Consider a one-sided search model with a two-rung job ladder. Let u, denote the unemployment rate...

 

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Consider a one-sided search model with a two-rung job ladder. Let u, denote the unemployment rate at time t. The remaining of labour force participants are employed at time t, either at a low-wage job or a high-wage job. Let e and et denote the fraction of the labour forced employed in low-wage and high-wage jobs. Note that ++ = 1. The (per period) unemployment benefit is denoted b, and the two wage levels are w and w satisfying b< <w. Unemployed workers only receive low-wage offers. Specifically, is the probability that an unemployed worker receives a wage offer in a given period. For simplicity, assume there is no exogenous separation of workers from low-wage jobs (i.e., & = 0). However, a low-wage worker may receive an offer to switch to a high-wage job. A high wage offer arrives with probability on each period, conditional on being employed at a low-wage job. A worker employed at wage Wh will experience an exogenous job separation with probability &, in which case they transition to unemployment. Workers make accept/reject decisions to maximize the present discounted expected value of lifetime income, where ß = 1/(1+r) is the discount factor and r> 0 is both the rate of time preference and the interest rate. In terms of notation, let V, V, and V denote the present discounted expected values of remaining lifetime income at time t conditional on being unemployed, employed at a low-wage job, and employed at a high-wage job. (i) Write down a system of Bellman equations that characterizes the sequence of values (V. V. V. Impose stationarity to obtain a system of three equa- tions with three unknowns (namely, V, V and V). (ii) Determine a necessary and sufficient condition (in terms of the exogenous pa- rameters of the model) under which a low-wage worker will optimally accept an offer of wh, rather than remaining employed at the low-wage job. (iii) Determine the stationary unemployment rate, assuming that high-wage job are accepted (i.e., under the condition derived in part (ii)). Show that the stationary unemployment rate would be lower if high-wage job offers are instead rejected by low-wage workers. Consider a one-sided search model with a two-rung job ladder. Let u, denote the unemployment rate at time t. The remaining of labour force participants are employed at time t, either at a low-wage job or a high-wage job. Let e and et denote the fraction of the labour forced employed in low-wage and high-wage jobs. Note that ++ = 1. The (per period) unemployment benefit is denoted b, and the two wage levels are w and w satisfying b< <w. Unemployed workers only receive low-wage offers. Specifically, is the probability that an unemployed worker receives a wage offer in a given period. For simplicity, assume there is no exogenous separation of workers from low-wage jobs (i.e., & = 0). However, a low-wage worker may receive an offer to switch to a high-wage job. A high wage offer arrives with probability on each period, conditional on being employed at a low-wage job. A worker employed at wage Wh will experience an exogenous job separation with probability &, in which case they transition to unemployment. Workers make accept/reject decisions to maximize the present discounted expected value of lifetime income, where ß = 1/(1+r) is the discount factor and r> 0 is both the rate of time preference and the interest rate. In terms of notation, let V, V, and V denote the present discounted expected values of remaining lifetime income at time t conditional on being unemployed, employed at a low-wage job, and employed at a high-wage job. (i) Write down a system of Bellman equations that characterizes the sequence of values (V. V. V. Impose stationarity to obtain a system of three equa- tions with three unknowns (namely, V, V and V). (ii) Determine a necessary and sufficient condition (in terms of the exogenous pa- rameters of the model) under which a low-wage worker will optimally accept an offer of wh, rather than remaining employed at the low-wage job. (iii) Determine the stationary unemployment rate, assuming that high-wage job are accepted (i.e., under the condition derived in part (ii)). Show that the stationary unemployment rate would be lower if high-wage job offers are instead rejected by low-wage workers.

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employed in a lowwage job or employed in a highwage job The model analyzes the probability that a lowwage worker will reject an offer for a higherpaying job Heres a breakdown of the relevant concepts ... View the full answer