Finding the Optimal Level of Consumption with a Finite Horizon

All of these calculations are based on the assumption that, trough a Roth IRA you can avoid all taxes on money that you sock away for the future, though you have to pay taxes on your labor earnings before you have money that can be put into the Roth IRA. If you have a regular IRA or retirement saving account instead, the biggest adjustment you need to make is to take the amount in your account statement and cut it in half or by some other factor to allow for all the taxes that will have to be paid on it when you withdraw the money. In other words, what looks like $1 million is probably only about $500,000 for the calculations below. Assuming you can save all you want in a tax-sheltered account (which may not be true) everything else (other than realizing that the money in your account needs to be cut in half or so because of taxes) should work the same in the ultimate calculation with a regular retirement saving account instead of a Roth one, but explaining why is complex. 

To find the optimal level of consumption, first, compute the present discounted value of your lifetime resources as best possible. This present discounted value of lifetime resources is called Modigliani full wealth. Key things to include are 

1. Put in your current financial wealth as is; since it is already there in the present, its present value is whatever your account statements say. That is, unless you will owe taxes on withdrawals. In that case, its after-tax present value is something like ½ what your account statement says.
2. Find the present discounted value of your labor income (salary and wages). If your future labor income is uncertain, doing things exactly right is very hard. As a rule of thumb, my best guess right now is that you should put in the 10th percentile outcome for your labor income to allow for risk. That is, put in a level of labor income where there is a 90% chance that you will do better than that. Make sure to allow for labor income taxes (including the income tax, the social security tax and the medicare tax), and only calculate the value of your after-tax labor income.   
3. Find the present value of your future social security benefits. 
4. Subtract the present value of future bequests and gifts to your kids that you want to be sure to make. (Remember to discount: given reasonable interest rates, $1 million 70 years from now is a lot easier to come by than $1 million now.)

Once you have calculated your Modigliani full wealth, multiply it by the optimal finite horizon propensity to consume B_T, where T is the number of years you have left to live (or estimate that you have left to live).  

The Formula for the Optimal Finite-Horizon Propensity to Consume:

B_T = B_infinity/( 1 - e^{-B_infinity T} )

To calculate the optimal finite-horizon propensity to consume, first find the optimal infinite-horizon propensity to consume. Figure out how many doublings you have at rate B_infinity for T years, or whatever else you need to do to calculate e^{-B_infinity T}.  Divide B_infinity by (1 - e to the B_infinity times T) to get B_infinity/( 1 - e^{-B_infinity T} ) .

In the problems below, you are asked to find the optimal finite horizon propensity to consume using the rule of 70.  That is, treat ln(2) as if it were equal to exactly .7.  

If B_infinity is negative, you will find that the formula for B_T still works fine.  If B_infinity is zero, L'Hopital’s rule gives the answer that B_T = 1/T

Exception to Formula Above: If the formula above would give 0/0, then the answer is 1/T.

Practice Problems: Find B_T Given T and B_infinity

Answers: