Question

The demand for cigarettes is given by \(P=500-.2 Q\). Cigarettes are manufactured at a constant marginal \(\operatorname{cost}\) of 50 and sold in a competitive market. a. What is the quantity of cigarettes sold in equilibrium? b. If cigarettes generate a marginal external cost of \(\mathrm{MEC}=.1 Q,\) what is the socially optimal level of cigarettes? c. Draw a diagram illustrating the private and social MC curves and the demand curve. Point out the private market quantity, the socially optimal quantity, and the social welfare cost.

Short answer

a. The equilibrium quantity of cigarettes sold in the market is 2250 units. b. The socially optimal level of cigarettes considering the external cost is 1500 units.

Step-by-step solution

Find The Equilibrium Quantity

Set the price equal to the marginal cost (MC) which is 50. So, the equation becomes: \(50 = 500 - 0.2Q\). Let's solve this for Q to find the equilibrium quantity. Solving this equation results in \(Q = \frac{500-50}{0.2} = 2250\). Therefore, the equilibrium quantity (Q) in the competitive market is 2250 units.

Find The Socially Optimal Quantity

The socially optimal quantity occurs when the price equals the sum of the MC and the Marginal External Cost (MEC). Thus, we have \(50 + 0.1Q = 500 - 0.2Q\). Solving this equation for Q, we get \(Q = \frac{500-50}{0.3} = 1500\). Therefore, the quantity (Q) that is socially optimal is 1500 units.

Draw The Diagram

In the diagram, the horizontal axis (x-axis) represents the Quantity of cigarettes, and the vertical axis (y-axis) represents the Price. Draw the demand curve using the equation \(P = 500 - 0.2Q\). Draw the private MC as a horizontal line at the price of 50. The social MC curve will be the private MC plus the MEC, which will be a line with a positive slope starting at 50. The private market quantity (2250) is where the demand curve intersects the private MC, and the socially optimal quantity (1500) is where the demand curve intersects the social MC. The social welfare cost is the area between the private and social MC from the private quantity to the socially optimal quantity.

Key concepts

Marginal External Cost

The concept of marginal external cost (MEC) plays a significant role in understanding market dynamics, particularly when externalities are present. These costs are not borne by the producer or consumer, but by third parties or society in general. In simpler terms, MEC reflects the cost of the negative effects produced by an additional unit, such as pollution or health issues from cigarette smoking. When the production of a good involves externalities, the true cost of production is not just the private cost but this also includes the MEC.

In our example, the MEC of cigarettes is represented by the equation \(\mathrm{MEC} = 0.1Q\). This indicates that with every additional unit of cigarettes produced, there is a cost of 0.1 times the quantity imposed on society. Without considering MEC, the market is not truly reflecting the entire cost incurred. Thus, producers do not have an incentive to reduce the production to the socially optimal level unless policy interventions, like taxes or regulations, are in place to account for these external costs.

Socially Optimal Quantity

Achieving a socially optimal quantity means balancing both the private and external costs involved in producing a good. To reach this optimal point, it's important to consider both the marginal cost of production and the marginal external cost. The socially optimal quantity is lower than the market equilibrium quantity in cases where negative externalities exist, like the ones generated by cigarette consumption.

In our exercise, when taking into account the MEC, the socially optimal level of cigarette consumption is obtained by setting the supply price equal to the sum of the marginal cost (MC) and the MEC. Mathematically, this is expressed as \(50 + 0.1Q = 500 - 0.2Q\), leading us to conclude that the socially optimal quantity is 1500 units. By reaching this quantity, it ensures that the societal welfare is maximized by minimizing the excess costs from negative externalities.

Demand and Supply Curves

The demand and supply curves are fundamental tools in economics that illustrate the relationship between the price and quantity of goods in a market. The demand curve demonstrates how much consumers are willing to buy at various prices, while the supply curve shows how much producers are willing to sell.

In the context of cigarettes, the demand curve is given by the equation \(P = 500 - 0.2Q\), which signals that as the price decreases, the quantity demanded increases. The supply curve, in our context, is initially simplified to be a horizontal line at the MC value of 50, showing that producers are willing to sell any amount at this constant price in a competitive market.

To account for externalities, we adjust the supply curve to reflect the social marginal cost (SMC), incorporating the marginal external cost. Thus, the modified supply curve becomes upward sloping from the MC, indicating increased costs with additional production. This modification is crucial for highlighting contrasts like the private market quantity (2250 units) versus the socially optimal quantity (1500 units), thus showing the importance of adjustments to address externalities.