## Abstract

We study a moral hazard model in which the agent receives a compensation package made up of multiple commodities. We allow for the possibility that commodities are traded on the market and consider two scenarios. When trade in commodities is verifiable, the agent cannot reshuffle the compensation package prescribed by the principal and simply selects the hidden action which is optimal given that package. When trade in commodities is, instead, not verifiable, the agent can reshuffle the prescribed package by trading it for another one and can select a different action. We prove that an optimal contract (i.e., a contract which maximizes the principal’s expected payoff) when trade is verifiable remains optimal when trade is not verifiable if agent’s preferences for commodities are independent of the action performed. When, instead, preference independence fails, we show it is always possible to find prices of commodities such that an optimal contract under trade verifiability cannot be optimal under nonverifiability.

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## Notes

- 1.
In what follows, we refer to the principal as “she” and to the agent as “he”.

- 2.
For instance, the US Supplemental Nutrition Assistance Program (SNAP) allows purchases of breads, cereals, fruits, vegetables, meats and fish, while beer, wine, liquor, cigarettes, or tobacco cannot be bought (USDA website, https://www.fns.usda.gov/snap/eligible-food-items, last accessed January 15, 2019).

- 3.
Gregory and Deb (2015) provide evidence that SNAP recipients seek less medical care but have more medical checkups than comparable non-recipients; this suggests that the former invest more in their health.

- 4.
Fringe benefits usually consist in insurance policies, pension plans, housing, childcare, etc.

- 5.
For a recent contribution on optimal contracting in the related setting of multitask moral hazard, see Chen et al. (2019).

- 6.
For sake of completeness, we should mention two other strands of literature in which the effects of moral hazard and (re)trading when preferences depend on action and multiple commodities are investigated, namely the models of general equilibrium with moral hazard (see, e.g., Greenwald and Stiglitz 1986; Arnott et al. 1994; Lisboa 2001; Citanna and Villanacci 2002; Panaccione 2007; Kielnthong and Townsend 2011; Acemoglu and Simsek 2012) and the models of taxation with moral hazard (see, e.g., Arnott and Stiglitz 1983, Arnott and Stiglitz 1986, and Panaccione and Ruscitti 2010). These models, however, are different from the agency model we are considering and have different research questions, namely efficiency of competitive equilibrium in the former case and desirability of differential taxation in the latter case.

- 7.
The intermediate scenario in which the principal verifies the agent’s trade in some, but not all, commodities is discussed in the concluding section.

- 8.
The local non-satiation hypothesis rules out the possibility that indifference curves are “thick” for any \( a\in {\mathcal {A}}\) and the weak monotonicity one that commodities are perceived as “bads” by the agent when he chooses \(a\in {\mathcal {A}}\).

- 9.
Our analysis can be extended to consider commodity prices which are state contingent. This alternative hypothesis is more appropriate in situations, such as the executive compensation example, where the value of the compensation package components (e.g., equity-like income and deferred income for the executive compensation) depends on the states of the world (e.g., the short-run and long-run value of the firm).

- 10.
As it is customary, we assume that the agent accepts the contract whenever indifferent between acceptance and refusal.

- 11.
By definition, the agent cannot increase his utility by changing only his action

*a*given the compensation package*x*, or, as shown in the proof of Proposition 1, only his compensation package*x*given the action*a*. - 12.
To see this, pick two pairs of bundles \((x_{s},y_{s})\) and \((\hat{x}_{s}, \hat{y}_{s})\), with \((\hat{x}_{s},\hat{y}_{s})\ne (x_{s},y_{s})\), such that \(u(x_{s},a)>u(y_{s},a)\) and \(u(\hat{x}_s,\hat{a})<u(\hat{y}_s,\hat{a})\). Let \(z_{s}\) and \(t_{s}\) be two bundles such that \((z_{s}^{a},z_{s}^{\hat{a} })=(x_{s}^{a},\hat{x}_{s}^{\hat{a}})\), \((t_{s}^{a},t_{s}^{\hat{a} })=(y_{s}^{a},\hat{y}_{s}^{\hat{a}})\), and \(z_{sk}=t_{sk}=x_{sk}\) for \(k\in {\mathcal {L}}\backslash {\mathcal {L}}(a)\cup {\mathcal {L}}(\hat{a})\). It follows that \(u(z_{s},a)=u(x_{s},a)\) and \(u(z_{s},\hat{a})=u(\hat{x}_{s},\hat{a})\), while \(u(t_{s},a)=u(y_{s},a)\) and \(u(t_{s},\hat{a})=u(\hat{y}_{s},\hat{a})\). Therefore, \(u(z_{s},a)>u(t_{s},a)\) and \(u(z_{s},\hat{a})<u(t_{s},\hat{a})\).

