1. This problem walks you through a two-period deterministic consumption-savings model, where a consumer chooses the optimal amount of savings to maximize his/her life-time utility:

where is the parameter of time impatience. The higher is, the more pleasure the person derives from delaying consumption, and the more patient he/she is. The person has income at t=0, and at t=1. These incomes are predetermined exogenously. Given that the interest rate is r, the person determines how much he/she saves at t=0 in order to maximizes his/her life-time utility. The equilibrium condition is that the marginal rate of substitution of future consumption for current consumption equals one plus the interest rate

Suppose the person has log utility: .

1. Denote current savings by . Write down the budget equation at t=0 and t=1.
2. Derive the intertemporal budget equation, and explain the meaning intuitively.
3. Solve for equilibrium consumption and , and savings . Show that current consumption is a fixed fraction of the PV of the persons life-time income, which is PV(life-time income)=. (Hint: use the equilibrium condition and the intertemporal budget equation, and the fact that the first-order derivative of log utility is: u(x)=1/x. Express consumption and savings in terms of all the exogenous variables of and ).
4. Suppose the person has a change of taste: he/she becomes more impatient. Determine mathematically how he/she would adjust current consumption, savings, and future consumption. And then explain the results intuitively.
5. Under what condition are savings negative? How to interpret this?