Optimal Two-Part Pricing and Capacity Allocation with Multiple User Classes and Elastic Arrivals at Constrained Transportation Facilities Article First Online: 16 September 2008
Received: 04 April 2008
Accepted: 25 August 2008
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Abstract This paper focuses on the joint determination of optimal two-part prices and optimal capacity allocation at transportation facilities with elastic arrivals of multiple user classes. The model considers a general cost function and a two-part tariff comprised of an entrance fee and a dwell charge for use of the facility. Examples of this type of decision problems include: parking lots, airline seat allocation, and area congestion pricing schemes, among others. The results indicate that, in general, the optimal prices for the major pricing rules (i.e., profit maximization or welfare maximization) have three main components: one that captures the contributions of the willingness to pay and marginal costs, i.e., the classic solution; one that reflects the role of capacity constraint; and another that captures the role of elastic arrivals. Both welfare maximizing and second best prices have an additional term that accounts for the effect on consumer surplus produced by the dwell charges through the arrival rates. The models developed demonstrate the existence of a new dimension, namely cross-effects among the optimal prices and the underlying demand functions. The theoretical analyses in the paper were complemented with numerical calculations to provide a context for the discussion.
Keywords Price differentiation theory Inverse elasticity rule Optimal pricing Congestion pricing Intermodal transportation Elastic arrivals This is a preview of subscription content,
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Notes Acknowledgments This paper has benefited from insightful comments and suggestions made by a number of colleagues that, needless to say, are not to blame for any of its shortcomings: Professors Herbert Mohring (University of Minnesota), Ross Weiner (City College of New York), Kazuya Kawamura (University of Illinois at Chicago), and Kara Kockelman (University of Texas at Austin) for their invaluable comments and constructive criticisms; and to Mr. Anupam Narayan for helping out with the numerical computations. The research on which the paper was based was supported by the National Science Foundation grants CAREER 0245165 and CMS 0324380, the Fondecyt (Chile) Grant 1080140 and the Millennium Institute on Complex Engineering Systems.
Appendix I Notation used in the paper a) Variables:
R Vector of entrance fees R i
R * Vector of optimal entrance fees \(R_i^ * \)
P Vector of dwell charges P i
P Vector of optimal dwell charges \(P_i^ * \)
N Vector of space allocation N i
N i Number of spaces allocated to class i
N * Vector of optimal space allocation \(N_i^ * \)
T j = I j Q j Total demand for user class j
b) Functions of
P and
Q :
W Total welfare
CS Consumer surplus
Π Producer surplus
\(\Delta CS_{I_j }^Q = \left( {\frac{{\partial Q_j }}{{\partial P_j }}} \right)^{ - 1} \frac{{\partial \int\limits_{R_j }^\infty {I_j dR_j } }}{{\partial P_j }}\) Variation of consumer surplus with respect to Q
\(\Delta CS_{IQ_j }^I = \left( {\frac{{\partial I_j }}{{\partial R_j }}} \right)^{ - 1} \frac{{\partial \int\limits_{P_j }^\infty {I_j } Q_j dP_j }}{{\partial R_j }}\) Variation of consumer surplus with respect to I
c) Functions of prices:
I = f (P, R ) Vector of arrival rates, I i
\(I = \sum\limits_i {I_i } \) Total arrival rate
Q = g (P ) Vector of dwell times Q i
Q * Vector of optimal dwell times \(Q_i^ * \) corresponding to \(P_i^ * \)
T j = I j Q j Total demand for space corresponding to user class j
d) Functions of
Q and
I :
C = h (I, Q ) Total production cost
\(\eta _{Q_i }^P = \frac{{P_i }}{{Q_i }}\frac{{\partial Q_i }}{{\partial P_i }}\) Price elasticity of dwell time with respect to dwell charge
\(\eta _{I_i }^P = \frac{{P_i }}{{I_i }}\frac{{\partial I_i }}{{\partial P_i }}\) Price elasticity of number of arrivals with respect to the dwell charge
\(\eta _{I_i }^R = \frac{{R_i }}{{I_i }}\frac{{\partial I_i }}{{\partial R_i }}\) Price elasticity of number of arrivals with respect to the entrance fee
