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tl;dr: Prices are identical because firms are identical. Profits of firm $i$ depend on own price $p_i$ and on neighbors' prices $p_{i-1}$ and $p_{i+1}$. Prices are set simultaneously, therefore competition in this market is described by a (normal form) game. A strategy is a price, and your formula for $p_i$ is the best response function, showing firm $i$'s ...

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For a variable $X$, let $dX$ denote its total differential. Let $k$ be a constant, and $X$ and $Y$ variables. You'll need the following rules: $$dk = 0$$ (constant rule), $$d(X + Y) = dX + dY$$ (sum rule), $$d(XY) = Y \cdot dX + X \cdot dY$$ (product rule) and $$d\left(\frac{X}{Y}\right) = \frac{ Y \cdot dX - X \cdot dY }{ Y^2 }$$ (quotient rule). ...

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Here is a link to Schmalansee's classic paper. Use SSRI to look forward in time from this paper. Useful search terms in Econlit or Google Scholar would be "crowding the product space" or "product space congestion."

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In the strict classical Hotelling model, it is impossible to have an equilibrium with 3 agents over the segment (deviation is always profitable). Some modification have been done to keep the transportation costs+inelastic demand with $n$ firms (with multiple segments and a common point, like a flake). By the way, d'Aspremont et al. modified the cost ...

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For an industry, one indicator used by competition authorities is the Herfindahl–Hirschman Index measured as $\displaystyle \sum_{i=1}^N s_i^2$ where there are $N$ firms in the sector and the market share of the $i$th firm is $s_i$. This gives a value of $1$ when there is a single firm and $\frac1N$ when the $N$ firms have equal market shares, but also ...

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Interesting! My guess is that: A) Fast food places are in the business of solving the masses nutricional needs (low margins!), while other restaurants are more in the business of selling a special experience that you could do without (high margins!): Article mentioning lower margin breakfast foods. B) Because of the above, people will go to a fast food ...

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This question makes a hidden assumption: that all diners are on the same circadian rhythm. A significant portion of the US population works nights, spans time zones, etc. For example, truckers. That's one answer, that addresses the market. There's also the question of inventory logistics for the restaurant. Most eateries have to do extensive prep for a ...

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