The role of matching grants as a commitment device in the federation model with a repeated soft budget setting

Nobuo Akai; Motohiro Sato

doi:10.1007/s10101-019-00221-7

Economics of Governance

March 2019, Volume 20, Issue 1, pp 23–39 | Cite as

The role of matching grants as a commitment device in the federation model with a repeated soft budget setting

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Nobuo Akai
Motohiro Sato

Original Paper

First Online: 25 January 2019

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Abstract

This paper revisits the soft budget versus the hard budget constraint in federations. By extending Besfamille and Lockwood (Int Econ Rev 49:577–593, 2008), who examine a case where the soft budget is ex ante favorable, we consider a model that allows the federal government to use a matching grant as an ex post policy instrument. We establish that this instrument acts as a commitment device and may improve social welfare compared with the situation in which the government’s ex ante policy options are limited to a hard or soft budget.

Keywords

Soft budget Matching grant Repeated game Commitment device

JEL Classification

H71 H72 H73 H77

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Notes

Appendix

Proof of Lemma 3

We have the following:

$$ u_{H} (p_{H}^{*} ) \equiv p_{H}^{*} b_{g} - c_{1} - \Delta (p_{H}^{*} ) < p_{H}^{*} b_{g} + (1 - p_{H}^{*} )(b_{b} - (1 - m)\gamma c_{2} ) - c_{1} - \Delta (p_{H}^{*} ) < u_{m} (p_{m}^{*} ) $$

$ u_{m} (p_{m}^{*} ) \equiv p_{m}^{*} b_{g} + (1 - p_{m}^{*} )(b_{b} - (1 - m)\gamma c_{2} ) - c_{1} - \Delta (p_{m}^{*} ) < p_{m}^{*} b_{g} + (1 - p_{m}^{*} )b_{b} - c_{1} - \Delta (p_{m}^{*} ) < u_{S} (p_{S}^{*} ) $. The last inequality in each equation above results from the fact that $ p_{m}^{*} $ maximizes $ u_{m} (p) $ and $ p_{s}^{*} $ maximizes $ u_{S} (p) $.

Proof of Lemma 4

When $ \Delta (p) = p^{2} /2\alpha $, we have $ p_{m}^{*} = \alpha \left( {b_{g} - b_{b} + (1 - m)\gamma c_{2} } \right) $ and $ \frac{d}{dm}p_{m}^{*} = - \alpha \gamma c_{2} $.

In addition, we have $ \frac{\partial }{\partial m}v_{m} = (1 - p_{m}^{*} )(\gamma - 1)c_{2} - m\alpha \gamma (c_{2} )^{2} $. Therefore, we have

$$ \frac{{\partial^{2} }}{{\partial m^{2} }}v_{m} = - \left( {\frac{d}{dm}p_{m}^{*} } \right)(\gamma - 1)c_{2} - \alpha \gamma (c_{2} )^{2} = \alpha \gamma c_{2} (\gamma - 1)c_{2} - \alpha \gamma (c_{2} )^{2} = \alpha \gamma (\gamma - 2)(c_{2} )^{2} < 0. $$

Proof of Lemma 5

We prove that $ c_{2}^{*} < b_{b} $. Suppose that $ c_{2}^{*} > b_{b} $. Then, from the definition of $ c_{2}^{*} $, we have:

$$ p_{H}^{*} b_{g} - c_{1} - \Delta (p_{H}^{*} ) = p_{S}^{*} b_{g} + (1 - p_{S}^{*} )(b_{g} - c_{2}^{*} ) - c_{1} - \Delta (p_{S}^{*} ) < p_{S}^{*} b_{g} - c_{1} - \Delta (p_{S}^{*} ), $$

which contradicts that $ p_{H}^{*} $ is the optimal value to maximize $ pb_{g} - c_{1} - \Delta (p) $.

