Can Banning Spatial Price Discrimination Improve Social Welfare?

Abstract

We analyze a two-stage sequential-move model of location and pricing to identify firm’s location, output, and welfare. We consider two pricing regimes (mill pricing and spatial price discrimination) and, unlike previous literature, allow in each of them for a non-uniform population density, non-constant location costs (i.e., the setup costs, such rental costs and land prices, differ by firm’s location), and endogenous market boundaries. Under constant location costs, our results show the firm locates at the city center under both mill and discriminatory pricing, and that output is larger under spatial price discrimination. Welfare comparisons are, however, ambiguous. Under non-constant location costs, we find the optimal location can move away from the city center, and does not coincide across pricing regimes. Compared with mill pricing, spatial price discrimination generates a higher level of output. We also find that welfare is higher (lower) under mill than under discriminatory pricing when transportation rates are low (high, respectively).

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Notes

  1. 1.

    These contracts specify discounts which depend on the ability of customers to substitute toward cement produced by other firms. Most Portland cement is moved by truck, and purchasers are responsible for its transportation costs, which accounts for a substantial proportion of total costs since this cement is inexpensive relative to its weight. For more institutional details on this industry, see Miller and Osborne (2014).

  2. 2.

    In the European Union (EU), the Treaty on the Functioning of the European Union (TFEU) excludes any discrimination between producers or consumers with the Union (see Article 40), where any common price policy should be based on common criteria and uniform methods of calculation. In China, the Antimonopoly Law (2008) prohibits price discrimination in all markets, including final product markets (see Article 17).

  3. 3.

    Under mill pricing, the firm charges each consumer a delivery (total) price that is equal to the sum of a mill price and the transportation cost, while under spatial discriminatory pricing the firm sets location-specific delivered prices for consumers.

  4. 4.

    Location costs refer to firm’s setup costs, such as rental costs and land prices. In this paper, non-constant location costs mean that the location costs differ by firm’s location.

  5. 5.

    The market area (i.e., customers served), profit and output are larger under spatial price discrimination than under mill pricing; whereas, the welfare may be larger or lower under discriminatory pricing.

  6. 6.

    Before a firm starts operation, it must incur some location-dependent costs, such as renting a building as factory.

  7. 7.

    See Greenhut and Ohta (1972), Holahan (1975), Gronberg and Meyer (1982), Hobbs (1986), and Anderson et al. (1989) for the case of uniformly distributed population, and Beckmann (1976), Hwang and Mai (1990), and Cheung and Wang (1995) for the case of non-uniformly distributed population. All these papers assume constant location costs.

  8. 8.

    In addition, the price differential between city center and suburbs has significantly increased since 2000, as documented by Edlund et al. (2015).

  9. 9.

    Here, exogenous market area (or fixed market area) means that the number of markets is given and all markets are served. Correspondingly, endogenous market area refers to the case where the firm can decide how many markets to serve.

  10. 10.

    As argued by Berliant and Konishi (2000), the setup costs in a marketplace depend on location and are proportional to the number of consumers in the market, n.

  11. 11.

    This assumption conforms to the work by Greenhut and Ohta (1972), Beckmann (1976), Holahan (1975), Guo and Lai (2014), Chen and Hwang (2014), and Andree (2013).

  12. 12.

    Results are unaffected if we solve the profit-maximization problem by determining location and prices simultaneously.

  13. 13.

    Note that the second-order condition for a maximum is satisfied since \(\frac {\partial ^{2} \pi _{m}}{\partial {p_{m}^{2}}}=-2b{\int }_{s-\frac {a-bp_{m}}{bt}}^{s+\frac {a-bp_{m}}{bt}}n\phi (x)dx<0\), i.e., profits are concave in the mill price.

  14. 14.