\(\omega _{IQ_j } = \frac{{Q_j }}{{I_j }}{{\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/ {\vphantom {{\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} = {{\left\{ {\frac{{P_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \right\}} \mathord{\left/ {\vphantom {{\left\{ {\frac{{P_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \right\}} {\left\{ {\frac{{P_j }}{{Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}} \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {\frac{{P_j }}{{Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}} \right\}}} = \frac{{\eta _{I_j }^P }}{{\eta _{Q_j }^P }}\) tradeoff ratio between arrival rates and dwell times
\(m_i^Q = \frac{{\partial C}}{{\partial Q_j }}\) Marginal cost with respect to Q
\(m_i^I = \frac{{\partial C}}{{\partial I_j }}\) Marginal cost with respect to I
e) Constants:
N Total number of available spaces
ψ i Amount of capacity consumed by a unit of user from class i
Appendix 2 Profit maximizing prices The optimization problem is:
$${\text{MAX}}\;\Pi = \sum\limits_i {I_i P_i Q_i } + \sum\limits_i {I_i R_i } - C$$
(52)
subject to:
$$N \geqslant \sum\limits_i {\psi _i I_i Q_i } ,\forall i$$
(53)
First set of optimal prices: Dwell charges Applying the first KKT condition:
$$\frac{{\partial \Pi }}{{\partial P_j }} - \lambda \frac{{\partial g}}{{\partial P_j }} \leqslant 0,\forall j$$
(54)
and dividing by
\(I_j \frac{{\partial Q_j }}{{\partial P_j }}\) and grouping terms:
$$\begin{aligned}& P_j + P_j \left( {{{\frac{{Q_j }}{{P_j }}} \mathord{\left/{\vphantom {{\frac{{Q_j }}{{P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right) + P_j \left( {{{\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/{\vphantom {{\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right) + {{\frac{{R_j }}{{Q_j }}\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/{\vphantom {{\frac{{R_j }}{{Q_j }}\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} - {{\frac{1}{{I_j }}\frac{{\partial C}}{{\partial P_j }}} \mathord{\left/{\vphantom {{\frac{1}{{I_j }}\frac{{\partial C}}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} \\& \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \lambda \psi _j - \lambda \psi _j \left( {{{\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/{\vphantom {{\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right) \leqslant 0 \\ \end{aligned} $$
(55)
Since
\(\frac{{\partial C}}{{\partial P_j }} = \frac{{\partial C_j }}{{\partial I_j }}\frac{{\partial I_j }}{{\partial P_j }} + \frac{{\partial C_j }}{{\partial Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}\) (because
C is a function of
I j and
Q j ), and letting
$$\omega _{IQ_j } = \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}\frac{{\partial P_j }}{{\partial Q_j }} = {{\left\{ {\frac{{P_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \right\}} \mathord{\left/{\vphantom {{\left\{ {\frac{{P_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \right\}} {\left\{ {\frac{{P_j }}{{Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}} \right\}}}} \right.\kern-\nulldelimiterspace} {\left\{ {\frac{{P_j }}{{Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}} \right\}}} = \frac{{\eta _{I_j }^P }}{{\eta _{Q_j }^P }},$$
\(m_j^Q = \frac{{\partial C}}{{\partial Q_j }}\) and
\(m_j^I = \frac{{\partial C}}{{\partial I_j }}\) the complementary slackness condition becomes:
$$P_j \left( {P_j - P_j \frac{1}{{\left| {\eta _{Q_j }^P } \right|}} + P_j \omega _{IQ_j } + R_j \frac{{\omega _{IQ_j } }}{{Q_j }} - \frac{{m_j^Q }}{{I_j }} - m_j^I \frac{{\omega _{IQ_j } }}{{Q_j }} - \lambda \psi _j \left( {1 + \omega _{IQ_j } } \right)} \right) = 0$$
(56)
The optimal prices
P j derived from the complementary slackness condition, (valid for
P j > 0) are:
$$\frac{{P_j - \frac{{m_j^Q }}{{I_j }} - m_j^I \frac{{\omega _{IQ_j } }}{{Q_j }}}}{{P_j }} = \frac{1}{{\left| {\eta _{Q_j }^P } \right|}} + \frac{{\lambda \psi _j }}{{P_j }} - \left( {1 - \frac{{\lambda \psi _j }}{{P_j }}} \right)\omega _{IQ_j } - \frac{{\omega _{IQ_j } }}{{Q_j }}\frac{{R_j }}{{P_j }}$$
(57)
Second set of optimal prices: Entrance fees Using the first KKT condition:
$$\frac{{\partial \Pi }}{{\partial R_j }} - \lambda \frac{{\partial g}}{{\partial R_j }} \leqslant 0,\forall j$$
(58)
and dividing by
\(\frac{{\partial I_j^{} }}{{\partial R_j^{} }}\) and grouping terms:
$$P_j Q_j + R_j + R_j {{\frac{{I_j }}{{R_j }}} \mathord{\left/{\vphantom {{\frac{{I_j }}{{R_j }}} {\frac{{\partial I_j }}{{\partial R_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial I_j }}{{\partial R_j }}}} - {{\frac{{\partial C}}{{\partial R_j }}} \mathord{\left/{\vphantom {{\frac{{\partial C}}{{\partial R_j }}} {\frac{{\partial I_j }}{{\partial R_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial I_j }}{{\partial R_j }}}} - \lambda \psi _j Q_j \leqslant 0$$
(59)
Letting
