Urgent!

Ageing Pensioner problem A consumer decides to retire at t=0 and has an initial level of assets given by a0>0 and will not receive any income in the future. Her instantanous utility at time t if is alive is given by u(ct); where ct is the consumption at time t. The consumer is perfectly patient and values equally utility in the future compared to utility today. Nevertheless, the individual has every period the probability of dying before consuming. The probability of dying is given by s if the individual is sick and h if the individual is healthy. The individual is initially healthy at t=0 but every period has the probability to get sick and then she will never go back to be healthy. If the individual dies then his instantanous utility is zero for the current and all the subsequent periods. We assume u is concave and bounded. Also that the pensioner cannot get indebted. The interest rate on savings is given by r. 1. Write the problem of the pensioner in recursive form. 2. Show that the operator T given in the recursive form is a contraction mapping. 3. Find the Euler equation and explain how different is the consumption decision of two pensioner with the same assets at period t but with different health status. Ageing Pensioner problem A consumer decides to retire at t=0 and has an initial level of assets given by a0>0 and will not receive any income in the future. Her instantanous utility at time t if is alive is given by u(ct); where ct is the consumption at time t. The consumer is perfectly patient and values equally utility in the future compared to utility today. Nevertheless, the individual has every period the probability of dying before consuming. The probability of dying is given by s if the individual is sick and h if the individual is healthy. The individual is initially healthy at t=0 but every period has the probability to get sick and then she will never go back to be healthy. If the individual dies then his instantanous utility is zero for the current and all the subsequent periods. We assume u is concave and bounded. Also that the pensioner cannot get indebted. The interest rate on savings is given by r. 1. Write the problem of the pensioner in recursive form. 2. Show that the operator T given in the recursive form is a contraction mapping. 3. Find the Euler equation and explain how different is the consumption decision of two pensioner with the same assets at period t but with different health status