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 (0, 6). On the graph, illustrate a situation in which the monopolist serves both individuals. In other words, sketch the marginal cost curve such that the monopolist maximizes profits by serving both individuals.
To illustrate the situation where the monopolist serves both individuals and maximizes profits based on the given information, we need to sketch the marginal cost curve.
The monopolist's total cost of production is given by TC(q) = cq, where c is a constant in the range (0,6). To find the marginal cost (MC) curve, we need to calculate the derivative of the total cost function with respect to quantity (q).
MC(q) = d/dq (TC(q)) = d/dq (cq) = c
Since the marginal cost is a constant value given by c, the MC curve will be a horizontal line on the graph.
Here is a sketch of the graph:
^
| MC
|
|---------------------->
|
The x-axis represents the quantity (q) and the y-axis represents the price (p). The MC curve is a horizontal line, indicating that the marginal cost is constant.
To determine the monopolist's profit-maximizing quantity and price, you need to consider the individual inverse demand curves. Since we have two individuals with different inverse demand functions, we need to find the intersection point of these inverse demand curves.
Setting the two inverse demand functions equal to each other:
6 - 2Qd = 3 - Qd
Solving for Qd:
2Qd = 3
Qd = 1.5
Substituting Qd back into the inverse demand function:
pd = 3 - Qd
pd = 3 - 1.5
pd = 1.5
So, at the profit-maximizing quantity of 1.5, the price will be 1.5.
Therefore, to serve both individuals and maximize profits, the monopolist would produce a quantity of 1.5 and charge a price of 1.5 per unit.