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Simple monopoly price theory in a spatial market

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Abstract

One of main conclusions drawn by prior studies is that in a spatial market, the shape of the demand function can fully determine whether one simple pricing policy is superior to another either on the basis of the firm’s preference or social desirability while there is one assumption that is generally stipulated, namely, that the fixed market area assumption under which the market area is exogenously determined and remains the same under alternative pricing policies. In order to fully understand the impact of a demand function, this paper attempts to reexamine the relative economic advantages between two simple spatial pricing policies in a world with variable market area, that is, the market area is endogenously determined the price charged. We show that the fixed market area assumption is valid only where demand is linear, but no longer holds where demand is nonlinear. Moreover, and more importantly, we show that in a world with variable market area, some conclusions drawn by prior studies on the relative economic benefits of two pricing policies cannot remain valid. The main conclusion of this paper is that even the relative economic benefits of two simple spatial pricing policies is concerned, the impact of economic space is significant.

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Fig. 1a
Fig. 1b
Fig. 2

Notes

  1. 1.

    They are mill pricing and uniform pricing. Under mill or constant F.O.B. pricing, a consumer pays for transportation costs incurred, while the firm is responsible for such expenses under uniform or constant C.I.F. pricing. Both belong to simple pricing since the firm will charge a constant mill or delivered price while different prices will be charged under another category of pricing policy, namely, spatial price discrimination.

  2. 2.

    This relation can also be derived from the profit-maximization with respect to the market area To see this, note first that since under mill pricing, \(\pi _{f} = {\left( {m - c} \right)}Q - F\), where \(Q = {\int_0^B {\;q{\left( x \right)}dx} }\). Thus, \({\partial \pi _{f} } \mathord{\left/ {\vphantom {{\partial \pi _{f} } {\partial B}}} \right. \kern-\nulldelimiterspace} {\partial B} = {\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial B}}} \right. \kern-\nulldelimiterspace} {\partial B} = q{\left( B \right)}\), from which we see that as long as the quantity demanded is positive at the boundary B, the marginal profit increases with the market area, and equals zero while the relation (6) holds. Moreover, the marginal profit is always negative for those sites beyond the boundary of the market area B. It follows that the optimal size of the market area must satisfy Eq. 6 since beyond that site B, the delivered price is higher than the reservation price.

  3. 3.

    Similar to mill pricing, this relation can be derived from the profit-maximization with respect to the market area rather than by imposing it as an assumption. To see this, note first that under delivered pricing, \(\pi _{u} = f{\left( p \right)}{\int_0^B {\;{\left( {p - c - tx} \right)}dx} }\), and thus, \({\partial \pi _{u} } \mathord{\left/ {\vphantom {{\partial \pi _{u} } {\partial B}}} \right. \kern-\nulldelimiterspace} {\partial B} = {\left( {p - c - tB} \right)}f{\left( p \right)}\) since \(\pi _{u} = f{\left( p \right)}{\left[ {{\left( {p - c} \right)}{B - tB^{2} } \mathord{\left/ {\vphantom {{B - tB^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} \right]}\). Here we see that the marginal profit under uniform pricing equals zero at the market site satisfying Eq. 7. Moreover, the marginal profit is always negative for those sites beyond the boundary of the market area B, and it increases with the market area as long as \({\left( {p - c - tx} \right)} > 0\).

  4. 4.

    In an earlier version of this paper, one of readers suggested that the examination of the second-order optimal conditions for profit-maximization maybe helpful for the relevant analysis. The formal examination is presented in Appendix. The finding hereupon is that our discussion in Section 2 shows that the value of v must be greater than zero for the slope of a demand being negative, and therefore, the second-order optimal conditions for two optimization problems of this paper are always satisfied.

  5. 5.

    Figure 1a and b are drawn using Microsoft Excel with a=b=1, c=0.5, t=0.1.

  6. 6.

    To see this, note first that the difference between the two delivered prices at the firm’s site is given by \(p_{f} {\left( 0 \right)} - p_{u} = {\left( {a/b - c} \right)}{\left[ {{{\left( { - v} \right)}} \mathord{\left/ {\vphantom {{{\left( { - v} \right)}} {{\left( {1 + 2v} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + 2v} \right)}}} \right]}\), where \(p_{f} {\left( x \right)} = m_{f} + tx\)= the optimal delivered price under mill pricing. It follows that p f (0)<p u since v>0.

  7. 7.

    The data set for Fig. 2 is computed using Microsoft Excel with the value of v ranging from 0.2 to 3, and is available upon request. Moreover, since the values of π f /π u and Q f /Q u are the same, only one curve is drawn for them.

