Abstract
We introduce a distinction between interregional and intraregional transportation costs in a footloose capital model. This allows assessing more precisely the effects of different types of transport policies on the spatial distribution of activities. From a normative point of view, we find that, in the absence of regulation, the concentration of industrial activity is too high in the center. We show what set of interregional and intraregional transport policies improves the equilibrium.
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Acknowledgments
Paul Chiambaretto would like to thank Thierry Mayer for his useful comments on the first draft of this paper. Jacques Thisse also provided useful comments on the topic treated here. We also benefited from the comments of Elijah DePalma. André De Palma and Stef Proost acknowledge the support of the French PREDIT research programme no 09 MT CV 14 and of the Sustaincity project (FP7- EU). The third author thanks also FWO Flanders.
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Appendices
Appendix 1: Proof of Proposition 1
We want to prove that if \(\phi _\mathrm{AB} <\left( {1-\theta } \right) \phi _\mathrm{BB} \), then \(\lambda \in [0,1]\)
1.1 Proof of \(\lambda >0\)
We know that: \(\left( {\phi _\mathrm{AA} -\phi _\mathrm{AB} } \right) \left( {\phi _\mathrm{AB} -\phi _\mathrm{BB} } \right) <0\). If \(\lambda >0\), then we must have \(\Psi \le 0\). We use proof by contradiction. If \(\Psi >0\), then we have
Since \(1-\theta <\theta \) and \(\phi _\mathrm{BB} -\phi _\mathrm{AB} <\phi _\mathrm{AA} -\phi _\mathrm{AB} \) and since \(\phi _\mathrm{AB} <\phi _\mathrm{BB} \), the previous line cannot be true. Thus, \(\Psi \le 0\), so that \(\lambda \ge 0\).
1.2 Proof of \(\lambda <1\)
We know that \(\left( {\phi _\mathrm{AA} -\phi _\mathrm{AB} } \right) \left( {\phi _\mathrm{AB} -\phi _\mathrm{BB} } \right) <0\). If \(\lambda \le 1\), then we must have:
Since\(\left( {1-\theta } \right) \left( {\phi _\mathrm{BB} -\phi _\mathrm{AB} } \right) >0\) and \(\left( {\phi _\mathrm{AB} -\phi _\mathrm{AA} } \right) <0\), we must have \(\theta \phi _\mathrm{BB} +\phi _\mathrm{AB} -\phi _\mathrm{BB} <0,\) which is true if \(\phi _\mathrm{AB} <\left( {1-\theta } \right) \phi _\mathrm{BB}.\)
1.3 Proof that if \(\phi _\mathrm{AB} >\left( {1-\theta } \right) \phi _\mathrm{BB} \) , then \(\lambda >1\)
We know that: \(\left( {\phi _\mathrm{AA} -\phi _\mathrm{AB} } \right) \left( {\phi _\mathrm{AB} -\phi _\mathrm{BB} } \right) <0\). If \(\lambda >1\), then we must have:
Since \(\phi _\mathrm{AB} >\left( {1-\theta } \right) \phi _\mathrm{BB} \), then \(-\phi _\mathrm{AB} <\left( {\theta -1} \right) \phi _\mathrm{BB} \), so that we have
Moreover, we note that: \(\phi _\mathrm{AA} -\theta \phi _\mathrm{AB} <\theta \left( {\phi _\mathrm{AA} -\phi _\mathrm{AB} } \right) \). Thus, we conclude that: \(\left( {\phi _\mathrm{BB} -\phi _\mathrm{AB} } \right) \left( {\phi _\mathrm{AA} -\theta \phi _\mathrm{AB} } \right) <\theta \left( {\phi _\mathrm{AA} -\phi _\mathrm{AB} } \right) \phi _\mathrm{BB} \), which implies that if \(\phi _\mathrm{AB} >\left( {1-\theta } \right) \phi _\mathrm{BB} \), then \(\lambda >1\).
Appendix 2: Proof of Proposition 2
Impact of \(\phi _\mathrm{AB} \): Improving the quality of the interregional infrastructure is equivalent to an increase in \(\phi _\mathrm{AB} \).
We have \(\frac{\partial \lambda }{\partial \phi _\mathrm{AB} }>0, \)if the condition \(\frac{\phi _\mathrm{AA} }{\phi _\mathrm{AB} }>2\theta \) is respected (which always holds under our assumptions). Using this hypothesis, we observe that the improvement of the infrastructure between regions will strengthen the concentration in the center.