- 13.
Assumption D is satisfied in the example discussed in Sect. 2.2.

- 14.
Recall that \(u_{\circ }>\max _{a\in A}\ v(0,a)\), in which \(0\in X^{S}\) is the null contingent commodity bundle.

- 15.
The notation for commodity prices is consistent with the partition of the commodity set; therefore, \(p^{a}\in {\mathbb {R}}_{++}^{L(a)}\) denotes the prices of commodities in \({\mathcal {L}}(a)\).

- 16.
For a systematic analysis of (in)dependence of preferences over consumption commodity baskets, see Blackorby et al. (1978).

- 17.
The notation \(0_{s}\) stands for \((0,\ldots ,0,\ldots 0)\in X\).

- 18.
This reasoning hinges on the fact that \(X={\mathbb {R}}_{+}^{L}\) and \(p\in {\mathbb {R}}_{++}^{L}\).

- 19.
Since \(\hat{x}_{s}=0_{s}\) implies \(\hat{x}_{s}\leqslant x_{s}^{*}\) and \(\hat{ x}_{s}\ne x_{s}^{*}\) and, therefore, \(u(\hat{x}_{s},a^{*})\leqslant u(x_{s}^{*},a^{*})\).

- 20.
Given a bundle \(x=(x_{1},\ldots ,x_{s},\ldots ,x_{S})^{\prime }\in X^{S}\), the notation \(x_{-s}\) stands for the bundle obtained from

*x*by deleting \(x_{s}\) , that is \((x_{1},\ldots ,x_{s-1},x_{s+1},\ldots x_{S})^{\prime }\in X^{S-1}\). Clearly, \(x=(x_{s},x_{-s})\). - 21.
The notation \(p\otimes x\) stands for \((p\cdot x_{1},\ldots ,p\cdot x_{S})^{\prime }\in {\mathbb {R}}_{+}^{S}\) for \(x\in X^{S}\) and \(p\in {\mathbb {R}} _{++}^{L}\).

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## Acknowledgements

Funding was provided by Libera Università di Bolzano (Grant No. WW2074).

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### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the editor Nicholas Yannelis, the associate editor and two anonymous reviewers for their useful and constructive comments and insights. We also thank Gaetano Bloise, Giorgia Piacentino, Francesco Ruscitti, Ernesto Savaglio, the seminar participants at the Department of Economics, Università G. D’Annunzio (Pescara, Italy) and the 2014 ASSET conference (Aix-en-Provence, France) for valuable suggestions. Alessandro Fedele gratefully acknowledges financial support from Free University of Bolzano (research project WW2074). Luca Panaccione gratefully acknowledges financial support from MIUR (PRIN 2011 and PRIN 2015).

## Appendix

### Appendix

### Proof of Proposition 1

As explained in Sect. 3, we prove the result in three steps. In these steps, we will use the fact that Assumption I is equivalent to assuming that there exist functions \( f\,:\,X\,\rightarrow \,{\mathbb {R}}\) and \(g\,:\,{\mathcal {R}}(f)\times \mathcal {A }\,\rightarrow \,{\mathbb {R}}\) (where \({\mathcal {R}}(f)\) is the range of *f*) such that \(u(x_{s},a)=g(f(x_{s}),a)\). Therefore, under Assumption I, a bundle \(x_{s}\in X\) can be assigned a sub-utility level \(f(x_{s})\) independent of *a*. Given the assumptions on the utility function *u*, it follows that *g* is strictly increasing in *f*.^{Footnote 16}

*Step* 1: If \(\left( x^{*},a^{*}\right) \in X^{S}\times {\mathcal {A}}\) is a solution to (P1), i.e., \(\left( x^{*},a^{*}\right) \in \beta (p,u_{\circ })\), then \(x_{s}^{*}\) is a solution to (A4) given \(a^{*}\) and \({\bar{u}}_{s}=u(x_{s}^{*},a^{*})\); that is it must be true that \(x_{s}^{*}\in h_{s}(p,u(x_{s}^{*},a^{*}),a^{*})\) for \(s=1,\ldots ,S\).

