2 Problems 1. Consider a three-period version of the one-sided search model with a tworung job ladder. Let at denote the unemployment rate at time t = 0,1,2. The remaining of labour force participants are employed, either at a low-wage job or a high-wage job at time t = 0, l, 2. Let cf, and 8? denote the fraction of the labour forced employed in lowwage and high-wage jobs. Note that at + cf, + of,\" = l. The unemployment benet is denoted b, and the two wage levels are w; and 11);, satis- fying b 0 is the discount rate. In terms of notation, let l4\(i) Consider the situation of an unemployed worker at time 1. Write down an equa tion for their present discounted expected value of remaining lifetime income, 1/3\". Be sure to include the accept / reject decision problem in this equation. Im pose the optimal accept /reject decision to write Vf' in terms of the parameters of the model (i.e., the discount factor, the probabilities, and income levels I), w; and tub). [3 points] (ii) Consider the situation of a worker in a low-wage job at time 1. Using an approach similar to what you did in part (i), derive an expression for V13 in terms of the parameters of the model. [3 points] (iii) Write down an expression for V1" in terms of the parameters of the model. [3 points] (iv) Now consider the time 0 situation of a worker in a low-wage job. Write down an equation for their present discounted expected value of lifetime income, V03. Determine conditions under which the lowwage worker will accept an offer to work for wage \"wk if it is offered at time 0, and when they will not. Briefly explain these conditions. [3 points] (v) Suppose b = 1/2, w; = 1 and 10;, = 2. The exogenous probabilities are c3; = 3/ 10, 933;, = 1/ 10, and (5;, = 1/20. The discount factor is ,8 = 0.99. Finally, the initial distribution of workers across employment status is no = 1/ 10, 83 = 3/10 and 83 = 6/10. Solve for the unemployment rate in periods 1 and 2. [3 points]