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Bundling versus unbundling: asymmetric information on information externalities

Abstract

This paper addresses the benefits of bundling two sequential activities in the context of public–private partnerships (PPPs). The paper introduces a source of asymmetric information in the form of an externality parameter that links the building stage with subsequent operational activity. Within this framework, bundling allows the government to extract private information about the magnitude of the externality parameter. The framework also implies a higher degree of asymmetric information related to the operational stage than unbundling does when the contract is written. Our results indicate that the use of bundled contracts allows PPPs to be commitment devices that force governments to define ex-ante more coherent and informed plans, thereby improving investments and reducing unexpected cost overruns. However, because of the presence of asymmetric information, bundling makes any cost-reducing effort suboptimal during the operational phase.

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Fig. 1

Notes

  1. 1.

    In most cases, their implementation led to the achievement of satisfactory and efficient outcomes in terms of cost overruns, execution time, and quality of goods/services provided (Saussier and Phuong Tra 2012; Koontz and Thomas 2012; Raisbeck et al. 2010; Hodge and Greve 2007).

  2. 2.

    Martimort and Pouyet (2008) expand their basic model to allow for general schemes in which the builder’s payment depends on the operator’s cost, more complete contracts, and the introduction of an adverse-selection issue concerning operating costs. The authors conclude that, with a benevolent decision-maker and a privately informed operator, bundling is still the optimal organizational form when the externality is positive.

  3. 3.

    Since the shadow cost of public funds is higher in developing countries than it is in developed countries, optimal choices between regulation, on one hand, and outsourcing or privatization strategies, on the other, are expected to differ between those countries (Auriol and Picard 2009b).

  4. 4.

    For instance, an automated metro system may reduce the need for drivers (positive externality). However, while innovative designs and materials for the construction of sustainable public buildings can increase the social surplus, they can also increase maintenance costs (negative externality).

  5. 5.

    This cost function follows the structure proposed by Laffont and Tirole (1986). In the context of PPPs, the setting of our model is related to Hoppe and Schmitz (2013), where the agent in charge of the building task can come up with an innovation, while the agent in charge of the operation task can implement adaptations to improve the service provision whose cost depends on the first stage innovation.

  6. 6.

    This hypothesis is believable in real world cases where is the operator that is informed about the magnitude of his own maintenance and management costs for a given quality of the infrastructure. The information setting of the model is particularly suitable for some sectors, such as the transport sector, where the operator is entitled of most of the relevant information and, in practice, has to manage all relevant risks (Roumboutsos 2015).

  7. 7.

    Starting from this case, there is always the possibility of decentralizing through a menu of linear contracts \(T=a-bC\) (Laffont and Tirole 1993).

  8. 8.

    This IC condition gives the agent no incentive to deviate and ensures truth-telling. This necessary condition is also sufficient if the marginal cost decreases as the positive externality increases. For a detailed proof, see the online appendix (Lemma 1).

  9. 9.

    For a detailed proof, see the online appendix (Lemma 2).

  10. 10.

    This IC condition gives the agent no incentive to deviate and ensures truth-telling. This necessary condition is also sufficient if the marginal cost increases as the negative externality increases. For a detailed proof, see the online appendix (Lemma 3).

  11. 11.

    For a detailed proof, see the online appendix (Lemma 4).

  12. 12.

    The introduction of the parameters p and h allows for more general results. For the purpose of this analysis, it is assumed that \(p>0\), \(h>0\) and \(hp<1\).

  13. 13.

    In the presence of a positive externality, the function \(\sigma _{\theta }^{w}\) is concave and reaches a maximum for a value of \(\lambda \) between 0 and 1 unless the value of \(\theta \) is very high.

  14. 14.

    In standard projects, information on management costs is generally common knowledge for both parties, while in R&D investments, future information on costs or project outcomes is uncertain for both the public and the private sectors.

  15. 15.

    All projects that involve new applications of existing innovations are suitable examples.