*Proof of Step* 1: Suppose there exists \( (x^{*},a^{*})\in \beta (p,u_{\circ })\) such that \(x_{s}^{*}\notin h_{s}(p,u(x_{s}^{*},a^{*}),a^{*})\) for some *s*. Observe that it must be \(x_{s}^{*}\ne 0_{s}\),^{Footnote 17} since \(x_{s}^{*}=0_{s}\) would imply \(p\cdot x_{s}^{*}\leqslant p\cdot x_{s}\) for every \( x_{s}\in X\) and, *a fortiori*, for all \(x_{s}\in X\) such that \( u(x_{s},a^{*})\geqslant u(x_{s}^{*},a^{*})\).^{Footnote 18} Pick \(\hat{x}_{s}\in h_{s}(p,u(x_{s}^{*},a^{*}),a^{*})\), so that \(p\cdot \hat{x}_{s}<p\cdot x_{s}^{*}\). Suppose that \(u(\hat{x}_{s},a^{*})>u(x_{s}^{*},a^{*})\). Because of monotonicity, \(\hat{x}_{s}\ne 0_{s}\).^{Footnote 19} Consider the bundle \(\alpha \hat{x}_{s}\) for \(\alpha \in (0,1)\). Clearly, \(\alpha \hat{x}_{s}\in X\) and, since \(\hat{x}_{s}\ne 0_{s}\), \(p\cdot (\alpha \hat{x}_{s})<p\cdot \hat{x}_{s}\). Furthermore, when \( \alpha \) is close enough to 1, \(u(\alpha \hat{x}_{s},a^{*})\geqslant u(x_{s}^{*},a^{*})\) follows from the continuity of the utility function. These conditions, however, would contradict the fact that \(\hat{x} _{s}\in h_{s}(p,u(x_{s}^{*},a^{*}),a^{*})\). Therefore, it must be that \(u(\hat{x}_{s},a^{*})=u(x_{s}^{*},a^{*})\). This equality implies that \(f(\hat{x}_{s})=f(x_{s}^{*})\). Therefore^{Footnote 20}\(\pi _{s}(a)g(f(\hat{x}_{s}),a)+\sum _{s^{ \prime }\ne s}\pi _{s^{\prime }}(a)g(f(x_{s^{\prime }}^{*}),a)=v(\hat{x} _{s},x_{-s}^{*},a)\) is equal to \(\pi _{s}(a)g(f(x_{s}^{*}),a)+\sum _{s^{\prime }\ne s}\pi _{s^{\prime }}(a)g(f(x_{s^{\prime }}^{*}),a)=v(x_{s}^{*},x_{-s}^{*},a)\) for every \(a\in {\mathcal {A}}\). Hence, it follows that \(a(\hat{x}_{s},x_{-s}^{*})=a(x_{s}^{*},x_{-s}^{*})\). Since \(a^{*}\in a(x_{s}^{*},x_{-s}^{*})\), this last equality implies \(a^{*}\in a(\hat{x}_{s},x_{-s}^{*})\). It follows that \((\hat{x}_{s},x_{-s}^{*},a^{*})\) satisfies (P1.a)-(P1.c), and therefore, it must be true that \(\sum _{s}\pi _{s}(a^{*})(\varrho _{s}-p\cdot x_{s}^{*})\geqslant \pi _{s}(a^{*})(\varrho _{s}-p\cdot \hat{x}_{s})+\sum _{s^{\prime }\ne s}\pi _{s}(a^{*})(\varrho _{s^{\prime }}-p\cdot x_{s^{\prime }}^{*})\). However, this inequality implies that \(\varrho _{s}-p\cdot x_{s}^{*}\geqslant \varrho _{s}-p\cdot \hat{x}_{s}\), hence that \(p\cdot x_{s}^{*}\leqslant p\cdot \hat{x}_{s}\), which is a contradiction.

*Step* 2: If \(\left( x^{*},a^{*}\right) \in X^{S}\times {\mathcal {A}}\) is a solution to (P1), i.e., \(\left( x^{*},a^{*}\right) \in \beta (p,u_{\circ })\), then \(x_{s}^{*}\) is a solution to (A3) with \(\omega _{s}=p\cdot x_{s}^{*}\) given \(a^{*}\); that is, it must be true that \(x_{s}^{*}\in x_{s}(p,p\cdot x_{s}^{*},a^{*})\) for \(s=1,\ldots ,S\).

