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Dynamic Games and Applications

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21.05.2016

Cooperative and Noncooperative Extraction in a Common Pool with Habit Formation

verfasst von: Sébastien Rouillon

Erschienen in: Dynamic Games and Applications | Ausgabe 3/2017

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Abstract

The present paper considers the exploitation of a common-property, nonrenewable resource, by individuals subject to habit formation. We formalize their behavior by means of a utility function, depending on the difference between the individuals’ current consumption and the consumption level which they aspire, the latter being a weighted average of past consumptions in the population. We derive and compare the benchmark cooperative solution and a noncooperative Markov-perfect Nash equilibrium of the differential game. We investigate how the intensity, persistence and initial level of habits shape the cooperative and noncooperative solutions. We prove that habit formation may either mitigate or worsen the tragedy of the commons.

Vorheriger Artikel Large Population Aggregative Potential Games

Nächster Artikel The Continuous Time Infection–Immunization Dynamics

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In fact, the model based on latter assumption can be seen as limit cases of our model, when the persistence of habits tends to zero.

Rouillon [27] proposes a general formulation, which embeds both assumptions.

Alternatively, the parameter \(\alpha \) also gives the marginal rate of substitution between consumption and aspiration.

In particular, \(u\left( q\right) =\ln q\) and \(u\left( q\right) =\left( q^{1-\mu }-1\right) /\left( 1-\mu \right) \).

Constantinides [10] and Pollak [24] explicitly use the same condition. This condition is implicit in Ljungqvist and Uhlig [18], and Long and McWhinnie [21]. Campbell and Cochrane [7] avoid the problem by means of a formalization of habits preventing the aspiration level to become larger than the rate of consumption.

Here, \(q_{i}\left( t\right) =\left( c_{i}\left( t\right) -\alpha y\left( t\right) \right) /\left( 1-\alpha \right) \).

Corollary 1 will be satisfied for example if \(u(q)=(q^{1-\mu }-1)/(1-\mu )\), with \(\mu >0\).

Remember that \(z=x-\frac{\alpha }{1-\alpha }\frac{y}{\beta }\).

Here and below, in saying that \(q^{\circ }\) is decreasing in \(\sigma \left( \cdot \right) \), we mean that when comparing two problems only differing with respect to their elasticity of marginal utility, \(\sigma _{1}\left( \cdot \right) \) and \(\sigma _{2}\left( \cdot \right) \) (say), the optimal consumption \(q^{\circ }\) will be smaller in the second problem when \(\sigma _{1}\left( q\right) <\sigma _{2}\left( q\right) \), for all q.

Again, here, \(\sigma _{1}\left( \cdot \right) \) is said smaller than \(\sigma _{2}\left( \cdot \right) \) if \(\sigma _{1}\left( q\right) <\sigma _{2}\left( q\right) \), for all q.

Recall that \(q^{\circ }<\underline{c}^{\circ }\). If \(y\le \underline{c}^{\circ }\), as \(c^{\circ }=\left( 1-\alpha \right) q^{\circ }+\alpha y\), we have \(c^{\circ }\le \underline{c}^{\circ }\); otherwise, if \(y>\underline{c}^{\circ }\), we get \(c^{\circ }\gtreqless \underline{c}^{\circ }\) if and only if \(\alpha \gtreqless \left( \underline{c}^{\circ }-q^{\circ }\right) /\left( y-q^{\circ }\right) \) .

Here, \(q_{i}\left( t\right) =\left( c_{i}\left( t\right) -\alpha y\left( t\right) \right) /\left( 1-\alpha \right) \).

As \(\left( n-1\right) /n<1\), for all n, examples of standard utility functions satisfying this condition are \(u\left( c\right) =\ln \left( c\right) \) and \(\left( c^{1-\mu }-1\right) /\left( 1-\mu \right) \), with \(\mu >1\).

Remember that \(z=x-\frac{\alpha }{1-\alpha }\frac{y}{\beta }\).

Again, \(\sigma _{1}\left( \cdot \right) \) is said smaller than \(\sigma _{2}\left( \cdot \right) \) if \(\sigma _{1}\left( q\right) <\sigma _{2}\left( q\right) \), for all q.

