Dynamic performance measurement with intangible assets

Abstract

The increasing importance of intangible assets in modern economies is driving companies to include measures of intangible assets in managerial performance evaluations. For the multiperiod principal-agent model analyzed in this paper, a manager must be motivated to invest in intangible assets like customer satisfaction or product quality. The intangible asset is not verifiable for contracting purposes, but the parties can rely on a noisy indicator of the current asset value. I derive a class of value added performance measures, which effectively aggregate the current cash flow and consecutive realizations of the noisy indicator of the intangible asset. This class of performance measures is shown to be optimal for different scenarios regarding contract commitment and observability of the actual investment decisions. Long-term contracts are examined as a baseline. However, in practice firms usually adopt shorter medium-term contracts that are periodically renegotiated. I show that this more realistic contracting scenario yields the same investment patterns and efficiency levels as those obtained under long-term commitment.

Appendix

Proof of Lemma 1

The derivation of this expression is based on Dutta and Reichelstein (1999a). The proof follows the same steps and only differs in the fact the performance measure π _t is contingent on two noisy signals every period: the cash flow c _t and the NFPI y _t . Given the following expressions for the expected value and variance of the compensation at a period k given the information at date t − 1,

$$ E\left[ {s_{k} |{\bf{I}}_{t - 1} } \right] = E\left[ {\alpha_{k} + \beta_{k} \cdot \left( {c_{k} + u_{k} \cdot y_{k} } \right)|{\bf{I}}_{t - 1} } \right] = e_{k} - h\left( {b_{k} } \right) + v \cdot A_{k - 1} + u_{k} \cdot A_{k} , $$

$$ Var\left[ {s_{k} |{\bf{I}}_{t - 1} } \right] = Var\left[ {\alpha_{k} + \beta_{k} \cdot \left( {c_{k} + u_{k} \cdot y_{k} } \right)|{\bf{I}}_{t - 1} } \right] = \beta_{k}^{2} \cdot \left( {\sigma^{2} + u_{k}^{2} \cdot \mu^{2} } \right), $$

deriving the certainty equivalent expression is a mechanical application of the steps described in Dutta and Reichelstein (1999a, b).

Proof of Proposition 1′

At time t the realization of the NFPI in the previous period, y _t−1, is common knowledge. Therefore, in the value added measure, $VA_{t} = c_{t} + m_{t} \cdot \omega \cdot \left[ {y_{t} - \left( {1 - \delta } \right)y_{t - 1} } \right] - v \cdot y_{t - 1},$ only the cash flow variable c _t and the current NFPI variable y _t are uncertain. Since in the short-term setting the fixed payment cancels any dependency from previous realizations, the agent is not affected by incentives provided in future periods. Therefore, the value added measure reduces to the memoryless performance measure minus a constant term, $\left[ {m_{t} \cdot \omega \cdot \left( {1 - \delta } \right) + v} \right] \cdot y_{t - 1}.$ This constant term only affects the fixed payment. The rest of the results obtained for the memoryless performance measure remain valid. That is, the product m _t  · ω plays the same role as the weight coefficient, m _t  · ω = u _t , the optimal slope coefficient and the effort and investment levels remain the same but the fixed payment becomes larger to compensate for the constant amount subtracted in the performance measure.

Proof of Proposition 2

Since the argument for the two last periods was already outlined in the main text, I focus the proof on completing the induction by assuming the results hold for all periods $t \in \left\{ {k + 1, \ldots ,\,T} \right\}$ and showing they also hold for t = k. In this proof, I do not invoke the RPP. Instead, I allow renegotiation. As will become apparent, the derivation of the renegotiation-proof contract is a particular case of the argument given here. In period k, when the long-term contract S ^k is signed, both parties anticipate that this contract is renegotiated and a new contract S ^k+1 is signed at the beginning of period k + 1. This contract is in turn renegotiated and a new contract S ^k+2 is signed at the beginning of period k + 2, and so on, until period T. The agent requires the certainty equivalent that he obtains with the new contract S ^k and subsequent renegotiations to be at least as large as the one that he would obtain with the old contract S ^k−1. The principal cannot observe investments. Therefore, she conjectures past investment decisions and makes the participation constraint binding given these conjectures:

$$ CE\left( {S_{k}^{k} ,\,S_{k + 1}^{k + 1} , \ldots ,\,S_{T}^{T} {|I}_{k - 1} ,\,A_{k - 1}^{c} } \right) = CE\left( {S_{k}^{k - 1} ,\,S_{k + 1}^{k - 1} , \ldots ,\,S_{T}^{k - 1} {|I}_{k - 1} ,\,A_{k - 1}^{c} } \right). $$

