Question 6: Search, Unemployment, and Matching. Consider the search model developed in class and continue using the same notation:

s: the amount of time spent searching for a job

p: the probability that a unit of time searching for a job finds suitable employment. Hence, the probability of not finding a job is 1 - p

b: the "unemployment benefit" received by each unsuccessful searcher

c: consumption

w: wage per unit of time spent at work

t: average tax rate

The representative individual's utility function is given by u(c) - h(n, p, s) = ln [(1 - p)s + n]

The budget constraint is: c = (1 - t)wn + (1 - p)sb

The job find constraint is: n = ps

1. Set up the Lagrangian for the consumer's optimization problem and compute the FOC with respect to c, n and s
2. Based on the three FOC you computed above, and the definition of "labour force participation" (LFP): lfp = (1 - p)s + n, construct this framework's "consumption-labour optimality condition". Hint: Eliminate Lagrange multipliers by combining all three FOC.
3. Starting from the "consumption-labour optimality condition" you obtained in part b, suppose for part c only that p =1 or b = 0. How does your solution compare to the classical consumption/labour model?
4. Using the expression you obtained in part b, qualitatively sketch a diagram that contains p on the vertical axis and lfp on the horizontal axis. (Note: all that matters for the qualitative diagram is whether the function is upward-sloping, downward-sloping, completely horizontal, or completely vertical). Provide brief economic interpretation.