Question

1. Consider a market with inverse demand P (Q) = 100 Q and two firms with cost function C(q) = 20q.

(A) Find the Stackelberg equilibrium outputs, price and total profits (with firm 1 as the leader).

(B) Compare total profits, consumer surplus and social welfare under Stackelberg and Cournot (just say which is bigger).

(C) Are the comparisons intuitively expected?

2. Consider the infinite repetition of the n-firm Bertrand game. Find the set of discount factors for which full collusion (i.e. monopoly pricing) can be sustained by reversion to the one-shot Bertrand equilibrium as a threat.

Answer #1

**1.**

**SOLUTION :-**

**Consider a market with inverse demand P(Q) = 100 - Q and
two firms with cost function C(q) = 20q**

**a) Q=Q1+Q2**

**P(Q)=100-(Q1+Q2)**

**TR1=P(Q)Q1=100Q1-Q1 ^{2}-Q1Q2**

**Or, MR1=100-2Q1-Q2**

**C(q)=20q**

**MC=20**

**Applying MR1=MC we get,**

**100-2Q1-Q2=20**

**Or, 2Q1+Q2=80--------------(1)**

**TR2=P(Q)Q2= 100Q2-Q1Q2+Q2 ^{2}**

**MR2=100-Q1-2Q2**

**MR2=MC we get,**

**100-Q1-2Q2=20**

**Or, Q1+2Q2=80-------------(2)**

**Multipling (1) by 2 we get,**

**4Q1+2Q2=160**

**Q1+2Q2=80**

**3Q1=80 Or, Q1=80/3=26.67**

**Q2=80-26.67/2=26.67**

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