\(\eta _{I_j }^R = \frac{{R_j }}{{I_j }}\frac{{\partial I_j }}{{\partial R_j }}\) and
\(m_j^I = \frac{{\partial C}}{{\partial I_j }}\) , the complementary slackness condition becomes:
$$R_j \left( {P_j Q_j + R_j - \frac{{R_j }}{{\left| {\eta _{I_j }^R } \right|}} - m_j^I - \lambda \psi _j Q_j } \right) = 0$$
(60)
From Eq. (
60 ) for
R j > 0:
$$\frac{{R_j - m_j^I }}{{R_j }} = \frac{1}{{\left| {\eta _{I_j }^R } \right|}} + \frac{{\lambda \psi _j Q_j }}{{R_j }} - P_j \frac{{Q_j }}{{R_j }}$$
(61)
Appendix 3 Welfare maximizing prices The corresponding optimization problem is:
$${\text{MAX}}\;W = \sum\limits_i {\int\limits_{P_i }^\infty {I_i Q_i } } \operatorname{d} P_i + \sum\limits_i {\int\limits_{R_i }^\infty {I_i } } \operatorname{d} R_i + \sum\limits_i {I_i P_i Q_i } + \sum\limits_i {I_i R_i } - C$$
(62)
subject to:
$$N \geqslant \sum\limits_i {\psi _i I_i Q_i } ,\forall i$$
(63)
First set of optimal prices: Dwell charges Applying the first KKT condition:
$$\frac{{\partial W}}{{\partial P_j }} - \lambda \frac{{\partial g}}{{\partial P_j }} \leqslant 0,\forall j$$
(64)
$$\frac{\partial }{{\partial P_j }}\left\{ {\sum\limits_i {\int\limits_{P_i }^\infty {I_i Q_i } } \operatorname{d} P_i + \sum\limits_i {\int\limits_{R_i }^\infty {I_i } } \operatorname{d} R_i + \sum\limits_i {I_i P_i Q_i } + \sum\limits_i {I_i R_i } - C} \right\} - \lambda \frac{\partial }{{\partial P_j }}\left\{ {\sum\limits_i {\psi _i I_i Q_i } - N} \right\} \leqslant 0$$
(65)
which after some manipulation yields:
$$\begin{aligned}& \left[ {I_j Q_j } \right]_{P_j }^\infty + I_j Q_j + I_j P_j \frac{{\partial Q_j }}{{\partial P_j }} + Q_j P_j \frac{{\partial I_j }}{{\partial P_j }} + \frac{\partial }{{\partial P_j }}\int\limits_{R_j }^\infty {I_j \operatorname{d} R_j } + R_j \frac{{\partial I_j }}{{\partial P_j }} - \frac{{\partial C}}{{\partial P_j }} \\& \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \lambda \psi _j \left\{ {I_j \frac{{\partial Q_j }}{{\partial P_j }} + Q_j \frac{{\partial I_j }}{{\partial P_j }}} \right\} \leqslant 0 \\ \end{aligned} $$
(66)
Since C is an implicit function of I j and Q j :
\(\frac{{\partial C}}{{\partial P_j }} = \frac{{\partial C}}{{\partial Q_j }}\frac{{\partial Q_j }}{{\partial P_j }} + \frac{{\partial C}}{{\partial I_j }}\frac{{\partial I}}{{\partial P_j }}\) and dividing by
\(I_j^{} \frac{{\partial Q_j }}{{\partial P_j^{} }}\) $$\begin{aligned}P_j + {{P_j \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/{\vphantom {{P_j \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} + {{\frac{1}{{I_j }}\frac{{\partial \int\limits_{R_j }^\infty {I_j \operatorname{d} R_j } }}{{\partial P_j }}} \mathord{\left/{\vphantom {{\frac{1}{{I_j }}\frac{{\partial \int\limits_{R_j }^\infty {I_j \operatorname{d} R_j } }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} + {{R_j \frac{1}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/{\vphantom {{R_j \frac{1}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} - {{\frac{{\partial C}}{{\partial Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}\frac{1}{{I_j }}} \mathord{\left/{\vphantom {{\frac{{\partial C}}{{\partial Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}\frac{1}{{I_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} \\\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - {{\frac{{\partial C}}{{\partial I_j }}\frac{{\partial I_j }}{{\partial P_j }}\frac{1}{{I_j }}} \mathord{\left/{\vphantom {{\frac{{\partial C}}{{\partial I_j }}\frac{{\partial I_j }}{{\partial P_j }}\frac{1}{{I_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} - \lambda \psi _j \left\{ {1 + {{\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/{\vphantom {{\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right\} \leqslant 0 \\ \end{aligned} $$
(67)
Since
\(
\begin{array}{*{20}l}
{{m^{Q}_{j} = {\frac{{\partial C}}
{{\partial P_{j} }}} \mathord{\left/
{\vphantom {{\frac{{\partial C}}
{{\partial P_{j} }}} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}}} \right.
\kern-\nulldelimiterspace} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}} \hfill} & {{m^{I}_{j} = {\frac{{\partial C}}
{{\partial P_{j} }}} \mathord{\left/
{\vphantom {{\frac{{\partial C}}
{{\partial P_{j} }}} {\frac{{\partial I_{j} }}
{{\partial P_{j} }}}}} \right.
\kern-\nulldelimiterspace} {\frac{{\partial I_{j} }}
{{\partial P_{j} }}}} \hfill} & {{\omega _{{IQ_{j} }} = {\frac{{Q_{j} }}
{{I_{j} }}\frac{{\partial I_{j} }}
{{\partial P_{j} }}} \mathord{\left/
{\vphantom {{\frac{{Q_{j} }}
{{I_{j} }}\frac{{\partial I_{j} }}
{{\partial P_{j} }}} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}}} \right.
