The Agglomeration-Differentiation Tradeoff in Spatial Location Choice

Abstract

Retailers often co-locate spatially to draw consumers, even though it increases price competition. The paper develops a structural model of entry and location choice that isolates the agglomeration benefit of co-location, after controlling for pure differentiation rationales for co-location such as (1) high demand and/or low cost at the location, (2) zoning restrictions, and (3) format differentiation that minimizes the need for spatial differentiation. We augment the entry and location choice data used in the literature with revenue and price data to help identify the agglomeration effect. We introduce a new approach to obtaining zoning data across a large number of markets that should be of general interest for a large stream of spatial location applications. We find that agglomeration benefits explain a significant fraction of observed co-location. While zoning restrictions have a little direct impact on co-location, in combination with the agglomeration benefit, they explain a surprisingly large fraction of observed co-location.

Appendices

Appendix 1

Expected total number of competing stores in a location:

$$E\left[{N}_{l}\right]={N}^{m}\sum_{f}{p}_{fl}$$

(33)

The expectation that a location will have stores with other formats besides format-f:

$$\begin{array}{c}E\left[{I}_{fl}^{\mathrm{OF}}\right]=1-{\mathrm{p}}{\mathrm{r}}{\mathrm{o}}{\mathrm{b}}\left(\mathrm{location has only format} - f\mathrm{ stores}\right)\\ =1-{p}_{fl}\displaystyle\prod_{{f}^{^{\prime}}\ne f}\left(1-{p}_{{f}^{^{\prime}}l}\right)\end{array}$$

(34)

Expected number of format-f’ rivals in distance band b around a format-f store that is in location l (interformat competition):

$$E\left[{N}_{{f}^{^{\prime}}bl}\right]=\left({N}^{m}\cdot \sum_{j\in {\mathcal{l}}_{lb}}{p}_{{f}^{^{\prime}}j}\right);{f}^{^{\prime}}\ne f$$

(35)

where ${\mathcal{l}}_{lb}$ is the set of locations in distance band b around location l.

Expected number of format-f rivals in distance band b around a format-f store that is in location l (intraformat competition):

When accounting for the number of rivals with the same format, we need to discount the choice probability of the focal firm, conditional on its decision to enter the market:

$$E\left[{N}_{fbl}\right]=\left({N}^{m}\cdot \sum_{j\in {\mathcal{l}}_{lb}}{p}_{fj}\right)-\left(1\cdot \sum_{j\in {\mathcal{l}}_{lb}}\left({p}_{fj}\left|f\mathrm{ Enters }m\right.\right)\right)$$

(36)

Note that the probability that an f-format firm enters the market is simply $\sum_{l=1}^{{l}_{m}}{p}_{fl}$. Hence, the probability $\left({p}_{fj}\left|f\mathrm{ Enters }m\right.\right)$ is given by ${p}_{fj}/\sum_{l=1}^{{l}_{m}}{p}_{fl}$ and Eq. (36) can be rewritten as:

$$E\left[{N}_{fbl}\right]=\left({N}^{m}\cdot \sum_{j\in {\mathcal{l}}_{lb}}{p}_{fj}\right)-\left(1\cdot \sum_{j\in {\mathcal{l}}_{lb}}{p}_{fj}/\sum_{l=1}^{{l}_{m}}{p}_{fl}\right)$$

(37)

Appendix 2

• Step 0: Initial population:

Generate a set of T vectors of starting values for retailers’ beliefs about rivals’ CCPs for location choices, $\left[{\overline{P} }_{0}^{1};{\overline{P} }_{0}^{2};...;{\overline{P} }_{0}^{T}\right]$ Also, create an initial guess for the parameter vector,$\theta \left(=\left\{\alpha ,\beta ,\gamma ,\sigma ,\rho \right\}\right)$.

