(30 points)
The demand for lithium is given by P = 500 – 0.2L, where L is the quantity of mined lithium and
P is the price. Lithium is mined at a constant marginal cost of $50 and sold in a competitive
market.
a) What is the quantity of lithium sold in equilibrium?
b) If lithium mining generates a marginal external cost of $0.1L, what is the socially optimal
level of lithium?
c) Draw a diagram illustrating the private and social marginal cost curves, and the demand curve.
Determine and place on the diagram the private market quantity, the socially optimal quantity,
and the social welfare cost.
To answer these questions, we need to understand the concepts of equilibrium quantity, socially optimal level, and how to draw a diagram illustrating the private and social marginal cost curves and the demand curve. Let's go step by step:
a) To find the quantity of lithium sold in equilibrium, we need to equate the demand and supply. In a competitive market, the equilibrium occurs when the quantity demanded is equal to the quantity supplied.
Demand: P = 500 - 0.2L
Supply: P = Marginal Cost = $50
Setting the demand equal to supply:
500 - 0.2L = 50
Solving this equation for L:
450 = 0.2L
L = 450 / 0.2
L = 2250
Therefore, the quantity of lithium sold in equilibrium is 2250 units.
b) To find the socially optimal level of lithium, we need to consider the marginal external cost. The socially optimal level occurs when the social marginal cost (private marginal cost + marginal external cost) is equal to the price.
Social Marginal Cost = Private Marginal Cost + Marginal External Cost
= $50 + (0.1 * L)
Setting the social marginal cost equal to the price:
500 - 0.2L = $50 + (0.1 * L)
Simplifying this equation for L:
0.3L = 450
L = 450 / 0.3
L = 1500
Therefore, the socially optimal level of lithium is 1500 units.
c) To draw the diagram illustrating the private and social marginal cost curves and the demand curve, we will use the quantity (L) on the x-axis and the price (P) on the y-axis.
- The private marginal cost curve is a horizontal line at $50.
- The social marginal cost curve is the private marginal cost curve plus the marginal external cost, which is an upward sloping straight line. It starts from $50 (same as private marginal cost) and increases at a constant rate of $0.1 per unit of lithium.
- The demand curve is given by P = 500 - 0.2L. It is a downward sloping straight line.
On the diagram:
- Plot the private marginal cost curve as a horizontal line at $50.
- Plot the social marginal cost curve as an upward sloping straight line starting from $50.
- Plot the demand curve as a downward sloping straight line.
- Label the x-axis as "Quantity (L)" and the y-axis as "Price (P)".
- Mark the private market quantity (equilibrium quantity) at the intersection of the demand curve and the private marginal cost curve (2250 units).
- Mark the socially optimal quantity at the intersection of the demand curve and the social marginal cost curve (1500 units).
- To calculate the social welfare cost, find the area between the supply curve (private marginal cost curve) and the social marginal cost curve from the private market quantity to the socially optimal quantity.
I hope this explanation helps you understand how to answer these questions and draw the diagram.