(18)

Here, $CE\left( {S_{k}^{k} ,\,S_{k + 1}^{k + 1} , \ldots ,\,S_{T}^{T} {|I}_{k - 1} ,\,A_{k - 1}^{c} } \right)$ is the certainty equivalent of the manager from the principal’s perspective, with contract S ^k in period k, replaced by successively renegotiated contracts in subsequent periods; $CE\left( {S_{k}^{k - 1} ,\,S_{k + 1}^{k - 1} , \ldots ,\,S_{T}^{k - 1} {|I}_{k - 1} ,\,A_{k - 1}^{c} } \right)$ is the certainty equivalent of the manager from the principal’s perspective under contract S ^k−1 in periods k to T; and $A_{k - 1}^{c}$ is the principal’s conjecture about the intangible asset level at the beginning of period k.

The investment incentive compatibility constraint can be expressed as follows:

$$ \begin{gathered} b_{k}^{k} = \mathop {\arg \max }\limits_{{\hat{b}_{k} }} \{ CE(S_{k}^{k} ,\,S_{k + 1}^{k + 1} , \ldots ,\,S_{T}^{T} |{I}_{k - 1} ,\,A_{k - 1}^{c} )\} \hfill \\ \begin{array}{*{20}c} { = \mathop {\arg \max }\limits_{{\hat{b}_{k} }} \left\{ {\sum\nolimits_{t = k}^{T} {\gamma^{t - k} } \cdot [\alpha_{t}^{t} + \beta_{t}^{t} \cdot (e_{t}^{t} - h(\hat{b}_{t} ) + v \cdot A_{t - 1} + u_{t}^{t} \cdot A_{t} )} \right.} \\ {\left. { - g(e_{t} ) - \rho \cdot (\beta_{t}^{t} )^{2} \cdot (\sigma^{2} + (u_{t}^{t} )^{2} \cdot \mu^{2} )]} \right\}.} \\ \end{array} \hfill \\ \end{gathered} $$

Using the first order approach, this constraint can be replaced by the first order condition:

$$ \begin{gathered} \beta_{k}^{k} \cdot \left( { - h^{\prime}\left( {b_{k}^{k} } \right) + u_{k}^{k} } \right) + \theta \cdot \gamma \cdot \left( {v + \left( {1 - \delta } \right) \cdot u_{k + 1}^{ * * } } \right) \hfill \\ + \theta \cdot \sum\nolimits_{t = k + 2}^{T} {\gamma^{t - k} } \cdot \left( {1 - \delta } \right)^{t - 1 - k} \cdot \left( {v + \left( {1 - \delta } \right) \cdot u_{t}^{ * * } } \right) = 0 \hfill \\ \end{gathered} $$

The investment compatibility constraint in period k + 1 is

$$ \theta \cdot \left( { - h^{\prime}\left( {b_{k + 1}^{ * * } } \right) + u_{k + 1}^{ * * } } \right) + \theta \cdot \mathop \sum \limits_{t = k + 2}^{T} \gamma^{t - k - 1} \cdot \left( {1 - \delta } \right)^{t - 2 - k} \cdot \left( {v + (1 - \delta ) \cdot u_{t}^{ * * } } \right) = 0. $$

The incentive compatibility constraint in period k is obtained by combining the last expressions as follows:

$$ \beta_{k}^{k} \cdot \left( { - h^{\prime}\left( {b_{k}^{k} } \right) + u_{k}^{k} } \right) + \theta \cdot \gamma \cdot \left( {v + \left( {1 - \delta } \right) \cdot h^{\prime}\left( {b_{k + 1}^{ * * } } \right)} \right) = 0. $$

Solving for the fixed payment in the participation constraint (18) and replacing its expression in the principals’ objective function, the principal’s program can be reduced to

$$ \begin{array}{ll}\mathop {\text{Max}}\limits_{{\beta_{k}^{k},\,u_{k}^{k} ,\,b_{k}^{k} }} & X(\bar{e},\,\beta_{k}^{k} ) +NPV(b_{k}^{k} ) - RP(\beta_{k}^{k} ,\,u_{k}^{k} ) \\ {\hbox{s.t.}}& {\beta_{k}^{k} \ge \theta } \\ {}& {\beta_{k}^{k} \cdot ( -h'(b_{k}^{k} ) + u_{k}^{k} ) + \theta \cdot \gamma \cdot (\upsilon+ (1 - \delta ) \cdot h^{\prime}(b_{k + 1}^{**} )) = 0.}\\ \end{array} $$

