Market demand is composed of individuals 1 and 2. Individual inverse demands are π = 6 β 2π π andπ = 3 βπ π. The monopolistβs total cost of production is ππΆ(π) = ππ where
12 π β (0,6). Suppose that π β [2, 6). Find (ππ, ππ). Is anyone excluded?
To find the monopolist's equilibrium price and quantity, we need to determine the market demand by summing the individual demands.
The market demand at price p is equal to the sum of individual demands:
D(p) = D1(p) + D2(p)
D1(p) = 6 - 2Qd1
D2(p) = 3 - Qd2
Substituting the given inverse demand functions into the market demand equation:
D(p) = (6 - 2Qd1) + (3 - Qd2)
D(p) = 9 - 2(Qd1 + Qd2)
Now we need to find the monopolist's quantity by setting the market demand equal to the monopolist's marginal cost:
9 - 2(Qd1 + Qd2) = cq
9 - 2Qm = cq, where Qm represents the monopolist's quantity
Solving for Qm, we get:
Qm = (9 - cq) / 2
The monopolist's equilibrium price is determined by substituting the monopolist's quantity back into the market demand equation:
pM = 9 - 2Qm
pM = 9 - 2((9 - cq) / 2)
pM = 9 - (9 - cq)
pM = cq
Therefore, the monopolist's equilibrium price is equal to cq, and the monopolist's quantity is (9 - cq) / 2.
To determine if anyone is excluded, we need to check if the monopolist's quantity is positive and if the price exceeds the willingness to pay of anyone in the market.
The monopolist's quantity is positive if (9 - cq) / 2 > 0, which simplifies to cq < 9.
The price exceeds the willingness to pay of individual 1 if pM > 6 - 2Qd1.
cq > 6 - 2(6 - cq)
cq > 6 - 12 + 2cq
cq > 6 - 12 + 2cq
0 > 18 - 12 + 2cq
0 > 6 + 2cq
-6 > 2cq
-3 > cq
cq < -3
The price exceeds the willingness to pay of individual 2 if pM > 3 - Qd2.
cq > 3 - (3 - cq)
cq > 3 - 3 + cq
0 > 3 - 3 + cq
0 > 0 + cq
0 > cq
From the above inequalities, we can conclude that for c β [2, 6), the monopolist's equilibrium quantity is positive (9 - cq) / 2 > 0, and no one is excluded as cq < 9.