The Household Problem Is to Solve:
A) $\max _ { c _ { \text {tadey } } \cdot c _ { \text {Mur } } } U = u \left( c _ { \text {todap } } , c _ { \text {future } } \right)$

The household problem is to solve:

A) $\max _ { c _ { \text {tadey } } \cdot c _ { \text {Mur } } } U = u \left( c _ { \text {todap } } , c _ { \text {future } } \right)$
B) $\max _ { c _ { \text {todoy } } \cdot c _ { \text {Murr } } } c _ { \text {today } } + c _ { \text {future } } ( 1 + R ) = \bar { X } \text {, subject to } U = u \left( c _ { \text {today } } , c _ { \text {future } } \right)$
C) $\min _ { c _ { \text {itdoy } } \cdot c _ { \text {Murs } } } U = u \left( c _ { \text {toduy } } \right) + \beta u \left( c _ { \text {future } } \right) , \text { subject to } c _ { \text {todup } } + c _ { \text {future } } ( 1 + R ) = \bar { X }$
D) $\max _{c_{\text {todoby }}, c_{\text {Aurre }}} U=u\left(c_{\text {todav }}\right) +\beta u\left(c_{\text {future }}\right) , \text { subject to }$$c_{\text {today }}+\frac{c_{\text {futare }}}{1+R}=\bar{X}$
E) $\max _ { c _ { \text {tadoy } } \cdot c _ { \text {Murs } } } U = u \left( c _ { \text {today } } \right) + \beta u \left( c _ { \text {future } } \right) , \text { subject to } c _ { \text {today } } + c _ { \text {future } } ( 1 + R ) = y _ { \text {today } } + y _ { \text {future } } ( 1 + R )$