References

  1. Beckmann MJ (1968) Location theory, New York: Random House

  2. Beckmann MJ (1976) Spatial price policies revisited. Bell J Econ 7:619–30

  3. Cheung FK, Wang X (1996) Mill and uniform pricing: a comparison. J Reg Sci 36:129–143

  4. Greenhut J (1977) On the economic advantages of spatially discriminatory prices compared with F.O.B. prices. South Econ J 44:161–165

  5. Greenhut ML, Hwang M, Ohta H (1975) Observations on the shape and relevance of the spatial demand function. Econometrica 43:669–682

  6. Greenhut ML, Ohta H (1972) Monopoly output under alternative spatial pricing techniques. Am Econ Rev 62:705–713

  7. Heffley DR (1980) Pricing in an urban spatial monopoly: some welfare implications for policies which alter transport rates. J Reg Sci 20:207–225

  8. Holahan WL (1975) The welfare effect of spatial price discrimination. Am Econ Rev 65:498–503

  9. Hsu S-K (1979) Monopoly output under alternative spatial pricing techniques: comment. Am Econ Rev 69:678–679

  10. Hsu S-K (1983a) Pricing in an urban spatial monopoly: a general analysis. J Reg Sci 23:165–175

  11. Hsu S-K (1983b) Monopoly output and social welfare under third degree price discrimination. South Econ J 50:234–239

  12. Smithies A (1941) Monopolistic price policy in a spatial market. Econometrica 9:63–73

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Acknowledgements

The author would like to thank Professor Kim as well as three anonymous readers of this journal, and Professor Kan of the Institute of Economics at Academia Sinica, Taiwan for helpful comments; however, he alone is responsible for any remaining errors or deficiencies in this paper. Financial aids from National Science Council (Taiwan) are gratefully acknowledged.

Author information

Correspondence to Song-ken Hsu.

Appendix

Appendix

This appendix is to show that the second-order optimal conditions for two optimization problems of this paper are always satisfied since for the slope of a demand being negative, the value of v in Eq. 1 is greater than zero.

In the case of mill pricing, the second derivative of the firms’ profit with respect to the mill price is

$$ \begin{array}{*{20}l} {{{\partial ^{2} \pi } \mathord{\left/ {\vphantom {{\partial ^{2} \pi } {\partial m^{2} }}} \right. \kern-\nulldelimiterspace} {\partial m^{2} }} \hfill} & {{ = 2{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)} + {\left( {m - c} \right)}{\left( {{\partial ^{2} Q} \mathord{\left/ {\vphantom {{\partial ^{2} Q} {\partial m^{2} }}} \right. \kern-\nulldelimiterspace} {\partial m^{2} }} \right)}} \hfill} \\ {{} \hfill} & {{ = 2{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)} - {\left[ {Q \mathord{\left/ {\vphantom {Q {{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)}}} \right]}{\left( {{\partial ^{2} Q} \mathord{\left/ {\vphantom {{\partial ^{2} Q} {\partial m^{2} }}} \right. \kern-\nulldelimiterspace} {\partial m^{2} }} \right)}} \hfill} \\ {{} \hfill} & {{ = {\left[ {2{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)}^{2} - Q{\left( {{\partial ^{2} Q} \mathord{\left/ {\vphantom {{\partial ^{2} Q} {\partial m^{2} }}} \right. \kern-\nulldelimiterspace} {\partial m^{2} }} \right)}} \right]}{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)}} \hfill} \\ {{} \hfill} & {{ = K_{1} {\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)}} \hfill} \\ \end{array} $$
(A1)

where \(K_{1} = 2{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)}^{2} - Q{\left( {{\partial ^{2} Q} \mathord{\left/ {\vphantom {{\partial ^{2} Q} {\partial m^{2} }}} \right. \kern-\nulldelimiterspace} {\partial m^{2} }} \right)}\)and the second equality is based on that the first order condition for profit maximization is \({\partial \pi } \mathord{\left/ {\vphantom {{\partial \pi } {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m} = Q + {\left( {m - c} \right)}{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)} = 0\)and thus, \(m - c = - Q \mathord{\left/ {\vphantom {Q {{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {{\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m}} \right)}}\)Since \({\partial Q} \mathord{\left/ {\vphantom {{\partial Q} {\partial m}}} \right. \kern-\nulldelimiterspace} {\partial m} = - {\int {{\left( {b \mathord{\left/ {\vphantom {b v}} \right. \kern-\nulldelimiterspace} v} \right)}} }{\left( {a - bp} \right)}^{{1 \mathord{\left/ {\vphantom {1 {v - 1}}} \right. \kern-\nulldelimiterspace} {v - 1}}} dx < 0\)the fulfillment of the second-order optimal condition, in turn, requires that K 1 >0. Moreover, since \({\partial ^{2} Q} \mathord{\left/ {\vphantom {{\partial ^{2} Q} {\partial m^{2} }}} \right. \kern-\nulldelimiterspace} {\partial m^{2} } = {\int {{\left[ {{b^{2} {\left( {1 - v} \right)}} \mathord{\left/ {\vphantom {{b^{2} {\left( {1 - v} \right)}} {v^{2} }}} \right. \kern-\nulldelimiterspace} {v^{2} }} \right]}{\left( {a - bp} \right)}^{{{{\left( {1 - 2v} \right)}} \mathord{\left/ {\vphantom {{{\left( {1 - 2v} \right)}} v}} \right. \kern-\nulldelimiterspace} v}} dx} }\)we have