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Acknowledgements

I am grateful to Emmanuelle Auriol, Federico Boffa, Sara Calligaris, Eshien Chong, Stefano Galavotti, Luciano Greco, Antonio Nicoló, Elena Podkolzina, Paola Valbonesi and the participants at the SIEPI workshop (Napoli 2014), the workshop How do Governance Complexity and Financial Constraints affect Public–Private Contracts? Theory and Empirical Evidence (Padova 2014), the EACES workshop (Moscow 2013) and the 54 RSA SIE conference (Bologna 2013) for their valuable comments and suggestions on different versions of this paper. A special thank to Eva Hoppe for her revision and very useful advice. I would also like to thank participants of the Baraza seminar (Toulouse TSE) and Chaire EPPP members for their useful feedback. Any remaining errors are the author’s responsibility. I gratefully acknowledge the financial support of the the University of Padova (Grant No. CPDA121089). I thank two anonymous referees as well as the Managing Editor, Giacomo Corneo, for their helpful comments and suggestions.

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Correspondence to Marco Buso.

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Supplementary material 1 (pdf 220 KB)

Appendix

Appendix

This Appendix contains the proofs and the the comparative statistical analysis.

Proof of Proposition 1

The expected function that is used to perform the welfare analysis is the following:

$$\begin{aligned} \int _{\theta ^{l}}^{\theta ^{h}}\lbrace S_{0}+s I-(1+\lambda )[T_{b}+T_{o}]+(T_{b}-C_{b}(I))+(T_{o}-C_o(\theta , e)-\phi (e))\rbrace f(\theta )d\theta \end{aligned}$$

Positive externality (\(a=1\))

Using the new investment and effort functions, the first-order conditions in the bundling case become:

  • \(I^{B}=\frac{p}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }-\frac{p}{1-hp}\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\),

  • \(e^{B}=h I^B-\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\).

Substituting in the government’s objective formula, we obtain the value function under bundling:

$$\begin{aligned} V^{B}= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}+s I^{B}-(1+\lambda )\left[ \frac{(I^{B})^{2}}{2p}+\frac{(e^{B})^{2}}{2h}+F-(\theta +e^{B})I^{B}\right] \right. \\&\quad \left. -\,\lambda \frac{1-F(\theta )}{f(\theta )}\frac{e^{B}}{h}\right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}+I^{B}(s+(1+\lambda )(\theta +e^B))\right. \\&\quad \left. -\,(1+\lambda )\left[ \frac{(I^{B})^{2}}{2p}+\frac{(I^{B})^{2} h}{2}+\frac{1}{2h}\left( \frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) ^2 \right. \right. \\&\quad \left. \left. -\,\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}I^B +F \right] -\lambda \frac{1-F(\theta )}{f(\theta )}\left( I^B-\frac{1}{h}\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) \right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}-(1+\lambda ) F+ \frac{(1+\lambda )(1-hp)}{2p}(I^{B})^2\right. \\&\quad \left. +\,(1+\lambda )\frac{1}{2h}\left( \frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) ^2 \right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}-(1+\lambda ) F+ \frac{p (1+\lambda )}{2(1-hp)}\left( \frac{s+(1+\lambda )\theta }{1+\lambda }\right) ^2+\frac{p(1+\lambda )}{2(1-hp)}\left( \frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) ^2\right. \\&\quad \left. -\,\frac{(1+\lambda )p}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}+(1+\lambda )\frac{1}{2h}\left( \frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) ^2 \right\} f(\theta )d\theta \\= & {} S_{0}-(1+\lambda ) F+ \frac{p (1+\lambda )}{2(1-hp)}\left( \frac{s^2+(1+\lambda )^2({\overline{\theta }}+\sigma ^2_{\theta })+2s(1+\lambda ){\overline{\theta }}}{(1+\lambda )^2}\right) \\&\quad -\,\frac{\lambda }{h}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{1-F(\theta )}{f(\theta )} \left[ \frac{hp}{(1-hp)}\frac{s+(1+\lambda )\theta }{1+\lambda }-\right. \right. \\&\quad \left. \left. -\,\frac{hp}{2(1-hp)}\left( \frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) -\frac{1}{2}\left( \frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) \right] \right\} f(\theta )d\theta \\= & {} S_{0}-(1+\lambda ) F+ \frac{p (1+\lambda )}{2(1-hp)}\left( \frac{s+(1+\lambda ){\overline{\theta }}}{1+\lambda }\right) ^2+ \frac{p(1+\lambda )}{2(1-hp)}\sigma ^2_{\theta }\\&\quad -\,\frac{\lambda }{2h}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{1-F(\theta )}{f(\theta )}\left( e^B+\frac{hp}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }\right) \right\} f(\theta )d\theta \end{aligned}$$

