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International Tax and Public Finance

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04-01-2021

Voluntary provision of environmental offsets under monopolistic competition

Authors: Masatoshi Yoshida, Stephen J. Turnbull

Published in: International Tax and Public Finance | Issue 4/2021

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Abstract

In a general equilibrium model where individuals voluntarily provide offsets which compensate for degradation of environmental quality by consuming differentiated goods produced by monopolistically competitive firms, this paper examines how the population size affects the equilibrium levels of offsets and net contributions. The results depend on the specification of the utility function. However, when environmental quality converges to a finite level, the offsets are independent of this specification in a large economy with many individuals. Offsets are positive in the large economy, and “carbon neutrality” holds: Net contributions are zero. The comparative statics of parameters are also analyzed.

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For the voluntary provision problem of a normal, pure public good in a standard model without consumption externalities, see Chamberlin (1974), McGuire (1974), Warr (1983), Bergstrom et al. (1986), and Andreoni (1988).

One exception is Gans and Groves (2012). They assumed that consumers derive utility from their own contributions to public good provision which decrease carbon emissions caused by consumption of electricity through the market purchases of offsets and showed that the introduction of an offset market can induce a rise in net emissions in equilibrium when producers of dirty consumption goods have market power. However, they did not consider the problem of voluntary provision of public goods.

In monopolistically competitive models, Pecorino (2009), Mondal (2013), and Bag and Mondal (2014) considered how the aggregate voluntary provision level of a pure public good depends on the population size. On the other hand, Heijdra and van der Ploeg (1996), Yoshida and Kenmochi (2011), Yoshida and Turnbull (2015), and Yoshida and Turnbull (2019) considered the optimal supply problem of public goods financed by taxation. However, these researches did not consider the problem of voluntary provision of environmental offsets.

Since each monopolistic firm has fixed and constant marginal costs in this model, the profit maximization and zero-profit conditions of this firm by themselves determine the firm’s price and the output in equilibrium. For this reason, the number of monopolistic firms adjusts the social demand and supply of environmental quality rather than the monopolistic price.

The limit properties of environmental quality generalize those obtained by Chamberlin (1974) and McGuire (1974) in the standard model of pure public goods without consumption externalities. They reexamined the hypothesis of Olson (1965) that a public good will not be provided in large economies and showed that its equilibrium level converges to a finite positive number in a model with identical individuals and identical wealth endowments. Andreoni (1988) demonstrated that this limit property also holds in a generalized model with both preference and wealth heterogeneity.

A part of this model is based on Yoshida (1998). However, we do not consider human capital and overlapping generations in this paper, since we are interested in the voluntary provision problem of environmental offsets in a static model with monopolistic competition.

See Dixit and Stiglitz (1977) and Heijdra and van der Ploeg (1996) for the PFD effect.

For example, this ambient level could be the “natural” level in the Garden of Eden, or it could arise from externalities received from the economic activity of surrounding economies.

The demand function \(G_{j}^{d} = d(P,m_{j} )\) in the interior solution is derived from \(PC_{j} + G = m_{j}\), \(U_{C} = \lambda_{j} P\), and \(U_{G} = \lambda_{j}\). Note that \(U_{G} = \lambda_{j}\) implies that the marginal utility of income (\(\lambda_{j}\)) is the opportunity cost of contributing to environmental quality.

It follows from (7) and \(\mu_{i} = (P_{C} /p_{i} )^{\theta } /\sum\nolimits_{i = 1}^{N} {(P_{C} /p_{i} )^{\theta } }\) that \(\sum\nolimits_{i = 1}^{N} {\mu_{i} } q_{i} = \sum\nolimits_{i = 1}^{N} {q_{i} } x_{ij} /\sum\nolimits_{i = 1}^{N} {x_{ij} }\). Since \(y_{j} = 0\) in the boundary solution, the budget constraint (5) is represented as \(\sum\nolimits_{i = 1}^{N} {q_{i} } x_{ij} = 1\). Thus, it holds that \(\beta /\sum\nolimits_{i = 1}^{N} {\mu_{i} } q_{i} = \beta \sum\nolimits_{i = 1}^{N} {x_{ij} }\) in this solution.

We mark the equilibrium levels of endogenous variables in the model with an asterisk (\(*\)).

We do not consider the labor market equilibrium condition: \(\sum\nolimits_{i = 1}^{N} {l_{i}^{X} } + l^{Y} = L\) by Walras’s Law, since it is derived from the budget constraint (5), the production technologies, (1) and (10), the zero-profit condition of the monopolistic firms, and the market conditions: \(\sum\nolimits_{j = 1}^{L} {x_{ij} } = X_{i}\) and \(\sum\nolimits_{j = 1}^{L} {y_{j} } = Y\).

It holds that \(g_{j}^{ * } = g^{ * } = (G^{ * } - \bar{G})/L\) for all \(j\) in the interior equilibrium. Thus, if \(G^{ * } - \bar{G} > 0\) (\(< 0\)), then it holds that \(g^{ * } > 0\) (\(< 0\)), that is, if the equilibrium level of environmental quality is greater (smaller) than its ambient level, then net contributions are positive (negative).

