Lemma 1 — the unique pricing equilibrium is linear in the demand signal (production economy)

Let $\phi > 0$ and $0 < c_e < 2/(1+\phi)$, put $c = -c_e$, and let $a, b$ be reals. Let $(\theta, Y)$ satisfy the signal model with variance $\sigma^2$ and weight $\beta = \beta(t,\sigma)$. For every status profile $X$ the pricing game of §4.1 has an equilibrium, and, provided $c_e \ne c_e^*$ (see the note), in every equilibrium $(\rho, f)$, for each manufacturer $i$, almost surely

where $\bar w = \frac{(1+(1+\phi)c)a + (1+\phi)b}{2+\phi+(1+\phi)c}$, $\bar p = \frac{(3+\phi+2(1+\phi)c)a + (1+\phi)b}{2(2+\phi+(1+\phi)c)}$, and $\alpha_w$, $\alpha_p$ are the coefficients of Lemma 1 selected by $n$ and, when $n = 1$, by $X_i$:

1. $\alpha_w(0) = \alpha_w^U(1) = 0$, $\alpha_w^I(1) = \frac{1+(1+\phi)c}{(1+\phi)(2+(1+\phi)c)}\beta$, $\alpha_w(2) = \frac{1+(1+\phi)c}{2+\phi+(1+\phi)c}\beta$;
2. $\alpha_p(0) = \alpha_p^U(1) = \frac{\beta}{2}$, $\alpha_p^I(1) = \frac{3+2\phi+(1+\phi)(2+\phi)c}{2(1+\phi)(2+(1+\phi)c)}\beta$, $\alpha_p(2) = \frac{3+\phi+2(1+\phi)c}{2(2+\phi+(1+\phi)c)}\beta$.

An informed manufacturer's wholesale price responds to the signal, positively when $c_e < 1/(1+\phi)$ and negatively when $c_e > 1/(1+\phi)$; an uninformed one's does not. All ex ante profits of §4.2 follow from this equilibrium.

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). Correction of the paper. Uniqueness is asserted only for $c_e \ne$ $c_e^* = \frac{2+3\phi}{(1+2\phi)(1+\phi)}$. At $c_e^*$ the slope $\phi(1+(1+\phi)c)/[(1+\phi)(2+(1+\phi)c)]$ of the best response (2) equals $-1$, the linear system $w_i = \hat w_i(w_j)$ is singular, and every profile has a continuum of equilibria (for instance $w_{1,2} = \bar w \pm t$ when $n = 0$) with different profits; the proof of Lemma 1 (p. 260) overlooks this. Existence holds for every $c_e$.