The new efforts functions applied to the unbundling case yield, respectively:

  • \(I^{U}=p\frac{s+(1+\lambda ){\overline{\theta }}}{(1+\lambda )(1-hp)}\),

  • \(e^{U}=h I^U\).

Substituting in the government’s objective formula, we obtain the value function under unbundling:

$$\begin{aligned} V^{U}= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}+s I^{U}-(1+\lambda )\left[ \frac{(I^{U})^{2}}{2p}+\frac{(e^{U})^{2}}{2h}+F-I^{U}(\theta +e^{U})\right] \right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}+s I^{U}-(1+\lambda )\left[ \frac{(I^{U})^{2}}{2p}+\frac{(I^{U})^{2} h}{2}+F-I^{U}(\theta +I^{U} h)\right] \right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}-(1+\lambda ) F+ I^{U} (s+(1+\lambda )\theta )-(1+\lambda )\frac{(I^{U})^{2}}{2p}(1-hp)\right\} f(\theta )d\theta \\= & {} S_{0}-(1+\lambda ) F+ I^{U} (s+(1+\lambda ){\overline{\theta }})-(1+\lambda )\frac{(I^{U})^{2}}{2p}(1-hp)\\= & {} S_{0}-(1+\lambda ) F+ I^{U} \frac{I^U}{p}(1+\lambda )(1-hp)-(1+\lambda )\frac{(I^{U})^{2}}{2p}(1-hp)\\= & {} S_{0}-(1+\lambda ) F + (1+\lambda )\frac{(I^{U})^{2}}{2p}(1-hp)\\= & {} S_{0}-(1+\lambda ) F +\frac{p(1+\lambda )}{2(1-hp)}\left( \frac{s+(1+\lambda ){\overline{\theta }}}{1+\lambda }\right) ^2 \end{aligned}$$

The net welfare gain/loss of governments when using bundling is equal to:

$$\begin{aligned} V^{B}-V^{U}= & {} \frac{p(1+\lambda )}{2(1-hp)}\sigma ^2_{\theta }-\frac{\lambda }{2h}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{1-F(\theta )}{f(\theta )}\left( e^B+\frac{hp}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }\right) \right\} f(\theta )d\theta \end{aligned}$$

where \(e^B=\frac{hp}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }-\frac{hp}{1-hp}\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}-\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\).

As a result, bundling dominates unbundling if and only if:

$$\begin{aligned} \sigma _{\theta }^{2}\ge & {} \frac{\lambda (1-hp)}{hp(1+\lambda )}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{1-F(\theta )}{f(\theta )}\left( e^B+\frac{hp}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }\right) \right\} f(\theta )d\theta \\ \sigma _{\theta }^{2}\ge & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{\lambda (1-hp)}{hp(1+\lambda )}\frac{1-F(\theta )}{f(\theta )}\left( \frac{hp}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }-\frac{hp}{1-hp}\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right. \right. \\&\left. \left. -\,\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}+\frac{hp}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }\right) \right\} f(\theta )d\theta \end{aligned}$$

Proof of Corollary 2—positive externality

To study the sign of the derivative of \(\sigma _{\theta }^{w}\) varies with \(\lambda \), it is sufficient to study the derivative of the integrand in the previous inequality with respect to \(\lambda \) that is equal to:

