Intrinsic Motivation and Dynamic Agency Contract

Abstract

In conventional agency theory, the situation in which the principal lets the agents be motivated to choose the desirable action for him by giving a (monetary) reward is mainly considered. Such a motivation given from outside is called extrinsic motivation in psychology. However in reality, a worker in a firm would not necessarily work only for a reward. The worker has incentive produced internally through sense of accomplishment to be acquired from work and a feeling of self-affirmation to feel that work is useful for the firm and for society. We designate such an incentive intrinsic motivation and distinguish such motivations of two types. Actually, when the firm employs workers and manages them, it would be important for the firm to examine how to promote worker motivation for effective management while considering intrinsic motivation. In this model, in addition to an external motivation as a reward, we assume that an agent with intrinsic motivation contributes for the firm by working diligently to let the work succeed. For the firm, which means the principal, we consider how it should design the contract with the worker having intrinsic motivation. Especially, we analyze a dynamic case theoretically in which a short-term contract is updated and agency relations are continued for two periods.

Appendices

A Proof of Lemma 1.1

First, we show that the contract satisfying (ICS) and $(\mathrm{\overline{PCS}})$ satisfies participation constraint (PCS) (PCS) with strict inequality. By (ICS) and $(\mathrm{\overline{PCS}})$

$$\displaystyle\begin{array}{rcl} \underline{U}& \geq & \overline{w} -\frac{c} {2}(\overline{a}-\varDelta \theta )^{2} +\gamma (\overline{a}-\varDelta \theta ) {}\\ & \geq & \left \{\frac{c} {2}\overline{a}^{2} -\gamma \overline{a}\right \} -\left \{\frac{c} {2}(\overline{a}-\varDelta \theta )^{2} -\gamma (\overline{a}-\varDelta \theta )\right \}> 0. {}\\ \end{array}$$

As $d[\frac{c} {2}a^{2} -\gamma a]/da> 0$ by the assumption, the last inequality described above holds unbinding. Therefore, these constraints are arranged as $(\mathrm{\underline{ICS}}),\ (\mathrm{\overline{PCS}})$, and $(\mathrm{\overline{ICS}})$. Hereinafter, we verify that the optimal contract satisfying (ICS) and $(\mathrm{\overline{PCS}})$ satisfies $(\mathrm{\overline{ICS}})$.

Assuming that $(\mathrm{\overline{PCS}})$ holds unbinding under the optimal contract, then we can design new contracts that satisfy the other constraint (ICS) and improve the value of the firm’s purpose function merely by decreasing $\overline{w}$. This contradicts an original presumption. Consequently, under the optimal contract $(\mathrm{\overline{PCS}})$ holds binding, that is, $\overline{w} = \frac{c} {2}\overline{a}^{2} -\gamma \overline{a}$. Substituting $(\mathrm{\overline{PCS}})$ into (ICS) and arranging it, we have

(ICS) also holds binding under the optimal contract by reduction to absurdity as described above. Therefore, substituting $(\mathrm{\overline{PCS}})$ and (ICS) into the firm’s purpose function, we have

$$\displaystyle\begin{array}{rcl} & & \max _{\underline{a},\overline{a}}\ \nu [S -\underline{\theta } +\underline{a} -\frac{c} {2}\underline{a}^{2} -\frac{c} {2}\overline{a}^{2} + \frac{c} {2}(\overline{a}-\varDelta \theta )^{2} +\gamma (\underline{a}+\varDelta \theta )] {}\\ & & \qquad \quad + (1-\nu )[S -\overline{\theta } + \overline{a} -\frac{c} {2}\overline{a}^{2} +\gamma \overline{a}]. {}\\ \end{array}$$

Calculating the first-order condition with respect to $\underline{a},\ \overline{a}$ and arranging it, we obtain that $\underline{a}_{s} = a^{fb} = \frac{1+\gamma } {c},\ \overline{a}_{s} = \frac{1+\gamma } {c} - \frac{\nu } {1-\nu }\varDelta \theta$.

