Managerial Payoff and Gift-Exchange in the Field

Abstract

We conduct a field experiment where we vary both the presence of a gift-exchange wage and the effect of the worker’s effort on the manager’s payoff. Results indicate a strong complementarity between the initial wage-gift and the agent’s ability to “repay the gift”. We control for differences in ability and reciprocal inclination and show that gift-exchange is more effective with more reciprocal agents. We present a principal-agent model with reciprocal subjects that motivates our findings. Our results help to reconcile the conflicting evidence on the efficacy of gift-exchange outside the lab.

Appendices

Appendix 1: Additional Material

See Fig. 2.

Fig. 2

Example Page Prussian Census of 1849

Appendix 2: Derivations

1.1 Comparative Statics

On the assumption that the contract is generous, the worker’s optimal effort choice $a^*$ for a given contract is implicitly defined by the first order condition

$$\begin{aligned} \frac{\partial U\left( {\tilde{w}}, a,{\hat{a}}\right) }{\partial a}= - c^{\prime }(a^*) + \eta \left( u({\tilde{w}}) - c({\hat{a}})- {\bar{u}}\right) M \cdot ER^{\prime }(a^*)=0 . \end{aligned}$$

Applying the implicit function theorem we can derive the relevant comparative statics w.r.t. ${\tilde{w}}$ and M:

$$\begin{aligned} \frac{\partial ^2 U\left( {\tilde{w}}, a,{\hat{a}}\right) }{\partial a^2}= & {} - c^{\prime \prime }(a^*) + \eta \left( u({\tilde{w}}) - c({\hat{a}})- {\bar{u}}\right) M \cdot ER^{\prime \prime }(a^*)<0\\ \frac{\partial ^2 U\left( {\tilde{w}}, a,{\hat{a}}\right) }{\partial a \partial M }= & {} \eta \left( u({\tilde{w}}) - c({\hat{a}})- {\bar{u}}\right) ER^{\prime }(a^*)>0\\ \frac{\partial ^2 U\left( {\tilde{w}}, a,{\hat{a}}\right) }{\partial a \partial {\tilde{w}}}= & {} \eta M \cdot u^\prime ({\tilde{w}}) \cdot ER^{\prime \prime }(a^*)>0\\ \end{aligned}$$

Hence it follows that

$$\begin{aligned} \frac{\partial a^*}{\partial M} = -\frac{\frac{\partial ^2 U\left( {\tilde{w}}, a,{\hat{a}}\right) }{\partial a \partial M }}{\frac{\partial ^2 U\left( {\tilde{w}}, a,{\hat{a}}\right) }{\partial a^2}}> 0 ; \qquad \frac{\partial a^*}{\partial {\tilde{w}}} = -\frac{\frac{\partial ^2 U\left( {\tilde{w}}, a,{\hat{a}}\right) }{\partial a \partial {\tilde{w}}}}{\frac{\partial ^2 U\left( {\tilde{w}}, a,{\hat{a}}\right) }{\partial a^2}}> 0 ; \qquad \text {and} \qquad \frac{\partial ^2 a^*}{\partial M \partial w} > 0. \end{aligned}$$

1.2 A Simple Example

To provide a concrete example assume that

$$\begin{aligned} ER(a)= a; \qquad u({\tilde{w}}) = {\tilde{w}}; \qquad \text {and} \qquad c(a) = \frac{a^2}{2} . \end{aligned}$$

Then the worker’s utility can be written as

$$\begin{aligned} U\left( {\tilde{w}}, a,{\hat{a}}\right) = {\tilde{w}}- \frac{a^2}{2} + \eta \left( {\tilde{w}}- \frac{{\hat{a}}^2}{2} - {\bar{u}}\right) M \left( a-{\tilde{w}}\right) , \end{aligned}$$

the first order condition is given by

$$\begin{aligned} -a + \eta \left( {\tilde{w}}- \frac{{\hat{a}}^2}{2} - {\bar{u}}\right) M =0, \end{aligned}$$

and can be rearranged to define explicitly $a^*$

$$\begin{aligned} a^* = \eta \left( {\tilde{w}}- \frac{{\hat{a}}^2}{2} - {\bar{u}}\right) M. \end{aligned}$$

Now taking explicit derivatives gives

$$\begin{aligned} \frac{\partial a^*}{\partial M} = \eta \left( {\tilde{w}}- \frac{{\hat{a}}^2}{2} - {\bar{u}}\right)> 0; \qquad \frac{\partial a^*}{\partial {\tilde{w}}} = \eta M> 0; \qquad \text {and} \qquad \frac{\partial ^2 a^*}{\partial M\partial {\tilde{w}}} = \eta > 0. \end{aligned}$$

Appendix 3: Additional Analysis

See Table 7.

Table 7 Regression: performance (10 min. Periods) incl.