Abstract
This paper develops a unified imperfectly competitive macroeconomic model, and uses it to analyze optimal fiscal policies in the presence of market imperfections. The salient feature of the model is that it is able to deal with three distinct types of market structure, including constant monopoly firms, endogenous monopoly firms and endogenous overhead costs. Several findings emerge from the analysis. First, the extent of production externalities will vary under the three distinct market structures, and in turn lead the government to implement different labor and capital tax policies to correct the different extents of the distortions. Second, the optimal ratio of government expenditure is determined solely by the extent of productive government spending if the number of firms is constant, while the optimal ratio is related to the internal increasing returns to scale and production specialization if the number of firms is determined endogenously. Finally, free entry in the competitive equilibrium may result in over entry or under entry, depending upon the relative degree between monopoly power and production specialization.
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Notes
According to Hornstein (1993, p. 301), the overhead cost refers to expenses “for administration purposes to keep production going, and this is independent of how much output is produced.” Equipped with this definition, the overhead cost could be viewed as the expenses per period that the firms incur to stay in business, such as advertising, marketing, and R&D and training expenditures.
To simplify the notation, in what follows the time subscript of all variables is omitted except in cases where it should be brought to the reader’s attention.
One point involving the specification in Eq. (1) should be mentioned. If all intermediate goods are hired in the same quantities \(y\), then final output is given by \(Y = N^{\alpha + 1} y\). Thus, the increase in the final goods production is proportionally more than the increase in the number of firms if \(\alpha > 0\). In their paper, Devereux et al. (1996) and Chang et al. (2007) specify that the production function of final output has the following form: \(Y = (\int_{\,0}^{\,N} {y_{i}^{\lambda } \,di} )^{{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}}\), where monopoly power and increasing returns to specialization are characterized by the same parameter \(\lambda\). Thus, the salient feature of Eq. (1) is that it allows us to clearly separate increasing returns to specialization from monopoly power.
Several studies provide a similar specification in the intermediate goods production. For example, Devereux et al. (1996), Weder (2000), Dos Santos Ferreira and Lloyd-Braga (2008), Xiao (2008) and Brito and Dixon (2013) introduce internal increasing returns to scale in the production functions of individual firms without productive government spending, i.e., \(\gamma \ge 1\) and \(\chi = 0\). In addition, Eicher and Turnovsky (2000), Chu et al. (2015) and Chang (2019) introduce productive government spending in the production functions of individual firms with internal constant returns to scale, i.e., \(\gamma = 1\) and \(\chi > 0\). By simultaneously considering the features of internal increasing returns to scale and productive government spending in this study, we are able to understand how the optimal fiscal policies are related to each of these factors.
See Costa and Dixon (2011) for a detailed discussion of this feature.
It is worthwhile mentioning that in our analysis the public services are subject to pure aggregate congestion, implying that congestion increases with the aggregate level of capital in the economy. Although public services may be subject to different types of congestion, we cannot find a compelling argument to support this. In particular, the public services function that we employ can be used to stress the important connection between optimal fiscal policies and the extent of congestion, as we will show later. Furthermore, owing to the fact that the public services function that we employ could achieve our objective as well as enable us to better understand the relationship between the degree of congestion and optimal fiscal policies, we thus specify the public services function in simple form. For a more complicated public services function, see, for example, Eicher and Turnovsky (2000) and Chang and Lai (2017).
For simplicity and without loss of generality, the depreciation rate of physical capital is set to zero.
Two points regarding tax tools should be mentioned. First, owing to the fact that two of the three market structures are characterized by the zero-profit condition of the intermediate goods firm, we thus do not consider a separate profit tax in this study. Second, in their earlier study, Liu and Turnovsky (2005) show that the capital income tax should be utilized to remedy production externalities, while the consumption tax can be used to correct the consumption externalities. Since this study focuses on how the government implements optimal fiscal policies under different market structures embedded with production externalities, consumption externalities are omitted for simplicity, and therefore, we do not consider consumption tax in this study. We are grateful to an anonymous referee for bringing this point to our attention.
We omit the overhead cost to simplify the model here (i.e., \(\phi= 0\)) since we do not need to impose the zero-profit condition of the intermediate goods firm to endogenously derive the number of firms or the overhead cost. However, we would derive the same result even if the overhead cost were to be introduced.
It should be noted that the profit maximization problem for the intermediate goods firm is not well-defined in a steady state if \(\Pi\) is negative (i.e., \(1 - \lambda \gamma < 0\)). Based upon this supposition, in the following analysis we impose the constraint \(1 - \lambda \gamma > 0\) throughout this paper.
To satisfy the common feature of a positive marginal productivity of capital, the restriction \((\beta \gamma - \sigma \chi ) > 0\) is imposed throughout this paper.
It should be mentioned that, in their previous studies, Turnovsky (1996, 1999) and Eicher and Turnovsky (2000) focus on the situation where returns to production specialization are absent (\(\alpha = 0\)) and the firm’s production technology exhibits internal constant returns to scale (\(\gamma = 1\)). As a result, their analysis finds that the optimal ratio of government expenditure is equal to the extent of the production externalities, i.e., \(g^{*} = \chi\). In addition, as documented by Turnovsky (2009, p. 50), “the optimal fraction of output claimed by the government … should equal the elasticity of output with respect to the government input. This optimality condition is standard across all models. It obtains both in the fixed-employment and elastic labor supply closed economy models, as well as in the fixed-employment small open economy model.”