In the boundary equilibrium, since \(G^{ * }\) and \(g^{ * }\) are given by \(G^{ * } = \bar{G} - \beta L/\bar{q}\) and \(g^{ * } = - \beta /\bar{q}\), we obtain the comparative static results as follows: (i) \(G_{{\bar{G}}}^{ * } = 1\) and \(g_{{\bar{G}}}^{ * } = 0\), (ii) \(G_{\beta }^{ * } = - G_{b}^{ * } < 0\) and \(g_{\beta }^{ * } = - g_{b}^{ * } < 0\), (iii) \(G_{a}^{ * } = 0\) and \(g_{a}^{ * } = 0\), and (iv) \(G_{L}^{ * } < 0\), \(g_{L}^{ * } = 0\), and \(G^{ * } \to - \infty\) as \(L \to \infty\).

In the CD case (\(\sigma = 1\)), it follows from \(c = (1 - \alpha )/\alpha\) that \(\partial G_{j}^{d} /\partial P_{C} = 0\).

The expenditure ratio of the composite good to environmental quality is different from the good–quality ratio (\(s \equiv C/G^{d}\)) because of \(s = r/P_{C}\). In the equilibrium, without the PFD effect, \(s^{ * }\) is constant, but with this effect, it depends on an endogenous variable \(N^{ * }\).

When \(\theta = \sigma\), it follows from (19) that the social demand function is independent of \(N^{ * }\), so that this function is shown by the horizontal line in the \((N{}^{ * },G^{ * } )\) space. Thus, the unique equilibrium levels of \(G^{ * }\) and \(N^{ * }\) are given by \(\hat{G}^{ * } = \bar{p}\bar{X}/\bar{r}L\) and \(\hat{N}^{ * } = (\bar{G} + L - \hat{G}^{ * } )/\bar{p}\bar{X}\), respectively.

The LHS and the RHS in (20) represent the supply and demand sides of environmental quality, respectively. When \(\bar{G} = 0\) and \(\beta = 0\), (20) is the same as (18) in Bag and Mondal (2014).

In the CD case, (20) in the CES case with the PFD effect reduces to (22) in the case without it.

The social demand function (21) without the PFD effect is shown by the line, but the function (19) with it is shown by the strictly concave curve. The reason is that with the PFD effect, \(C^{ * }\) increases due to the negative effect of a rise in \(N^{ * }\) on \(P_{C}^{ * }\) so that \(G^{ * }\) decreases, but without it, there exists no such indirect effect through a change in the price of the composite good.

When \(\beta = 0\), (23) is the same as \(G^{{\prime }} (S) = \lambda = \theta (\theta /\beta )^{\theta } [L(1 - \theta )/\alpha ]^{1 - \theta }\) which is derived from (4) and (14) in Pecorino (2009).

If it is assumed that \(H(G) = \ln G\), (24) is written as \(G^{ * } = (a\theta /L)N^{ * }\). Since this is the same as the social demand function in the case of the Cobb–Douglas function: \(u{}_{j} = C_{j}^{1/2} G^{1/2}\), the Krugman–Mondal function (18) may be interpreted as a special case of the CES function (16).

Although the composite good in the KM case is the same as the CES aggregation (3) with the PFD effect (\(\rho = \theta /(\theta - 1) > 1\)), this effect does not affect the equilibrium level of environmental quality. When \(\bar{G} = 0\), (25) is the same as Eq. (15) in Mondal (2013).

We mark the equilibrium levels of variables in the partial equilibrium with double asterisks (\(* *\)).

The comparative statics of \((L,\bar{G},\beta ,a,b)\) on the number of monopolistic firms are: (i) \(N_{L}^{ * } > 0\) in all cases, (ii) \(N_{{\bar{G}}}^{ * } > 0\) in all cases, (iii) \(N_{\beta }^{ * } = 0\) in the CD and KM cases, but \(N_{\beta }^{ * } < 0\) (\(N_{\beta }^{ * } > 0\)) in the KP case and the CES case when environmental quality and the composite good are substitutes (complements), (iv) \(N_{a}^{ * } < 0\) in all cases, (v) the sign of \(N_{b}^{ * }\) in each case is the same as that of \(N_{\beta }^{ * }\).

Since \(G_{L}^{ * }\) can be also represented as \(G_{L}^{ * } = (1 - \bar{r}G^{ * } )/(1 + \bar{r}L)\), if \(\bar{r}^{ - 1} > G^{ * } > \bar{G}\) (\(\bar{r}^{ - 1} < G^{ * } < \bar{G}\)), then it holds that \(G_{L}^{ * } > 0\) (\(< 0\)).

When the ambient level of environmental quality is zero, its equilibrium level is increasing in the population size and approaches a finite positive number in all cases. They confirm the classic results of Chamberlin (1974) and McGuire (1974).

The methods of Yoshida (1998) and Ihori (1999), who examined a related problem of voluntary provision of offsets in a dynamic model of environmental capital, may be applicable here.

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Title: Voluntary provision of environmental offsets under monopolistic competition
Authors: Masatoshi Yoshida
Stephen J. Turnbull
Publication date: 04-01-2021
Publisher: Springer US
Published in: International Tax and Public Finance / Issue 4/2021
Print ISSN: 0927-5940
Electronic ISSN: 1573-6970
DOI: https://doi.org/10.1007/s10797-020-09630-5

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