Question 1 With probability :7, an independent agent loses the agent's clients, and becomes unemployed at the beginning of each period. In this case, the agent does not receive the income for an independent agent, y, in the period. Instead, the agent is matched with an employer with probability 11 within the same period, like the other unemployed agents. Assume I] > t, so that an independent agent has more chance to become unemployed than an employed agent. With probability 1 1], an independent agent remains independent, receiving the income y in the period, with no chance to change the agent's state until the next period. Each agent's lifetime utility is defined by the expected present discounted value of future income. The discount rate is denoted by r (i.e., the time discount factor is 1/(1 + r)). Question 1 in the following questions, assume r = 0.1, w = 1, y = 0.5. A=QL=QLn=Qz a. What is the minimum value ofy that makes an employed agent choose to be independent when it is possible? (Assume that if an agent is indifferent between being employed and becoming independent, the agent always choose to be independent.) Derive the answer with 2 decimal places (i.e., if the minimum value ofy is 1.6875, only answer 1.68). b. If the value of y takes the minimum value derived above, what is the expected lifetime utility for an employed agent? Derive the answer with 2 decimal places. Question 1 Time is discrete. There is a continuum of agents. In each period, each agent is in one of three states: unemployed. employed. and independent. In each period. each unemployed agent is matched with an employer with probability y. In a match between an unemployed agent and an employer, the unemployed agent receives a wage offer from the employer. The value of the offered wage is constant, denoted by w. If the unemployed agent accepts the offered wage. the agent becomes employed, and receives the wage from the next period. With probability 1 3.1, an unemployed agent is not matched with any employer, and remains unemployed in the period, with no chance to change the agent's state until the next period. The agent's income in the period is zero in this case. Question 1 With probability A, an employed agent loses employment, and becomes unemployed at the beginning of each period. In this case, the agent does not receive the wage in the period. Instead, the agent is matched with an employer with probability y within the same period, like the other unemployed agents. With probability 4), an employed agent has chance to be independent. If the agent chooses to be independent, then the agent receives the wage w in the period, and a constant income, y, from the next period. Assume y > w. With probability 1 2L (i), an employed agent remains employed, receiving the wage w in the period, with no chance to change the agent's state until the next period