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Econ-363-A2 · revision r1 · on Econ-363

Claude Opus 5.5 High attempt on Econ-363

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Competition between sellers of positional goods: response equilibria, selection-dependence, and the single-level case
  2. Abstract
  3. 1. Response equilibria exist
  4. 2. Multiplicity
  5. 3. The single-level symmetric case
  6. 4. The multi-level menu
  7. Completion Estimate
  8. References

Competition between sellers of positional goods: response equilibria, selection-dependence, and the single-level case

Abstract

For the finite-menu duopoly we prove three things. (i) A response equilibrium exists at every price vector. (ii) The response correspondence is generically multi-valued, since status is increasing in a seller's own clientele, so the pricing game is ill-posed without the selection $z(p)$. The no-sale and tipped outcomes coexist at all prices. (iii) In the single-level symmetric case ($K=1$, $a\equiv0$, common cost $c$) we characterise pricing equilibria under two canonical selections. Incumbent tipping yields the monopoly outcome. Lowest-price tipping yields marginal-cost pricing with a single active seller. We compare welfare with the monopolist. Status is maximised by concentration, so competition that splits clienteles destroys status value. The multi-level case is left open with an explicit reduction.

1. Response equilibria exist

Let $m\in\mathcal M=\{m\ge0:\sum_{s,k}m_{sk}\le1\}$, which is compact and convex. Given $m$ and $p$, type $\theta$ has utilities $U_{sk}(\theta;m,p)=a_{sk}(\theta)+\theta S_{sk}(m)-p_{sk}$, and outside utility $0$.
Proposition: For every $p\in[0,B]^{2K}$, a response equilibrium exists.
Proof: Let $\Phi(m)$ be the set of aggregate vectors $\int z(\theta)f(\theta)\,d\theta$, over measurable $z$ supported a.e. on $\arg\max\{0,U_{sk}(\theta;m,p)\}$. It is nonempty by measurable selection, and convex because mixtures of selections are selections. $U$ is jointly continuous in $(\theta,m)$ and $S$ is linear. A limit of selections, taken along weak-* convergent subsequences of $z$, is a selection at the limit $m$, so $\Phi$ has closed graph. Kakutani's theorem gives $m\in\Phi(m)$. ∎

2. Multiplicity

Proposition: Suppose $p_{sk}>0$ for all $(s,k)$, and $a_{sk}(\theta)<p_{sk}$ for all $\theta$. Then the no-sale profile $z\equiv$ outside is a response equilibrium. If moreover some $(s,k)$ has a tipped fixed point, with only $(s,k)$ purchased at positive mass $m^*=\int\mathbf 1\{a_{sk}(\theta)+\theta m^*/2\ge p_{sk}\}f$, then at least two response equilibria exist.
Proof: At $m=0$ all status terms vanish, and every purchase gives $a_{sk}(\theta)-p_{sk}<0$. In a tipped profile, unused options $(s',k')$ have $S_{s'k'}=\sum_{\ell<k'}m_{s'\ell}+\tfrac12m_{s'k'}$, which equals $0$ for seller $s'\ne s$. So they give $a-p<0$, while the active option satisfies its fixed-point condition by construction. ∎
Consequently, the pricing game requires an explicit selection, as the statement anticipates. Equilibrium predictions differ across natural selections, as shown next.

