Question $2.$ Consider a household whose utility is determined by its consumption in periods $0$ and $1.$ Let Co and c$$ denote the consumption in periods $0$ and $1,$ respectively. The utility of this household can be represented by a utility function U$($CO$,$C$1)=$ u$($co$)+$ Bu$($ci$).=(2)=$ Assume further that the per$-$period utility u$($c$)$ is given by u$($c$)=$ log c$,$ and the discount factor B is given by B $=$ jo$.$ In periods $0$ and $1,$ this household is endowed with incomes yo $=190$ and y$=380,$ respectively. Importantly, this household can save or borrow in period $0$ at the interest $.=$ rater $=($a$)$ Check if the per$-$period utility function u$($c$)$ satisfies i$)$ u$\text{'}($c$)>0$ and ii$)$ u$"($c$)<0,$ where u$\text{'}($c$)$ denotes du$/$dc and u$"($c$)$ denotes d$?$u$/$dc$2.$ Describe the economic meaning of these conditions. Show that when the per$-$period utility function u$($c$)$ satisfies the two conditions above, house$-$ holds' total utility U$($CO$,$ C$1)$ satisfies i$)$ Uc $>0,$ Ucz $>0$ and ii$)$ Uco,co $<0,$ Uc$1,41<0,$ where Uc and Uc$,$ denote au$/$aco and du$/$dc$1,$ respectively, and Uco,co and Ucz,c$$ denote $22$u$/$acz and $22$u$/$acz$,$ respectively. $($b$)$ Write down this household's intertemporal optimization problem using sequential budget con$-$ straints. Indicate which term captures the saving or borrowing of this household in the sequential budget constraints. Using this term, describe when this household saves and when borrows. $($c$)$ Derive the intertemporal budget constraint and explain its economic meaning $($using the con$-$ cept of the present discounted value$).($d$)$ Rewrite this household's intertemporal optimization problem using the intertemporal budget constraint. Explain why the gross interest rate $(1+$r$)$ can be interpreted as a relative price be$-$ tween current and future consumption by comparing this optimization problem with the static optimization problem over two goods specified in Q$1($b$).($e$)$ Set up a Lagrangian equation and derive the optimal conditions. $($f$)$ Derive the Euler equation and provide an economic reason why this equation has to hold at the optimum.