Proof of Lemma 6

First, the first-order condition for the optimization is given by:

$$ \frac{{dv_{m} }}{dm} = (1 - p_{m} )(\gamma - 1)c_{2} + \frac{{\partial v_{m} }}{{\partial p_{m} }}\frac{{dp_{m} }}{dm} = \left( {(1 - p_{m} )(\gamma - 1) - \tilde{m}\alpha \gamma c_{2} } \right)c_{2} = 0, $$

using $ p_{m}^{*} = \alpha \left( {b_{g} - b_{b} + (1 - m)\gamma c_{2} } \right) $ and $ \partial v_{m} /\partial p_{m} = mc_{2} $. Using the first-order condition, we have $ \frac{{\partial^{2} v_{m} }}{{\partial m\partial c_{2} }} < 0 $ and $ \frac{{\partial^{2} v_{m} }}{\partial m\partial m} < 0 $, which implies $ d\tilde{m}/dc_{2} < 0 $. Using the envelop theorem and the first-order condition, we can also establish:

$$ \frac{{dv_{m} }}{{dc_{2} }} = - (1 - p_{m} )(m + \gamma (1 - m)) + \frac{{\partial v_{m} }}{{\partial p_{m} }}\frac{{dp_{m} }}{{dc_{2} }} = - (1 - p_{m} ) < 0. $$

Second, from Eq. (3’) and $ p_{m} > p_{S} $, we have (iii).

Finally, at $ \tilde{m} = \underline{m} $, $ {p_m} < {p_H} $ and $ v_{m} = p_{m}^{*} b_{g} - c_{1} - \Delta (p_{m}^{*} ) $. Then, (ii) is immediate.

Proof of Lemma 7

As long as $ \hat{m}(\beta ,c_{2} ,\gamma ) < 1 $, the first part of Proposition 4 is straightforward, as shown in Fig. 2. Given that $ v_{m} $ is concave from Lemma 6, the figure also implies that if the absolute value of the slope of $ \frac{\beta }{1 - \beta }\left( {v_{M} (m,c_{2} ,\gamma ) - v_{S} (c_{2} )} \right) $ is larger than $ (\gamma - 1)(1 - m)c_{2} $, we have $ \hat{m}(\beta ,c_{2} ,\gamma ) < 1 $. To derive the condition for this, we define the following:

$$ \varPsi \equiv \frac{\beta }{1 - \beta }\left( {v_{M} (m,c_{2} ,\gamma ) - v_{S} (c_{2} )} \right) - (\gamma - 1)(1 - m)c_{2} . $$

Differentiating the above with respect to m and evaluating it at m = 1 yields:

$$ \left. {\frac{\partial }{\partial m}\varPsi } \right|_{m = 1} = c_{2} \left( {\frac{\beta }{1 - \beta }\left( {\left( {1 - \alpha (b_{g} - b_{b} )} \right)(\gamma - 1) - \alpha \gamma c_{2} } \right) + (\gamma - 1)} \right). $$

Therefore, for $ c_{2} > \underline{c}_{2}^{*} $, we have $ \left. {\frac{\partial }{\partial m}\varPsi } \right|_{m = 1} \le 0 $, which means that $ \hat{m}(\beta ,c_{2} ,\gamma ) < 1 $.

Proof of Lemma 8

We have the following:

$$ \frac{{\partial \hat{m}}}{\partial \gamma } = - \frac{{\varOmega_{\gamma } }}{{\varOmega_{m} - \varPi_{m} }} > 0,\;\frac{{\partial \hat{m}}}{\partial \beta } = - \frac{{\varOmega_{\beta } }}{{\varOmega_{m} - \varPi_{m} }} < 0. $$

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Authors and Affiliations

Nobuo Akai
- 1
Email author
Motohiro Sato
- 2

1.Osaka School of International Public PolicyOsaka UniversityToyonakaJapan
2.Hitotsubashi UniversityKunitachiJapan

The role of matching grants as a commitment device in the federation model with a repeated soft budget setting

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