(40 marks) Suppose household preferences are described by the utility function U(c, ) = 60 + anf) where c is the consumption of market goods, if is the leisure hours for the household, and 6, > 0 are constant parameter values. (a) 0)) Let the market (and real) wage in the economy be to and the total number of hours available to the household be h. Use the information to nd the optimal choice of leisure and consumption as functions of 6, [3, h and to. For ease, assume that the household does not have any nonmarket income and no tax on the household. Based on your answer from part (a) what is the optimal labour supply N S for the household? Does the labour supply function make sense, i.e. how does it depend on to? How does the labour supply function depend on the parameter value [7' (for example what happens lim N S or lim N 3)? ,8>0 ;8>oo Let the output Y be produced by competitive rm having a Cobb-Douglas tech- nology: Y = Z K1/ 2 L1/ 2 where K is the (xed) supply of capital, N is the amount of labour, and 2: is the total factor productivity. Given this information, nd the rm's labour demand curve N d as a function of K, 2, and to. Use your answer from part (b) and (c) to nd the market-clearing wage rate as a function of 9, ,8, h, K, and 2. How does the wage rate depend on 3? Is this consistent with your answer in part (b)? Which of the following would increase the overall wage rate more: an increase in z or and increase in K? Explain why this is the case. b :l: x/bv2 Alec Reminder: Quadratic formula if (L592 + bx + c = 0, then m = 2a