Note that I did not state the incentive compatibility constraints for the effort and investment choices under contract S ^k in periods $t \in \left\{ {k + 1, \ldots ,\,T} \right\}.$ Since contract S ^k is renegotiated at the beginning of period k + 1, all coefficients and managerial decisions under this contract in periods $t \in \left\{ {k + 1, \ldots ,\,T} \right\},$ do not affect the principal’s objective function. The coefficients $\left\{ {\alpha_{t}^{k} ,\,\beta_{t}^{k} ,\,u_{t}^{k} } \right\}_{t = k + 1}^{T}$ jointly satisfy the participation constraint in period k + 1 and otherwise are individually undetermined. As such, the principal can choose to make the contract renegotiation-proof by setting these coefficients equal to the coefficients on the equilibrium path $\left\{ {\alpha_{t}^{ * * } ,\,\beta_{t}^{ * * } ,\,u_{t}^{ * * } } \right\}_{t = k + 1}^{T}.$

The Lagrangian of the principal’s program is

$$ \begin{gathered} L = \omega \cdot b_{k}^{k} - h\left( {b_{k}^{k} } \right) - \rho \cdot \left( {\beta_{k}^{k} } \right)^{2} \cdot \left( {\sigma^{2} + \left( {u_{k}^{k} } \right)^{2} \cdot \mu^{2} } \right) - \lambda_{e} \cdot \left( {\theta - \beta_{k}^{k} } \right) \hfill \\ + \lambda_{b} \cdot \left( {\beta_{k}^{k} \cdot \left( { - h^{\prime}\left( {b_{k}^{k} } \right) + u_{k}^{k} } \right) + \theta \cdot \gamma \cdot \left( {v + \left( {1 - \delta } \right) \cdot h^{\prime}\left( {b_{k + 1}^{ * * } } \right)} \right)} \right) \hfill \\ \end{gathered} $$

The corresponding Kuhn-Tucker conditions are

$$ \tfrac{dL}{{db_{k}^{k} }} = \omega - h^{\prime}\left( {b_{k}^{k} } \right) - \lambda_{b} \cdot \beta_{k}^{k} \cdot h^{\prime\prime}\left( {b_{k}^{k} } \right) = 0 $$

$$ \tfrac{dL}{{d\beta_{k}^{k} }} = - 2 \cdot \rho \cdot \left( {\beta_{k}^{k} } \right) - \left(\sigma^{2} + \left(u_{k}^{k}\right)^{2} \cdot {\mu^{2}}\right) +\lambda_{e} + \lambda_{b} \cdot \left( { - h^{\prime}\left( {b_{k}^{k} } \right) + u_{k}^{k} } \right) = 0, $$

$$ \tfrac{dL}{{du_{k}^{k} }} = - 2 \cdot \rho \cdot \left( {\beta_{k}^{k} } \right)^{2} \cdot u_{k}^{k} \cdot \mu^{2} + \lambda_{b} \cdot \beta_{k}^{k} = 0, $$

$$ \tfrac{dL}{{d\lambda_{b} }} = \beta_{k}^{k} \cdot ( - h^{\prime}(b_{k}^{k} ) + u_{k}^{k} ) + \theta \cdot \gamma \cdot (v + (1 - \delta ) \cdot h^{\prime}(b_{k + 1}^{ * * } )) = 0, $$

$$ \lambda_{e} \cdot \left( {\theta - \beta_{k}^{k} } \right) = 0\,\,{\text{and}}\,\,\beta_{k}^{k} \ge \theta . $$

With $\tfrac{dL}{{d\beta_{k}^{k} }} = 0$ and $\tfrac{dL}{{du_{k}^{k} }} = 0$ the following expressions are obtained for the multipliers:

$$ \lambda_{e} = 2 \cdot \rho \cdot \beta_{k}^{k} \cdot \left( {\sigma^{2} + u_{k}^{k} \cdot \mu^{2} \cdot h^{\prime}\left( {b_{k}^{k} } \right)} \right),\,\quad \quad \lambda_{b} = 2 \cdot \rho \cdot \beta_{k}^{k} \cdot u_{k}^{k} \cdot \mu^{2} . $$

Assume, first, that the $\beta_{k}^{k} > \theta$ and λ _e  = 0. From the multiplier expression, this implies that $u_{k}^{k} = - \tfrac{{\sigma^{2} }}{{\mu^{2} \cdot h^{\prime}(b_{k}^{k} )}} < 0.$ With this expression for $u_{k}^{k},$ we can use the incentive compatibility condition $\tfrac{dL}{{d\lambda_{b} }} = 0$ to derive the following expression for the slope coefficient, $\beta_{k}^{k} = \theta \cdot \gamma \cdot \tfrac{{v + (1 - \delta ) \cdot h^{\prime}\left( {b_{k + 1}^{ * * } } \right)}}{{\sigma^{2} + \mu^{2} \cdot h^{\prime}\left( {b_{k}^{k} } \right)^{2} }} \cdot \mu^{2} \cdot h^{\prime}\left( {b_{k}^{k} } \right).$ From the condition $\beta_{k}^{k} > \theta$ this expression implies, $\gamma \cdot \left( {v + \left( {1 - \delta } \right) \cdot h^{\prime}\left( {b_{k + 1}^{ * * } } \right)} \right) > h^{\prime}\left( {b_{k}^{k} } \right)$. Since there is underinvestment in period k + 1 it follows that $h^{\prime}\left( {b_{k + 1}^{ * * } } \right) < \omega.$ Therefore, $\gamma \cdot \left( {v + \left( {1 - \delta } \right) \cdot \omega } \right) = \omega > h^{\prime}\left( {b_{k}^{k} } \right).$ Thus, there is underinvestment in period k. However, replacing the expressions for λ _b and $u_{k}^{k}$ in equation $\tfrac{dL}{{db_{k}^{k} }} = 0$ yields $\omega - h^{\prime}(b_{k}^{k} ) + 2 \cdot \rho \cdot \beta_{k}^{k} \cdot \tfrac{{\sigma^{2} }}{{h^{\prime}(b_{k}^{k} )}} \cdot h^{\prime\prime}(b_{k}^{k} ) = 0,$ which clearly implies that $\omega < h^{\prime}(b_{k}^{k} ).$ That is a contradiction and therefore it must be the case that $\beta_{k}^{k} = \theta$ and λ _e  > 0.

We can now recalculate the Kuhn-Tucker conditions with the expressions obtained for the multipliers and $\beta_{k}^{k} = \theta$ and obtain the equations:

$$ \omega - h^{\prime}(b_{k}^{k} ) - 2 \cdot \rho \cdot \theta^{2} \cdot u_{k}^{k} \cdot \mu^{2} \cdot h^{\prime\prime}(b_{k}^{k} ) = 0 , $$

$$ - h^{\prime}(b_{k}^{k} ) + u_{k}^{k} + \gamma \cdot (v + (1 - \delta ) \cdot h^{\prime}(b_{k + 1}^{ * * } )) = 0. $$

The first of these two equations already yields the expression for $h^{\prime}(b_{k}^{k} )$ described in the discussion of Proposition 2, $h^{\prime}(b_{k}^{k} ) = \omega - 2 \cdot \rho \cdot \theta^{2} \cdot u_{k}^{k} \cdot \mu^{2} \cdot h^{\prime\prime}(b_{k}^{k} ).$ Solving for $u_{k}^{k}$ in these two equations we obtain the expression:

$$ u_{k}^{ * * } = \frac{{\omega - \gamma \cdot (v + (1 - \delta ) \cdot h^{\prime}(b_{k + 1}^{ * * } ))}}{{1 + 2 \cdot \rho \cdot \theta^{2} \cdot \mu^{2} \cdot h^{\prime\prime}(b_{k}^{k} )}} . $$

For the next period, k + 1, we can write, $h^{\prime}(b_{k + 1}^{ * * } ) = \omega - 2 \cdot \rho \cdot \theta^{2} \cdot u_{k + 1}^{k + 1} \cdot \mu^{2} \cdot h^{\prime\prime}(b_{k + 1}^{ * * } ).$ Substituting this value into the expression for $u_{k}^{k},$ we obtain:

$$ u_{k}^{ * * } = u_{k + 1}^{ * * } \cdot \gamma \cdot (1 - \delta ) \cdot \frac{{2 \cdot \rho \cdot \theta^{2} \cdot \mu^{2} \cdot h^{\prime\prime}(b_{k + 1}^{ * * } )}}{{1 + 2 \cdot \rho \cdot \theta^{2} \cdot \mu^{2} \cdot h^{\prime\prime}(b_{k}^{k} )}} . $$