Finally, we show that these optimal solutions satisfy $(\mathrm{\overline{ICS}})$.

$$\displaystyle\begin{array}{rcl} & & \ \overline{U} -\left \{\underline{w}_{S} -\frac{c} {2}(\underline{a}_{S}+\varDelta \theta )^{2} +\gamma (\underline{a}_{ S}+\varDelta \theta )\right \} {}\\ & & = 0 -\left \{\frac{c} {2}\underline{a}_{S}^{2} + \frac{c} {2}\overline{a}_{S}^{2} -\frac{c} {2}(\overline{a}_{S}-\varDelta \theta )^{2} -\gamma (\underline{a}_{ S}+\varDelta \theta )\right \} + \frac{c} {2}(\underline{a}_{S}+\varDelta \theta )^{2} -\gamma (\underline{a}_{ S}+\varDelta \theta ) {}\\ & & = [\underline{a}_{S} -\overline{a}_{S}+\varDelta \theta ]c\varDelta \theta> 0. {}\\ \end{array}$$

Here the last inequality holds using $\underline{a}_{S}> \overline{a}_{S}$.

B Derivation of $(\mathrm{\overline{ICD^{'}}})$

First, we demonstrate that the contract satisfying (ICD) and $(\mathrm{\overline{PCD}})$ satisfies the participation constraint (PCD) with strict inequality. By (ICD), $(\mathrm{\overline{PCD}})$, and $\delta \underline{U}(\overline{\nu }) \geq 0$, we have

$$\displaystyle\begin{array}{rcl} \underline{w}_{1} -\frac{c} {2}(\underline{a}_{1})^{2} +\gamma \underline{ a}_{ 1} +\delta \underline{ U}(\underline{\nu })& \geq & \overline{w}_{1} -\frac{c} {2}(\overline{a}_{1}-\varDelta \theta )^{2} +\gamma (\overline{a}_{ 1}-\varDelta \theta ) +\delta \underline{ U}(\overline{\nu }) {}\\ & \geq & \overline{w}_{1} -\frac{c} {2}(\overline{a}_{1}-\varDelta \theta )^{2} +\gamma (\overline{a}_{ 1}-\varDelta \theta ) {}\\ & \geq & \left \{\frac{c} {2}(\overline{a}_{1})^{2} -\gamma \overline{a}_{ 1}\right \} -\left \{\frac{c} {2}(\overline{a}_{1}-\varDelta \theta )^{2} -\gamma (\overline{a}_{ 1}-\varDelta \theta )\right \} {}\\ &>& 0. {}\\ \end{array}$$

Because $d[\frac{c} {2}a^{2} -\gamma a]/da> 0$ by assumption, the last inequality described above holds unbinding.

Next we show that $(\mathrm{\overline{PCD}})$ holds binding. Assuming that $(\mathrm{\overline{PCD}})$ holds unbinding under the optimal contract, then we define a new wage contract as reducing $\underline{w}_{1},\ \overline{w}_{1}$ slightly each and satisfying (ICD) and $(\mathrm{\overline{ICD}})$. This contract improves the firm’s purpose function. Therefore, it contradicts the assumption that $(\mathrm{\overline{PCD}})$ holds unbinding under the optimal contract. The optimal contract makes $(\mathrm{\overline{PCD}})$ hold binding, that is, $\overline{w}_{1} = \frac{c} {2}(\overline{a}_{1})^{2} -\gamma \overline{a}_{ 1}$.

