Abstract
We consider a two-stage voluntary provision model where individuals in a family contribute to a pure public good and/or a household public good, and an altruistic parent makes a non-negative income transfer to his or her child. The subgame perfect equilibrium derived in the model is analyzed using two evolutionary dynamics games (i.e., replicator dynamics and best response dynamics). As a result, the equilibria with ex-post transfers and pre-committed transfers coexist, and are unstable in the settings of replicator dynamics as well as best response dynamics, whereas the monomorphic states (i.e., all families undertake either ex-post or pre-committed transfers) are stable. An income redistribution policy does not alter the real allocations in the settings of both evolutionary dynamics games, because the resulting real allocations depend on only the total income of society and not on the distribution of individual income.
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Notes
- 1.
- 2.
Equation (11) are constantly \(\partial U_{k}^{i}/\partial g_{k}^{i}=0\) because \(d\pi ^{i}(g_{k}^{i})/dg_{k}^{i}=1\) and \( dg_{p}^{i}(g_{k}^{i})/dg_{k}^{i}=-1\) for \(i=1,2,\) by (9) and (10). Then, (11) holds for any value \(g_{k}^{i}>0\). Then indeterminacy occurs for transfers and contributions to public goods.
- 3.
We calculate the area for \(b-d<0\) and \(c-a<0,\) the white region in Fig. 1. By simple calculation, we find the tendency that the blue region \(b-d>0\) shrinks and the red region \(c-a>0\) as \(\alpha \) increases.
- 4.
From the OECD data, many countries have adopted both the gift tax, and the estate and inheritance tax. This implies that both ex-post transfer and pre-committed transfer coexist. OECD data suggest that our model should be modified or it has not reached a stationary point. It is a future challenge.
References
Cones, R.C., Itaya, J., Tanaka, A.: Private provision of public goods between families. J. Popul. Econ. 25, 1451–1480 (2012)
Fudenberg, D., Levine, D.K.: The Theory of Learning in Games, 2nd edn. MIT Press, Cambridge (1999)
Samuelson, L.: Evolutionary Games and Equilibrium Selection, 1st edn. MIT Press, Cambridge (1998)
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Tanaka, A., Itaya, Ji. (2017). An Evolutionary Game Model of Families’ Voluntary Provision of Public Goods. In: Elaydi, S., Hamaya, Y., Matsunaga, H., Pötzsche, C. (eds) Advances in Difference Equations and Discrete Dynamical Systems. ICDEA 2016. Springer Proceedings in Mathematics & Statistics, vol 212. Springer, Singapore. https://doi.org/10.1007/978-981-10-6409-8_17
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DOI: https://doi.org/10.1007/978-981-10-6409-8_17
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