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The Annals of Regional Science

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01.11.2014 | Original Paper

Land use and rent gradients with a monopoly vendor and two central business districts

verfasst von: Fu-Chuan Lai, David Merriman, Jyh-Fa Tsai

Erschienen in: The Annals of Regional Science | Ausgabe 3/2014

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Abstract

Dispersed consumer amenities such as shopping and cultural attractions greatly influence land use patterns and rent gradients. Lai and Tsai (J Urban Econ 63:536–543, 2008) generalize the traditional Alonso–Mills–Muth model by introducing a monopoly vendor and show that the vendor will choose a boundary location. We generalize their model by allowing for inter-city shopping, which is very common in the real world. The location choice of a monopoly vendor and the consequent changes of urban configuration in two adjacent cities are discussed. This paper shows that as the distance between two CBDs decreases, the vendor may choose to locate at an inner city boundary, at the mid-point between the two cities, and at one of the outer boundaries, respectively. In many cases, a core-periphery urban structure will result.

Vorheriger Artikel Sectoral productivity gains in two regional economies: key sectors from a supply-side perspective

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Tsai and Lai (2012) generalize the Lai and Tsai (2008) model to include labor market competition between the firms in CBD and the vendor. The urban configuration in their model is very similar to the above edge city literature.

Recently, Takahashi (2014) constructs a model with two firms strategically choosing a location game in an Alonso–Mills–Muth city. However, the product price is fixed in his model, while our monopoly vendor endogenously determines product price, which highly affects the land use and rent gradient in the cities.

Note that the CBD in our model corresponds to a job center in the real world. One could think of these job centers as being parts of a metropolitan area (e.g., edge cities) or as being smaller provincial cities (e.g., Eau Claire, Stevens Point, Racine or Kenosha, Wisconsin).

For example, a large retailer (Walmart or K-Mart) could easily serve only Racine, only Kenosha, or both Racine and Kenosha. In fact, many smaller US cities are served by essentially a single shopping mall for durable goods. Moreover, if the assumption that the composite goods are provided ubiquitously is still kept, then the monopoly vendor in our model can be replaced by a large public facility such as a hospital, library, airport, or sports stadium to which residents travel on a regular basis, and all results will be similar to that in the current paper.

In a much more general case, development and zoning authorities approve only those developments that will adequately serve the population. In these cases, political forces are certainly involved and developments that do not adequately serve the entire population are unlikely to garner approval. Even when there are two separate jurisdictions, regional planning agencies, or state (or provincial) or national governments may become involved in issuing permits to the vendor. Generalizations in which a vendor endogenously chooses to serve a smaller portion of the population are beyond the scope of this paper but may be fruitful areas for future research.

If $t=k$, then land rent at the vendor’s point may be higher than that of in two CBDs. For example, if the vendor chooses to locate at the mid-point of these two cities, then the distance to shopping is longer than the distance to the office for remote consumers. Therefore, the highest rent ($y-p$) will emerge at the vendor’s location, instead of the original two CBDs. This situation is less likely happen in the real world, because a traditional CBD has many functions other than a job center, such as a traffic center, culture center, or financial center.

For example, when $x\le z$ and $x\le 0$, from (1), we have the zero rent condition: $r(x)=y-p-k(z-x)-t(0-x)=0$. Solving this equation yields $x=-(\frac{y-p-kz}{k+t})$. Similarly, when $x> z$ and $x\le 0$, we have the zero rent condition: $r(x)=y-p-k(x-z)-t(0-x)=0$. Solving this equation yields $x=(\frac{y-p+kz}{k-t})$. Detailed calculations underlying Table 1 are available upon request to the authors.

Since both cities are identical in the absence of the vendor, the analysis of the case in which the vendor chooses the right part of city 1 (inner city area) is completely analogous to the case in which the vendor chooses the left part of city 2, the only difference being that the city in which the vendor locates will be bigger. Similarly, analysis of the case in which the vendor chooses the left part of city 1 (outer city area) is analogous to the case in which it selects the right part of city 2.

If the vendor has two stores, one store serves one city. Then, this situation can be directly applied to the scenario in Lai and Tsai (2008).

If we restrict $z\in [{\overline{x}_1^L ,0}]$, then one of the boundaries $z=\overline{x}_1^L $ or $z = 0$ will be chosen, but profits will be less than when $z=\overline{x}_1^R $. Detailed calculations are available upon request to the authors.

Equation (12) means that $\pi ^{N*}_r>\pi ^0$ when $d<\hat{d}$ or $d>\check{d}$. However, if $d\ge \check{d}$, it will violate the condition that the composite good price must be positive. Because when $d>\underline{d}=2y/k$, then $p<0$. Checking $\check{d}-\underline{d}=\frac{y\sqrt{2\,t(t-k)}}{kt}>0$, it is then $d\ge \check{d}$ is infeasible. Numerically, if $y=100, t=1$, and $k=1/5$, then $\hat{d}=367.54$.

For $d<\hat{d}$, $\partial \,p/\partial \,d=-\frac{k}{4}<0$, and $\partial \,\pi _r^{N*}/\partial \,d=\frac{kt(dk-2y)}{2(t+k)(t-k)}<0$, if $d<\frac{2y}{k}$. Since $\hat{d}-\frac{2y}{k}=-\frac{y\sqrt{2t(t-k)}}{kt}<0$, therefore, in the feasible region of $d$, the larger $d$, the lower equilibrium price and the lower equilibrium profit. It is straightforward that $d$ is a negative factor to the profit of the vendor.

We are grateful to one of the referees for pointing out the possibility of $d_t$ $>(=,<) \hat{d}$.

In case of $z\in [x_1^L,0]$, then the bid rent function for the left side of city 1 is

$$\begin{aligned} r_1^{LR}=y-p+tx-k(x-z), \nonumber \\ r_1^{LL}=y-p+tx-k(z-x). \nonumber \end{aligned}$$

and the bid rent for the right side of city 1 is

$$\begin{aligned} r_1^R=y-p-tx-k(x-z). \nonumber \end{aligned}$$

When $d$ is inside the unequal twin cities range, $\partial \,p/\partial \,d=\frac{t}{4}>0$ and $\partial \,\pi _l^{O*}/\partial \,d>0$. The larger $d$ result in a larger equilibrium price and equilibrium profit. This is because the larger $d$ represent a larger market size in this subcase.

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Titel: Land use and rent gradients with a monopoly vendor and two central business districts
verfasst von: Fu-Chuan Lai
David Merriman
Jyh-Fa Tsai
Publikationsdatum: 01.11.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: The Annals of Regional Science / Ausgabe 3/2014
Print ISSN: 0570-1864
Elektronische ISSN: 1432-0592
DOI: https://doi.org/10.1007/s00168-014-0643-z

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