Proof of Proposition 3

From Proposition 2 recall that in the derivation of the optimal contract signed in period k, the coefficients $\{ \alpha_{t}^{k} ,\,\beta_{t}^{k} ,\,u_{t}^{k} \}_{t = k + 1}^{T}$ are not retained on the equilibrium path. These coefficients belong to periods in which contract S ^k does not apply anymore because new contracts have been chosen at successive renegotiation stages. Therefore, these coefficients only affect the participation constraint in period k + 1 and otherwise are individually undetermined. As long as there is at least one period left to renegotiate, the length of each contract does not affect the actions the agent takes nor the coefficients the principal applies in the equilibrium path. That is, the sequence of medium-term contracts obtains the same performance as the optimal renegotiation-proof long-term contract. Nevertheless, shortening the length of each contract can potentially open opportunities for the parties to deviate profitably from the equilibrium path. In the following exposition, I show that there is a simple way to obtain a sequence of contracts that precludes any deviation. For expositional purposes, I use the memoryless performance measure, but an analogous proof is trivially obtained with the value added performance measure.

Let the principal follow the following strategy,

Strategy: at the beginning of any period t < T offer the agent the contract $S^{t} = \{ S_{t}^{t} ,\,S_{t + 1}^{t} \},$ where:

• $S_{t}^{t} = \{ \alpha_{t}^{t} ,\,\beta_{t}^{ * * } ,\,u_{t}^{ * * } \}.$ That is, S ^t _t contains the slope and weight coefficients corresponding to period t of the optimal renegotiation-proof long-term contract. $\alpha_{t}^{t}$ is set to satisfy the renegotiation participation constraint,

  $$ CE(S_{t}^{t} , \ldots ,\,S_{T}^{T} {|I}_{t - 1} ,\,A_{t - 1}^{c} ) = CE(S_{t}^{t - 1} {|I}_{t - 1} ,\,A_{t - 1}^{c} ). $$

  (19)

• $S_{t + 1}^{t} = \{ 0,\,\beta_{t + 1}^{t} = \beta_{t + 1}^{ * * } ,\,u_{t + 1}^{t} = h^{\prime}(b_{t + 1}^{ * * } )\}$

Let’s first show that, if the principal uses the strategy described above, the agent has no incentive to deviate. Proceeding by backward induction, start at the last period, T. At the beginning of period T, the contract S ^T−1 is in place, and the principal offers the agent a new one-period contract S ^T. It is easy to see that the strategy used by the principal makes the contract S ^T−1 renegotiation-proof by setting in period T the corresponding coefficients of the optimal renegotiation-proof long-term contract, $S_{T}^{T - 1} = \{ 0,\,\beta_{T}^{ * * } ,\,u_{T}^{ * * } \}.$

At the beginning of period T − 1, the principal offers the agent to replace the period left in the contract in place, $S_{T - 1}^{T - 2},$ with a new two-period contract S ^T−1. To accept the contract, the agent requires being at least as well off with the new contract as with the old one. Since the principal cannot observe investment decisions, she conjectures past investments and, as prescribed by the above strategy, sets the fixed payment $\alpha_{T - 1}^{T - 1}$ to make the participation constraint binding given these conjectures:

$$ CE(S_{T - 1}^{T - 1} ,\,S_{T}^{T - 1} |I_{T - 2} ,\,A_{T - 2}^{c} ) = CE(S_{T - 1}^{T - 2} |I_{T - 2} ,\,A_{T - 2}^{c} ) . $$

(20)

If the agent accepts contract S ^T−1, then he chooses the actions the principal prescribes in the remaining periods because it is incentive compatible for him to do so. Furthermore, observe that if the agent executed the principal’s prescribed actions in all previous periods, then in period T − 1 the agent has no incentive to deviate by rejecting the contract S ^T−1. This is the case because the principal’s conjectures are then true and the participation constraint IR T − 1 makes the manager indifferent between accepting the new contract and rejecting it.

