Abstract
This paper develops a general-equilibrium model which features the Home Market Effect and land use for production in the sector of increasing returns to scale. The land rent in the larger region is higher. Meanwhile, the larger region holds a more-than-proportionate share of firms, the so called HME in terms of firm share. These two aspects of spatial inequalities are shown to be equivalent. Moreover, the industrial distribution in the larger region and the land rent differential form bell-shaped patterns in economic integration. We demonstrate that the welfare in the larger region is higher and both regions may benefit from trade liberalization.
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Notes
It was then empirically tested by Wang and Xu (2015) using Chinese panel data 1980–2012.
See Gaspar (2018) for a more complete review.
Source: Statistisches Bundesamt 2008, http://www.destatis.de.
Source from European Quarterly, 2017,http://www.knightfrank.co.uk.
Land use only for consumption (housing) is considered by Helpman (1998), showing that agglomeration of economic activity is strongest at high trade costs, successively lower as trade costs are reduced and completely dispersed when trade costs are low enough. If land use for both consumption and production is considered, as in PT, for limited range of parameters, a bell-shaped curve emerges.
The land is assumed to be equally owned by jurisdictional residents in this paper. In some of the literature, land is owned by absentee landlords (e.g., Tabuchi 1998), which are criticized by PT on the grounds that it is not a general equilibrium one.
Strictly speaking, in the short-run, the capital income in region i should be \(K \lambda _i [c_i r_i+(1-c_i) r_j]\) where \(c_i\) is the share of capital employed domestically. However, since the capital return rate equals in equilibrium, we denote it as \(r_i\) to simplify the notation.
The Cobb–Douglas function is used in, e.g., Beckman (1972) and Lucas and Rossi-Hansberg (2002) in which land is a variable factor and enters the variable costs along with labor. For the cases of land use in the IRS sector, WY assume a variable input of Cobb–Douglas composite of land and labor. PT allow land to enter not only the variable costs, but also the fixed costs. Wrede (2013) assumes land enters the production along with labor and intermediate good in Cobb–Douglas form.
Alternatively, we can also assume a Leontief production technology which combines labor and land as inputs. However, adoption of such assumption does not change our main results on the role of land use in manufacturing production and its impacts on industrial agglomeration and welfare, but makes the expressions complicated. To keep the model simple but without loss of intuitions, we focus on land use in manufacturing and treat the homogeneous good production technology as simple as possible.
By normalizing the units of capital and land, we let \(K=S=L\) to reduce the mathematical expressions. Please note that the normalizations in the Dixit–Stiglitz sector do not reduce the generalities of the model (See Baldwin et al. 2003, Chapter 2, p. 14).
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Acknowledgements
The idea of studying the productive role of land in the IRS sector and its impacts on economic agglomeration was first generated during my doctoral study. The author is grateful to Asao Ando, Tatsuhito Kono, Dao-Zhi Zeng and other committee members. The author thanks the editor and two anonymous referees for helpful comments and suggestions. The usual caveat applies. Financial support from National Natural Science Foundation of China (71663023, 71773042, 71863011, 71863010) is acknowledged.
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Appendices
Appendices
1.1 Appendix 1: Proof of Proposition 1
By dividing Eq. (6) with Eq. (7) and plugging Eqs. (1), (3), (5) and (11) into it, we derive an equation which can be defined as
where
which is a quadratic function in \(\phi \). Remember that r is a linear function of \(s_1\) and \(s_2\), solved in Eq. (12). Meanwhile, we can solve N from Eqs. (8) and (9):
Plugging the expression of N together with Eqs. (11) and (12) into Eq. (10) generates another equation defined as
On ther other hand, by dividing Eq. (8) by Eq. (9), we define
Three endogenous variables, \(s_1\), \(s_2\) and k, are implicitly determined by three equations \(\mathcal{F}_1(s_1, s_2, k, \phi )=0\), \(\mathcal{F}_2(s_1, s_2, k)=0\) and \(\mathcal{F}_3(s_1, s_2, k)=0\). At \(\phi =0\) and \(\phi =1\), we solve
With these solutions, we derive
and
where the inequalities come from \(\sigma >1\), \(1>\gamma >0\), \(1>\alpha >0\) and \(1>\lambda >1/2.\)
For given k, variables \(s_1\) and \(s_2\) are simultaneously determined by \(\mathcal{F}_2(s_1, s_2, k)=0\) and \(\mathcal{F}_3(s_1, s_2, k)=0\). Note that the parameter \(\phi \) is independent of functions \(\mathcal{F}_2(s_1, s_2, k)\) and \(\mathcal{F}_3(s_1, s_2, k)\). For simplicity, we do not consider the possibility of multiple solutions. Therefore, we can rewrite equation \(\mathcal{F}_1(s_1, s_2, k, \phi )=0\) as \(\mathcal{F}_1(s_1(k), s_2(k), k, \phi )=0,\) which is a quadratic function in terms of \(\phi \). For any \(k^\sharp \), equation \(\mathcal{F}_1(s_1(k^\sharp ), s_2(k^\sharp ), k^\sharp ,\phi )=0\) has at most two solutions of \(\phi \). In the \(\phi \)-k plane, the curve \(k(\phi )\) crosses any horizontal line \(k=k^\sharp \) at most twice. Together with inequalities (16), (17) and \(k=\lambda \) at \(\phi =0\) and \(\phi =1\), we conclude that \(k>\lambda \) and \(k(\phi )\) evolves in a bell-shaped pattern in terms of \(\phi \in (0,1)\). \(\square \)
1.2 Appendix 2: Proof of Proposition 2
Differentiating \(s_1/s_2\) w.r.t. \(\phi \) generates
On the other hand, by Eq. (13), we have
where the inequality comes from \(\lambda <1\) and \(\gamma <1\). It implies that the sign of \(\partial (s_1/s_2)/\partial \phi \) is identical to that of \(\partial k/\partial \phi \). Because k evolves in a bell-shaped pattern in terms of \(\phi \), as shown by Appendix 1. Meanwhile, at \(\phi =0\) and \(\phi =1\), Appendix 1 indicates \(s_1/s_2=1\). Therefore, we have \(s_1/s_2>1\) and \(s_1/s_2\) evolves in a bell-shaped pattern in \(\phi \in (0,1)\).
