need step by step explanation thank you

1. Consider the model of consumption under uncertainty from the module 2 lectme notes. Let the per period utility function be quadratic: n[c} =c E. 2 The household's discount factor is ,3 = 1H1 + ,0}. Future income is uncertain, but evolves according to 3H=J+Hy+et where :5 2::- l}, I] 4:; H c: 1, and s' is a random variable 1with mean zero. Assume that parameter values are such that the present value of the consumer's lifetime income is positive for all possible realizations of 5'. The household has rational expectations. Suppose there is a lump sum tax that reduces the household's income in the current period {i.e., after tax income in the rst period equals 3; T, where 1' is the tax] The government announces that there will be a tax increase in the future period. Specically, future taxes 1will be a multiple, call it ,u, of current taxes: 1"" = ,ur, where p. 2:- 1. Let's assume that the government's announcement is perfectly credible. {i} Formally 1write down the household's optimisation problem. 'Write down the Lagrangian function. [3 points] {ii} Derive and interpret the rst order conditions. [3 points] {iii} Assume that p = 1r. Solve for the optimal value of c. [3 points] {iv} Derive a. Show that it diam. from 4%. Explain. [a points] 3': v Derive . Interpret your expression. 3 points 39