The consumer's lifetime utility is given as $c _ { \text {today } } + c _ { \text {ftuture } } ( 1 + R ) = f _ { \text {today } } + y _ { \text {today } } + y _ { \text {ftuture } } ( 1 + R )$

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The equation seems to be incorrectly written. The consumer's lifetime utility should be expressed in terms of present value, typically as $c_{\text{today}} + \frac{c_{\text{future}}}{1+R} = y_{\text{today}} + \frac{y_{\text{future}}}{1+R}$, where $R$ is the interest rate.