    Recall that \(\frac {\partial \pi _{m}}{\partial p_{m}}\) is decreasing in p m . This condition, together with the fact that \(\frac {\partial \pi _{m}}{\partial p_{m}}|_{p_{m}= 0}={\int }_{s+a/(bt)}^{s-a/(bt)}n\phi (x)[a-bt|x-s|]dx>0\) and \(\frac {\partial \pi _{m}}{\partial p_{m}}|_{p_{m}=a/(2b)}=-{\int }_{s+a/(2bt)}^{s-a/(2bt)}n\phi (x)bt|x-s|dx<0\), implies that, using the mean value theorem, the optimal price \(p_{m}^{*}\), which is determined by \(\frac {\partial \pi _{m}}{\partial p_{m}}= 0\), must be unique and at an interior point of the interval \((0,\frac {a}{2b})\).

  15. 15.

    Similar arguments are made in the work by Holahan (1975) and Cheung and Wang (1995).

  16. 16.

    Note that the second-order condition for a maximum is satisfied since \(\frac {\partial ^{2} NR_{d}(p_{d},x)}{\partial {p_{d}^{2}}}=-2bn\phi (x)<0\).

  17. 17.

    A direct ranking of the two marginal revenues is infeasible at this general stage of the model; but several numerical simulations are provided at the end of the section.

  18. 18.

    For simplicity, we set a = b = t = σ = σ F = 1 and A = 0.15. More details about the simulation can be found in Appendix E.

  19. 19.

    Venkatesh and Kamakura (2003) generate a population of 90,000 consumers in their simulation to study bundling strategies and pricing patterns under a monopoly. We use 100,000 consumers in our simulation so that the population sample is closer to a normal distribution. In addition, when n is large enough and the population approximates normal distribution, the value of n does not affect performance comparisons between mill pricing and discriminatory pricing. For example, the sign of Q d Q m is not affected by the value of n.

  20. 20.

    For instance, for the parameter values considered in our simulation, customers at a distance \(r \in [0,\frac {1-p_{m}^{*}}{t}]\) are served under both pricing regimes, and customers \(r \in (\frac {1-p_{m}^{*}}{t},1]\) are only served under discriminatory pricing. In Fig. 3, while intercepts a/b and a/2b become 1 and 1/2 in our parametric example, all remaining intercepts are still functions of transportation rate t and mill price \(p_{m}^{*}\). We could not obtain an analytical expression for such a price, hence we need to rely on numerical simulations.

  21. 21.

    When t ≈ 0.538, locations essentially coincide under both pricing regimes, \(s_{m}^{*} \approx s_{d}^{*}\). This finding is in line with Hwang and Mai (1990) and Cheung and Wang (1995).

  22. 22.

    Quantities demanded decrease in delivery prices, which are increasing in the transportation rate (see expressions (23) and (24)).

  23. 23.

    We also find that spatial price discrimination results in a larger market area, higher profit, and larger output than mill pricing. However, compared with mill pricing, social welfare under spatial price discrimination is higher (lower) when transportation rate is low (high, respectively).

  24. 24.

    While Miller and Osborne (2014) find that banning price discrimination would increase consumer surplus, they do not evaluate profit losses, and thus cannot conclude whether social welfare increases or decreases. Our results, hence, help identify under which contexts price discrimination has welfare improving effects.

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Acknowledgments

We gratefully acknowledge the constructive comments of Professors Jia Yan, Ana Espinola-Arredondo, and Dr. PakSing Choi.

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Correspondence to Félix Muñoz-García.

Appendices

Appendix A: Proof of Lemma 1

Consider a uniform population density, ϕ(x) = v at each location x. In Case 1, location costs are constant for all s, i.e., F(s,n) = A n.

Second Stage: Pricing Decisions

Mill Pricing

In this case, the first order condition for optimal price under mill pricing (expression (7)) becomes \(2{\int }_{0}^{s+\frac {a-bp_{m}}{bt}}nv[a-2bp_{m}-bt|x-s|]dx= 0\). Solving for \(p_{m}^{*}\), we obtain \(p_{m}^{*}=\frac {a}{3b}\). Thus, under mill pricing, we obtain the market boundaries \(R_{m}=s \pm \frac {2a}{3bt}\), aggregate output \(Q_{m}=\frac {4nva^{2}}{9bt}\), profit \({\Pi }_{m}=\frac {4nva^{3}}{27b^{2}t}-An\), and social welfare \(W_{d}=\frac {nva^{3}}{4b^{2}t}-An\).

