ECON 1110: Intermediate Macro

Consumption and Savings

Here we are looking a two period model. The consumer has income $y$ today and $y^{\prime}$ in the next period. Similarly, taxes are $t$ today and $t^{\prime}$ in the next period. In order to shift consumption between the two periods, the consumer can save or borrow at a common rate $r$. This choice is represented by the variable $s$, which is positive in the case of savings and negative in the case of borrowing.

Consumption today and tomorrow are represented by $c$ and $c^{\prime}$. The budget constraints for the two periods are then c + s = y - t \\ c^{\prime} = y^{\prime} - t^{\prime} + (1+r) s We can eliminate $s$ to combine these two into a single present value budget constraint. \Rightarrow\ \align s = y - t - c \\ \Rightarrow\ \align c^{\prime} = y^{\prime} - t^{\prime} + (1+r) (y-t-c) \\ \Rightarrow\ \align (1+r) c + c^{\prime} = (1+r) (y-t) + y^{\prime}-t^{\prime} \\ \Rightarrow\ \align c + \frac{c^{\prime}}{1+r} = y-t + \frac{y^{\prime}-t^{\prime}}{1+r} \equiv we The last term is what we'll call wealth, since it is the present value of the income stream. It determines the size of the budget set, while $1+r$ determines the slope. Two consumers with the same wealth will make the same consumption-savings choices, even if $y$ and $y^{\prime}$ differ.

Now let's solve for the consumer's optimal choice. In general a consumer's preferences are represented by U(c,c^{\prime}) = u(c) + \beta u(c^{\prime}) for some per-period utility function $u$ and discount rate $\beta \in (0,1)$. Usually, we will look at the case where $u(c) = \log(c)$, though as we saw in assignment 4, there are other options. Then we have U(c,c^{\prime}) = \log(c) + \beta \log(c^{\prime}) We can use the budget constraints to formulate the consumer's problem as purely one of choosing the optimal $s$ \max_s \ \log(y-t-s) + \beta \log(y^{\prime}-t^{\prime}+(1+r)s) To find this, we simply take the derivative with respect to $s$ \align \frac{-1}{y-t-s} + \frac{\beta(1+r)}{y^{\prime}-t^{\prime}+(1+r)s} = 0 \\ \Rightarrow\ \align \frac{1}{y-t-s} = \frac{\beta(1+r)}{y^{\prime}-t^{\prime}+(1+r)s} \\ \Rightarrow\ \align y^{\prime} - t^{\prime} + (1+r)s = \beta (1+r) (y-t-s) \\ \Rightarrow\ \align \beta (1+r) s + (1+r) s = \beta (1+r) (y-t) - (y^{\prime}-t^{\prime}) \\ \Rightarrow\ \align (1+\beta) (1+r) s = \beta (1+r) (y-t) - (y^{\prime}-t^{\prime}) \\ \Rightarrow\ \align s^{\ast} = \frac{\beta(1+r)(y-t)-(y^{\prime}-t^{\prime})}{(1+\beta)(1+r)} We can also use the second line of the above derivation to see that \Rightarrow\ \align \frac{y^{\prime}-t^{\prime}+(1+r)s}{y-t-s} = \beta (1+r) \\ \Rightarrow\ \align \frac{c^{\prime}}{c} = \beta (1+r) Thus we can see that if $\beta(1+r)=1$, then there will be perfect consumption smoothing, $c^{\prime} = c$.