EX-7

Bertrand competition with asymmetric costs. Demand $P = 100 - Q$. Firm 1: $MC_1 = 10$. Firm 2: $MC_2 = 80$. Tie-breaking: if $p_1 = p_2$, all consumers buy from Firm 1.

What is the maximum price Firm 1 can charge while still excluding Firm 2 from the market?

Is $p_1 = 80$ profit-maximising for Firm 1?

Now $MC_2$ drops to 50 (with $MC_1 = 10$ unchanged). What is the Nash Equilibrium?

With $MC_2 = 50$, suppose Firm 1 must pay a fixed entry cost $F$ to enter the market. What is the maximum $F$ for which Firm 1 still enters?

Bertrand with two cost regimes. Demand $Q = 10 - P$. Firm 1: $C_1(q_1) = 6 q_1$ ($MC_1 = 6$). Firm 2: $C_2(q_2) = c q_2$ ($MC_2 = c$). Tie-breaking: if $p_1 = p_2$, all demand goes to Firm 2. Neither firm prices below its own MC.

Find the Nash Equilibrium when $2 < c < 6$.

Same setup as 2a but now $c < 2$. Find the Nash Equilibrium.

Monopolistic competition (differentiated Bertrand). Two firms with demand curves $q_1 = 168 - 2 P_1 + P_2$ and $q_2 = 168 - 2 P_2 + P_1$. Zero production costs. Firms set prices simultaneously.

Find the Nash Equilibrium prices, quantities, and profits.

Would a merger of the two firms (creating a multi-product monopoly) benefit both? Show the price, quantity, and profit comparison.

Price competition with complementary goods (the road problem). Two firms each charge a toll for a segment of a road. Demand for completing the trip: $Q = 10 - P$ where $P = p_1 + p_2$. Zero costs. Firms set tolls simultaneously.

Find the Nash Equilibrium tolls, total price, quantity, and profits.

Same road setup as 4a, but Firm 1 owns 75% of the road and Firm 2 owns 25%. Does the ownership split change the Nash Equilibrium?

Same road setup, but now $N$ firms each own $1/N$ of the road. Derive the symmetric Nash Equilibrium and discuss what happens as $N \to \infty$.