We dedicate Sect. 5 to deepen this important question.

Remember that \(c^{*}\) and \(q^{*}\) vary parallely.

Recall that \(q^{*}<\underline{c}^{*}\). If \(y_{0}\le \underline{c}^{*}\), as \(c^{*}=\left( 1-\alpha \right) q^{*}+\alpha y_{0}\), we have \(c^{*}\le \underline{c}^{*}\); otherwise, if \(y_{0}>\underline{c}^{*}\), we get \(c^{*}\gtreqless \underline{c}^{*}\) if and only if\(\alpha \gtreqless \left( \underline{c}^{*}-q^{*}\right) /\left( y_{0}-q^{*}\right) \) .

When \(q\left( t\right) =f\left( z\left( t\right) \right) \), \(\mu \left( t\right) =-\frac{\alpha }{\left( 1-\alpha \right) \beta }u^{\prime }\left( f\left( z\left( t\right) \right) \right) \) and \(\eta \left( t\right) =0\), (8) simplifies to \(\dot{\mu }\left( t\right) =\delta \mu \left( t\right) \).

When \(q\left( t\right) =0\), \(\mu \left( t\right) =-\frac{\alpha }{\left( 1-\alpha \right) \beta }\mathrm{e}^{\delta \left( t-T\right) }u^{\prime }\left( 0\right) \) and \(\eta \left( t\right) =\left( \mathrm{e}^{\delta \left( t-T\right) }-1\right) u^{\prime }\left( 0\right) \), (8) simplifies to \(\dot{\mu }\left( t\right) =\delta \mu \left( t\right) \) .

Here, we use \(\partial s_{j}^{*}\left( x,y\right) /\partial x=\left( 1-\alpha \right) g'\left( x-\frac{\alpha }{\left( 1-\alpha \right) \beta }y\right) \) and \(\partial s_{j}^{*}\left( x,y\right) /\partial y=\frac{\alpha }{\beta }\big (\beta -g'\big (x-\frac{\alpha }{(1-\alpha )\beta }y\big )\big )\).

When \(q_{i}\left( t\right) =g\left( z\left( t\right) \right) \), \(\lambda _{i}\left( t\right) =u'\left( g\left( z\left( t\right) \right) \right) \), \(\mu _{i}\left( t\right) =-\frac{\alpha }{\left( 1\,-\,\alpha \right) \beta }u'\left( g\left( z\left( t\right) \right) \right) \) and \(\eta _{i}\left( t\right) =0\), (14) and (15) simplify to \(\dot{\lambda }_{i}\left( t\right) =\left( \delta +\left( n-1\right) g'\left( z\left( t\right) \right) \right) \lambda _{i}\left( t\right) \) and \(\dot{\mu }_{i}\left( t\right) =\left( \delta +\left( n-1\right) g'\left( z\left( t\right) \right) \right) \mu _{i}\left( t\right) \).

When \(q_{i}\left( t\right) =0\), \(\lambda _{i}\left( t\right) =\mathrm{e}^{\rho \left( t-T\right) }u^{\prime }\left( 0\right) \), \(\mu _{i}\left( t\right) =-\frac{\alpha }{\left( 1\,-\,\alpha \right) \beta }\mathrm{e}^{\rho \left( t-T\right) }u^{\prime }\left( 0\right) \) and \(\eta _{i}\left( t\right) =\left( \mathrm{e}^{\rho \left( t-T\right) }-1\right) u^{\prime }\left( 0\right) \), (14) and (15) simplify to \(\dot{\lambda }_{i}\left( t\right) =\left( \delta +\left( n-1\right) g'\left( z\left( t\right) \right) \right) \lambda _{i}\left( t\right) \) and \(\dot{\mu }_{i}\left( t\right) =\left( \delta +\left( n-1\right) g'\left( z\left( t\right) \right) \right) \mu _{i}\left( t\right) \).

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Titel: Cooperative and Noncooperative Extraction in a Common Pool with Habit Formation
verfasst von: Sébastien Rouillon
Publikationsdatum: 21.05.2016
Verlag: Springer US
Erschienen in: Dynamic Games and Applications / Ausgabe 3/2017
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI: https://doi.org/10.1007/s13235-016-0192-4

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