\kern-\nulldelimiterspace} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}} \hfill} \\
\end{array}
\) and letting
\(\Delta CS_{I_j }^Q = \left( {\frac{{\partial Q_j }}{{\partial P_j }}} \right)^{ - 1} \frac{{\partial \int\limits_{R_j }^\infty {I_j \operatorname{d} R_j } }}{{\partial P_j }}\) $$P_j + P_j \omega _{IQ_j } + \frac{{\Delta CS_{I_j }^Q }}{{I_j }} + \frac{{R_j \omega _{IQ_j } }}{{Q_j }} - \frac{{m_j^Q }}{{I_j }} - \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } - \lambda \psi _j - \lambda \psi _j \omega _{IQ_j } \leqslant 0$$
(68)
$$P_j \left( {1 + \omega _{IQ_j } } \right) + \frac{{\Delta CS_{I_j }^Q }}{{I_j }} + \frac{{R_j \omega _{IQ_j } }}{{Q_j }} - \frac{{m_j^Q }}{{I_j }} - \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } - \lambda \psi _j \left( {1 + \omega _{IQ_j } } \right) \leqslant 0$$
(69)
From the complementary slackness condition:
$$P_j \left( {\left( {1 + \omega _{IQ_j } } \right)P_j + \frac{{\Delta CS_{I_j }^Q }}{{I_j }} + \frac{{R_j \omega _{IQ_j } }}{{Q_j }} - \frac{{m_j^Q }}{{I_j }} - \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } - \lambda \psi _j \left( {1 + \omega _{IQ_j } } \right)} \right) = 0$$
(70)
Since
P j > 0
$$\left( {1 + \omega _{IQ_j } } \right)P_j + \frac{{\Delta CS_{I_j }^Q }}{{I_j }} + \frac{{R_j \omega _{IQ_j } }}{{Q_j }} - \frac{{m_j^Q }}{{I_j }} - \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } - \lambda \psi _j \left( {1 + \omega _{IQ_j } } \right) = 0$$
(71)
Finally:
$$P_j = \left\{ {\frac{{m_j^Q }}{{I_j }} + \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } + \lambda \psi _j \left( {1 + \omega _{IQ_j } } \right) - R_j \frac{{\omega _{IQ_j } }}{{Q_j }} - \frac{{\Delta CS_{I_j }^Q }}{{I_j }}} \right\}\frac{1}{{\left( {1 + \omega _{IQ_j } } \right)}}$$
(72)
Second set of optimal prices: Entrance fees Applying the first KKT condition:
$$\frac{{\partial W}}{{\partial R_j }} - \lambda \frac{{\partial g}}{{\partial R_j }} \leqslant 0,\forall j$$
(73)
Dividing by
\(\frac{{\partial I_j }}{{\partial R_j }}\) $${{\frac{{\partial \int\limits_{P_j }^\infty {I_j } Q_j \operatorname{d} P_j }}{{\partial R_j }}} \mathord{\left/{\vphantom {{\frac{{\partial \int\limits_{P_j }^\infty {I_j } Q_j \operatorname{d} P_j }}{{\partial R_j }}} {\frac{{\partial I_j }}{{\partial R_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial I_j }}{{\partial R_j }}}} + Q_j P_j + R_j - {{\frac{{\partial C}}{{\partial R_j }}} \mathord{\left/{\vphantom {{\frac{{\partial C}}{{\partial R_j }}} {\frac{{\partial I_j }}{{\partial R_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial I_j }}{{\partial R_j }}}} - \lambda \psi _j Q_j \leqslant 0$$
(74)
Since
\(
\begin{array}{*{20}l}
{{m^{Q}_{j} = {\frac{{\partial C}}
{{\partial R_{j} }}} \mathord{\left/
{\vphantom {{\frac{{\partial C}}
{{\partial R_{j} }}} {\frac{{\partial I_{j} }}
{{\partial R_{j} }}}}} \right.
\kern-\nulldelimiterspace} {\frac{{\partial I_{j} }}
{{\partial R_{j} }}}} \hfill} & {{\omega _{{IQ_{j} }} = {\frac{{Q_{j} }}
{{I_{j} }}\frac{{\partial I_{j} }}
{{\partial P_{j} }}} \mathord{\left/
{\vphantom {{\frac{{Q_{j} }}
{{I_{j} }}\frac{{\partial I_{j} }}
{{\partial P_{j} }}} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}}} \right.
\kern-\nulldelimiterspace} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}} \hfill} \\
\end{array}
\) and letting
\(\Delta CS_{IQ_j }^I = \left( {\frac{{\partial I_j }}{{\partial R_j }}} \right)^{ - 1} \frac{{\partial \int\limits_{P_j }^\infty {I_j } Q_j dP_j }}{{\partial R_j }}\) $$R_j - m_j^I - \lambda \psi _j Q_j + Q_j P_j + \Delta CS_{IQ_j }^I \leqslant 0$$
(75)
From the complementary slackness condition:
$$R_j \left( {R_j - m_j^I - \lambda \psi _j Q_j + Q_j P_j + \Delta CS_{IQ_j }^I } \right) = 0$$
(76)
Since
R j > 0
$$R_j = m_j^I + \lambda \psi _j Q_j - Q_j P_j - \Delta CS_{IQ_j }^I $$
(77)
Appendix 4 Second best pricing The optimization problem is:
$${\text{MAX}}\;W = \sum\limits_i {\int\limits_{P_i }^\infty {I_i Q_i } } \operatorname{d} P_i + \sum\limits_i {\int\limits_{R_i }^\infty {I_i } } \operatorname{d} R_i + \sum\limits_i {I_i P_i Q_i } + \sum\limits_i {I_i R_i } - C$$
(78)
subject to the revenue and capacity constraints:
$$\sum\limits_i {\psi _i I_i Q_i } - N \leqslant 0,\forall i\;\left( {\lambda _1 } \right)$$
(79)
$$C - \sum\limits_i {I_i P_i Q_i } - \sum\limits_i {I_i R_i } \leqslant 0,\forall i\;\left( {\lambda _2 } \right)$$
(80)
First set of optimal prices: Dwell charges Applying the first KKT condition:
$$\frac{{\partial W}}{{\partial P_j }} - \lambda _1 \frac{{\partial g_1 }}{{\partial P_j }} - \lambda _2 \frac{{\partial g_2 }}{{\partial P_j }} \leqslant 0,\forall j$$
(81)
One could obtain, after some manipulation:
$$\begin{aligned}& \left[ {I_j Q_j } \right]_{P_j }^\infty + \frac{\partial }{{\partial P_j }}\int\limits_{R_j }^\infty {I_j \operatorname{d} R_j } + I_j Q_j + I_j P_j \frac{{\partial Q_j }}{{\partial P_j }} + Q_j P_j \frac{{\partial I_j }}{{\partial P_j }} + R_j \frac{{\partial I_j }}{{\partial P_j }} - \frac{{\partial C}}{{\partial P_j }} \\& \quad \quad - \lambda _1 \psi _j \left\{ {I_j \frac{{\partial Q_j }}{{\partial P_j }} + Q_j \frac{{\partial I_j }}{{\partial P_j }}} \right\} - \lambda _2 \left\{ {\frac{{\partial C}}{{\partial P_j }} - I_j Q_j - I_j P_j \frac{{\partial Q_j }}{{\partial P_j }} - P_j Q_j \frac{{\partial I_j }}{{\partial P_j }} - R_j \frac{{\partial I_j }}{{\partial P_j }}} \right\} \leqslant 0 \\ \end{aligned} $$
(82)
Since C is an implicit function of I j and Q j :
\(\frac{{\delta C}}{{\delta P_j }} = \frac{{\delta C}}{{\delta Q_j }}\frac{{\delta Q_j }}{{\delta P_j }} + \frac{{\delta C}}{{\delta I_j }}\frac{{\delta I_j }}{{\delta P_j }}\) and dividing by
\(I_j \frac{{\partial Q_j }}{{\partial P_j }}\) $$\begin{aligned} \quad \frac{1}{{I_j }}\left( {\frac{{\partial Q_j }}{{\partial P_j }}} \right)^{ - 1} \frac{\partial }{{\partial P_j }}\int\limits_{R_j }^\infty {I_j \operatorname{d} R_j } + P_j + {{P_j \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/ {\vphantom {{P_j \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} + {{R_j \frac{1}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/ {\vphantom {{R_j \frac{1}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} \\ \quad - {{\frac{{\partial C}}{{\partial Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}\frac{1}{{I_j }}} \mathord{\left/ {\vphantom {{\frac{{\partial C}}{{\partial Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}\frac{1}{{I_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} - {{\frac{{\partial C}}{{\partial I_j }}\frac{{\partial I_j }}{{\partial P_j }}\frac{1}{{I_j }}} \mathord{\left/ {\vphantom {{\frac{{\partial C}}{{\partial I_j }}\frac{{\partial I_j }}{{\partial P_j }}\frac{1}{{I_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} - {{\lambda _2 \frac{{\partial C}}{{\partial Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}\frac{1}{{I_j }}} \mathord{\left/ {\vphantom {{\lambda _2 \frac{{\partial C}}{{\partial Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}\frac{1}{{I_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} - {{\lambda _2 \frac{{\partial C}}{{\partial I_j }}\frac{{\partial I_j }}{{\partial P_j }}\frac{1}{{I_j }}} \mathord{\left/ {\vphantom {{\lambda _2 \frac{{\partial C}}{{\partial I_j }}\frac{{\partial I_j }}{{\partial P_j }}\frac{1}{{I_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} \\ - \lambda _1 \psi _j - {{\lambda _1 \psi _j \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/ {\vphantom {{\lambda _1 \psi _j \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} + {{\lambda _2 Q_j } \mathord{\left/ {\vphantom {{\lambda _2 Q_j } {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} + \lambda _2 P_j + {{\lambda _2 P_j \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/ {\vphantom {{\lambda _2 P_j \frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} - {{\lambda _2 R_j \frac{1}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/ {\vphantom {{\lambda _2 R_j \frac{1}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}} \leqslant 0 \\ \end{aligned} $$
(83)
Since
\(
\begin{array}{*{20}l}
{{m^{Q}_{j} = {\frac{{\partial C}}
{{\partial P_{j} }}} \mathord{\left/
{\vphantom {{\frac{{\partial C}}
{{\partial P_{j} }}} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}}} \right.
\kern-\nulldelimiterspace} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}} \hfill} & {{m^{I}_{j} = {\frac{{\partial C}}
{{\partial P_{j} }}} \mathord{\left/
{\vphantom {{\frac{{\partial C}}
{{\partial P_{j} }}} {\frac{{\partial I_{j} }}
{{\partial P_{j} }}}}} \right.
\kern-\nulldelimiterspace} {\frac{{\partial I_{j} }}
{{\partial P_{j} }}}} \hfill} & {{\omega _{{IQ_{j} }} = {\frac{{Q_{j} }}
{{I_{j} }}\frac{{\partial I_{j} }}
{{\partial P_{j} }}} \mathord{\left/
{\vphantom {{\frac{{Q_{j} }}
{{I_{j} }}\frac{{\partial I_{j} }}
{{\partial P_{j} }}} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}}} \right.
\kern-\nulldelimiterspace} {\frac{{\partial Q_{j} }}
{{\partial P_{j} }}}} \hfill} \\
\end{array}
\) and
\(\Delta CS_{I_j }^Q = \left( {\frac{{\partial Q_j }}{{\partial P_j }}} \right)^{ - 1} \frac{{\partial \int\limits_{R_j }^\infty {I_j dR_j } }}{{\partial P_j }}\) :
$$\begin{aligned}& \frac{{\Delta CS_{I_j }^Q }}{{I_j }} + P_j + P_j \omega _{IQ_j } + \frac{{R_j }}{{Q_j }}\omega _{IQ_j } - \frac{{m_j^Q }}{{I_j }} - \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } - \lambda _2 \frac{{m_j^Q }}{{I_j }} - \lambda _2 \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } \\& \quad - \lambda _1 \psi _j \omega _{IQ_j } - \lambda _1 \psi _j - \lambda _2 P_j \frac{1}{{\left| {\eta _{Q_j }^P } \right|}} + \lambda _2 P_j + \lambda _2 P_j \omega _{IQ_j } + \lambda _2 \frac{{R_j }}{{Q_j }}\omega _{IQ_j } \leqslant 0 \\ \end{aligned} $$
(84)
$$\left( {1 + \lambda _2 } \right)\left( {P_j - \frac{{m_j^Q }}{{I_j }} - \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } } \right) + \left( {1 + \lambda _2 } \right)P_j \omega _{IQ_j } + \left( {1 + \lambda _2 } \right)\frac{{R_j }}{{Q_j }}\omega _{IQ_j } - \lambda _2 P_j \frac{1}{{\left| {\eta _{Q_j }^P } \right|}} - \left( {1 + \omega _{IQ_j } } \right)\lambda _1 \psi _j + \frac{{\Delta CS_{I_j }^Q }}{{I_j }} \leqslant 0$$
(85)
Finally, from the complementary slackness condition:
$$\frac{{P_j - \frac{{m_j^Q }}{{I_j }} - \frac{{m_j^I }}{{Q_j }}\omega _{IQ_j } }}{{P_j }} = \frac{{{\raise0.7ex\hbox{${\lambda _2 }$} \!\mathord{\left/{\vphantom {{\lambda _2 } {\left( {1 + \lambda _2 } \right)}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\left( {1 + \lambda _2 } \right)}$}}}}{{\left| {\eta _{Q_j }^P } \right|}} + \frac{{{\raise0.7ex\hbox{${\lambda _1 }$} \!\mathord{\left/{\vphantom {{\lambda _1 } {\left( {1 + \lambda _2 } \right)}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\left( {1 + \lambda _2 } \right)}$}}}}{{P_j }}\psi _j \left( {1 + \omega _{IQ_j } } \right) - \omega _{IQ_j } - \frac{{R_j }}{{P_j }}\frac{{\omega _{IQ_j } }}{{Q_j }} - \frac{{\Delta CS_{I_j }^Q }}{{I_j P_j \left( {1 + \lambda _2 } \right)}}$$
(86)
Second set of optimal prices: Entrance fees Applying the first KKT condition:
$$\frac{{\partial W}}{{\partial R_j }} - \lambda _1 \frac{{\partial g_1 }}{{\partial R_j }} - \lambda _2 \frac{{\partial g_2 }}{{\partial R_j }} \leqslant 0,\forall j$$
(87)
After some manipulation:
$$\begin{aligned}& \frac{\partial }{{\partial R_j }}\int\limits_{P_j }^\infty {I_j Q_j \operatorname{d} P_j } + \left. {I_j } \right|_{R_j }^\infty + Q_j P_j \frac{{\partial I_j }}{{\partial R_j }} + I_j + R_j \frac{{\partial I_j }}{{\partial R_j }} - \frac{{\partial C}}{{\partial R_j }} - \lambda _1 \psi _j Q_j \frac{{\partial I_j }}{{\partial R_j }} \\& \quad \quad \quad \quad \quad \quad \quad \quad \quad - \lambda _2 \left\{ {\frac{{\partial C}}{{\partial R_j }} - P_j Q_j \frac{{\partial I_j }}{{\partial R_j }} - I_j - R_j \frac{{\partial I_j }}{{\partial R_j }}} \right\} \leqslant 0 \\ \end{aligned} $$
(88)
Dividing by
\(\frac{{\partial I_j }}{{\partial R_j }}\) :
$$\begin{aligned}& \left( {\frac{{\partial I_j }}{{\partial R_j }}} \right)^{ - 1} \frac{\partial }{{\partial R_j }}\int\limits_{P_j }^\infty {I_j Q_j \operatorname{d} P_j } + Q_j P_j + R_j - {{\frac{{\partial C}}{{\partial R_j }}} \mathord{\left/{\vphantom {{\frac{{\partial C}}{{\partial R_j }}} {\frac{{\partial I_j }}{{\partial R_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial I_j }}{{\partial R_j }}}} - \lambda _1 \psi _j Q_j \\& \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \lambda _2 \left\{ {\frac{{\partial C}}{{\partial I_j }} - P_j Q_j - {{I_j } \mathord{\left/{\vphantom {{I_j } {\frac{{\partial I_j }}{{\partial R_j }}}}} \right.\kern-\nulldelimiterspace} {\frac{{\partial I_j }}{{\partial R_j }}}} - R_j } \right\} \leqslant 0 \\ \end{aligned} $$
(89)
Since
\(\omega _{IQ_j } = {{\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} \mathord{\left/ {\vphantom {{\frac{{Q_j }}{{I_j }}\frac{{\partial I_j }}{{\partial P_j }}} {\frac{{\partial Q_j }}{{\partial P_j }}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial Q_j }}{{\partial P_j }}}}\) ,
