# 7030, Micro II, Spring 2022 First Test Q1
## Quesiton
Suppose that individuals are characterized by two parameters. These are a private value, α, for one unit of a good, and a spill-over value, β. Let α and β be independently and uniformly distributed on [0,1]. Think of the good as a vaccine. The private value describes how much the individual values her own consumption of the good (protection from disease) while the spill-over value tells us the externality cost imposed on society from an unvaccinated individual of type β.

Assume that these individuals do not interact with each other – the private value does not depend on how many people are vaccinated, and the spillover does not depend on the vaccination decisions of others within the group. Let the cost of the good (vaccine) be 𝑐 ∈ (1/2 , 1) per dose.
a. Draw the type space with β on the vertical axis and α on the horizontal axis. If the good is
priced at cost, c, indicate which individuals will buy it.
b. If the good is priced at 𝑝, what is the demand for the good?
c. Suppose that the social planner can identify individuals precisely by their types. Which types
should receive the good? Draw the corresponding set on your graph.
d. Now suppose that the planner can only set the price of the good. Is the socially optimal price
above or below c? Why?
e. Find the socially optimal price. It may help you to first determine the spill-over cost from
unvaccinated individuals given the consumer purchase decision. What is the tax or subsidy
per unit?

## Suggested solution for e
Suppose there is a continum of population $1$. If the government set price $p$, people who have $\alpha>p$ will take vaccine, that is $1-p$ population.
The social planner's problem is

\begin{align}
\max_p \int \alpha_i + \beta_i -c(1-p)di\\
\text{s.t. } \alpha_i \ge p
\end{align}

The social welfare from $\alpha$ is

$$
\frac{1-p^2}{2}
$$


Because $\alpha$ is not correlated with $\beta$, the expected value of $\beta$ is $1/2$. The social welfare from $\beta$ is

$$
(1-p) \cdot \frac{1}{2}
$$

The problem becomes,

$$
\max_p \quad \frac{1-p^2}{2} + (1-p) \cdot \frac{1}{2} -c(1-p)
$$

By the FOC, we can get

$$
p= \frac{2c-1}{2} = c - \frac{1}{2}
$$

It's a reasonable result since the government should subsidise the behavior with positive externality, and the price will increase as cost.