1. Consider a three-period version of the one-sided search model with a two-rung job ladder. Let u; 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, 1, 2. Let ei and 6? denote the fraction of the labour forced employed in low-wage and high-wage jobs. Note that at + 8% + e? = 1. The unemployment benet is denoted b, and the two wage levels are w; and 10;, satis- fying b 0 is the discount rate. In terms of notation, let V,\(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, V1\". Be sure to include the accept / rej ect decision problem in this equation. Im- pose the Optimal accept / reject decision to write V1\" in terms of the parameters of the model (i.e., the discount factor, the probabilities, and income levels I), w; and wh). [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 V1; 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, V05. Determine conditions under which the low-wage worker will accept an offer to work for wage 10;, if it is offered at time 0, and when they will not. Briey explain these conditions. [3 points] (v) Suppose b = 1/2, w; = 1 and 10;, = 2. The exogenous probabilities are 351 = 3/10, 925;, = 1/10, and 6;, = 1/20. The discount factor is )8 = 0.99. Finally, the initial distribution of workers across employment status is 11.0 = 1/ 10, e[, = 3/ 10 and 63 = 6/10. Solve for the unemployment rate in periods 1 and 2. [3 points]