Discriminatory Pricing

Given ϕ(x) = v and F(s,n) = A n, we find market boundaries \(R_{d}=s \pm \frac {a}{bt}\), aggregate output \(Q_{d}=\frac {nva^{2}}{bt}\), profit \({\Pi }_{d}=\frac {nva^{3}}{6b^{2}t}-An\), and social welfare \(W_{d}=\frac {nva^{3}}{4b^{2}t}-An\).

First Stage: Location Decisions

Under mill pricing, profit function is \({\Pi }_{m}=\frac {4nva^{3}}{27b^{2}t}-An\), which is independent on s.

Under discriminatory pricing, profit function is \({\Pi }_{d}=\frac {nva^{3}}{6b^{2}t}-An\), which is also independent on s.

Hence, both first order conditions hold for all s, indicating the monopolist obtains the same profits at any location. The market radius, output, and social welfare are not affected by s, either. Finally, we can easy show that Q d > Q m and W d > W m .

Appendix B: Proof of Lemma 2

Second Stage: Pricing Decisions

Mill Pricing

Under a uniformly distributed population density ϕ(x) = v, the first order condition for optimal price under mill pricing (expression (7)) becomes

$$2{\int}_{0}^{s+\frac{a-b p_{m}}{bt}}nv[a-2b p_{m}-bt|x-s|]dx= 0 $$

Solving for \(p_{m}^{*}\), we obtain \(p_{m}^{*}=\frac {a}{3b}\). Thus, under mill pricing, the market boundaries (expression (5)), aggregate output (expression (8)), profit (expression (9)) and social welfare (expression (11)) become

$$\begin{aligned} &R_{m}=s \pm \frac{2a}{3bt}\\ &Q_{m}=\frac{4nva^{2}}{9bt}\\ &{\Pi}_{m}=\frac{4nva^{3}}{27b^{2}t}-F(s,n)\\ &W_{m}=\frac{20nva^{3}}{81b^{2}t}-F(s,n) \end{aligned} $$

Discriminatory Pricing

Still under a uniformly distributed population, the expressions for market boundaries, aggregate output, profit, and social welfare (Eqs. 1620) become

$$\begin{aligned} &R_{d}=s\pm \frac{a}{bt}\\ &Q_{d}=\frac{nva^{2}}{bt}\\ &{\Pi}_{d}=\frac{nva^{3}}{6b^{2}t}-F(s,n)\\ &W_{d}=\frac{nva^{3}}{4b^{2}t}-F(s,n) \end{aligned} $$

First Stage: Location Decisions

Under mill pricing, profit function is \({\Pi }_{m}=\frac {4nva^{2}}{27b^{2}t}-F(s,n)\). Taking first order condition with respect to s, we obtain

$$\frac{\partial {\Pi}_{m}}{\partial s}=-\frac{\partial F(s,n)}{\partial s}= 0 $$

Under discriminatory pricing, profit function is \({\Pi }_{d}=\frac {nva^{2}}{6b^{2}t}-F(s,n)\). Taking first order condition with respect to s, we find

$$\frac{\partial {\Pi}_{d}}{\partial s}=-\frac{\partial F(s,n)}{\partial s}= 0 $$

Hence, both first order conditions indicate that the monopolist will locate at the location where the location cost is minimum, which implies \(s_{d}^{*}=s^{*}_{m}=s^{*}\).

Appendix C: Proof of Proposition 1

Under mill pricing, the profit function is shown in expression (9). Taking derivative with respect to firm’s location, we get

$$\frac{\partial {\Pi}_{m}}{\partial s}=btnp_{m}^{*}\left[{\int}_{s}^{s+\frac{a-bp_{m}^{*}}{bt}}\phi(x)dx-{\int}_{s-\frac{a-bp_{m}^{*}}{bt}}^{s}\phi(x)dx \right] $$

Given the normal density function ϕ(x) in Eq. 1, we can find \(\frac {\partial {\Pi }_{m}}{\partial s}= 0\) for s = 0 and \(\frac {\partial {\Pi }_{m}}{\partial s}<0\) for s > 0. Thus, the unique optimal location under mill pricing and constant location cost is the city center, \(s_{m}^{*}= 0\).