Substituting $(\mathrm{\overline{PCD}})$ into (ICD) and $(\mathrm{\overline{ICD}})$ and arranging it, we obtain

Here we assume that (ICD) holds unbinding. We can design a new contract that reduces w slightly and which satisfies another constraint $(\mathrm{\overline{ICD}})$. This contract improves the firm’s purpose function. It contradicts the assumption presented above. Therefore, under the optimal contract (ICD) holds binding, i.e., $\underline{w}_{1} = \frac{c} {2}(\underline{a}_{1})^{2} + \frac{c} {2}(\overline{a}_{1})^{2} -\frac{c} {2}(\overline{a}_{1}-\varDelta \theta )^{2} -\gamma (\underline{a}_{ 1}+\varDelta \theta ) +\delta [\underline{U}(\overline{\nu }) -\underline{ U}(\underline{\nu })]$. Substituting (ICD) into $(\mathrm{\overline{ICD}})$ and arranging it, we have

C Derivation of Eq. (1.9)

Differentiating expected profit W _D ^A with respect to x ^A, we obtain the following equation:

$$\displaystyle\begin{array}{rcl} \frac{dW_{D}^{A}} {d\underline{x}^{A}} & =& \frac{\partial W_{D}^{A}} {\partial \underline{x}^{A}} + \frac{\partial W_{D}^{A}} {\partial \underline{a}_{1}^{A}} \frac{\partial \underline{a}_{1}^{A}} {\partial \underline{x}^{A}} + \frac{\partial W_{D}^{A}} {\partial \overline{a}_{1}^{A}} \frac{\partial \overline{a}_{1}^{A}} {\partial \underline{x}^{A}} + \frac{\partial W_{D}^{A}} {\partial \underline{U}(\overline{\nu }^{A})} \frac{\partial \underline{U}(\overline{\nu }^{A})} {\partial \overline{\nu }^{A}} \frac{\partial \overline{\nu }^{A}} {\partial \underline{x}^{A}} {}\\ & +& \frac{\partial W_{D}^{A}} {\partial W_{s}(\overline{\nu }^{A})} \frac{\partial W_{s}(\overline{\nu }^{A})} {\partial \overline{\nu }^{A}} \frac{\partial \overline{\nu }^{A}} {\partial \underline{x}^{A}}. {}\\ \end{array}$$

By the envelope theorem, the above equation can be simplified as shown below:

$$\displaystyle{ \frac{dW_{D}^{A}} {d\underline{x}^{A}} = \frac{\partial W_{D}^{A}} {\partial \underline{x}^{A}} +\bigg (\frac{\partial W_{D}^{A}} {\partial \underline{U}(\overline{\nu }^{A})} \frac{\partial \underline{U}(\overline{\nu }^{A})} {\partial \overline{\nu }^{A}} + \frac{\partial W_{D}^{A}} {\partial W_{s}(\overline{\nu }^{A})} \frac{\partial W_{s}(\overline{\nu }^{A})} {\partial \overline{\nu }^{A}} \bigg) \frac{\partial \overline{\nu }^{A}} {\partial \underline{x}^{A}}. }$$

Therefore, we have

$$\displaystyle{ \frac{dW_{D}^{A}} {d\underline{x}^{A}} \bigg\vert _{\underline{x}^{A}=0} =\bigg [\frac{\partial W_{D}^{A}} {\partial \underline{x}^{A}} +\bigg (\frac{\partial W_{D}^{A}} {\partial \underline{U}(\overline{\nu }^{A})} \frac{\partial \underline{U}(\overline{\nu }^{A})} {\partial \overline{\nu }^{A}} + \frac{\partial W_{D}^{A}} {\partial W_{s}(\overline{\nu }^{A})} \frac{\partial W_{s}(\overline{\nu }^{A})} {\partial \overline{\nu }^{A}} \bigg) \frac{\partial \overline{\nu }^{A}} {\partial \underline{x}^{A}}\bigg]_{\underline{x}^{A}=0}. }$$

