1. Problem 1 (Utility maximization problem]. Consider a consumer who lives for two period: young and old. The consumer works when young and retires when old. The consumer supplies 1 unit of time to labor market and earns 10 units of income when young. The consumer chooses a sequence of consumption to maximize the lifetime utility maximization problem as follows: 610 (,10' V ,R 1 + '2 (1e ) 311%? { 1 a 1 0' at. C2 4+7 "1 R 'U). o (1.1) Find the analytical solution for optimal consumption and saving. o (1.2) Assume that 0 = 2, m = 5 and R = 1.04. Solve the household problem numerically with the fsolve function. Report the optimal choices. 0 (1.3) Now add labor-leisure choice to the problem 8 19 10. 10 _ ((3131 ) C2 V(w,R) 2%{ 10 +10' 3.35. C2 . 7 = 11 f1_i R 'LU( I): where 11 is leisure and (1 11) is labor supply. Assume that or = 3, 6 = %, w = 5 and R = 1.04. Solve the household problem numerically with the fsolve function. Vary the wage rate between 1 and 10 and plot the labor supply. Is the labor supply curve upward slopping? Why