Under discriminatory pricing, the profit function is shown in expression (18), taking derivative with respect to s, we obtain

$$\frac{\partial {\Pi}_{d}}{\partial s}={\int}_{s}^{s+\frac{a}{bt}}tn\phi(x)\frac{a-bt(x-s)}{2}dx-{\int}_{s-\frac{a}{bt}}^{s}tn\phi(x)\frac{a-bt(s-x)}{2}dx $$

Now let r = |xs|, where r is the distance from the firm’s location and \(r\in [0,\frac {a}{bt}]\). Each consumer at distance r has a demand \(q_{r}=\frac {a-btr}{2}\geq 0\) under price policy (14). Then \(\frac {\partial {\Pi }_{d}}{\partial s}\) becomes

$$\frac{\partial {\Pi}_{d}}{\partial s}=t{\int}_{0}^{\frac{a}{bt}}n[\phi(s+r)-\phi(s-r)]q_{r}dr $$

Given the normal density function ϕ(x), it follows \(\frac {\partial {\Pi }_{d}}{\partial s}= 0\) for s = 0 and \(\frac {\partial {\Pi }_{d}}{\partial s}<0\) for s > 0. Thus, similarly to mill pricing, the unique optimal location under discriminatory pricing and constant location cost is the city center, \(s_{d}^{*}= 0\).

Appendix D: Comparison of equilibrium outcomes in Case 3

Based on Proposition 1, we know \(s_{m}^{*}=s_{d}^{*}= 0\). We next compare market radius, profits, output, and welfare in mill and discriminatory pricing.

Market Radius

Using Eqs. 5 and 16, the market radius under mill pricing and spatial price discrimination are \(radius_{m}=|R_{m}-s_{m}^{*}|=\frac {a-bp_{m}^{*}}{bt}\) and \(radius_{d}=|R_{d}-s_{d}^{*}|=\frac {a}{bt}\). Because \(p_{m}^{*} \in \left (0,\frac {a}{2b}\right )\), it follows that \(radius_{m}<\frac {a}{bt}\), so the market area is larger under discriminatory pricing than under mill pricing when the firm’s location is Given.

Profits

Under discriminatory pricing, the market area can be divided into three regions \(x \in \left [-\frac {a}{bt},-\frac {a-bp_{m}^{*}}{bt}\right )\), \(x \in \left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]\), and \(x \in \left (\frac {a-bp_{m}^{*}}{bt}, \frac {a}{bt}\right ]\). For any market x in the market interval \(\left [-\frac {a}{bt},-\frac {a-bp_{m}^{*}}{bt}\right )\) and \(\left (\frac {a-bp_{m}^{*}}{bt},\frac {a}{bt}\right ]\), the firm can make positive net revenue above location cost under discriminatory pricing, while zero net revenue above location cost under mill pricing since the demand in this market interval is zero. Under mill pricing, for any market x in \(\left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]\), the net revenue above location cost (4) is maximized at \(p_{m}=\frac {a-bt|x|}{2b}\) with a value of \(\frac {n\phi (x)(a-bt|x|)^{2}}{4b}\), which is the optimal net revenue above location cost (12) under price discrimination. Since \(p_{m}^{*}\) is a constant mill price, \(p_{m}^{*}\) cannot be equal to \(\frac {a-bt|x|}{2b}\) and maximize the net revenue for every market \(x \in \left [-\frac {a-bp_{m}^{*}}{bt}, \frac {a-bp_{m}^{*}}{bt}\right ]\). Thus, discriminatory pricing yields higher aggregate revenue than under mill pricing. Given the same location cost of the given location, discriminatory pricing is more profitable than mill pricing.

