Proposition 7 — equilibrium information sharing under concurrent and sequential contracting (production economy)

For every $\phi > 0$ there are thresholds $c_e^C, c_e^S \in (1/(1+\phi), 2/(1+\phi))$, depending only on $\phi$, such that the following holds. Let $0 < c_e < 2/(1+\phi)$, put $c = -c_e$, and let $a, b$ be reals. Let $(\theta, Y)$ satisfy the signal model with variance $\sigma^2$ and weight $\beta = \beta(t,\sigma)$, let $E$ be any family of pricing equilibria with payoff table $P$, and let $n_e^Z$ be the number of informed manufacturers under concurrent ($Z = C$) or sequential ($Z = S$, for either first mover) information contracting. Then

in the two-clause form: on each region the stated value of $n$ is attained by some outcome, and on the interior of the region it is the only value attained (at a boundary point the retailer is indifferent between outcomes). Precisely, with $O$ the set of values of $n$ over all concurrent outcomes (resp. all SPEs with first mover $k$): $O = \{0\}$ for $c_e < 1/(1+\phi)$; $2 \in O$ on $[1/(1+\phi), c_e^Z)$ and $O = \{2\}$ on its interior; $1 \in O$ on $[c_e^Z, 2/(1+\phi))$ and $O = \{1\}$ on its interior.

Under production economy the retailer shares information exactly when the economy is large relative to competition, whichever contracting sequence is used.

Formalization Note The production economy model takes $c = -c_e$ with $0 < c_e < 2/(1+\phi)$ (the Assumption of p. 251). The ex ante profits are those of an equilibrium of the pricing game on the signal model, not the closed forms: the closed forms are the content of the §4.2 milestones. The production cost is the uncapped quadratic $bq - c_e q^2$ (the paper's cap at $\bar q = b/(2c_e)$ is assumed never to bind, footnote 11, p. 251). At $c_e = 1/(1+\phi)$ every profit is independent of $n$, so all three values are attained there. Correction of the paper. The single value $c_e^* = \frac{2+3\phi}{(1+2\phi)(1+\phi)}$ (which lies in $(1/(1+\phi), 2/(1+\phi))$) is excluded: there the slope $\phi(1+(1+\phi)c)/[(1+\phi)(2+(1+\phi)c)]$ of the best response (2) equals $-1$, so the pricing stage has a continuum of equilibria with different ex ante profits, and Lemma 1's uniqueness claim, on which the payoff table rests, fails.

Retired. Retired because the concurrent-outcome concept of Def_InfoSharing_Economy_IsConcurrentOutcome (supremum over Pareto-optimal equilibria) is not attained where π_M^I(1) − π_M(0) = π_M^I(2) − π_M^U(1), which refutes the closed threshold clauses (accepted disproof); replaced by InfoSharing.Economy.proposition_7_v2 on the corrected module _v2.