\(m_j^I = \frac{{\partial C}}{{\partial I_j }}\) ;
\(\eta _{Q_j }^P = \frac{{P_j }}{{Q_j }}\frac{{\partial Q_j }}{{\partial P_j }}\) ; and
\(\Delta CS_{IQ_j }^I = \left( {\frac{{\partial I_j }}{{\partial R_j }}} \right)^{ - 1} \frac{{\partial \int\limits_{P_j }^\infty {I_j Q_j dP_j } }}{{\partial R_j }}\) :
$$\Delta CS_{IQ_j }^I + P_j Q_j + R_j - m_j^I - \lambda _1 \psi _j Q_j - \lambda _2 m_j^I + \lambda _2 P_j Q_j + \lambda _2 \frac{{I_j }}{{R_j }}\frac{{\partial R_j }}{{\partial I_j }}R_j + \lambda _2 R_j \leqslant 0$$
(90)
$$\left( {1 + \lambda _2 } \right)R_j - \left( {1 + \lambda _2 } \right)m_j^I + \left( {1 + \lambda _2 } \right)P_j Q_j - \lambda _1 \psi _j Q_j - \frac{{\lambda _2 }}{{\left| {\eta _{I_j }^R } \right|}}R_j + \Delta CS_{IQ_j }^I \leqslant 0$$
(91)
Therefore:
$$\frac{{R_j - m_j^I }}{{R_j }} = \frac{{{\raise0.7ex\hbox{${\lambda _2 }$} \!\mathord{\left/{\vphantom {{\lambda _2 } {\left( {1 + \lambda _2 } \right)}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\left( {1 + \lambda _2 } \right)}$}}}}{{\left| {\eta _{I_j }^R } \right|}} + \frac{{{\raise0.7ex\hbox{${\lambda _1 }$} \!\mathord{\left/{\vphantom {{\lambda _1 } {\left( {1 + \lambda _2 } \right)}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\left( {1 + \lambda _2 } \right)}$}}}}{{R_j }}\psi _j Q_j - \frac{{P_j }}{{R_j }}Q_j - \frac{{\Delta CS_{IQ_j }^I }}{{\left( {1 + \lambda _2 } \right)R_j }}$$
(92)
Appendix 5: Profit maximizing prices for C = C(IQ) and I(R + PQ) This appendix considers a particular case of the profit maximization problem in which the cost function is of the form
C (
IQ ) =
C (
I 1 Q 1 ,
I 2 Q 2 ....
I K Q K ), and the arrival rates,
I j , are a function of the total expenditure per container (
R j +
P j Q j ). From the complementary slackness conditions shown in Appendix
2 , for both dwell charge and entrance fee:
$$P_j \left( {P_j - P_j \frac{1}{{\left| {\eta _{Q_j }^P } \right|}} + P_j \omega _{IQ_j } + R_j \frac{{\omega _{IQ_j } }}{{Q_j }} - \frac{{m_j^Q }}{{I_j }} - m_j^I \frac{{\omega _{IQ_j } }}{{Q_j }} - \lambda \psi _j \left( {1 + \omega _{IQ_j } } \right)} \right) = 0$$
(93)
$$R_j \left( {P_j Q_j + R_j - \frac{{R_j }}{{\left| {\eta _{I_j }^R } \right|}} - m_j^I - \lambda \psi _j Q_j } \right) = 0$$
(94)
From Eq. (
93 ):
$$P_j - P_j \frac{1}{{\left| {\eta _{Q_j }^P } \right|}} + P_j \omega _{IQ_j } + R_j \frac{{\omega _{IQ_j } }}{{Q_j }} - \frac{{m_j^Q }}{{I_j }} - m_j^I \frac{{\omega _{IQ_j } }}{{Q_j }} - \lambda \psi _j \left( {1 + \omega _{IQ_j } } \right) = 0$$
(95)
$$\lambda \psi _j = \frac{1}{{\left( {1 + \omega _{IQ_j } } \right)}}\left\{ {P_j - P_j \frac{1}{{\left| {\eta _{Q_j }^P } \right|}} + P_j \omega _{IQ_j } + R_j \frac{{\omega _{IQ_j } }}{{Q_j^ * }} - \frac{{m_j^Q }}{{I_j }} - m_j^I \frac{{\omega _{IQ_j } }}{{Q_j }}} \right\}$$
(96)
From Eq. (
94 ):
$$P_j Q_j + R_j - \frac{{R_j }}{{\left| {\eta _{I_j }^R } \right|}} - m_j^I - \lambda \psi _j Q_j = 0$$
(97)
$$\lambda \psi _j = \frac{1}{{Q_j }}\left\{ {P_j Q_j + R_j - \frac{{R_j }}{{\left| {\eta _{I_j }^R } \right|}} - m_j^I } \right\} = 0$$
(98)
Since Eq. (
96 ) = Eq. (
98 )
$$\frac{1}{{\left( {1 + \omega _{IQ_j } } \right)}}\left\{ {P_j - P_j \frac{1}{{\left| {\eta _{Q_j }^P } \right|}} + P_j \omega _{IQ_j } + R_j \frac{{\omega _{IQ_j } }}{{Q_j }} - \frac{{m_j^Q }}{{I_j }} - m_j^I \frac{{\omega _{IQ_j } }}{{Q_j }}} \right\} = \frac{1}{{Q_j }}\left\{ {P_j Q_j + R_j - \frac{{R_j }}{{\left| {\eta _{I_j }^R } \right|}} - m_j^I } \right\}$$
(99)
which translates into:
$$\frac{{ - P_j }}{{\left| {\eta _{Q_j }^P } \right|}} + \frac{{R_j }}{{Q_j }}\left( {\frac{{\left( {1 + \omega _{IQ_j } } \right)}}{{\left| {\eta _{I_j }^R } \right|}} - 1} \right) - \frac{{m_j^Q }}{{I_j }} + \frac{{m_j^I }}{{Q_j }} = 0$$
(100)
$$\frac{{R_j }}{{Q_j }}\left( {\frac{{\left( {1 + \omega _{IQ_j } } \right)}}{{\left| {\eta _{I_j }^R } \right|}} - 1} \right) + \frac{{m_j^I }}{{Q_j }} = \frac{{P_j }}{{\left| {\eta _{Q_j }^P } \right|}} + \frac{{m_j^Q }}{{I_j }}$$
(101)
Since for the cost function of the form
C (
IQ ) =
C (
I 1 Q 1 ,
I 2 Q 2 ....
