the independent contractor, $(a^\star_{1C}(b), a^\star_{2C}(b))$ $-$ allows the parties to achieve greater total surplus."
The discussion so far has relied on an implicit assumption that the value of the asset is not contractible and therefore the owner of the asset receives its value. Since the asset's value is not contractible, putting ownership in the hands of the agent provides him with incentives that cannot be replicated via a contract. But providing the agent with the incentive to increase the value of the asset may or may not help the principal control the agent's incentives via contract. That is, if the agent owns the asset, he has two sources of incentives, the asset's post-production value and the contracted for performance. Without ownership, he concentrates solely on the contracted for performance. Integration would be efficient, that is, having the principal own the asset is efficient, when having the agent do so hurts the principal's efforts to create incentives via contract.

In sum, the distinctive point of the incentive-system approach to the firm is that asset ownership can be one instrument in a multi-task incentive problem. Asset ownership has two sets of effects; one it provides incentives via the value of the asset itself and two it provides incentives via changes induced in the optimal incentive contract. Joint optimization over asset ownership and contract parameters illustrates the system approach to incentive problems. in line with the property-rights approach, the incentive-system theory has the advantage of providing a unified account of both the costs and benefits of integration. In addition the incentive-system theory counters one problems of the property-rights theory in that workets now face incentives, they are no longer act like robot drones.

There are some problems with this approach however. First, most employees are not governed by formal incentive contracts. We just don't see, in the real world, the type of contract the incentive system suggest should be written. Second, and perhaps even more importantly, the elemental incentive system theory omits one of the central and appealing aspects of the rent-seeking and property-rights theories: control. That is, in the elemental incentive-system theory, whether the agent owns the asset affects only the agent's payoff function, it does not effect the agent's action space.

References:
• Gibbons, Robert (2005). 'Four formal(izable) theories of the firm?', Journal of Economic Behavior and Organization, 58(2) October: 200-45.
• Holmstrom, Bengt (1999). 'The Firm as a Subeconomy', Journal of Law, Economics, and Organization, 15(1) April: 74-102.
• Holmstrom, Bengt and Paul Milgrom (1991). 'Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design', Journal of Law, Economics, and Organization, 7(Special Issue): 24-52.
• Holmstrom, Bengt and Paul Milgrom (1994). 'The Firm as an Incentive System', American Economic Review, 84(4) September: 972-91.
• Holmstrom, Bengt and Jean Tirole (1991). 'Transfer Pricing and Organizational Form', Journal of Law, Economics, and Organization 7(2) Fall: 201-28.