On the other hand, by equations \(\mathcal{F}_1(s_1, s_2, k, \phi )=0\), \(\mathcal{F}_2(s_1, s_2, k)=0\) and \(\mathcal{F}_3(s_1, s_2, k)=0\), at \(\phi =0\), we solve
whereas at \(\phi =1\), we have
where the inequalities above come from \(\alpha , \gamma , \beta \in (0,1)\), \(\lambda \in (1/2,1)\) and \(\sigma >1\).
Moreover, for given \(s_1\) (resp. \(s_2\)), variables \(s_2\) (resp. \(s_1\)) and k are simultaneously determined by \(\mathcal{F}_2(s_1, s_2, k)=0\) and \(\mathcal{F}_3(s_1, s_2, k)=0\). Note that the parameter \(\phi \) is independent of functions \(\mathcal{F}_2(s_1, s_2, k)\) and \(\mathcal{F}_3(s_1, s_2, k)\). Therefore, we can rewrite equation \(\mathcal{F}_1(s_1, s_2, k, \phi )=0\) as \(\mathcal{F}_1(s_1, s_2(s_1), k(s_1), \phi )=0\) (resp. \(\mathcal{F}_1(s_1(s_2), s_2, k(s_2), \phi )=0\)) which is a quadratic function in terms of \(\phi \). For any \(s_1^\sharp \) (resp. \(s_2^\sharp \)), equation \(\mathcal{F}_1(s_1^\sharp , s_2(s_1^\sharp ), k(s_1^\sharp ),\phi )=0\) (resp. \(\mathcal{F}_1(s_1(s_2^\sharp ), s_2^\sharp , k(s_2^\sharp ),\phi )=0\)) has at most two solutions of \(\phi \). In the \(\phi \)-\(s_1\) (resp. \(\phi \)-\(s_2\)) plane, the curve \(s_1(\phi )\) (resp. \(s_2(\phi )\)) crosses any horizontal line \(s_1=s_1^\sharp \) (resp. \(s_2=s_2^\sharp \)) at most twice. Together with inequalities \(\frac{\partial s_1}{\partial \phi }|_{\phi =0}>0\), \(\frac{\partial s_1}{\partial \phi }|_{\phi =1}<0\), \(\frac{\partial s_2}{\partial \phi }|_{\phi =0}<0\) and \(\frac{\partial s_2}{\partial \phi }|_{\phi =1}>0\), we conclude that \(s_1\) evolves in a bell-shaped pattern, while \(s_2\) evolves in a U-shaped pattern in \(\phi \in (0,1)\). \(\square \)
1.3 Appendix 3: Proof of Lemma 1
By Eq. (15), we have \(\mathcal{C}_2<0\) and \(\mathcal{C}_0<0\) where the first inequality comes from \(k>\lambda >1/2\) and \(s_1>s_2,\) while the second inequality is generated by using Eq. (13) and
We therefore have
where the second inequality comes from \(\mathcal{C}_1>0\) and \(s_2^{\gamma }<s_1^{\gamma }\). It further derives
where the second inequality comes from \(s_1/s_2>1\). We therefore have \(k s_1^{(1-\sigma )\beta }>(1-k)s_2^{(1-\sigma )\beta }. \)\(\square \)
1.4 Appendix 4: Proof of Proposition 4
Lemma 1 implies that the price index in the larger region is lower. Meanwhile, individual income in the larger region is higher because \(r_1>r_2\). Therefore, the welfare in the larger region is always higher. On the other hand, with total differentiate \(\omega _i\) w.r.t. \(\phi \) at \(\phi =0\), we obtain
where
In particular, the second inequality comes from the fact that, if \(\lambda +\alpha \lambda -1<0\), we have
On the other hand, at \(\phi =1\), we have
\(\square \)
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Zhou, Y. Home market effect, land rent, and welfare. Asia-Pac J Reg Sci 3, 561–580 (2019). https://doi.org/10.1007/s41685-018-00103-6
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DOI: https://doi.org/10.1007/s41685-018-00103-6