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 u . 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 - u, 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.

With probability λ, 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 u within the same period, like the other                unemployed agents.

With probability o, 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 - λ - o, 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.

With probability A, 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 u within the same period, like the other unemployed agents. Assume A > λ, so  that an independent agent has more chance to become           unemployed than an employed agent.

With probability 1 - A, 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)).

In the following questions, assume r = 0.1, w = 1, u = 0.5, λ = 0.1, o = 0.1, A = 0.2.

a. What is the minimum value of y 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 of y 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.

Time is discrete. There is a continuum of agents indexed by   i e [0, 1]. Consider the following utility maximization problem for each agent:

Eó ┌ pt(ln ci ﹐t + y ln ki ﹐t+1)┐

s.t. ki ﹐t+1 + + ci﹐ t  = (ai ﹐t + 1 - 6)ki﹐ t + bi﹐ t

where p e (0, 1); y > 0; 6 e (0, 1); rt  is the competitive     interest rate, which is taken as given by each agent; ci﹐ t           denotes consumption in period t; ki ﹐t+1 and bi ﹐t+1 are the    amounts of capital stock and the credit balance, respectively, held at the end of period t (and thus at the beginning of      period t + 1); and ai﹐ t  is the productivity of capital stock in period t .

Assume that

ai ﹐t+1 = ,0(a)

with with

probability u

probability 1 - u

for all i and t, where a > 0. Each agent can learn the value of ai ﹐t+1  in period t .

In equilibrium, the credit market must clear in each period:

1

bi ﹐t+1 di = 0

ó

For the initial condition, assume bi ﹐ó  = 0 and ki ﹐ó  > 0 for all i .