$$ \begin{array}{*{20}l} {{K_{1} } \hfill} & {{ = 2{\int {{\left( {b/v} \right)}^{2} {\left( {a - bp} \right)}^{{2 \mathord{\left/ {\vphantom {2 {v - 2}}} \right. \kern-\nulldelimiterspace} {v - 2}}} dx} } - {\int {{\left( {{b^{2} } \mathord{\left/ {\vphantom {{b^{2} } v}} \right. \kern-\nulldelimiterspace} v} \right)}{\left( {1 \mathord{\left/ {\vphantom {1 {v - 1}}} \right. \kern-\nulldelimiterspace} {v - 1}} \right)}{\left( {a - bp} \right)}^{{{{\left( {2 - 2v} \right)}} \mathord{\left/ {\vphantom {{{\left( {2 - 2v} \right)}} v}} \right. \kern-\nulldelimiterspace} v}} dx} }} \hfill} \\ {{} \hfill} & {{ = {\left( {1 + v} \right)}{\int {{\left[ {{\left( {b \mathord{\left/ {\vphantom {b v}} \right. \kern-\nulldelimiterspace} v} \right)}{\left( {a - bp} \right)}^{{1 \mathord{\left/ {\vphantom {1 {v - 1}}} \right. \kern-\nulldelimiterspace} {v - 1}}} } \right]}^{2} dx} }} \hfill} \\ \end{array} $$
(A2)

Here we see that while v>0, K 1 >0, and therefore, \(\partial ^{2} \pi /\partial m^{2} < 0\)

In the case of delivered pricing, the firm’s profit can be written as \(\pi _{u} = f{\left( p \right)}{\left[ {{\left( {p - c} \right)}B - {tB^{2} } \mathord{\left/ {\vphantom {{tB^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} \right]}\)and thus, the marginal profit with respect to the delivered price is

$$ \begin{array}{*{20}l} {{{\partial \pi } \mathord{\left/ {\vphantom {{\partial \pi } {\partial p}}} \right. \kern-\nulldelimiterspace} {\partial p}} \hfill} & {{ = f\prime {\left( p \right)}{\left[ {{\left( {p - c} \right)}B - {tB^{2} } \mathord{\left/ {\vphantom {{tB^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} \right]} + f{\left( p \right)}{\left[ {{B + {\left( {p - c} \right)}} \mathord{\left/ {\vphantom {{B + {\left( {p - c} \right)}} t}} \right. \kern-\nulldelimiterspace} t - {tB} \mathord{\left/ {\vphantom {{tB} t}} \right. \kern-\nulldelimiterspace} t} \right]}} \hfill} \\ {{} \hfill} & {{ = {\left[ {f\prime {\left( p \right)}{\left( {p - c - {tB} \mathord{\left/ {\vphantom {{tB} 2}} \right. \kern-\nulldelimiterspace} 2} \right)} + f{\left( p \right)}} \right]}B} \hfill} \\ {{} \hfill} & {{ = {\left[ {f\prime {\left( p \right)}{\left( {{tB} \mathord{\left/ {\vphantom {{tB} 2}} \right. \kern-\nulldelimiterspace} 2} \right)} + f{\left( p \right)}} \right]}B} \hfill} \\ \end{array} $$
(A3)

where the third equality is based on that \(B = {{\left( {p - c} \right)}} \mathord{\left/ {\vphantom {{{\left( {p - c} \right)}} t}} \right. \kern-\nulldelimiterspace} t\)Thus, we have from the first order optimal condition that

$${tB} \mathord{\left/ {\vphantom {{tB} 2}} \right. \kern-\nulldelimiterspace} 2 = - {f{\left( p \right)}} \mathord{\left/ {\vphantom {{f{\left( p \right)}} {f\prime {\left( p \right)}}}} \right. \kern-\nulldelimiterspace} {f\prime {\left( p \right)}}$$
(A4)