We calculate each term of the right-hand side in the above equation. The first term is the following:

$$\displaystyle\begin{array}{rcl} \frac{\partial W_{D}^{A}} {\partial \underline{x}^{A}} \bigg\vert _{\underline{x}^{A}=0}& =& -\nu \bigg[S -\underline{\theta } +a^{fb} -\frac{c} {2}(a^{fb})^{2} -\frac{c} {2}(\overline{a}^{s})^{2} + \frac{c} {2}(\overline{a}^{s}-\varDelta \theta )^{2} +\gamma (a^{fb}+\varDelta \theta ) {}\\ & -& \delta \underline{U}(\nu ) +\delta W_{s}(1)\bigg] +\nu \bigg [S -\overline{\theta } + \overline{a}^{s} -\frac{c} {2}(\overline{a}^{s})^{2} +\gamma \overline{a}^{s} +\delta W_{ s}(\nu )\bigg]. {}\\ \end{array}$$

Here W _s(1),  W _s(ν),  U(ν) can be written respectively as shown below:

$$\displaystyle\begin{array}{rcl} W_{s}(1)& =& S -\underline{\theta } +a^{fb} -\frac{c} {2}(a^{fb})^{2}, {}\\ W_{s}(\nu )& =& S -\nu \bigg [\underline{\theta }+a^{fb} -\frac{c} {2}(a^{fb})^{2} -\frac{c} {2}(\overline{a}^{s})^{2} + \frac{c} {2}(\overline{a}^{s}-\varDelta \theta )^{2} +\gamma (a^{fb}+\varDelta \theta )\bigg] {}\\ & -& (1-\nu )\bigg[\overline{\theta } -\overline{a}^{s} + \frac{c} {2}(\overline{a}^{s})^{2} -\gamma \overline{a}^{s}\bigg], {}\\ \underline{U}(\nu )& =& \bigg(1 -\frac{c} {2} \frac{1+\nu } {1-\nu }\varDelta \theta \bigg)\varDelta \theta. {}\\ \end{array}$$

Arranging these equations, we have

$$\displaystyle{ \frac{\partial W_{D}^{A}} {\partial \underline{x}^{A}} \bigg\vert _{\underline{x}^{A}=0} = - \frac{\nu c(\varDelta )^{2}} {2(1-\nu )}\bigg( \frac{1} {1-\nu }+\delta \bigg). }$$

The sign of the above equation becomes negative. We can interpret the sign in the first term as follows: The possibility exists that efficient type reports non-efficiency type falsely in the first. Therefore, when the contract for the non-efficiency type is implemented in the first, the firm designs the second contract under incomplete information because it updates the probability of the efficient type to positive in the second beginning. Consequently, this situation reduces the firm’s expected profit in two terms. Next we calculate the second term and have

$$\displaystyle{ \bigg[\bigg(\frac{\partial W_{D}^{A}} {\partial \underline{U}(\overline{\nu }^{A})} \frac{\partial \underline{U}(\overline{\nu }^{A})} {\partial \overline{\nu }^{A}} + \frac{\partial W_{D}^{A}} {\partial W_{s}(\overline{\nu }^{A})} \frac{\partial W_{s}(\overline{\nu }^{A})} {\partial \overline{\nu }^{A}} \bigg) \frac{\partial \overline{\nu }^{A}} {\partial \underline{x}^{A}}\bigg]_{\underline{x}^{A}=0} = \frac{\nu (1-\nu )\delta c(\varDelta )^{2}} {2(1-\nu )^{3}}. }$$

The sign of the expression above becomes positive because the second term operates the effects by which the second information rent decreases and the second firm’s profit increases with the former independently. Therefore we have

$$\displaystyle{ \frac{dW_{D}^{A}} {d\underline{x}^{A}} \bigg\vert _{\underline{x}^{A}=0} = - \frac{\nu c(\varDelta )^{2}} {2(1-\nu )}\bigg( \frac{1} {1-\nu }+\delta \bigg) + \frac{\nu (1+\nu )\delta c(\varDelta )^{2}} {2(1-\nu )^{3}}. }$$