• Step 1: Locally contractive, q-NPL iteration:

For the likelihood maximization, set up an internal loop to do the following for each of the T CCP vectors:

Given the current parameter values, pick a large number of Halton draws of price, revenue, and cost shocks for all retail locations. Obtain the location choice probabilities (Eqs. 20–22) and the market-specific cost parameters (Eqs. 26 and 27). Next, calculate the price indices of firms, sans the unobserved component, for the chosen locations and with the observed configuration of stores in 2001. Compare the price estimates with the price data to obtain the price shocks at the chosen locations of the store chain for which we have price data,$\left({\overline{\omega }}_{\mathrm{obv}}^{\mathrm{pr}}\left|\theta \right.\right)$. Also, calculate the revenues of stores by integrating over the distribution of unobserved price shocks, sans the unobserved revenue component, for the chosen locations and with the observed store configuration in 2008. Compare the revenue estimates with the revenue data for all stores to obtain the revenue shocks of firms in their chosen locations,$\left({\overline{\omega }}_{\mathrm{obv}}^{r}\left|\theta \right.\right)$. We now have all the components of the likelihood function.²⁷

Maximize the pseudo likelihood (Eq. 28) to obtain a set of T vectors of parameter estimates: ${\Theta }_{n}^{t}=\underset{\Theta }{\mathrm{argmax}}\left(L\left({\overline{P} }_{n-1}^{t},\Theta \right)\right)$ and a new population of CCPs using the q-NPL operator:${\widehat{\overline{P}} }_{n}^{t}={\Lambda }^{q}\left({\overline{P} }_{n-1}^{t},{\Theta }_{n}^{t}\right)$.

Within each market, normalize the CCPs for each store format so that the CCPs of all formats add up to one. Essentially, for each format-f, and market location l, we have:

$${\widehat{\overline{P}} }_{fln}^{t}={\Lambda }^{q}\left({\overline{P} }_{fln-1}^{t},{\Theta }_{n}^{t}\right)/\sum_{f=1}^{F}\sum_{l=1}^{{l}_{m}}{\Lambda }^{q}\left({\overline{P} }_{fln-1}^{t},{\Theta }_{n}^{t}\right)$$

(38)

• Step 2: Selection of Parents:

Based on their fitness, draw, with replacement, T “mother” CCP vectors and T “father” CCP vectors from the set, $\left[{\widehat{\overline{P}} }_{n}^{1};{\widehat{\overline{P}} }_{n}^{2};...;{\widehat{\overline{P}} }_{n}^{T}\right]$ and form couples or Parents. CCPs with high likelihood values, $L\left({\widehat{\overline{P}} }_{n}^{t},{\Theta }_{n}^{t}\right)$, and those closer to convergence (Absolute value of $\left({\widehat{\overline{P}} }_{n}^{t}-{\overline{P} }_{n-1}^{t}\right)$ closer to zero) are considered more fit to continue. In our problem, we use the following fitness criterion:

$$h\left({\hat{\bar{P_{n}^{t}}} }\right)={\lambda }_{1}\mathrm{ln}\left[L\left({\hat{\bar{P_{n}^{t}}} },{\Theta }_{n}^{t}\right)\right]-{\lambda }_{2}\Vert {\hat{\bar{P_{n}^{t}}} }-{\overline{P} }_{n-1}^{t}\Vert$$

(39)

where, ${\lambda }_{1}$ and ${\lambda }_{2}$ are small positive constants. The tth CCP vector gets selected with the probability:

$${~}^{S^t=\exp\left(h\left(\hat{\bar P_n^t}\right)\right)}\!\left/ \!{~}_{\sum_{j=1}^T\exp\left(h\left(\hat{\bar P_n^j}\right)\right)}\right.$$

(40)

Now, we have the set of couples: $\left[\left({\hat{\bar{P_{n}^{{1}^{^{\prime}}}}} },{\hat{\bar{P_{n}^{{1}^{^{\prime\prime} }}}} }\right);\left({\hat{\bar{P_{n}^{{2}^{^{\prime}}}}} },{\hat{\bar{P_{n}^{{2}^{^{\prime\prime} }}}} }\right);...;\left({\hat{\bar{P_{n}^{{T}^{^{\prime}}}}} },{\hat{\bar{P_{n}^{{T}^{^{\prime\prime}}}}} }\right)\right]$

• Step 3: Crossover and mutation

Obtain an offspring from each couple as follows:

$$\overline P_n^t=D\bullet\left(\hat{\bar{P_n^{t'}}}+Z_n\bullet\delta_n\bullet\hat{\bar{P_n^{t'}}}\right)+\left(1-D\right)\bullet\left(\hat{\bar{P_n^{t''}}}+Z_n\bullet\delta_n\bullet\hat{\bar{P_n^{t''}}}\right)$$

