Final answer:
The competitive equilibrium occurs at a quantity of 440 and a price of $15, while the socially optimal equilibrium, factoring in pollution costs, occurs at a quantity of 410 and a price of $30.
Step-by-step explanation:
The competitive equilibrium in the coffee market is found where the demand curve P=500−Q intersects the private marginal cost curve MCp=20+Q, which occurs at a quantity (Q) of 440 and a price (P) of $15.
However, when the marginal external harm from pollution MCt=Q is taken into account, we need to find a new equilibrium where the social marginal cost (SMC), which is the sum of MCp and MCt, intersects with the demand curve.
This results in a socially optimal equilibrium with a higher price of $30, a lower quantity of 410, and less pollution.
The competitive equilibrium occurs when the quantity demanded is equal to the quantity supplied. To find this, we set the inverse demand curve equal to the private marginal cost curve:
P = 500 - Q = 20 + Q
By solving this equation, we find that the competitive equilibrium quantity is Q = 240 and the price is P = $260.
The social optimum equilibrium occurs when the quantity takes into account the external cost of pollution. So, we need to subtract the marginal external harm from the private marginal cost curve:
P = 20 + Q - Q = 20
By solving this equation, we find that the social optimum quantity is Q = 0 and the price is P = $20.