Question

The market demand (MPB) and supply (MPC) for vaccinations ( Q) are given by the following expressions:

MPB ( Q ) = 12 − 1/4 Q

MPC ( Q ) = 1/4 Q

Assume no externalities arise from the production of the private good. The consumption of the private good does result in a positive externality in the form of health benefits. Assume that the health benefits are directly proportional to the amount of consumed. The total external benefit from the production of is given by the following expression:

TEB ( Q ) = 1/16 Q²

1. What is the competitive (unregulated) level of output?

2. What is the socially optimal level of output?

3. What subsidy for vaccinations would need to be introduced to arrive at the socially optimal level of output?

Answer 1

1. Competitive level of output is determined where MPB = MPC.
So, 12 - (1/4)Q = (1/4)Q
So, (1/4)Q + (1/4)Q = (2/4)Q = 12
So, Q = 12*(4/2)
So, Q = 24

2. MEB = d(TEB)/dQ = 2*(1/16)Q = (1/8)Q
So, MSB = MPB + MEB = 12 - (1/4)Q + (1/8)Q = 12 - 0.25Q + 0.125Q = 12 - 0.125Q
Socially optimal level of output is determined where MSB = MPC.
So, 12 - 0.125Q = (1/4)Q = 0.25Q
So, 0.125Q + 0.25Q = 0.375Q = 12
So, Q = 12/.375
So, Q = 32

3. Subsidy = MEB at socially optimal output.
MEB = (1/8)Q = (1/8)*32
So, MEB = 4