Consider the dynamic programming problem that leads to Figure $8\mathrm{.}4.$ This problem asks you to solve the problem numerically with one change: preferencesare logarithmic, so that u$($C$)=$ In C$.$ Specifically, it asks you to approximate thevalue function by value$-$function iteration, along the lines of equation $(8\mathrm{.}73),$ with V$\backslash$deg $($X$)$ assumed to equal zero for all X$.($a$)$ As a preliminary, explain why V$\text{'}($X$)=$ In X$.($b$)$ Since it is not literally possible to find V$"($X$)$ for every X from $0$ to infin$-$ity, proceed by discretizing the problem. Choose an N$,$ and define e $=100/$N$.$Now, assume that Y can take on only the values e$,3$e$,5$e$,...,200-$ e$,$ each with probability $1/$N$.$ Likewise, assume that C can only take on the values e$,3$e$,5$e$,...,$ and find the V$"($X$)\text{'}$s only for X equal to e$,3$e$,5$e$,...,$ up to someupper bound B that you choose $($and assume that V$"($X$)=$ V$"($B$)$ for X $>$ B$).$Finally, only do some finite number of iterations. $($Use whatever programming language or software you wish; MATLAB is a natural candidate.$)$ Plot or sketch the resulting V$).($c$)$ Comment briefly on the process of solving the problem numerically. For ex$-$ample, explain why you chose the values of N$,$ B$,$ and the number of iterations that you did. Did you encounter anything unexpected?$($d$)$ Using the value function you found, find C$(),$ and plot or sketch that.$($e$)$ Compare the C$()$ you found with that in Figure $8\mathrm{.}4.$ What are the main similarities? The main differences?