Find out the optimal consumption path, taking the interest rate as given and then to find out the equilibrium interest rate that will eliminate demand for saving and investment, taking the consumption...

Find out the optimal consumption path, taking the interest rate as given and then to find out the equilibrium interest rate that will eliminate demand for saving and investment, taking the consumption path as given (Chapter 11 of GLS). Both times, the utility function will be the same:

U = ln(ct) + 0.9ln(ct+1)

So, the future counts 90% as much as the present. In Part 1, income each period is 100, and the interest rate is 20%. In Part 2, consumption in each period is 100–in other words, income each period is 100, but income isn’t storable (it’s “manna”), so you have to consume it or lose it. Answer the following question.

What is the equilibrium interest rate between these two periods? If the equilbrium interest rate were lower than that level, would that create a surplus or a shortage?