Let \(PMP_{K}\) denote the “private” marginal product of capital in the decentralized economy, and \(SMP_{K}\) denote the “social” marginal product of capital in the centralized economy. Based on Eqs. (17c) and (18c), we can infer that \(PMP_{K} = {{(1 - \tau_{k} )\lambda \beta \gamma Y} \mathord{\left/ {\vphantom {{(1 - \tau_{k} )\lambda \beta \gamma Y} K}} \right. \kern-\nulldelimiterspace} K}\) and \(SMP_{K} = {{(\beta \gamma - \sigma \chi )Y} \mathord{\left/ {\vphantom {{(\beta \gamma - \sigma \chi )Y} K}} \right. \kern-\nulldelimiterspace} K}\). As is evident, the presence of internal increasing returns to scale leads to a higher value of \(SMP_{K}\) and drives a wedge between the private and social marginal product of capital. Therefore, a capital subsidy is set to encourage production, given that internal increasing returns to scale exist.
Our analysis with free entry of firms in this section shares three features with Bénassy (1998), Devereux et al. (2000), Kim (2004), Chang et al. (2007), Pavlov and Weder (2012) and Chang et al. (2018). First, homogeneous firms can freely enter the market by paying a fixed overhead cost. Second, the number of firms is endogenously determined by a zero-profit condition due to free entry and exit. Third, the overhead cost is paid per period by all intermediate goods firms. Even though the consideration regarding the dynamic entry and exit of firms helps to improve the performance in replicating the variability of the business cycle (we will highlight this point later in Sect. 5), our paper does not deal with the transitional dynamics of firm entry. The reason for this ignorance is that our paper mainly focuses on the normative analysis rather than the positive analysis, and hence, for the sake of convenient comparison, our paper instead examines the differences in optimal fiscal policies among three different market structures. With regard to the fiscal effect on firm entry and exit in a heterogeneous firm model, see, for example, Chang et al. (2020).
At first glance, there would be confusion as to why the zero-profit condition of the final goods sector determines the price of the intermediate goods \(p = N^{\alpha }\). In fact, in the symmetric equilibrium, from the demand function for the intermediate goods of the final goods firm \(y = p_{{}}^{1/(\lambda - 1)} N^{\lambda (\alpha + 1 - 1/\lambda )/(1 - \lambda )} Y\) reported in Eq. (2), we can infer that other things being equal, \(y\) has a one-to-one relationship with \(p\). Accordingly, the price of the intermediate goods \(p = N^{\alpha }\) is associated with a specific output level of intermediate goods \(y\). Then, based on the production function of final output in the symmetric equilibrium \(Y = N^{\alpha + 1} y\) reported in Eq. (1), we can further infer the specific final output level \(Y\). Accordingly, from the perspective of the final goods firm, the determination of the price of the intermediate goods \(p = N^{\alpha }\) is equivalent to the determination of its final output level.
In line with existing studies, we introduce overhead costs to pin down the equilibrium number of firms from the zero-profit condition. Note that when \(\phi = 0\), the solution in association with this situation would be degenerate. See Chang and Lai (2012) for a detailed discussion.
See Chang et al. (2007) for detailed discussions.
There are increasing returns to an expansion of variety by inspecting Eq. (22). As a result, increases in the number of firms will enhance the effect of government spending on productivity. That is, the effects of government spending are magnified by a coefficient \(\alpha + 1\) owing to increasing returns to specialization. See also Chang et al. (2007).
From Eqs. (29a) and (29b), we can derive the results: \({{\partial \tau_{l}^{ * } } \mathord{\left/ {\vphantom {{\partial \tau_{l}^{ * } } {\partial \lambda }}} \right. \kern-\nulldelimiterspace} {\partial \lambda }} = {{\partial \tau_{k}^{ * } } \mathord{\left/ {\vphantom {{\partial \tau_{k}^{ * } } {\partial \lambda }}} \right. \kern-\nulldelimiterspace} {\partial \lambda }} = 0\).
By developing a partial equilibrium framework, Bénassy (1996) shows that in the presence of a consumer’s taste for variety, under entry may result in a decentralized economy compared to the social optimum.
See also Goerke (2017) for this issue.
See also Bilbiie et al. (2014) for this issue.
By using highly disaggregated trade data at the CN-8 level from Eurostat, Mohler and Seitz (2012) structurally estimate the gains from variety for the 27 countries within the European Union. Their empirical study finds that members of the European Union exhibit high gains from newly increased varieties.
It should be noted that the number of firms remains constant in this structure. This is because the zero-profit condition here is used to determine the overhead cost (rather than the number of firms).
As before, we may refer to \(\tau_{l}^{*}\), \(\tau_{k}^{*}\) and \(g^{*}\) as the Pareto (sub)optimal policies.
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We are deeply grateful to an anonymous referee for their instructions and comments that substantially improved the paper. We are also indebted to Juin-Jen Chang, Been-Lon Chen, Kuan-Jen Chen, Nan-Kuang Chen, Chi-Ting Chin, Hsun Chu, Hsiao-Wen Hung, Chun-Chieh Huang, Fu-Sheng Hung, Chih-Hsing Liao, and seminar participants at Fu-Jen Catholic University, Annual Meeting of the Taiwan Economic Association, who provided us with helpful suggestions in relation to earlier versions of this article. Any remaining errors or shortcomings are, however, the authors’ responsibility
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Chang, Cw., Lai, Cc. Optimal fiscal policies and market structures with monopolistic competition. Int Tax Public Finance 28, 1385–1411 (2021). https://doi.org/10.1007/s10797-020-09645-y
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DOI: https://doi.org/10.1007/s10797-020-09645-y