*Proof of Step* 2: Suppose there exists \( (x^{*},a^{*})\in \beta (p,u_{\circ })\) such that \(x_{s}^{*}\notin x_{s}(p,p\cdot x_{s}^{*},a^{*})\) for some *s*. Observe that it must be \(x_{s}^{*}\ne 0_{s}\), since \(x_{s}^{*}=0_{s}\) would imply \(p\cdot x_{s}^{*}=0\) and, therefore, \(x_{s}(p,p\cdot x_{s}^{*},a^{*})=\{x_{s}^{*}\}\). Pick \(\hat{x}_{s}\in x_{s}(p,p\cdot x_{s}^{*},a^{*})\), so that \(u(\hat{x}_{s},a^{*})>u(x_{s}^{*},a^{*})\) and, because of monotonicity, \(\hat{x}_{s}\ne 0_{s}\). Consider the bundle \(\alpha \hat{x}_{s}\) for \(\alpha \in (0,1)\). Clearly, \( \alpha \hat{x}_{s}\in X\) and, since \(\hat{x}_{s}\ne 0_{s}\), \(p\cdot (\alpha \hat{x}_{s})<p\cdot \hat{x}_{s}=p\cdot x_{s}^{*}\). Furthermore, when \( \alpha \) is close enough to 1, \(u(\alpha \hat{x}_{s},a^{*})\geqslant u(x_{s}^{*},a^{*})\) follows from the continuity of the utility function. These conditions, however, would contradict the fact that \( x_{s}^{*}\in h_{s}(p,u(x_{s}^{*},a^{*}),a^{*})\), as established in the previous step.

*Step* 3: For any \((x^{*},a^{*})\in X^{S}\times {\mathcal {A}}\) and for every \(s=1,\ldots ,S\), if \(a^{*}\) is a solution to (A1) given \(x^{*}\), that is \(a^{*}\in a(x^{*})\), and \(x_{s}^{*}\) is a solution to (A3) with \(\omega _{s}=p\cdot x_{s}^{*}\) given \(a^{*}\), that is \(x_{s}^{*}\in x_{s}(p,p\cdot x_{s}^{*},a^{*})\), then \((x^{*},a^{*})\) is a solution to (A2) with \( \omega _{s}=p\cdot x_{s}^{*}\) for \(s=1,\ldots ,S\); that is \((x^{*},a^{*})\in \varphi (p,p\otimes x^{*})\).^{Footnote 21}

*Proof of Step* 3: We prove the result by contraposition, i.e., we show that if \((x^{*},a^{*})\notin \varphi (p,p\otimes x^{*})\) then either \(a^{*}\notin a(x^{*})\) or \( x_{s}^{*}\notin x_{s}(p,p\cdot x_{s}^{*},a^{*})\) for at least one *s*. Clearly, if \(a^{*}\notin {\mathcal {A}}\), then a fortiori \(a^{*}\notin a(x^{*})\). Therefore, assume \(a^{*}\in {\mathcal {A}}\). Since \( (x^{*},a^{*})\notin \varphi (p,p\otimes x^{*})\), it follows that there exist \((\hat{x},\hat{a})\) such that \(\hat{a}\in {\mathcal {A}}\), \(p\cdot \hat{x}_{s}\leqslant p\cdot x_{s}^{*}\) for every *s*, and \(v(\hat{x}, \hat{a})>v(x^{*},a^{*})\). Recall that Assumption I holds. Suppose \( f(x_{s}^{*})\geqslant f(\hat{x}_{s})\) for every *s*. In this case, \( u(x_{s}^{*},\hat{a})\geqslant u(\hat{x}_{s},\hat{a})\) for every *s*, hence \(v(x^{*},\hat{a})\geqslant v(\hat{x},\hat{a})\). This implies \( v(x^{*},\hat{a})>v(x^{*},a^{*})\) and therefore \(a^{*}\notin a(x^{*})\). Suppose instead that \(f(x_{s}^{*})<f(\hat{x}_{s})\) for some *s*. In this case, \(u(x_{s}^{*},a^{*})<u(\hat{x}_{s},a^{*})\) and, therefore, \(x_{s}^{*}\notin x_{s}(p,p\cdot x_{s}^{*},a^{*})\). \(\square \)