Output

Since p m∗ solves the first order condition (7), this means \({\int }_{-\frac {a-bp_{m}^{*}}{bt}}^{\frac {a-bp_{m}^{*}}{bt}}n\phi (x)(a-2b p_{m}^{*}-bt|x|)dx= 0\). Using Eqs. 8 and 17, we can calculate the output difference between the two pricing systems:

$$\begin{array}{llllll} Q_{d}-Q_{m} &= {\int}_{-\frac{a}{bt}}^{-\frac{a-bp_{m}^{*}}{bt}}\frac{n\phi(x) (a-bt|x|)}{2}dx +{\int}_{\frac{a-bp_{m}^{*}}{bt}}^{\frac{a}{bt}}\frac{n\phi(x) (a-bt|x|)}{2}dx \\[-1pt] & \quad -\frac{1}{2}{\int}_{-\frac{a-bp_{m}^{*}}{bt}}^{\frac{a-bp_{m}^{*}}{bt}} n \phi(x)[a-2b p_{m}^{*}-bt|x|]dx\\[-1pt] &= {\int}_{-\frac{a}{bt}}^{-\frac{a-bp_{m}^{*}}{bt}}\frac{n\phi(x) (a-bt|x|)}{2}dx +{\int}_{\frac{a-bp_{m}^{*}}{bt}}^{\frac{a}{bt}}\frac{n\phi(x) (a-bt|x|)}{2}dx\\[-1pt] &>0 \end{array} $$

Thus, the output of the monopolist is higher under spatial price discrimination than mill pricing. From above equation, we can clearly see that the output difference between discriminatory and mill pricing is equal to the output gain from the extra market area under spatial price discrimination.

Social Welfare

Using Eqs. 11 and 20, we can calculate the social welfare difference between the two pricing regimes:

$$\begin{array}{@{}rcl@{}} W_{d}-W_{m}&=&\underbrace{{\int}_{-\frac{a}{bt}}^{-\frac{a-bp_{m}^{*}}{bt}}\frac{3n \phi(x)(a-bt|x|)^{2}}{8b}dx+{\int}_{\frac{a-bp_{m}^{*}}{bt}}^{\frac{a}{bt}}\frac{3n \phi(x)(a-bt|x|)^{2}}{8b}dx}_{>0, \text{Welfare gain from extra market regions}~ \left[-\frac{a}{bt},-\frac{a-bp_{m}^{*}}{bt}\right) \text{and} \left( \frac{a-bp_{m}^{*}}{bt},\frac{a}{bt}\right]}\\ && +\underbrace{\frac{1}{2}\left( {\Pi}_{m|x \in \left[-\frac{a-bp_{m}^{*}}{bt},\frac{a-bp_{m}^{*}}{bt}\right]}-{\Pi}_{d|x \in \left[-\frac{a-bp_{m}^{*}}{bt},\frac{a-bp_{m}^{*}}{bt}\right]}\right)}_{<0, \text{Welfare loss in the nearby market interval}~ \left[-\frac{a-bp_{m}^{*}}{bt},\frac{a-bp_{m}^{*}}{bt}\right]}\\ & \gtreqqless 0 \end{array} $$

where \({\Pi }_{m|x \in \left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]}\) and \({\Pi }_{d|x \in \left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]}\) are the profits under mill pricing and discriminatory pricing in market area \(\left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]\), respectively. As we argued previously, \({\Pi }_{m|x \in \left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]}<{\Pi }_{d|x \in \left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]}\). This implies that in market area \(\left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]\), spatial price discrimination reduces welfare. Relative to mill pricing, discriminatory pricing regime serves extra market regions \(\left [-\frac {a}{bt},-\frac {a-bp_{m}^{*}}{bt}\right )\) and \(\left (\frac {a-bp_{m}^{*}}{bt},\frac {a}{bt}\right ]\), where the welfare increases. The sign of W d W m depends on the welfare gain from \(\left [-\frac {a}{bt},-\frac {a-bp_{m}^{*}}{bt}\right )\) and \(\left (\frac {a-bp_{m}^{*}}{bt},\frac {a}{bt}\right ]\) and the welfare loss from \(\left [-\frac {a-bp_{m}^{*}}{bt},\frac {a-bp_{m}^{*}}{bt}\right ]\). Thus, the welfare therefore may be higher or lower under discrimination than under mill pricing.

