Economic Equation With D-link

p_{st} = \alpha _s + \sum\limits_{j = 0}^{J + 1} {\beta _{js} c_{t - j} } + \sum\limits_{k = 1}^{K + 1} {\gamma _{ks} p_{s,t - k} } + u_{st}

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E(p_{st} |\left\{ {c_t } \right\} = \overline c ) = \alpha _s + \overline c \sum\limits_{j = 0}^{J + 1} {\beta _{js} } + \sum\limits_{k = 1}^{K + 1} {\gamma _{ks} E(p_{s,t - k} |\left\{ {c_t } \right\} = \overline c )}

= \alpha _s + \sum\limits_{j = 0}^{J + 1} {\beta _{js} + E(p_{s,t} |\left\{ {c_t } \right\} = \overline c )} \sum\limits_{k = 1}^{K + 1} {\gamma _{ks} }

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E(p_{st} |\left\{ {c_t } \right\} = \overline c ) = \frac{{\alpha _s }}{{1 - \sum\limits_{k = 1}^{K + 1} {\gamma _{ks} } }} + \frac{{\sum\limits_{j = 0}^{J + 1} {\beta _{js} } }}{{1 - \sum\limits_{k = 1}^{K + 1} {\gamma _{ks} } }}\overline c

= \widetilde{\alpha _s } + \theta _s \overline c .

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Delta p_{st} = \sum\limits_{j = 0}^J {\widetilde{\beta _{sj} }} \Delta c_{t - j} + \sum\limits_{k = 1}^K {\widetilde{\gamma _{sk} }} \Delta p_{,t - k} + \lambda _s \left( {p_{,t - 1} - \widetilde{\alpha _s } - \theta _s c_{t - 1} } \right) + u_{st}

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\Delta p_{st} = \sum\limits_{j = 0}^J {\Delta c_{t - j} \left( {\widetilde\beta _{sj}^ - + \widetilde\beta _{sj}^ + {\rm I}{\rm I}\left( {\Delta c_{t - j} > 0} \right)} \right)} + \sum\limits_{k = 1}^K {\Delta p_{s,t - k} \left( {\widetilde\gamma _{sk}^ - + \widetilde\gamma _{sk}^ + {\rm I}{\rm I}\left( {\Delta p_{s,t - k} > 0} \right)} \right)} + \lambda _s (p_{s,t - 1} - \widetilde{\alpha _s } - \theta _s c_{t - 1} ) + u_{st}

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p_{s,t + 1} + \widetilde\beta _{s1}^ - + \Delta p_{s,t + 1} \left( {\widetilde\gamma _{s1}^ - + \widetilde\gamma _{s1}^ + {\rm I}{\rm I}\left( {\Delta p_{s,t + 1} > 0} \right)} \right) + \lambda _s \left( {p_{s,t + 1} - \theta } \right)

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CRF_{s,t + f}^ - = p_{s,t + f - 1} + \widetilde\beta _{s,f - 1}^ - + \sum\limits_{}^{} \Delta p_{s,t + f - k} \left( {\widetilde\gamma _{s,k}^ - + \widetilde\gamma _{s,k}^ + {\rm I}{\rm I}\left( {\Delta p_{s,t + f - k} > 0} \right)} \right) + \lambda \left( {p_{s,t + f - 1} - \theta } \right)

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A_{s,t + f} = CRF_{s,t + f}^ + - CRF_{s,t + f}^ -

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\widetilde{\beta _s } = \left( {\widetilde\beta _{0,s}^ - ,\widetilde\beta _{0,s}^ + ...\widetilde\beta _{J,s}^ - ,\widetilde\beta _{J,s}^ + ,\widetilde\gamma _{1,s}^ - ,\widetilde\gamma _{1,s}^ + ,...\widetilde\gamma _{K,s}^ - ,\widetilde\gamma _{K,s}^ + } \right)^\prime

\ind~\N\left( {\left( {I_{J + K} \otimes w_s } \right)\beta ^* ,\sum _\beta } \right)

\lambda _s \ind~\N\left( {w_s \lambda ^* ,\sigma _\lambda ^2 } \right)

\theta _s\ind~\N\left( {w_s \theta ^* ,\sigma _\lambda ^2 } \right)

\tilde \alpha = \left( {\tilde \alpha _1 ,...,\tilde \alpha _S } \right)^\prime \sim N\left( {W\alpha ^* ,A} \right),

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\begin{array}{l}
A_{ij} = \left\{ {\begin{array}{*{20}c} {\sigma _\alpha ^2 \exp \left( { - \varphi _\alpha d_{ij} } \right)} \\ {\tau _\alpha ^2 + \sigma _\alpha ^2 } \\ \end{array}} \right.\begin{array}{*{20}c} {{\rm{ if }}i \ne j} \\ {{\rm{ if }}i = j}
\end{array}}. \\
\end{array}

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y_{st} = - \frac{{\Delta p_{st} - x_{st} \beta _s - \lambda _s p_{s,t - 1} }}{{\lambda _s }} - \theta _s c_{t - 1}

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The graph shows the relative performance of Retail versus Spot gasoline prices over 2003. It begins with a nearly +$.50 disparity (Retail/Spot) and narrows during the summer months of May-September. The narrowing appears to coincide with the San Francisco refinery outages and Texas-to-Arizona pipeline rupture. Sharp price increases for both Retail and Spot products occur during the months of March and August, with a more mild increase in June.

