The consumer's problem is
\[\max_{c,c',s}\log c + \beta \log c'\]
\[s.t. c\geq 0;c'\geq 0\]
\[c+s\leq y-t\]
\[c'\leq (1+r)s+y'-t'\]
Utility is strictly increasing, so the budget constraint will bind with equality.
Solve for \(s\) in the second period, and then plug it into the first to get
\[s={c' -y' +t' \over 1+r}\]
\[c+{c'\over 1+r}= y+{y' \over 1+r} - t - {t' \over 1+r}\equiv we\]
If the consumer is behaving optimally, they will choose consumption such that their marginal rate of substitution is equal to the exchange rate between past and future consumption:
\[(1+r)=MRS={{1/ c} \over {\beta / c'}}={c\over \beta c'}\]
Therefore \(c'=(1+r)\beta c\) and
\[c+{(1+r)\beta c \over 1+r}=c(1+\beta)=we\]
\[c={we\over 1+\beta}\]
At this point, we have found the formulas for all the endogenous variables, and can plug in the exogenous parameters to find the result.