Appendix E: Simulation description

Under mill pricing, the two-stage model can be formulated as constrained optimization problem

$$ \begin{array}{llllll} \max_{p_{m},s}\quad &\pi_{m}(p_{m},s)={\int}_{s-\frac{a-bp_{m}}{bt}}^{s+\frac{a-bp_{m}}{bt}}n\phi(x)p_{m}[a-b(p_{m}+t|x-s|)]dx-F(s,n)\\[-1pt] \text{s.t.}\quad & {\int}_{s-\frac{a-bp_{m}}{bt}}^{s+\frac{a-bp_{m}}{bt}}n\phi(x)[a-2bp_{m}-bt|x-s|]dx= 0 \end{array} $$
(25)

The two-stage model under discriminatory pricing can be formulated as constrained optimization problem

$$ \begin{array}{llllll} \max_{p_{d},s}\quad & \pi_{d}(p_{d},s)={\int}_{s-\frac{a}{bt}}^{s+\frac{a}{bt}}n\phi(x)p_{d}[a-b(p_{d}+t|x-s|)]dx-F(s,n)\\ \text{s.t.}\quad & p_{d}=\frac{a-bt|x-s|}{2b} \end{array} $$
(26)

The integrals can be approximated with Monte Carlo Simulation. Generally, suppose q(x) is density function of x and that we want to compute \(\int g(x)q(x)dx\). We can simulate N draws (x 1,...,x N ) from q(x), and let \(N^{-1}{\sum }_{i = 1}^{N} g(x_{i})\) be the approximation of \(\int g(x)q(x)dx\). In practice, many researchers adopt this technique to approximate integral in their studies (Berry et al. 1995; Dubé et al. 2012; Lee and Seo 2015).

We set a = b = σ = σ F = 1, and A = 0.15. Now we simulate n = 100, 000 artificial consumers drawn from ϕ(x). We only analyze the case where s ≥ 0. Analogous results apply when s ≤ 0. In footnote Section 3.1, we also show that \(p_{m} \in (0,\frac {a}{2b})\). Thus, the constrained optimization problem under mill pricing becomes

$$ \begin{array}{llll} \max_{p_{m},s}\quad & \pi_{m}(p_{m},s)\,=\,\frac{1}{n}{\sum}_{i = 1}^{n}\left( 1\left( s\,-\,\frac{a-bp_{m}}{bt}\!\leq\! x_{i}\!\leq\! s\,+\,\frac{a-bp_{m}}{bt}\right)np_{m}[a\,-\,b(p_{m}\,+\,t|x_{i}\,-\,s|)]\right)\,-\,F(s,n)\\ \text{s.t.}\quad & \frac{1}{n}{\sum}_{i = 1}^{n} \left( 1\left( s-\frac{a-bp_{m}}{bt}\leq x_{i} \leq s+\frac{a-bp_{m}}{bt}\right)n[a-2bp_{m}-bt|x_{i}-s|]\right)= 0\\ & 0<p_{m}<\frac{a}{2b},s\geq 0 \end{array} $$
(27)

where indicator function \(1\left (s-\frac {a-bp_{m}}{bt}\leq x_{i} \leq s+\frac {a-bp_{m}}{bt}\right )\) takes 1 if x i is in the interval \(\left (s-\frac {a-bp_{m}}{bt}, s+\frac {a-bp_{m}}{bt}\right )\) and 0 otherwise.

Under discriminatory pricing, the constrained optimization problem becomes

$$ \begin{array}{lllll} \max_{s} \quad & \pi_{d}(s)=\frac{1}{n}{\sum}_{i = 1}{n}\left( 1\left( s-\frac{a}{bt}\leq x_{i}\leq s+\frac{a}{bt}\right)\frac{n(a-bt|x_{i}-s|)^{2}}{4b}\right)-F(s,n)\\ \text{s.t.} \quad & s\geq 0 \end{array} $$
(28)

where \(1\left (s-\frac {a}{bt}\leq x_{i}\leq s+\frac {a}{bt}\right )\) takes 1 if x i is in the interval \(\left (s-\frac {a}{bt},s+\frac {a}{bt}\right )\) and 0 otherwise.

