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The Annals of Regional Science

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01-01-2014 | Original Paper

Spatial price discrimination and location choice with labor markets

Authors: Wen-Chung Guo, Fu-Chuan Lai

Published in: The Annals of Regional Science | Issue 1/2014

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Abstract

This paper generalizes Hwang and Mai’s (Am Econ Rev 80:567–575, 1990) model to include labor markets and shows that their results are still valid when the difference of inverse demand slopes is large enough, while, when this difference is small, a monopoly firm always chooses the same (boundary) location under different pricing scenarios. In the latter situation, total outputs are identical in different pricing scenarios, while the social welfare under discriminatory pricing is definitely less than that of the mill pricing. Our results are robust, no matter whether the commuting costs are paid by the firm or workers, and no matter whether wages are discriminatory or uniform.

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In this section, it is assumed that workers cannot move between these two cities and wage discrimination is thus feasible. It is not rare that an employer pays different wages to its workers coming from different cities. For example, many Taiwanese-owned firms in China pay different wages to Taiwanese workers and local workers. We will relax this assumption in Sect. 4 to consider the case of a uniform wage.

Practically, employees may be compensated for the commuting costs. For example, firms may provide company cars to pick up their employees, and company-provided parking and gasoline subsidies are also popular. However, our model can be applied to the case in which commuting costs are paid by workers (see Sect. 5 for details).

The second-order conditions are satisfied, because the bordered Hessian matrix is

$$\begin{aligned} \left[ \begin{array}{cccc} {\partial ^{2}L/\partial \lambda ^{2}} &{} {\partial ^{2}L/\partial \lambda \partial w_1}&{} {\partial ^{2}L/\partial \lambda \partial w_2}&{} {\partial ^{2}L/\partial \lambda \partial m} \\ {\partial ^{2}L/\partial w_1 \partial \lambda }&{} {\partial ^{2}L/\partial w_1^2}&{} {\partial ^{2}L/\partial w_1 \partial w_2}&{} {\partial ^{2}L/\partial w_1 \partial m} \\ {\partial ^{2}L/\partial w_2 \partial \lambda }&{} {\partial ^{2}L/\partial w_2 \partial w_1}&{} {\partial ^{2}L/\partial w_2^2}&{} {\partial ^{2}L/\partial w_2 \partial m} \\ {\partial ^{2}L/\partial m\partial \lambda }&{} {\partial ^{2}L/\partial m\partial w_1}&{} {\partial ^{2}L/\partial m\partial w_2}&{} {\partial ^{2}L/\partial m^{2}} \end{array}\right]&= \left[ \begin{array}{cccc} 0&{} {\theta _1}&{} {\theta _2} &{} {(\beta +b)} \\ {\theta _1}&{} {-2\theta _1} &{} 0&{} 0 \\ {\theta _2}&{} 0&{} {-2\theta _2}&{} 0 \\ {(\beta +b)}&{} 0&{} 0&{} {-2(\beta +b)} \\ \end{array} \right] ,\\\hbox {and}\,\, \det \left[ \begin{array}{ccc} 0&{} {\theta _1}&{} {\theta _2} \\ {\theta _1}&{} {-2\theta _1}&{} 0 \\ {\theta _2}&{} 0 &{} {-2\theta _2} \\ \end{array}\right]&= 2\theta _1 \theta _2 (\theta _1 +\theta _2)>0, \\ \det \left[ \begin{array}{cccc} 0&{} {\theta _1}&{} {\theta _2}&{} {(\beta +b)} \\ {\theta _1}&{} {-2\theta _1}&{} 0&{} 0 \\ {\theta _2}&{} 0&{} {-2\theta _2}&{} 0 \\ {(\beta +b)}&{} 0&{} 0&{} {-2(\beta +b)} \\ \end{array}\right]&= -4\theta _1 \theta _2 (\beta +b)(b+\beta +\theta _1 +\theta _2)<0. \end{aligned}$$

The denominator is obviously positive. However, the numerator of Eq. (9) can be denoted by $K(\theta _2)$. Since $\partial K(\theta _2)/\partial \theta _2 =(-\beta -r \beta +b+br)^{2}+4\theta _1 (b+\beta )>0$, then $K(\theta _2)\ge K(\theta _2 =\theta _1)=r^{2}(b-\beta )^{2}+(b+\beta )^{2} +2\theta _1 (b+\beta )>0$. Therefore, Eq. (9) is positive.

Note that $b^{S}$ is unrelated to $a$. This is because the demand intercept is identical in our model.

Similarly, the second-order conditions are satisfied.

Since we have assumed $a$ is large enough such that there exist interior solutions of wages and prices for all locations, Eq. (18) yields $w_1^D (x=1)=\frac{2a-r\beta -\beta -b-\theta _2}{2(\beta +b+\theta _2 +\theta _1)}>0$. Therefore, $2a-\beta -\beta r>0$ and thus $b^{D}>\beta $.

Since the solutions of $\hbox {W}^{S}$ and $\hbox {W}^{D}$ are very messy, we will not show the whole equations in the text. They are available upon request from the authors.

We assume the per distance commuting cost equals one. When $r=1$, the input transport rate is identical to the output transport rate. In this situation, the range between $b^{S}$ and $b^{D}$ is relatively large.

If all living conditions are identical in these two cities, then any wage difference will be eliminated by the free mobility of workers.

If these two rates are different, then there exist many subcases depending on the relative size of these two rates. A full discussion for these subcases is very tedious. Overall speaking, the results are basically the same as the discussion in Sect. 3.

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Title: Spatial price discrimination and location choice with labor markets
Authors: Wen-Chung Guo
Fu-Chuan Lai
Publication date: 01-01-2014
Publisher: Springer Berlin Heidelberg
Published in: The Annals of Regional Science / Issue 1/2014
Print ISSN: 0570-1864
Electronic ISSN: 1432-0592
DOI: https://doi.org/10.1007/s00168-013-0576-y

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