The second derivative of the firms’ profit with respect to the delivered price under delivered pricing is

$$ \begin{array}{*{20}l} {{{d^{2} \pi _{u} } \mathord{\left/ {\vphantom {{d^{2} \pi _{u} } {dp^{2} }}} \right. \kern-\nulldelimiterspace} {dp^{2} }} \hfill} & {{ = {\left[ {f\prime \prime {\left( p \right)}{\left( {{tB} \mathord{\left/ {\vphantom {{tB} 2}} \right. \kern-\nulldelimiterspace} 2} \right)} + {f\prime {\left( p \right)}} \mathord{\left/ {\vphantom {{f\prime {\left( p \right)}} 2}} \right. \kern-\nulldelimiterspace} 2 + f\prime {\left( p \right)}} \right]}B + {{\left[ {f\prime {\left( p \right)}{\left( {{tB} \mathord{\left/ {\vphantom {{tB} 2}} \right. \kern-\nulldelimiterspace} 2} \right)} + f{\left( p \right)}} \right]}} \mathord{\left/ {\vphantom {{{\left[ {f\prime {\left( p \right)}{\left( {{tB} \mathord{\left/ {\vphantom {{tB} 2}} \right. \kern-\nulldelimiterspace} 2} \right)} + f{\left( p \right)}} \right]}} t}} \right. \kern-\nulldelimiterspace} t} \hfill} \\ {{} \hfill} & {{ = {\left\{ { - {{\left[ {f\prime \prime {\left( p \right)}f{\left( p \right)}} \right]}} \mathord{\left/ {\vphantom {{{\left[ {f\prime \prime {\left( p \right)}f{\left( p \right)}} \right]}} {f\prime {\left( p \right)}}}} \right. \kern-\nulldelimiterspace} {f\prime {\left( p \right)}} + {f\prime {\left( p \right)}} \mathord{\left/ {\vphantom {{f\prime {\left( p \right)}} 2}} \right. \kern-\nulldelimiterspace} 2 + f\prime {\left( p \right)}} \right\}}B + {{\left[ { - f{\left( p \right)} + f{\left( p \right)}} \right]}} \mathord{\left/ {\vphantom {{{\left[ { - f{\left( p \right)} + f{\left( p \right)}} \right]}} t}} \right. \kern-\nulldelimiterspace} t} \hfill} \\ {{} \hfill} & {{ = K_{2} {\left[ {B \mathord{\left/ {\vphantom {B {f\prime {\left( p \right)}}}} \right. \kern-\nulldelimiterspace} {f\prime {\left( p \right)}}} \right]}} \hfill} \\ \end{array} $$
(A5)

where \(K_{2} = { - f{\left( p \right)}f{\left( p \right)} + 3{\left[ {f\prime {\left( p \right)}} \right]}^{2} } \mathord{\left/ {\vphantom {{ - f{\left( p \right)}f{\left( p \right)} + 3{\left[ {f\prime {\left( p \right)}} \right]}^{2} } 2}} \right. \kern-\nulldelimiterspace} 2\)and the second equality is based on Eq. A4. Note in addition that

$$ \begin{array}{*{20}l} {{K_{2} = } \hfill} & {{{\left( {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2} \right)}{\left( {b \mathord{\left/ {\vphantom {b v}} \right. \kern-\nulldelimiterspace} v} \right)}^{2} {\left( {a - bp} \right)}^{{2 \mathord{\left/ {\vphantom {2 {v - 2}}} \right. \kern-\nulldelimiterspace} {v - 2}}} - {\left( {1 - v} \right)}{\left( {b \mathord{\left/ {\vphantom {b v}} \right. \kern-\nulldelimiterspace} v} \right)}^{2} {\left( {a - bp} \right)}^{{{{\left( {1 - 2v} \right)}} \mathord{\left/ {\vphantom {{{\left( {1 - 2v} \right)}} v}} \right. \kern-\nulldelimiterspace} v}} {\left( {a - bp} \right)}^{{1 \mathord{\left/ {\vphantom {1 v}} \right. \kern-\nulldelimiterspace} v}} } \hfill} \\ {{} \hfill} & {{ = {\left( {v + 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} \right)}{\left[ {{\left( {b \mathord{\left/ {\vphantom {b v}} \right. \kern-\nulldelimiterspace} v} \right)}{\left( {a - bp} \right)}^{{1 \mathord{\left/ {\vphantom {1 {v - 1}}} \right. \kern-\nulldelimiterspace} {v - 1}}} } \right]}^{2} } \hfill} \\ \end{array} $$
(A6)

It follows that while v>0, K 1 >0, and therefore, \(\partial ^{2} \pi _{u} /\partial p^{2} < 0\) since the term \({\left[ {B \mathord{\left/ {\vphantom {B {f\prime {\left( p \right)}}}} \right. \kern-\nulldelimiterspace} {f\prime {\left( p \right)}}} \right]}\) in Eq. A5 is always negative.

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Hsu, S. Simple monopoly price theory in a spatial market. Ann Reg Sci 40, 531–544 (2006). https://doi.org/10.1007/s00168-006-0075-5

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  • R32