2 Problems 1. Consider the model of consumption under uncertainty from the module 2 lecture notes. Let the per period utility function be quadratic: on2 a(c) c ?. The household's discount factor is = 1/(1 + p). Future income is uncertain, but evolves according to y'=6+9y+5r, where 5 > 0, 0 0. 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; 1', Where 1' is the tax). The tax is known to be temporary. In fact, the government announces that there will be a subsidy paid to the household in the future period. Specically, after tax income in the nal period is y' + 3', where 3" > [I is the subsidy. Let's assume that the government's announcement is perfectly credible. (i) Formally write down the household's optimization problem. Write down the Lagrangian function. [3 points] (ii) Derive and interpret the rst order conditions. [3 points] (iii) Assume that p = 1'. Solve for the optimal value of c. [3 points] (iv) Derive %. Show that it differs from Tf,. Explain. [3 points] (v) Suppose the subsidy is proportional to the tax. That is, suppose a\" = #1", where p. > [I is a constant. Under this assumption, is it possible that current oonsumption is unaffected by changes in 1'? [3 points]