5. Differentiated Duopoly with Asymmetric Information There are
two firms 1 and 2 that sell different products, and engage in Bertrand
competition. The demand function of each firm is given by:
$\Pi_i(p_i, p_j) = a - bp_i + dp_j$
where $0 < d < b$. The cost function of firm 2 is $C_2(q) = c_2q$; the cost
function of firm 1 $C_1(q) = c_1q$ is known to 1, but not to 2. 2 knows that
$c_1$ is either $c_1^L$ or $c_1^H$, and prescribes probability $x$ to the case $c_1 = c_1^L$ and
$(1 - x)$ to the case $c_1 = c_1^H$. Let $c_1^L < c_1^H$.
(a) Fix 2's action to $p_2$. Solve for 1's profit maximizing price if it is the
low type, call this $p_1^L$. Do the same for $p_1^H$.
(b) Let $p_1^e = xp_1^L + (1 - x)p_1^H$ be 1's expected action. Write down firm
2's expected profit function in terms of $p_2$, $x$, $p_1^L$, and $p_1^H$ (and the
other parameters $a$, $b$, and $d$). Simplify this so that it is in terms of
$p_2$ and $p_1^e$.
(c) Solve for 2's profit maximizing price in terms of $p_1^e$ (and the other
parameters).
(d) Interpret the fact that 2 best responds to $p_1^e$.
(e) When 1 is the high type, the outcome is $(p_1^H, p_2^*)$, and 1's profit is
$\Pi_1(p_1^H, p_2^*) = (p_1^H - c_1^H)(a - bp_1^H + dp_2^*)$. Show that if 1 reveals to 2
that $c_1 = c_1^H$, then 1's profit increases.
