-+ 80% 1. Consider the model of consumption under uncertainty from the module 2 lecture notes. Let the per period utility function be quadratic: u(c) = c- 2 The household's discount factor is B = 1/(1 + p). Future income is uncertain, but evolves according to y' = 6 + by + =', where 6 > 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 first period equals y - 7, where 7 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. Specifically, after tax income in the final period is y' + s', where s' > 0 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 first order conditions. [3 points] (iii) Assume that p = r. Solve for the optimal value of c. [3 points] iv) Derive or ac. Show that it differs from - 9. Explain. [3 points] (v) Suppose the subsidy is proportional to the tax. That is, suppose s' = AT, where u > 0 is a constant. Under this assumption, is it possible that current consumption is unaffected by changes in 7? [3 points]