Consoder a monolist facing a linear demand Q=60-P/3. The monolist products with constant marginal costs c=3 and no fixed costs. Its marginal revenue function is MR=60-6Q
A) plot demand function in a picture together with the marginal revenue curve and marginal cost curve. Indicate on the picture the monolist profit.
B) find the monolist price, the monopolist quantity.
C) compute the consumer surplus and the deadweight
To answer these questions, let's go step by step:
A) To plot the demand function, marginal revenue curve, and marginal cost curve, we need to determine the range of quantity values.
Given the demand function: Q = 60 - P/3
To plot this, we need to isolate P in terms of Q:
P = 180 - 3Q
Next, let's plot the three curves on a graph:
- Demand curve: Q = 60 - P/3
- For simplicity, let's consider a quantity range from 0 to 60.
- Substituting the values, we get P = 180 - 3Q.
- Plot several points, e.g., (0, 180), (20, 120), (40, 60), (60, 0), and draw a line smoothly connecting these points.
- Marginal revenue curve: MR = 60 - 6Q
- This is a linear function with a negative slope of 6, which intersects the quantity axis at Q = 10.
- Plot points (0, 60), (10, 0), and draw a straight line.
- Marginal cost curve: MC = 3
- Since the marginal cost is constant at 3, plot a horizontal line at y = 3.
Now, on the graph, find the quantity level where the marginal cost curve intersects the marginal revenue curve. This is the monopolist quantity. Draw a vertical line from that point to the demand curve and note the corresponding price. Lastly, identify the area representing the monopolist's profit.
B) To find the monopolist's price and quantity, we need to equate marginal cost (MC) with marginal revenue (MR):
MC = MR
Substituting the corresponding equations:
3 = 60 - 6Q
Solving for Q:
6Q = 57
Q = 9.5
Now, substitute the value of Q back into the demand function to find the price:
P = 180 - 3Q
P = 180 - 3(9.5)
P = 180 - 28.5
P = 151.5
Therefore, the monopolist's price is 151.5 and the quantity is 9.5.
C) To compute the consumer surplus, we first need to find the area between the demand curve and the price line at Q = 9.5.
To calculate this, we can use the formula for the area of a triangle:
Consumer Surplus = 0.5 * (Q * P)
Substituting the values:
Consumer Surplus = 0.5 * (9.5 * 151.5)
To compute the deadweight loss, we need to determine the area between the monopolist's quantity and the perfectly competitive equilibrium quantity.
Deadweight Loss = Consumer Surplus - Monopolist Revenue
Since the monopolist's revenue is given by the equation P * Q, we can calculate it by substituting the price and quantity values:
Monopolist Revenue = 151.5 * 9.5
Finally, the deadweight loss can be computed as:
Deadweight Loss = Consumer Surplus - Monopolist Revenue
I hope this helps you solve the problem step by step.