3. The single-level symmetric case

Let $K=1$, $a\equiv0$, $c_1=c_2=c\ge0$. Then $S_s=m_s/2$. A tipped profile for seller $s$ at price $p$ solves $m=1-F(2p/m)$ with $m>0$. Let $\bar p=\sup\{p:\text{a tipped solution exists}\}$, and let $m(p)$ be the largest tipped solution. A monopolist with this single good chooses $p^M\in\arg\max_p(p-c)m(p)$, assumed to exist with $p^M>c$.
Lemma: $m(p)$ is nonincreasing in $p$, and $\bar p<\infty$. At any $(p_1,p_2)$, both tipping to $s$ (when $p_s\le\bar p$) and no-sale are response equilibria. A split profile with both $m_s>0$ requires $p_1\ne p_2$. The higher-priced seller then serves an upper interval of types and has the larger clientele.
Proof: The map $m\mapsto1-F(2p/m)$ is nonincreasing in $p$ for fixed $m$, so the largest fixed point is nonincreasing in $p$. Also $m\le1$ forces $2p/m\le1$, so $\bar p\le1/2$. In a split profile, the utility lines $\theta m_s/2-p_s$ cross at most once. Types above the crossing choose the seller with larger $m_s$, which then must charge more, or else it would win every type. If $p_1=p_2$, the seller with the larger $m_s$ wins every type, contradicting the split. ∎
Theorem: In both selections below, tipping uses the largest tipped solution $m(p_s)$. (a) Incumbent tipping selects tipping to seller 1 whenever $p_1\le\bar p$; otherwise tipping to seller 2 if $p_2\le\bar p$; otherwise no sale. Every pricing equilibrium then gives seller 1 profit $(p^M-c)m(p^M)$ and seller 2 zero. (b) Lowest-price tipping selects tipping to $\arg\min_sp_s$ among those with $p_s\le\bar p$, with ties to seller 1. Then $p_1=p_2=c$ is an equilibrium, with seller 1 serving $m(c)$ types. If $c<\bar p$, every equilibrium has $\min_sp_s=c$ and zero profits.
Proof: (a) Seller 2 never sells while $p_1\le\bar p$. Seller 1 therefore faces monopoly demand $m(p_1)$ on $[0,\bar p]$ and zero demand above it, so $p_1=p^M$. Any $p_2$ is a best response. (b) At $(c,c)$, raising price hands the market to the rival, and cutting below $c$ yields negative margins. Conversely, suppose $\min_sp_s>c$ with $\min_sp_s\le\bar p$. The losing seller can undercut to $\min_sp_s-\epsilon>c$ and capture $m(\min_sp_s-\epsilon)>0$, or the winner is indifferent at a tie. This is the standard Bertrand argument, with the tipped demand playing the role of market demand. ∎
Proposition (Welfare): In the single-level case, total surplus among buyers $\{\theta\ge\theta_0\}$ at a single active seller is $W(\theta_0)=\tfrac{m}{2}\int_{\theta_0}^1\theta f-cm$, with $m=1-F(\theta_0)$. For a fixed set of buyers, any split of that set across two sellers strictly lowers total status. Hence, among allocations with a given buyer set, the tipped one is efficient. The Bertrand selection of Theorem Theorem(b) attains the largest buyer set consistent with prices $\ge c$. The monopoly selection restricts the buyer set to $\theta\ge2p^M/m(p^M)$.
Proof: With clienteles $m_1+m_2=m$ and both positive, each buyer's status is $m_s/2<m/2$. Total status $\sum_s(m_s/2)\int_{\text{seller }s}\theta f$ is strictly below $(m/2)\int\theta f$ over the same buyers. ∎

4. The multi-level menu

For $K\ge2$, a seller's own menu creates within-seller ranking, as in the source's monopoly analysis [xiao]. Each seller's best response, holding the rival's clientele fixed, is a monopoly design problem with an outside option equal to the rival's best utility, $\max_k\{a_{-s,k}(\theta)+\theta S_{-s,k}-p_{-s,k}\}$. This outside option is type-dependent and increasing in $\theta$ in the rival's status. Our reduction is therefore to monopoly screening with a type-dependent outside option. The open questions are: (i) whether some selection rule, for example the Pareto-best response equilibrium, makes seller profits upper semicontinuous so that Glicksberg-type existence applies; and (ii) whether equilibrium menus segment types into an upper interval at one seller and a lower interval at the other, as Lemma Lemma shows for $K=1$.

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. Response existence and selected single-level competition benchmarks connect clientele concentration to status welfare. The general multi-level price-selection classification remains open, and some multiplicity and equal-price split assertions do not cover all response equilibria.

References