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This map displays the distribution of gasoline stations throughout South Orange County. High station concentrations are located in and around San Clemente along the Pacific Coast, bordering San Diego County. Other high concentrations are in El Toro and Mission Viejo. There are small concentrations in Laguna Beach and Capistrano Beach.

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This graph shows an approximately $.80 disparity between Average gasoline prices and Spot prices. Recording the months between September and May, it highlights the rise in prices during the month of March of nearly $.50 for both Average and Spot prices, followed by a sharp decrease.

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Figure 4a displays the effects of a cost shock on probability intervals for two cumulative response functions (CRFs)-representing both positive and negative cost shocks. For the first 8 weeks there is an increasing effect on retail prices. The graph finally shows prices plateau toward equilibrium in subsequent weeks. Week 2 is a significant exception, however, and it is where the probability intervals for the two CRFs overlap at the long-run average effect.

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Figure 4b is a positive asymmetry function and it shows a majority of the mass centered above 0, showing the positive effects on price responsiveness following both positive and negative wholesale cost shocks. The sharp decrease in week 2 represents the temporary yet significant hesitance of station owners to respond to cost increases. A negative cost shock elicits the same behavior in week 3.

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Figure 5a shows the predicted marginal pricing effect for just Salary-Operated (company-owned) stations and Figure 5b shows the same function, but after the salary stations' status is changed to that of Lessee-Dealers. For both, a break in pricing response occurs between week 2 and 3 after a negative shock, until finally rising again to a new plateau around $1.375. The time it takes to adjust to equilibrium is a bit more pronounced in (b) than in (a).

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Figure 6a separates the predicted price-response asymmetry by company. The line representing Chevron, Shell and Unocal appear to have the highest response following a cost shock, whereas Mobil, moving to extremes within the first 3 weeks, appears to have been affected the least. Figure 6b simply reduces the brand element of Figure 6a into a more simple Branded/Unbranded approach. The graph shows that for the first 9 weeks, the branded stations had far greater asymmetrical pricing response than the unbranded ones. Eventually, however, differences between the two converged to zero.

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Figure 7a tests for the proximity of rivals and its effects on the predicted asymmetry following a cost shock. It appears that gasoline stations with no nearby rivals have a stronger response to a shock than those with at least one nearby rival (added through the pooling asymmetry process), with a stronger decrease during week 2, followed by stronger subsequent increases. Figure 7b creates the difference effect between the two classes, which shows visually how the difference is greater for stations with no rivals for at least the first 9 weeks, before converging to zero.

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Figure 8 continues the examination of geographical factors with respect to the predicted asymmetrical pricing response of gasoline stations. Figure 8a isolates the difference in mileage from a major clientele source (a freeway) and the resulting graph is an interesting pure rise in pricing response until week 2, before declining slightly and ultimately converging to zero. Figures 8b and 8c show the more usual sharp decline until week 2, followed by an equally sharp rise, before eventual convergence to zero. Figure 8d, which tracks the difference in standard pumps per acre, another metric for spatial differentiation, shows an unusual, almost uniform decrease over the period, with a small plateau between weeks 3 and 4, before eventually converging to zero.

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Figure 9 factors in different proxies for market power and demographics. Unlike many of the previous graphs, there is no initial upward tend in any of the 4 graphs in Figure 9. All have sharp declines until week 2, and all except for 9c continue the decline until week 3. Figure 9c has a sharp recovery in pricing response post-shock; it corroborates the hypothesis that significant geographic differentiation will give a store greater market power. Figure 9d, however, shows a decrease in pricing response relative to stores located next to a demographic with higher marginal incomes, a peculiar result given the expectation that this proximity to higher rents would encourage a greater price response.

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\Delta p_{st} = \tilde \beta _{0s} \Delta c_t + \tilde \beta _{1s} \Delta c_{t - 1} + \tilde \beta _{2s} \Delta c_{t - 2} + \tilde \gamma _{1s} \Delta p_{s,t - 1} + \tilde \gamma _{2s} \Delta p_{s,t - 2} + \lambda _s \left( {p_{s,t - 1} - \tilde \alpha _s - \theta _s c_{t - 1} } \right) + u_{st} ,

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\tilde \beta _{0s} = \beta _{0s} ,\tilde \beta _{1s} = - \beta _{2s} - \beta _{3s} ,\tilde \beta _{2s} = - \beta _{3s} ,\tilde \gamma _{1s} = - \gamma _{2s} - \gamma _{3s} ,\tilde \gamma _{2s} = - \gamma _{3s} ,\lambda _s = \gamma _{1s} + \gamma _{2s} + \gamma _{3s} - 1,\tilde \alpha _s = - \alpha _s /\lambda _s ,

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Updated June 25, 2015

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