In this paper, we solve the Mathematical Program with Equilibrium Constraints (MPEC) with KNITRO optimization solver (Su and Judd 2012; Dubé et al. 2012). After we find the equilibrium \(p_{m}^{*}\), \(s_{m}^{*}\) and \(s_{d}^{*}\), we can use Monte Carlo approximation to get the equilibrium profits, outputs, consumer surplus, and welfare under both pricing regimes. For example, equilibrium output under mill pricing can be approximated by \(Q_{m}^{*} = \frac {1}{n} {\sum }_{i = 1}^{n} (1(s_{m}^{*}-\frac {a-bp_{m}^{*}}{bt} \leq x_{i} \leq s_{m}^{*}+ \frac {a-bp_{m}^{*}}{bt}) [a- b (p_{m}^{*}+t|x_{i}-s_{m}^{*}|)] )\). We replicate the Monte Carlo simulation 1000 times and find the mean of each variable.

To get Fig. 2, we first generate a sequence of location (s 1,...,s n s ) and calculate MCL, M R L m , and M R L d at each location. For a given location s k , the marginal cost of location can be obtained by \(MCL(s_{k})=\frac {Ans_{k}}{\sqrt {2\pi }{\sigma _{F}^{3}}}e^{-\frac {{s_{k}^{2}}}{2{\sigma _{F}^{2}}}}\).

Under discriminatory pricing, the marginal revenue of location at s k can be approximated by

$$ \begin{array}{lllll} MRL_{d}(s_{k})=&\frac{1}{n}{\sum}_{i = 1}^{n}\left( 1\left( s_{k}-\frac{a}{bt}\leq x_{i} \leq s_{k}\right)\frac{nt(a-bt|x_{i}-s_{k}|)}{2}\right)\\ &-\frac{1}{n}{\sum}_{i = 1}^{n}\left( 1\left( s_{k}\leq x_{i} \leq s_{k}+\frac{a}{bt}\right)\frac{nt(a-bt|x_{i}-s_{k}|)}{2}\right) \end{array} $$
(29)

Under mill pricing, we need to find the optimal price at location s k . To achieve this, we solve

$$\begin{array}{llll} \max_{p_{m}} \pi_{m}(p_{m},s_{k})=&\frac{1}{n}{\sum}_{i = 1}^{n}\left( 1\left( s_{k}-\frac{a-bp_{m}}{bt}\leq x_{i} \leq s_{k}+\frac{a-bp_{m}}{bt}\right)np_{m}[a-b(p_{m}+t|x_{i}-s_{k}|)]\right)\\ &-F(s_{k},n)\\ \text{s.t.}\quad & 0<p_{m}<\frac{a}{2b} \end{array} $$

By solving above problem with KNITRO, we get the optimal price \(p_{m}^{*}\). Then we can approximate marginal revenue of location at s k under mill pricing by Monte Carlo simulation.

$$ \begin{array}{lllll} MRL_{m}(s_{k})=&\frac{1}{n}{\sum}_{i = 1}^{n}\left( 1\left( s_{k}-\frac{a-bp_{m}^{*}}{bt}\leq x_{i} \leq s_{k}\right)nbtp_{m}^{*}\right)\\ &-\frac{1}{n}{\sum}_{i = 1}^{n}\left( 1\left( s_{k}\leq x_{i} \leq s_{k}+\frac{a-bp_{m}^{*}}{bt}\right)nbtp_{m}^{*}\right) \end{array} $$
(30)

Finally, we can plot MCL, M R L m , and M R L d and obtain Fig. 2.

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Yang, Z., Muñoz-García, F. Can Banning Spatial Price Discrimination Improve Social Welfare?. J Ind Compet Trade 18, 223–243 (2018). https://doi.org/10.1007/s10842-017-0263-2

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Keywords

  • Monopoly spatial price discrimination
  • Non-uniform distribution
  • Location choice
  • Social welfare
  • Mill pricing
  • Non-constant location costs

JEL Classification

  • D42
  • D60
  • L12
  • L50
  • R32