$($Lehr #$4\mathrm{.}7)$ Interest$($ed$)$ in the Life Cycle. Example $4\mathrm{.}1$ assumes that Nia has both a discount

rate of zero and faces an interest rate of zero. These assumptions made calculating her constant

level of consumption expenditure of $$56,000$ fairly straightforward. When the discount rate and

interest rate are equal, but not necessarily zero, the constant per$-$period consumption expenditure

is $c^{**},$ given as follows:

$c^{**}=\frac{R-1}{R-\frac{1}{R^T}}{\displaystyle \underset{k=0}{\overset{T}{}}}\frac{y_k}{R^k}$

Compute $($using a computer$)$ Nia's constant annual consumption expenditure in the following

cases.

a$.$ Assume that $=r=0\mathrm{.}025.$

b$.$ Assume that $=r=0\mathrm{.}05.$

c$.$ Assume that $=r=0\mathrm{.}1.$

d$.$ Assume that $=r=0\mathrm{.}2.$

e$.$ Explain how increasing the interest and discount rates impacts Nia's annual consumption

expenditure. Provide intuition for your findings.