Consider the following model of consumption and health. A household lives for two periods and its preferences over consumption in the two periods are given by U(c; c') = u(c) + Bu(hc') where c is consumption of the household in the 1rst period, c' is consumption of the household in the second period, h is the health status of the household in the second period, and B is the discount factor. Health status affects utility because it determines how much the household enjoys its consumption level. In period 1, the household can decide how much to invest in health. If a household invests i units of the good in health in period 1, its health status in the second period will be h = i^n, where 0 < n < 1. The household receives income y in the first period and y' in the second period; there are no taxes. The household can borrow and save in a risk-free asset between the first and the second period at a gross interest rate R = 1 + r. 1. Write down the optimization problem of the household and form the Lagrangian associated with this problem. 2. Find the first-order conditions for the household's optimization problem and provide some economic intuition for the equations. 3. Assume that u(c) = ln(c). How does h change when the interest rate R increases? Give some intuition for your answer.

Consider the following model of consumption and health. A household lives for two periods and its preferences over consumption in the two periods are given by U(c; c') = u(c) + Bu(hc') where c is consumption of the household in the 1rst period, c' is consumption of the household in the second period, h is the health status of the household in the second period, and B is the discount factor. Health status affects utility because it determines how much the household enjoys its consumption level. In period 1, the household can decide how much to invest in health. If a household invests i units of the good in health in period 1, its health status in the second period will be h = i^n, where 0 < n < 1. The household receives income y in the first period and y' in the second period; there are no taxes. The household can borrow and save in a risk-free asset between the first and the second period at a gross interest rate R = 1 + r. 1. Write down the optimization problem of the household and form the Lagrangian associated with this problem. 2. Find the first-order conditions for the household's optimization problem and provide some economic intuition for the equations. 3. Assume that u(c) = ln(c). How does h change when the interest rate R increases? Give some intuition for your answer.