I K Q K ):
$$m_j = \frac{{\partial C}}{{\partial \left( {I_j Q_j } \right)}} = \frac{{m_j^Q }}{{I_j }} = \frac{{m_j^I }}{{Q_j }}$$
(102)
Equation (
101 ) reduces to:
$$\frac{{R_j }}{{Q_j }}\left( {\frac{{\left( {1 + \omega _{IQ_j } } \right)}}{{\left| {\eta _{I_j }^R } \right|}} - 1} \right) = \frac{{P_j }}{{\left| {\eta _{Q_j }^P } \right|}}$$
(103)
Assuming that
I j is a function of the total cost
T j =
R j +
P j Q j then:
$$\frac{{\partial T_j }}{{\partial P_j }} = Q_j + P_j \frac{{\partial Q_j }}{{\partial P_j }}$$
(104)
$$\frac{{\partial I_j }}{{\partial R_j }} = \frac{{\partial I_j }}{{\partial T_j }}\frac{{\partial T_j }}{{\partial R_j }} = \frac{{\partial I_j }}{{\partial T_j }}$$
(105)
Then:
$$\frac{{\partial I_j }}{{\partial P_j }} = \frac{{\partial I_j }}{{\partial T_j }}\frac{{\partial T_j }}{{\partial P_j }} = \frac{{\partial I_j }}{{\partial R_j }}\left( {Q_j + P_j \frac{{\partial Q_j }}{{\partial P_j }}} \right),\;{\text{which}}\;{\text{implies}}\;{\text{that}}:$$
(106)
$$\eta _{I_j }^P = \frac{{\partial I_j }}{{\partial P_j }}\frac{{P_j }}{{I_j }} = \frac{{\partial I_j }}{{\partial R_j }}\frac{{P_j }}{{I_j }}\frac{{R_j }}{{R_j }}\left( {Q_j + P_j \frac{{\partial Q_j }}{{\partial P_j }}} \right) = \eta _{I_j }^R \frac{{P_j Q_j }}{{R_j }}\left( {1 + \eta _{Q_j }^P } \right)$$
(107)
As a result:
$$\omega _{IQ_j } = \frac{{\eta _{I_j }^P }}{{\eta _{Q_j }^P }} = \eta _{I_j }^R \frac{{P_j Q_j }}{{R_j }}\left( {\frac{1}{{\eta _{Q_j }^P }} + 1} \right)$$
(108)
and
$$\frac{{\omega _{IQ_j } }}{{\eta _{I_j }^R }} = \frac{{P_j Q_j }}{{R_j }}\left( {\frac{1}{{\eta _{Q_j }^P }} + 1} \right)$$
(109)
Replacing the elasticities for the absolute values
$$\frac{{\omega _{IQ_j } }}{{\left| {\eta _{I_j }^R } \right|}} = - \frac{{P_j Q_j }}{{R_j }}\left( {1 - \frac{1}{{\left| {\eta _{Q_j }^P } \right|}}} \right)$$
(110)
Since from Eq. (
107 ):
$$\frac{{R_j }}{{Q_j }}\left( {\frac{1}{{\left| {\eta _{I_j }^R } \right|}} + \frac{{\omega _{IQ_j } }}{{\left| {\eta _{I_j }^R } \right|}} - 1} \right) = \frac{{R_j }}{{Q_j }}\left( {\frac{1}{{\left| {\eta _{I_j }^R } \right|}} - \frac{{P_j Q_j }}{{R_j }}\left( {1 - \frac{1}{{\left| {\eta _{Q_j }^P } \right|}}} \right) - 1} \right) = \frac{{P_j }}{{\left| {\eta _{Q_j }^P } \right|}}$$
(111)
$$\frac{{R_j }}{{Q_j }}\frac{1}{{\left| {\eta _{I_j }^R } \right|}} + \frac{{P_j }}{{\left| {\eta _{Q_j }^P } \right|}} - P_j - \frac{{R_j }}{{Q_j }} = \frac{{P_j }}{{\left| {\eta _{Q_j }^P } \right|}}$$
(112)
Finally:
$$\frac{{R_j^ * }}{{\left| {\eta _{I_j }^R } \right|}} = R_j^ * + P_j^ * Q_j^ * $$
(113)
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Authors and Affiliations 1. Rensselaer Polytechnic Institute Troy USA 2. Universidad de Chile Santiago Chile