Two-Period Endowment Economy

Exogenous Variables and Parameters:
\(y-t\):   \(y'-t'\):   \(r_s\):   \(r_b\):  

Solving the problem algebraically:

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.