Non-price spatial competition on retail market
Specificity of retail market formation in an arbitrary agglomeration in terms of optimal sizes and location retailers would choose. Social welfare analysis of a centralized planning of the retail market. The model of duopoly in the retail market.
| Рубрика | Маркетинг, реклама и торговля |
| Вид | дипломная работа |
| Язык | английский |
| Дата добавления | 28.11.2019 |
| Размер файла | 1,1 M |
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while (I.2) and (I.4) give the following equation:
Joining them, we derive:
that is the “matching strategies condition”.
Then, combining conditions (I.2) and (I.4) again, we get another one, and both might generate the following system:
Substituting in the second expression by we, after some trivial manipulations, gain the set:
According to logic, provided in the paper's body, it completes the analysis of the current case.
C.2:
We can realize manipulations with the expressions above, similar to the previous related to the asymmetric case. Precisely, a link of conditions (I.5) and (I.7) and a link of conditions (I.6) and (I.8) respectively lead to a couple of equations, which we can join into the system:
In turn, it provides a key expression:
that illustrates the matching of retailers' sizes. Indeed, as we naturally suppose as a condition to agents' location symmetry, it pushes the right part of the equation above to zero and, therefore, there is a necessity to take a nominator of the left one being zero, too:
Then, a system, satisfying the first retailer's optimal strategies, is derived in the research's body. Below I provide some ordinary computations up to the very solution for this system:
The second expression straightforwardly gives a solution for goods' variety variable,
Division of the first expression by the second one might simplify calculations for the optimal location,
Here we get a quadratic function by argument of which should be explored further by traditional discriminant analysis:
Keeping in mind an interval for a parameter of town's centricity, which is derived in the main paper's section, the discriminant's positivity becomes obvious for any set of values, included there. It leads to the following roots:
Notice that the term turns out to be impermissibly large on the range of acceptable values, being likely to make a root negative, that is forbidden by our model's framework. Therefore, albeit not strict, we take the other root fair and uniquely optimal.
At this point, calculations, related to the derivation of a symmetric equilibrium and missed in the paper's body, are run off.
Appendix D
The Appendix illustrates a solution for socially optimal variables determining the specificity of the retail market. Practically, below there are some lines going from the optimization problem expression, stated in the third section, to the Pareto-efficient outcome.
The latter straightforwardly gives the results:
As a continuum of consumers is distributed symmetrically, the other optimization problem, related to the right side of an abstract town, would reveal the following values:
Then, we have to derive the condition for matching of equilibrium and socially optimal locations, as follows:
Calculations, related to a condition of the matching in the other variable, appear to be too simple and, therefore, are directly presented in section 3.
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