(41)

where D is a vector of indicators for the identity of the parent who provides each element of the CCPs. Its elements are i.i.d. with $\mathrm{Pr}\left({D}_{j}=1\right)=0.5$ for the jth element. Z_n is another vector of indicators for the identity of the elements of the CCPs, which undergo mutation. Its elements are also i.i.d. with $\mathrm{Pr}\left({Z}_{jn}=1\right)=0.5/\sqrt{n}$. Hence, with multiple iterations, as we get closer to the global optimum, we allow the number of mutations to reduce to zero. Finally, ${\delta }_{n}$ is a vector whose elements represent the magnitude of a mutation. It is also defined such that its elements go to zero with multiple iterations. Specifically, we use: ${\delta }_{jn}\in U\left(-0.5/\sqrt{n},\mathrm{ 0.5}/\sqrt{n}\right)$

As with step 1, within each market, again normalize the CCPs so that the CCPs of all formats add up to one. Now, we have the new set of CCPs,$\left[{\overline{P} }_{n}^{1};{\overline{P} }_{n}^{2};...;{\overline{P} }_{n}^{T}\right]$.

Iterate steps 1–3 until the set of CCPs converges.

Appendix 3

Price index data are not critical for identification because our model parameters can be potentially identified with only revenue data as we are exploiting the variation in the spatial distribution of consumers around stores, the locations of stores relative to consumers, and the locations of stores relative to competitors. We illustrate the intuition using a stylized example.

Consider two markets (illustrated below), each with two consumers, C1 and C2, and two firms, F1 and F2. Consumers’ distances from F1 and F2 are the same in both markets $\left(\sqrt{6}\mathrm{ mi. and }\sqrt{10}\mathrm{ mi.}\right)$. So, any difference in the observed revenue of F1 in the two markets (similarly for F2) must be because of a difference in the price index of F1 in these markets, which in turn will be attributed to the difference in the location of F1 relative to F2 in these markets (2 mi. vs. 4 mi.).

This stylized example illustrates that even without any price data, the variations in observed revenue and the distances of stores from their respective rivals can potentially help identify the competition parameters in our price index function. Furthermore, format-specific scaling parameters (i.e., intrinsic ability parameters in the price index function and customer value parameters in the volume function) can also be identified from the magnitude of revenue and its response to variations in distances from rivals versus its response to variations in consumer characteristics or distances from consumers. Therefore, price index data may not be critical if we have a large number of sample markets so that the data are likely to be sufficiently rich in terms of the spatial distribution of consumers around stores, the locations of stores relative to consumers, and the locations of stores relative to competitors. But when working with a limited number of sample markets, augmenting the revenue data with even a limited amount of price index data can give us more precise estimates.

Appendix 4

The following steps explain the technical operations involved in extracting commercial land use pixel point data from NLCD. This is the authors’ original approach. However, a more efficient approach may be plausible.

• 1. Open NLCD data in ArcGIS

• 2. Zoom in to the interested market area and select the data frame for further processing

• 3. Change coordinate system to WGS 1984

• 4. Reclassify the raster data to show only commercial land pixel points

• 5. Convert the reclassified raster data into Point Features and save them as a Shapefile

• 6. Convert the saved Shapefile into a kml file using shp2kml software. The kml file can be opened in Google Earth (GE), allowing us to see the pixel point data on GE

• 7. Make a copy of the saved kml file and rename the file from “.kml” to “.xml.” This xml file can be opened in Excel, and the spreadsheet will show the coordinates (latitude and longitude) of each pixel point, which may be used for further analysis

• 8. The count of these pixel points within each 1-mi.² block market location gives a measure of the intensity of commercial activity in the location, and the mean of the coordinates of the pixel points within the location gives the commercial center of the location

In their classification of land types, NLCD 2001 combines high-density residential land with commercial land, but NLCD 1992 separates them. Hence, we match the two datasets using ArcGIS software to separate the pixel data for all residential land areas from land areas with commercial activity in 2001. We are able to do this separation because land areas that were high-density residential in 1992 are unlikely to convert to commercial land areas by 2001, and vice versa. In the rare instances where an area that was low-density residential in 1992 was classified as commercial land in the 2001 data, we do a quick visual inspection of the geographical area using Google Earth to confirm whether that area is truly commercial land or if it has converted into a high-density residential land.