Question

Suppose the interest rate declines on certain types of investment ac- counts commonly used by older Americans planning for retirement. Further, suppose that the decline was a surprise, but now that it has happened, it is expected to persist for the foreseeable future. Con- sider a 62 year old worker who lives alone and is still working, who is determining when in the future to enter retirement, and who has a retirement account impacted by the interest rate change. (a) How will the change in this interest rate affect this person's present value of lifetime income (PVLI) at different potential re- tirement ages? (b) How will the change in interest rates impact the worker's optimal retirement age. Appeal to income and substitution effects in your answer.

Answer 1

(a) Let's consider the 62 years old man who is planning to retire. Suppose the man wants to have certain amount of money each year to spend let's call it Yr and given the market intrest rate on the investment we can calculate the present the value that it must have in order to give the man Yr after the retirement.

The formula to calculate the present value,

= Yr/(1+i)^T

Here, Yr = amount of money that the retiree wants to have after retirement.

i = Interest rate on investment.

T = years after the retirement.

The present value formula tells us about the money that we need to have to today in order to have a certain amount of money in future given the interest rate.

And it is not hard see from the formula that the present value and the interest rates are inversely related. That is to say that when interest rates are high people will have to save less money in order to get a particular amount of money in future after their retirement. Or in other words the income after the retirement will have a lower present value if the interest rates are high.

And when interest rates decreases the present value of after retirement income increases which means now the man who is planning to retire will have to save more money to get the same amount of income after the retirement.

Conclusion : The present value and interest rates are inversely related, which means when interest rates decrease the retirees are required to save more today in order to get a certain amount on income after retirement and when interest rates increases they are required to save less to get the same amount of income after retirement.

b. Let's find out how the interest rate affects the optimal retirement age. When interest rates are high a low sum of money will have a higher future value so the man will think that his future income has increase due to increase in interest rates this is what we call income effect, when due to some reason our real income increases. So the person would be tempted to retuire early since his future income has been increased due to increased interest rates.

But there is another effect that comes into play. When interest rates increases, every dollar saved today will give a higher return in future. So retiring when interest rates are high will become costlier since the foregone income now has higher future value and if one chooses to retires he will lose the opportunity to earn and save more. So this what we call the substitution effect when retiring today becomes costlier.

As you can see the income and substitution effect of increased interest rates works in opposite direction. The income effect makes the person rich and induce him to retire early while the substitution effects makes the retirement costlier which induces the man to work more.

So what will happen will totally depend on the which force is greater, that whether income effect dominates the substitution effect in that case person's optimal retirement age will be lower. But if the substitution effect dominates the income effect then the person will choose to work more, in that case the optimal retirement age will be higher.