$[20$ points$]$ Consider Akerlof's lemon market model for this question and assume the

standard assumptions of the model. Suppose Patient $($P$)$ and Insurance Company's $($I$)$

utility functions are defined below.

$U_{\text{I}}={\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i+M$

$U_P={\displaystyle \underset{i=1}{\overset{n}{}}}\frac{5}{2}{x}_i+M$

Also, assume in this market, patient's health care expenditure $(x_i)$ is distributed

uniformly as $x_{i}U[0,500],$ and $M$ is the number of other goods the patient and company

consume.

$($a$)[5$ points$]$ Will there be any transactions if the insurance premium is priced at

$$700?$ Why or why not?

$($b$)[5$ points$]$ What happens if the insurance premium changes to $$550?$

$($c$)[5$ points$]$ Suppose the government mandates that all insurance companies sell

contracts at a maximum premium price of $$250.$ What will happen in this case?

Explain your answer.

$($d$)[5$ points$]$ Suppose a patient values insurance $150\%$ more than the insurance

company. What is the maximum price the patient is willing to pay for the

premium?