Capacity competition (Stackelberg-Spence-Dixit-Model)

1. Capacity competition (Stackelberg-Spence-Dixit-Model)

2. How does a firm become a first mover? • Innovation takes time. • Official licensing procedures

3. Strategic effects of sunk costs • Costs are rearranged from the stage of production to the investment period. • Capacity costs of first period are sunk in the second. • Sunk costs provide a precommitment possibility for firm 1.

4. Assumptions • Equal marginal costs: • Equal marginal capacity costs: • Firm 1 (time and cost leader) builds the capacity in stage 1. • Firm 2 builds its capacity in stage 2. • Both firms decide on production in stage 2.

5. Capacity and production costs • Cost functions: • Marginal costs:

6. Depicting the marginal costs firm 1 capacity leader firm 2 capacity follower

7. Simultaneous quantity competition (stage 2) • Firm 1’s profit function: • Firm 2’s profit function:

8. Leader’s reaction function with different marginal costs

9. Equilibria with different marginal costs

10. Leader’s reaction function with different capacities

11. Depicting the equilibria

12. The case of capacity shortage( ) • Marginal costs: • Firm 1’s first order condition: • Firm 1’s reaction function: • Equilibrium:

13. The case of overcapacity ( ) • Marginal costs: • Firm 1’s first order condition: • Firm 1’s reaction function: • Equilibrium:

14. Output level = capacity ( ) • Marginal costs: • Condition for : • Firm 1’s reaction function : • Equilibrium:

15. Entry Deterrence • Entry blockaded for each firm: • Entry blockaded for the follower: • Does deterrence pay?

16. Summarizing the equilibria • Equilibria: • Will the leader ever want to build a capacity above

17. The Cournot solution • Leader builds capacity • Cournot solution: • Is this solution an equilibrium in a capacity competition?

18. The Cournot solution is never optimal for firm 1 Cournot solution Stackelberg solution

19. The optimal capacity obeys ... By increasing the output level toward the Stackelberg output the leader realizes higher profits. Otherwise the same output level could be realized with a lower capacity.

20. Finding the optimal capacity • Firm 1’s reduced profit function: direct effect strategic effects 0 ? >0

21. The Stackelberg solution I • The Stackelberg output level is below the highest possible output: • This is equivalent to: • Quantities: • Hence, the leader can profit from high capacity costs.

22. Depicting the Stackelberg solution Cournot solution Stackelberg solution

23. The Stackelberg solution II • Leader’s optimal capacity: • Total output: • Price: • Leader’s profit: • Follower’s profit:

24. The small Stackelberg solution I • The Stackelberg output level is above the highest possible output: • This is equivalent to: • Output levels:

25. Depicting the small Stackelberg solution Cournot solution small Stackelberg solution Stackelberg solution

26. The small Stackelberg solution II • Leader’s optimal capacity: • Total output: • Price: • Leader’s profit: • Follower’s profit:

27. Deterrence versus optimal capacity Entry cannot be blockaded Entry can be blockaded for sufficiently high capacity r 0 small Stackelberg solution Stackelberg solution no supply

28. Summary • Capacity costs and capacity leadership allow commitment on high volume of output. • The output level of a Stackelberg leader can be obtained with sufficiently high capacity costs. • Even with low capacity costs the small Stackelberg solution can be realized. • Deterrence does not pay.

29. Exercise (capacity competition) • Inverse demand function: • Marginal costs: • Marginal capacity costs: • Calculate the output levels of each game. • Solution: