# Two-Period Endowment Economy

 $$we$$: test $$c$$: test $$c'$$: test $$s$$: test $$U$$: test
Exogenous Variables and Parameters:
 $$y$$: $$y'$$: $$t$$: $$t'$$: $$\beta$$: $$r$$:

### 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.