### Proof of Proposition 2

Fix a price vector \(\bar{ p}\in {\mathbb {R}}_{++}^{L}\). Select a pair of actions \(\hat{a}\) and \({\tilde{a}} \) and pick \(\hat{x}\in H(\hat{a},{\bar{p}},u_{\circ })\) and \({\tilde{x}}\in H( {\tilde{a}},{\bar{p}},u_{\circ })\). Choose a state of nature *s* such that \(\bar{ p}^{\hat{a}}\cdot \hat{x}_{s}^{\hat{a}}>0\), which is always possible because of property (1) in Remark 3. Take a price vector *p* which is related to \( {\bar{p}}\) as follows: \(p_{l}=\delta {\bar{p}}_{l}\) for every \(l\in {\mathcal {L}}( {\tilde{a}})\) and \(p_{l}={\bar{p}}_{l}\) for every \(l\in {\mathcal {L}}\backslash {\mathcal {L}}({\tilde{a}})\). Choose \(\delta \in (0,1]\) such that \(p^{\hat{a} }\cdot \hat{x}_{s}^{\hat{a}}>\delta \left( {\bar{p}}^{{\tilde{a}}}\cdot {\tilde{x}} _{s}^{{\tilde{a}}}\right) =p^{{\tilde{a}}}\cdot {\tilde{x}}_{s}^{{\tilde{a}}}\), which is possible since \(p^{\hat{a}}\cdot \hat{x}_{s}^{\hat{a}}={\bar{p}}^{ \hat{a}}\cdot \hat{x}_{s}^{\hat{a}}>0\). Note that \(H({\tilde{a}},p,u_{\circ })=H({\tilde{a}},{\bar{p}},u_{\circ })\) and \(H(\hat{a},p,u_{\circ })=H(\hat{a}, {\bar{p}},u_{\circ })\). Therefore, \(\hat{x}\in H(\hat{a},p,u_{\circ })\) and \( {\tilde{x}}\in H({\tilde{a}},p,u_{\circ })\). From properties (2) and (3) in Remark 3, it follows \(u_{\circ }=v(\hat{x},\hat{a})\) and also

By property (4) in Remark 3, \({\hat{x}}^{-\hat{a}}=0\) and \({{\tilde{x}}}^{- {\tilde{a}}}=0\) implies that \(p^{\hat{a}}\cdot {\hat{x}}_{s}^{\hat{a}}>p^{\tilde{a }}\cdot {{\tilde{x}}}_{s}^{{\tilde{a}}}\) is equivalent to \(p\cdot \hat{x} _{s}>p\cdot {\tilde{x}}_{s}\). Therefore, it must be that

The above inequality holds since the agent, conditional on choosing \({\tilde{a}}\), obtains higher utility with income equal to \(p\cdot \hat{x}_{s}\) than equal to \(p\cdot {\tilde{x}}_{s}\) in every *s*, since he can afford more in at least one state and his preferences are locally non-satiated on *X*. Finally, denote with \((x^{*},a^{*})\) a solution to (A2) with \(\omega _{s}=p\cdot \hat{x}_{s}\) in every *s*. It follows that

Finally, (4) and (5) imply that \(v(x^{*},a^{*})>u_{\circ }=v(\hat{x},\hat{a})\). This eventually means that, if the principal proposes the contract \((\hat{x},\hat{a})\) to the agent, he accepts it and, since trade in commodities is nonverifiable, is willing to reshuffle his compensation basket. In particular, he sells bundle \(\hat{x}\) and buys bundle \(x^{*}\) under the constraint that he can spend up to \(p\cdot \hat{x}_{s}\) in every state, and chooses action \( a^{*}\) rather than \(\hat{a}\). To conclude the proof, we must have that \(( \hat{x},\hat{a})\), with \(H(\hat{a},p,u)\), is a solution to (P1) at prices *p*. For this, it suffices to choose \((\varrho _{1},\ldots ,\varrho _{S})\) in such a way that

\(\square \)

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Fedele, A., Panaccione, L. Moral hazard and compensation packages: does reshuffling matter?.
*Econ Theory* **70, **223–241 (2020). https://doi.org/10.1007/s00199-019-01208-5

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### Keywords

- Moral hazard
- Compensation packages
- Reshuffling
- Independent preferences

### JEL Classification

- D82 (Asymmetric and private information • Mechanism design)
- D86 (Economics of contract: theory)
- J33 (Compensation packages • Payment methods)