1 An infinitely-lived household's lifetime utility can be expressed as follows: Uo = EB'In ( ct - yen! ) where In is the natural log operator, c is consumption and n, are hours worked. > 1 and denotes the parameter determining labour supply elasticity. Y, is a time- varying parameter determining the disutility of labour. y follows an AR(1) process In Yt = pin Y: 1 + v. v. is an lid shock with a zero mean. The representative household produces goods, yr, using a Cobb-Douglas technology in hours worked and the capital stock: It = Atkin) " The parameter o denotes the share of capital and lies between 0 and 1. A, denotes total factor productivity and follows an AR(1) process, ie In At = pln At-1 + et. In the steady state, Y = A =1. The capital stock, k, is pre-determined in period t and evolves a follows: Kt+1 = yt - Ct + (1 - 6) kt The firm belongs to the household, which chooses consumption, hours as well as next period's capital stock: (a) Set up the Lagrangian function and derive the first-order conditions for Ct, nt and Kt+1. (5 marks) (b ) () Combine the first order conditions for ct and me and carefully analyse the response of hours to an increase in At. (ii) Carefully analyse the response of hours worked to an increase in the disutility of labour. (7 marks) (c) Using the steady state of the model, show that impatient economies have lower capital-labour ratios. (7 marks)