D Proof of Proposition 1.5

In Case A, differentiating expected profit W _D ^A with respect to γ, we obtain the following equation:

$$\displaystyle{ \frac{dW_{D}^{A}} {d\gamma } = \frac{\partial W_{D}^{A}} {\partial \gamma } +\frac{\partial W_{D}^{A}} {\partial a^{fb}} \frac{\partial a^{fb}} {\partial \gamma } +\frac{\partial W_{D}^{A}} {\partial \overline{a}_{1}^{A}} \frac{\partial \overline{a}_{1}^{A}} {\partial \gamma } + \frac{\partial W_{D}^{A}} {\partial W_{s}(1)} \frac{\partial W_{s}(1)} {\partial \gamma } + \frac{\partial W_{D}^{A}} {\partial W_{s}(\overline{\nu }^{A})} \frac{\partial W_{s}(\overline{\nu }^{A})} {\partial \gamma }. }$$

By the envelope theorem, the equation above can be simplified as shown below:

$$\displaystyle\begin{array}{rcl} \frac{dW_{D}^{A}} {d\gamma } & =& \frac{\partial W_{D}^{A}} {\partial \gamma } + \frac{\partial W_{D}^{A}} {\partial W_{s}(1)} \frac{\partial W_{s}(1)} {\partial \gamma } + \frac{\partial W_{D}^{A}} {\partial W_{s}(\overline{\nu }^{A})} \frac{\partial W_{s}(\overline{\nu }^{A})} {\partial \gamma } {}\\ & =& \{\nu (1 -\underline{ x}^{A})(a^{fb}+\varDelta \theta ) + (\nu \underline{x}^{A} + 1-\nu )\overline{a}_{ 1}^{A}\} +\delta \nu (1 -\underline{ x}^{A})a^{fb} {}\\ & & \ \ \ +\delta (\nu \underline{x}^{A} + 1-\nu )\left [\overline{\nu }^{A}(a^{fb}+\varDelta \theta ) + (1 -\overline{\nu }^{A})\overline{a}^{s}(\overline{\nu }^{A})\right ] {}\\ & =& \left \{\nu (1 -\underline{ x}^{A})(a^{fb}+\varDelta \theta ) + (\nu \underline{x}^{A} + 1-\nu )\left (a^{fb} -\frac{\nu (1 -\underline{ x}^{A})} {\nu \underline{x}^{A} + 1-\nu }\varDelta \theta \right )\right \} {}\\ & +& \delta \nu (1 -\underline{ x}^{A})a^{fb} +\delta (\nu \underline{x}^{A} + 1-\nu )\left [\overline{\nu }^{A}(a^{fb}+\varDelta \theta ) + (1 -\overline{\nu }^{A})\left (a^{fb} - \frac{\overline{\nu }^{A}} {1 -\overline{\nu }^{A}}\varDelta \theta \right )\right ] {}\\ & =& a^{fb} +\delta \nu (1 -\underline{ x}^{A})a^{fb} +\delta (\nu \underline{x}^{A} + 1-\nu )a^{fb} = a^{fb} +\delta a^{fb}. {}\\ \end{array}$$

Next in Case B, differentiating expected profit W _D ^B with respect to γ, we obtain the following equation:

$$\displaystyle{ \frac{dW_{D}^{B}} {d\gamma } = \frac{\partial W_{D}^{B}} {\partial \gamma } +\frac{\partial W_{D}^{B}} {\partial \underline{a}_{1}^{B}} \frac{\partial \underline{a}_{1}^{B}} {\partial \gamma } +\frac{\partial W_{D}^{B}} {\partial \overline{a}_{1}^{B}} \frac{\partial \overline{a}_{1}^{B}} {\partial \gamma } + \frac{\partial W_{D}^{B}} {\partial W_{s}(\underline{\nu }^{B})} \frac{\partial W_{s}(\underline{\nu }^{B})} {\partial \gamma } + \frac{\partial W_{D}^{B}} {\partial W_{s}(\overline{\nu }^{B})} \frac{\partial W_{s}(\overline{\nu }^{B})} {\partial \gamma }. }$$