Impact of \(\phi _\mathrm{AA} \): We measure the effects of an improvement of infrastructures in the center. We anticipate that it will lead to a higher concentration in the center.
We obtain \(\frac{\partial \lambda }{\partial \phi _\mathrm{AA} }>0\). As anticipated, we conclude that the higher quality of infrastructure in the center will increase the concentration.
Impact of \(\phi _\mathrm{BB} \): Since the other actions on infrastructures have led to a higher concentration, we anticipate that the reduction in transport costs in the periphery will lead to a reduction in the concentration in the center.
We find that \(\frac{\partial \lambda }{\partial \phi _\mathrm{BB} }<0\) if we have \(\phi _\mathrm{AB} >\theta \phi _\mathrm{AA} \) (which holds under our hypotheses).The better quality of infrastructure in the periphery will lead to a relocation of firms from the center to the periphery.
Appendix 3: Remarks concerning the value of \(\zeta \)
First, we want to show that \(0<\zeta <1.\)
We can rewrite \(\zeta \) as:
Since \(0<\Omega <1\) and \(\phi _\mathrm{BB} <\phi _\mathrm{AA} \), we conclude that \(0<\zeta <1.\)
Second, we want to show that \(\zeta >1-\theta .\)
We use proof by contradiction. Assume that \(1-\theta >\zeta \), so that we must have:
Since \(\frac{1}{\alpha \mu +1}=\frac{\sigma -1}{\sigma -1-\mu }\) and since \(0<1-\theta <1\) we then have: \((1-\theta )^{\frac{1}{\alpha \mu +1}}>(1-\theta )^{\frac{1}{\alpha \mu +1}+1}.\)
We can then rewrite the inequality:
which is false. Thus, must have \(1-\theta <\zeta .\)
Third, with \(1-\theta <\zeta <1\), we then have \(\lambda ^\mathrm{o}\in [0,1]\)
We know that \(1-\theta <\zeta \). Since we have shown that \(\left( {1-\theta } \right) \phi _\mathrm{BB} >\phi _\mathrm{AB} \), we now have: \(\zeta \phi _\mathrm{BB} >\phi _\mathrm{AB}.\)
This allows us to deduce:
Appendix 4: Impact of the weights on the optimal value of industry share
We wish to assess the impact of the weights in the total utility function on the optimal value of the industry share. To do this, we normalize the weight for region A to 1 and give a weight \(\eta \) to region B. The indirect utility function to be maximized is then:
The first-order condition gives us the following value:
where \(\widetilde{\zeta }=\left[ {\frac{\eta \left( {1-\theta } \right) \left( {\phi _\mathrm{BB} -\phi _\mathrm{AB} } \right) }{\theta \left( {\phi _\mathrm{AA} -\phi _\mathrm{AB} } \right) }} \right] ^{\frac{-1}{\alpha \mu +1}}\).
It is easy to show that \(\frac{\partial \widetilde{\zeta }}{\partial \eta }<0\). Moreover, one can prove that \(\frac{\partial \lambda ^\mathrm{o}}{\partial \widetilde{\zeta }}>0\). Knowing the signs of these two derivatives, we conclude that an increase in \(\eta \) will lead to a reduction of \(\lambda ^\mathrm{o}\).
Appendix 5: Proof of Proposition 3
The spatial equilibrium is given by the value \(\lambda ^\mathrm{Eq}\):
We wish to compare \(\lambda ^\mathrm{Eq}\) with the optimal share of firms \(\lambda ^\mathrm{o}\) that can be rewritten as:
First, we compare the second parts of the two equations. Since \(\zeta \left( {\phi _\mathrm{BB} -\phi _\mathrm{AB} } \right) >0\) and \(\theta \phi _\mathrm{AB} >0\), then we observe that:
Second, we wish to compare the first parts of the equations. Let us prove by contradiction that
To do so, we make the hypothesis that:
which implies that:
This is a contradiction, since we know that \(0<\zeta <1\). Thus, we must have:
These two inequalities lead us to the conclusion that: \(\lambda ^{Eq}>\lambda ^\mathrm{o}\).
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Chiambaretto, P., De Palma, A. & Proost, S. A normative analysis of transport policies in a footloose capital model with interregional and intraregional transportation costs. Ann Reg Sci 51, 811–831 (2013). https://doi.org/10.1007/s00168-013-0563-3
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DOI: https://doi.org/10.1007/s00168-013-0563-3