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In your model, workers face a trade-off between spending time at work or enjoying time off

Economics

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In your model, workers face a trade-off between spending time at work or enjoying time off. Workers choose the share of time they spend in the labour force, ??, either at work, ?? or looking for a job, ??, and the amount of time they set aside to rest and to enjoy leisure, 1 - ??. When at work, workers earn the wage rate ??. However, labour income is taxed at the rate ??. Moreover, the more time workers spend in the labour force, the more exhausted they feel and, as a result, the more disutility they derive. Put formally, workers face the following optimization problem: ???????? (1 - ??)???????? - F ???? 1+?? 1 + ?? where ?? > 0 drives the elasticity of labour participation and F > 0 is a scale parameter governing the relative weight of the disutility of participation in the labour force. Participation in the labour force, 0 < ?? < 1, can take two forms: workers can either spend time actually working for a wage, which is denoted by ??, or spend time looking for a job, which is denoted by ??: 3 ???? = ???? + ???? Firms are modelled in the usual way. They use labour, ??, and capital, ??, to produce output according to the following Cobb-Douglas function ???? = ??¯???? ?????? (1-??) You will proceed by preparing answers to the following questions: Q1. The short(er) tun You start your analysis by focusing on the labour market alone and by making two simplifying assumptions. In particular, you will consider that the capital stock and unemployment are both taken as given, and equal to: ?? = ??¯ ?? = ??¯ ??¯+ ??¯ ?? where ??¯, ??¯> 0 are the separation and job-finding rates, respectively. a. Using these two simplifying assumptions, derive the labour demand and supply curves. Derive the expressions for the equilibrium level of ??, ??, and ??. Explain the relevant steps in your derivations.