Using the envelope theorem again, the equation above can be simplified as shown below:

$$\displaystyle\begin{array}{rcl} \frac{dW_{D}^{B}} {d\gamma } & =& \frac{\partial W_{D}^{B}} {\partial \gamma } + \frac{\partial W_{D}^{B}} {\partial W_{s}(\underline{\nu }^{B})} \frac{\partial W_{s}(\underline{\nu }^{B})} {\partial \gamma } + \frac{\partial W_{D}^{B}} {\partial W_{s}(\overline{\nu }^{B})} \frac{\partial W_{s}(\overline{\nu }^{B})} {\partial \gamma } {}\\ & =& [\nu (1 -\underline{ x}^{B}) + (1-\nu )\overline{x}^{B}]\left (\overline{a}_{ 1}^{B} + \frac{\delta [\underline{U}(\overline{\nu }) -\underline{ U}(\underline{\nu })]} {c\varDelta \theta } \right ) {}\\ & & \ \ \ + [\nu \underline{x}^{B} + (1-\nu )(1 -\overline{x}^{B})]\overline{a}_{ 1}^{B} {}\\ & & \ \ \ +\delta [\nu (1 -\underline{ x}^{B}) + (1-\nu )\overline{x}^{B}][\underline{\nu }^{B}(a^{fb}+\varDelta \theta ) + (1 -\underline{\nu }^{B})\overline{a}_{ s}(\underline{\nu }^{B})] {}\\ & & \ \ \ +\delta [\nu \underline{x}^{B} + (1-\nu )(1 -\overline{x}^{B})][\overline{\nu }^{B}(a^{fb}+\varDelta \theta ) + (1 -\overline{\nu }^{B})\overline{a}_{ s}(\overline{\nu }^{B})] {}\\ & =& \overline{a}_{1}^{B} + \frac{\delta [\underline{U}(\overline{\nu }) -\underline{ U}(\underline{\nu })]} {c\varDelta \theta } [\nu (1 -\underline{ x}^{B}) + (1-\nu )\overline{x}^{B}] {}\\ & & \ \ \ +\delta [\nu (1 -\underline{ x}^{B}) + (1-\nu )\overline{x}^{B}]\left [\underline{\nu }^{B}(a^{fb}+\varDelta \theta ) + (1 -\underline{\nu }^{B})\left (a^{fb} - \frac{\underline{\nu }^{B}} {1 -\underline{\nu }^{B}}\varDelta \theta \right )\right ] {}\\ & & \ \ \ +\delta [\nu \underline{x}^{B} + (1-\nu )(1 -\overline{x}^{B})]\left [\overline{\nu }^{B}(a^{fb}+\varDelta \theta ) + (1 -\overline{\nu }^{B})\left (a^{fb} - \frac{\overline{\nu }^{B}} {1 -\overline{\nu }^{B}}\varDelta \theta \right )\right ] {}\\ & =& a^{fb} -\frac{\delta [\underline{U}(\overline{\nu }) -\underline{ U}(\underline{\nu })]} {c\varDelta \theta } [\nu (1 -\underline{ x}^{B}) + (1-\nu )\overline{x}^{B}] {}\\ & & \ \ \ + \frac{\delta [\underline{U}(\overline{\nu }) -\underline{ U}(\underline{\nu })]} {c\varDelta \theta } [\nu (1 -\underline{ x}^{B}) + (1-\nu )\overline{x}^{B}] {}\\ & & \ \ \ +\delta [\nu (1 -\underline{ x}^{B}) + (1-\nu )\overline{x}^{B}]a^{fb} +\delta [\nu \underline{x}^{B} + (1-\nu )(1 -\overline{x}^{B})]a^{fb} {}\\ & =& a^{fb} +\delta a^{fb}. {}\\ \end{array}$$