In what follows I complete the induction by showing that if the agent does not deviate in periods $(t + 1, \ldots ,\,T),$ then it does not deviate in period t either. At the beginning of period t the principal offers the agent to replace the period left in the contract, $S_{t}^{t - 1},$ with a new two-period contract, S ^t. In doing so, the principal takes into account that the agent demands to be no worse off when accepting the new contract. That is, the fixed payment $\alpha_{t}^{t}$ is set to satisfy the participation constraint,

$$ CE(S_{t}^{t} , \ldots ,\,S_{T}^{T} |I_{t - 1} ,\,A_{t - 1}^{c} ) = CE(S_{t}^{t - 1} |I_{t - 1} ,\,A_{t - 1}^{c} ). $$

(21)

The agent can potentially deviate in two different ways:

• (a) The agent can reject contract S ^t and go on with the remaining period in S ^t−1.

• (b) The agent can accept contract S ^t and plan to reject a future contract S ^t+n, with $n \in \{ 1, \ldots ,\,T - t\}$, choosing his current actions accordingly.

Let’s refute each potential deviation in turn:

• (a) If the agent has performed all previous actions as prescribed by the principal, the principal’s conjectures are true. Therefore, the participation constraint in the principal’s is satisfied and, therefore, the agent has no incentive to reject the contract.

• (b) The principal avoids such deviations by choosing the second period coefficients of each contract appropriately. If in period t the agent plans to reject contract S ^t+n, he must take into account the coefficients of $S_{t + n}^{t + n - 1}.$ However, following the strategy above, these coefficients are chosen so that the decisions taken by the agent are the same whether he accepts or rejects contract S ^t+n. To see this recall from the long-term baseline scenario that the investment incentive compatibility condition at time t is:

  $$ \beta_{t}^{ * * } \cdot [ - h^{\prime}(b_{t} ) + u_{t}^{ * * } ] + \gamma \cdot \mathop \sum \limits_{k = t}^{T} \gamma^{k - t} \cdot \beta_{k + 1}^{ * * } \cdot (1 - \delta )^{k - t} \cdot [v + (1 - \delta ) \cdot u_{k + 1}^{ * * } ] = 0. $$

  (22)

However, if the agent plans to reject contract S ^t+n, his investment choice is taken according to the first order condition:

$$ \begin{gathered} \beta_{t}^{ * * } \cdot [ - h^{\prime}(b_{t} ) + u_{t}^{ * * } ] + \gamma \cdot \mathop \sum \limits_{k = t}^{t + n - 1} \gamma^{k - t} \cdot \beta_{k + 1}^{ * * } \cdot (1 - \delta )^{k - t} \cdot [v + (1 - \delta ) \cdot u_{k + 1}^{ * * } ] \\ + \gamma^{n} \cdot \beta_{t + n}^{ * * } \cdot (1 - \delta )^{n - 1} \cdot [v + (1 - \delta ) \cdot h^{\prime}(b_{t + n}^{ * * } )] = 0. \\ \end{gathered} $$

(23)

It is easy to see that these two equations yield the same investment decision at time t, b _t , by noticing that the first order condition for the investment decision $b_{t + n}^{ * * }$ is

$$ \beta_{t + n}^{ * * } \cdot [ - h^{\prime}(b_{t + n}^{ * * } ) + u_{t + n}^{ * * } ] + \gamma \cdot \mathop \sum \limits_{k = t + n}^{T} \gamma^{k - (t + n)} \cdot \beta_{k + 1}^{ * * } \cdot (1 - \delta )^{k - (t + n)} \cdot [v + (1 - \delta ) \cdot u_{k + 1}^{ * * } ] = 0. $$

(24)

Solving for $h^{\prime}(b_{t + n}^{ * * } )$ in (24) and plugging the resulting expression in (23), one can obtain expression (22) with some basic algebra. That is, the first order conditions of both investment decisions are identical. The principal can induce the same actions in period t regardless of whether the agent is planning to deviate in the future by rejecting a contract S ^t+n.³³ Since with this strategy the agent always takes the actions conjectured by the principal, the participation constraint of contract S ^t+n is satisfied, and the agent has no reason to reject the contract.

Let’s now show that the principal is always better off by renegotiating the current contract at the end of each period. By construction, the principal always offers the sequentially optimal contract to the agent given his information at that time and taking into account future renegotiations. Observe that, if at the beginning of period t the principal does not offer a new contract S ^t to the agent, the principal obtains a short-term outcome in the period left in the current contract, $S_{t}^{t - 1},$ and then the maximum outcome she can achieve is to sign a long-term contract for the remaining T − t periods with another agent. However, staying in the equilibrium path the principal achieves the efficiency of a long-term contract for all the T − t + 1 periods. This efficiency is higher than the one obtained by deviating. Therefore, the principal is always better off by renegotiating the contract.

