Consider a duopoly with homogenous goods where Firm 1 has the following production function: Q1 = F1(L,K) = L1/2 K1/2, where Q and K are measured in units and L in hours. Firm 2 uses labour and capital as well but has a different production function, given by Q2 = F2(L,K) = L1/3 K2/3. You may assume that the market for labour and capital is perfectly competitive and the current wage rate is £40 and the rental rate on capital is £10. Both firms sell their products on the same market with inverse demand function P = 52 – (Q1 + Q2), where P is measured in pound sterling. Which production function(s) exhibit(s) decreasing returns to scale? Suppose Firm 1 wishes to produce 6 units. What is the cost minimising input mix for Firm 1? Suppose Firm 2 wishes to produce 4 units. What is the cost minimising input mix for Firm 2
Consider a duopoly with homogenous goods where Firm 1 has the following production function:
Q1 = F1(L,K) = L1/2 K1/2,
where Q and K are measured in units and L in hours. Firm 2 uses labour and capital as well but has a different production function, given by
Q2 = F2(L,K) = L1/3 K2/3.
You may assume that the market for labour and capital is
P = 52 – (Q1 + Q2),
where P is measured in pound sterling.
- Which production function(s) exhibit(s) decreasing returns to scale?
- Suppose Firm 1 wishes to produce 6 units. What is the cost minimising input mix for Firm 1?
- Suppose Firm 2 wishes to produce 4 units. What is the cost minimising input mix for Firm 2?
Assume both firms now have the option to produce either 4 units or 6 units. We will consider the situation where both firms simultaneously, but independently, choose their strategies. Both firms are aiming to maximise their profits.
- For each player (firm), what are their strategies? Write the game in normal form.
- Find each player’s dominant strategy (if they exists).
- Find all Nash equilibria.
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