$$\begin{aligned}= & {} \frac{1-hp}{hp}\frac{1}{(1+\lambda )^2}\frac{1-F(\theta )}{f(\theta )}\left( e^B+\frac{hp}{1-hp}\frac{s+(1+\lambda )\theta }{1+\lambda }\right) \\&\quad +\,\frac{1-hp}{hp}\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\left( -\frac{hp}{1-hp}\frac{2s}{(1+\lambda )^2}-\frac{hp}{1-hp}\frac{1}{(1+\lambda )^2}\frac{1-F(\theta )}{f(\theta )}\right. \\&\quad \left. -\,\frac{1}{(1+\lambda )^2}\frac{1-F(\theta )}{f(\theta )}\right) \\= & {} 2\frac{1-hp}{hp}\frac{1}{(1+\lambda )^2}\frac{1-F(\theta )}{f(\theta )}\left( \frac{hp}{1-hp}\frac{s(1-\lambda )+(1+\lambda )\theta }{1+\lambda }-\frac{hp}{1-hp}\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right. \\&\quad \left. -\,\frac{\lambda }{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) \\= & {} 2\frac{1-hp}{hp}\frac{1}{(1+\lambda )^2}\frac{1-F(\theta )}{f(\theta )}\left( e^B-\frac{hp}{1-hp}\frac{\lambda s}{1+\lambda }\right) \end{aligned}$$

As a conclusion, the value of \(\sigma _{\theta }^{2}\) can increase or decrease with \(\lambda \). Precisely, the derivative is positive if \(\lambda =0\), while it may become negative as \(\lambda \) increases. The second derivative is equal to:

$$\begin{aligned}= & {} 4\frac{1-hp}{hp}\frac{1}{(1+\lambda )^3}\frac{1-F(\theta )}{f(\theta )}\left( -e^B+\frac{hp}{1-hp}\frac{\lambda s}{1+\lambda }\right) + \\&\quad +\,2\frac{1-hp}{hp}\frac{1}{(1+\lambda )^2}\frac{1-F(\theta )}{f(\theta )}\left( -\frac{hp}{1-hp}\frac{2s}{(1+\lambda )^2}\right. \\&\quad \left. -\,\frac{hp}{1-hp}\frac{1}{(1+\lambda )^2}\frac{1-F(\theta )}{f(\theta )}-\frac{1}{(1+\lambda )^2}\frac{1-F(\theta )}{f(\theta )}\right) \\= & {} -2\frac{1-hp}{hp}\frac{1}{(1+\lambda )^3}\frac{1-F(\theta )}{f(\theta )}\left( 2e^B+2\frac{hp}{1-hp}\frac{s(1-\lambda )}{1+\lambda }\right. \\&\quad \left. +\,\frac{hp}{1-hp}\frac{1}{1+\lambda }\frac{1-F(\theta )}{f(\theta )}+\frac{1}{1+\lambda }\frac{1-F(\theta )}{f(\theta )}\right) \le 0 \end{aligned}$$

Negative externality (\(a=-1\))

Using the new investment and effort functions, the first-order conditions in the bundling case become:

  • \(I^{B}=\frac{p}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }-\frac{p}{1-hp}\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\),

  • \(e^{B}=h I^B-\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\).

Substituting in the government’s objective formula, we obtain the value function under bundling:

$$\begin{aligned} V^{B}= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}+s I^{B}-(1+\lambda )\left[ \frac{(I^{B})^{2}}{2p}+\frac{(e^{B})^{2}}{2h}+F+(\theta -e^{B})I^{B}\right] -\lambda \frac{F(\theta )}{f(\theta )}\frac{e^{B}}{h}\right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}+I^{B}(s-(1+\lambda )(\theta -e^B))\right. \\&\quad \left. -\,(1+\lambda )\left[ \frac{(I^{B})^{2}}{2p}+\frac{(I^{B})^{2} h}{2}+\frac{1}{2h}\left( \frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) ^2 \right. \right. \\&\quad \left. \left. -\,\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}I^B +F \right] -\lambda \frac{F(\theta )}{f(\theta )}\left( I^B-\frac{1}{h}\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) \right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}-(1+\lambda ) F+ \frac{(1+\lambda )(1-hp)}{2p}(I^{B})^2+(1+\lambda )\frac{1}{2h}\left( \frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) ^2 \right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}-(1+\lambda ) F+ \frac{p (1+\lambda )}{2(1-hp)}\left( \frac{s-(1+\lambda )\theta }{1+\lambda }\right) ^2+\frac{p(1+\lambda )}{2(1-hp)}\left( \frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) ^2\right. \\&\quad \left. -\,\frac{(1+\lambda )p}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}+(1+\lambda )\frac{1}{2h}\left( \frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) ^2 \right\} f(\theta )d\theta \\= & {} S_{0}-(1+\lambda ) F+ \frac{p (1+\lambda )}{2(1-hp)}\left( \frac{s^2+(1+\lambda )^2({\overline{\theta }}+\sigma ^2_{\theta })-2s(1+\lambda ){\overline{\theta }}}{(1+\lambda )^2}\right) \\&\quad -\,\frac{\lambda }{h}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{F(\theta )}{f(\theta )} \left[ \frac{hp}{(1-hp)}\frac{s-(1+\lambda )\theta }{1+\lambda }-\right. \right. \\&\quad \left. \left. -\,\frac{hp}{2(1-hp)}\left( \frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) -\frac{1}{2}\left( \frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) \right] \right\} f(\theta )d\theta \\= & {} S_{0}-(1+\lambda ) F+ \frac{p (1+\lambda )}{2(1-hp)}\left( \frac{s-(1+\lambda ){\overline{\theta }}}{1+\lambda }\right) ^2+ \frac{p(1+\lambda )}{2(1-hp)}\sigma ^2_{\theta }\\&\quad -\,\frac{\lambda }{2h}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{F(\theta )}{f(\theta )}\left( e^B+\frac{hp}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }\right) \right\} f(\theta )d\theta \end{aligned}$$

The new efforts functions applied to the unbundling case yield, respectively:

  • \(I^{U}=p\frac{s-(1+\lambda ){\overline{\theta }}}{(1+\lambda )(1-hp)}\),

  • \(e^{U}=h I^U\).

Substituting in the government’s objective formula, we obtain the value function under unbundling:

$$\begin{aligned} V^{U}= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}+s I^{U}-(1+\lambda )\left[ \frac{(I^{U})^{2}}{2p}+\frac{(e^{U})^{2}}{2h}+F+I^{U}(\theta -e^{U})\right] \right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}+s I^{U}-(1+\lambda )\left[ \frac{(I^{U})^{2}}{2p}+\frac{(I^{U})^{2} h}{2}+F+I^{U}(\theta -I^{U} h)\right] \right\} f(\theta )d\theta \\= & {} \int _{\theta ^{l}}^{\theta ^{h}}\left\{ S_{0}-(1+\lambda ) F+ I^{U} (s-(1+\lambda )\theta )-(1+\lambda )\frac{(I^{U})^{2}}{2p}(1-hp)\right\} f(\theta )d\theta \\= & {} S_{0}-(1+\lambda ) F+ I^{U} (s-(1+\lambda ){\overline{\theta }})-(1+\lambda )\frac{(I^{U})^{2}}{2p}(1-hp)\\= & {} S_{0}-(1+\lambda ) F+ I^{U} \frac{I^U}{p}(1+\lambda )(1-hp)-(1+\lambda )\frac{(I^{U})^{2}}{2p}(1-hp)\\= & {} S_{0}-(1+\lambda ) F + (1+\lambda )\frac{(I^{U})^{2}}{2p}(1-hp)\\= & {} S_{0}-(1+\lambda ) F +\frac{p(1+\lambda )}{2(1-hp)}\left( \frac{s-(1+\lambda ){\overline{\theta }}}{1+\lambda }\right) ^2 \end{aligned}$$

The net welfare gain/loss of governments when using bundling is equal to:

$$\begin{aligned} V^{B}-V^{U}= & {} \frac{p(1+\lambda )}{2(1-hp)}\sigma ^2_{\theta }-\frac{\lambda }{2h}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{F(\theta )}{f(\theta )}\left( e^B+\frac{hp}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }\right) \right\} f(\theta )d\theta \end{aligned}$$

where \(e^B=\frac{hp}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }-\frac{hp}{1-hp}\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}-\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\).