Observe that, in the elaboration of this proof, I did not rely on the assumption of a fixed effort choice. In fact, Proposition 3 is general enough to include the case of an interior effort solution.

Proof of Proposition 4

Since the argument for the two last periods was already outlined in Sect. 5.2, I focus this proof on completing the induction by assuming the results hold for all periods $\{ t + 1, \ldots ,\,T\}$ and showing they also hold for period t.

In period t, when the long-term contract S ^t is signed, both parties anticipate that this contract is renegotiated and a new contract S ^t+1 is signed at the beginning of period t + 1. This contract is in turn renegotiated and a new contract S ^t+2 is signed at the beginning of period t + 2, and so on, until period T. The certainty equivalent of the agent at the beginning of period t can be shown to be

$$ CE(S_{t}^{t} , \ldots ,\,S_{T}^{T} |{\bf{I}}_{t - 1} ) = \mathop \sum \limits_{k = t}^{T} \gamma^{k - t} \cdot (\alpha_{k}^{k} + \beta_{k}^{k} \cdot E[VA_{k}^{k} |{\bf{I}}_{t - 1} ] - g(e_{k}^{k} ) - \rho \cdot \beta_{k}^{k} \cdot \sigma^{2} - RP_{k}^{t} ) $$

where,

• $E[VA_{t}^{t} |{\bf{I}}_{t - 1} ] = e_{t}^{t} + v \cdot A_{t - 1} - h(b_{t}^{t} ) + m_{t}^{t} \cdot \omega \cdot [A_{t} - (1 - \delta ) \cdot y_{t - 1}^{t} ] - v \cdot y_{t - 1}^{t}$

• $E[VA_{k}^{k} |{\bf{I}}_{t - 1} ] = e_{k}^{k} + m_{k}^{k} \cdot \omega \cdot b_{k}^{k} - h(b_{k}^{k} )$ for $k \in \{ t + 1, \ldots ,\,T\}$

• $RP_{k}^{k} = \rho \cdot [\beta_{k}^{k} \cdot m_{k}^{k} \cdot \omega - \beta_{k + 1}^{k} \cdot \gamma \cdot (v + (1 - \delta ) \cdot m_{k + 1}^{k} \cdot \omega )]^{2} \cdot \mu^{2}$ for $k \in \{ t, \ldots ,\,T - 1\}$

• $RP_{T}^{t} = \rho \cdot [\beta_{T}^{t} \cdot m_{T}^{t} \cdot \omega ]^{2} \cdot \mu^{2}$

Also, the participation constraint in period t + 1 requires the contract S ^t+1, anticipating future renegotiations, to yield the same certainty equivalent as the remaining period in the old contract S ^t. Therefore, the participation constraint is $CE(S_{t + 1}^{t + 1} , \ldots ,\,S_{T}^{T} {|I}_{t} ,\,A_{t} ) = CE(S_{t + 1}^{t} {|I}_{t} ,\,A_{t} ),$ where $CE(S_{t + 1}^{t + 1} , \ldots ,\,S_{T}^{T} {|I}_{t} ,\,A_{t} )$ is the agent’s certainty equivalent if he accepts contract S ^t+1 (and all future renegotiations) from the principal’s perspective, taking into account that she can observe the investment decisions. $CE(S_{t + 1}^{t} {|I}_{t} ,\,A_{t} )$ is the agent’s certainty equivalent obtained with the remaining period in contract S ^t, also from the principal’s perspective and taking into account that she can observe the investment decisions. Combining the resulting equation with the expression for $CE(S_{t + 1}^{t + 1} , \ldots ,\,S_{T}^{T} |{\bf {I}}_{t} ),$ it can be shown that the agent’s certainty equivalent at time t from the agent’s perspective reduces to

$$ \begin{aligned} CE(S_{t}^{t} , \ldots ,\,S_{T}^{T} |{\bf{I}}_{t - 1} ) = & \alpha_{t}^{t} + \beta_{t}^{t} \cdot E[VA_{t}^{t} |{\bf{I}}_{t - 1} ] - g(e_{t}^{t} ) - \rho \cdot \beta_{t}^{t} \cdot \sigma^{2} - RP_{t}^{t} \\ & + (\alpha_{t + 1}^{t} + \beta_{t + 1}^{t} \cdot E[VA_{t + 1}^{t} |{\bf{I}}_{t - 1} ] - g(e_{t + 1}^{t} ) - \rho \cdot \beta_{t + 1}^{t} \cdot \sigma^{2} - RP_{t + 1}^{t} \\ \end{aligned} $$

where, $E[VA_{t}^{t} |{\bf{I}}_{t - 1} ],$ and $RP_{k}^{k}$ are as defined above, and

• $E[VA_{t + 1}^{t} |{\bf{I}}_{t - 1} ] = e_{t + 1}^{t} + m_{t + 1}^{t} \cdot \omega \cdot b_{t + 1}^{t} - h(b_{t + 1}^{t} )$

• $RP_{t + 1}^{t} = \rho \cdot [\beta_{t + 1}^{t} \cdot m_{t + 1}^{t} \cdot \omega ]^{2} \cdot \mu^{2} .$

That is, the agent behaves as if contract S ^t was in effect in periods t and t + 1. Observe that $E[VA_{t + 1}^{t} |{\bf{I}}_{t - 1} ]$ does not depend on the current investment decision $b_{t}^{t}.$ Therefore, the investment incentive compatibility condition is, $m_{t}^{t} \cdot \omega = h^{\prime}(b_{t}^{t} ).$ The principal anticipates that renegotiation allows her to switch to a new contract at the beginning of period t + 1. Therefore, the coefficients that determine compensation in period t + 1 are the ones under contract S ^t+1, $(\alpha_{t + 1}^{t + 1} ,\,\beta_{t + 1}^{t + 1} ,\,u_{t + 1}^{t + 1} ).$ The current contract S ^t affects only the current managerial decisions. Hence, the principal can restrict attention to the effects of current decisions on the current and future periods. As a result, the principal’s program reduces to,

$$ \begin{array} {ll}\mathop {\text{Max}}\limits_{{\beta_{t}^{t} ,\,m_{t}^{t} ,\,\beta_{t + 1}^{t} ,\,m_{t + 1}^{t} ,\,b_{t}^{t} }} & X(\bar{e},\,\beta_{t}^{t} ) + NPV(b_{t}^{t} ) - RP_{t}^{t} (\beta_{t}^{t} ,\,m_{t}^{t} ,\,\beta_{t + 1}^{t} ,\,m_{t + 1}^{t} ) \\ {s.t.} & {\beta_{t}^{t} \ge \theta } \\ {} & {\,m_{t}^{t} \cdot \omega = h^{\prime}(b_{t}^{t} )} \\ \end{array}$$

This incentive compatibility condition shows that the principal can only rely on the current coefficient $m_{t}^{t}$ to motivate the agent to invest in period t. The risk premium expression, however, is contingent on both the current period coefficients and the next period coefficients:

$$ RP_{t}^{t} (\beta_{t}^{t} ,\,m_{t}^{t} ,\,\beta_{t + 1}^{t} ,\,m_{t + 1}^{t} ) = \rho \cdot [\beta_{t}^{t} \cdot m_{t}^{t} \cdot \omega - \beta_{t + 1}^{t} \cdot \gamma \cdot (v + (1 - \delta ) \cdot m_{t + 1}^{t} \cdot \omega )]^{2} \cdot \mu^{2} $$

Observe that the coefficients $\left\{ {\beta_{t + 1}^{t} ,\,m_{t + 1}^{t} } \right\}$ only affect the risk premium. This allows the principal to set the current coefficients $\{ \beta_{t}^{t} ,\,m_{t}^{t} \}$ at the benchmark levels and to use the coefficients in period t + 1, $\left\{ {\beta_{t + 1}^{t} ,\,m_{t + 1}^{t} } \right\}$ to eliminate the risk premium. The principal sets $\beta_{t}^{t} = \theta$ to induce effort $\bar{e},$ sets $m_{t}^{t} = 1$ to induce first-best investment and chooses the coefficients $\left\{ {\beta_{t + 1}^{t} ,\,m_{t + 1}^{t} } \right\}$ to eliminate the risk premium by satisfying:

$$ \theta \cdot \omega - \gamma \cdot \beta_{t + 1}^{t} \cdot \left[ {v + \left( {1 - \delta } \right) \cdot \omega \cdot m_{t + 1}^{t} } \right] = 0 $$

This condition is the only restriction imposed on the coefficients $\left\{ {\beta_{t + 1}^{t} ,\,m_{t + 1}^{t} } \right\}$.