Consider an economy with households that live for two periods and care only about current and future consumption. Their utility function is u = ln c1 + ln c2 where c1 is consumption in period 1 and c2 is consumption in period 2. The household has a fixed income of 2 units of consumption goods in the first period, and 1 unit in the second period (think of the household as being a middle aged couple who is currently earning a lot but its retirement -second period -income is small). The interest rate is denoted by r. 1. Write down the period budget constraints for the household. Then combine them to eliminate savings and write down the present value budget constraint for the household. 2. Write down the utility maximization problem of the household. 3. Solve for c!1and c!2as functions of the interest rate r. 4. Use the first period budget constraint to express savings s! as a function of r. 5. The competitive equilibrium interest rate r! must be such that s! is equal to zero. Solve for the equilibrium interest rate. 6. If you’ve done things correctly, you should have found out that the equilibrium interest rate is very low. Explain intuitively why this is so. (Hint: think about the household’s desire to smooth consumption over time)