As a result, bundling dominates unbundling if and only if:

$$\begin{aligned} \sigma _{\theta }^{2}\ge & {} \frac{\lambda (1-hp)}{hp(1+\lambda )}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{F(\theta )}{f(\theta )}\left( e^B+\frac{hp}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }\right) \right\} f(\theta )d\theta \\ \sigma _{\theta }^{2}\ge & {} \frac{\lambda (1-hp)}{hp(1+\lambda )}\int _{\theta ^{l}}^{\theta ^{h}}\left\{ \frac{F(\theta )}{f(\theta )}\left( \frac{hp}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }-\frac{hp}{1-hp}\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right. \right. \\&\quad \left. \left. -\,\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}+\frac{hp}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }\right) \right\} f(\theta )d\theta \end{aligned}$$

Proof of Corollary 2—negative externality

To study the sign of the derivative of \(\sigma _{\theta }^{w}\) varies with \(\lambda \), it is sufficient to study the derivative of the integrand in the previous inequality with respect to \(\lambda \) that is equal to:

$$\begin{aligned}= & {} \frac{1-hp}{hp}\frac{1}{(1+\lambda )^2}\frac{F(\theta )}{f(\theta )}\left( e^B+\frac{hp}{1-hp}\frac{s-(1+\lambda )\theta }{1+\lambda }\right) \\&\quad +\,\frac{1-hp}{hp}\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\left( -\frac{hp}{1-hp}\frac{2s}{(1+\lambda )^2}\right. \\&\quad \left. -\,\frac{hp}{1-hp}\frac{1}{(1+\lambda )^2}\frac{F(\theta )}{f(\theta )}-\frac{1}{(1+\lambda )^2}\frac{F(\theta )}{f(\theta )}\right) \\= & {} 2\frac{1-hp}{hp}\frac{1}{(1+\lambda )^2}\frac{F(\theta )}{f(\theta )}\left( \frac{hp}{1-hp}\frac{s(1-\lambda )-(1+\lambda )\theta }{1+\lambda }\right. \\&\quad \left. -\,\frac{hp}{1-hp}\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}-\frac{\lambda }{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) \\= & {} 2\frac{1-hp}{hp}\frac{1}{(1+\lambda )^2}\frac{F(\theta )}{f(\theta )}\left( e^B-\frac{hp}{1-hp}\frac{\lambda s}{1+\lambda }\right) \end{aligned}$$

As a conclusion, the value of \(\sigma _{\theta }^{2}\) can increase or decrease with \(\lambda \). Precisely, the derivative is positive if \(\lambda =0\), while it is negative if \(\lambda =1\). The second derivative is equal to:

$$\begin{aligned}= & {} 4\frac{1-hp}{hp}\frac{1}{(1+\lambda )^3}\frac{F(\theta )}{f(\theta )}\left( -e^B+\frac{hp}{1-hp}\frac{\lambda s}{1+\lambda }\right) + \\&\quad +\,2\frac{1-hp}{hp}\frac{1}{(1+\lambda )^2}\frac{F(\theta )}{f(\theta )}\left( -\frac{hp}{1-hp}\frac{2s}{(1+\lambda )^2}\right. \\&\quad \left. -\,\frac{hp}{1-hp}\frac{1}{(1+\lambda )^2}\frac{F(\theta )}{f(\theta )}-\frac{1}{(1+\lambda )^2}\frac{F(\theta )}{f(\theta )}\right) \\= & {} -2\frac{1-hp}{hp}\frac{1}{(1+\lambda )^3}\frac{F(\theta )}{f(\theta )}\left( 2e^B+2\frac{hp}{1-hp}\frac{s(1-\lambda )}{1+\lambda }\right. \\&\quad \left. +\,\frac{hp}{1-hp}\frac{1}{1+\lambda }\frac{F(\theta )}{f(\theta )}+\frac{1}{1+\lambda }\frac{F(\theta )}{f(\theta )}\right) \le 0 \end{aligned}$$

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Buso, M. Bundling versus unbundling: asymmetric information on information externalities. J Econ 128, 1–25 (2019). https://doi.org/10.1007/s00712-018-0642-0

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Keywords

  • Bundling/unbundling
  • Agency theory
  • Information externality
  • Public–private partnership

JEL Classification

  • D86
  • H11
  • H57