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# Taxes and Retirement Accounts

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Hi Kim,

The program is correct and the reports indicate this. It's true that with higher inflation and higher nominal rates of return, the effective tax on asset income is higher because our tax system taxes not the real, but the nominal return on regular assets. So higher inflation would make having money in retirement accounts a better deal from a tax advantage compared to having regular assets whose return gets slammed annually by the higher effective asset income tax.

But you are rightly wondering why the agent, if he has all his money in his retirement account, should have higher spending. With all the money in the retirement account, the person just spends his smooth annual retirement account withdrawals, which are taxed no different from a stream of wages that rose with the inflation rate and continued until death. I.e., the higher effective capital income tax doesn't come to bear.

However, you'll rightly point out that with a 16 percent nominal return and a 13 percent inflation rate, the real pre-tax return is 1.16/1.13 - 1, which is lower than 1.06/1.03 -1. So you are asking why is the stream of withdrawals coming out of the account larger when the real return is lower. This, again, isn't a matter of taxes, because we're talking about the withdrawals before they are subject to taxation.

The answer is that the initial real balance in the retirement account is larger in the 16 percent nominal return case than in the 6 percent nominal return case. The reason is that when we ask for the initial asset balances we are considering them to be end of 2015 balances. We then assume that they earn the nominal rate of return you enter, i.e., 16 percent in the second case. The initial assets times 1 plus the nominal rate of return is the stock of real assets at the end of 2016 before spending and taxes and also before income. We measure all amounts in end of 2016 dollars. We assume that all flows occur at the end of 2016. So when we ask about wages in 2016, we assume they arrive at the end of the 2016. We also assume that taxes in 2016 are paid at the end of the year. Spending in 2016 occurs at the end of 2016. And we assume that retirement account withdrawals occur at the end of the year.

So you effectively made person 2 richer in real terms when you assumed a higher nominal rate of return. You can see this clearly in the detailed retirement account balance sheet. The end of 2016 assets are larger when you assume a higher inflation rate.

The accounting of the retirement account balance works like this (where i is the nominal interest rate, p is the inflation rate and r is the real interest rate. Note that (1+i) = (1+r) x (1+p), which is called Fisher's Equation in honor of Irving Fisher who discovered the relationship between the real return, the nominal return, and the inflation rate.

A2016 = A2015 x (1+i) - W2016 = A2015 x (1+r) x (1+p) - W2016 = A2015 x (1+p) + r x (1+p) x A2015 - W2016

= A2015 x (1+p) + [ (1+i)/(1+p) - 1]x (1+p) A2015 - W2016 = A2015 x (1+p) + [(1+i) - (1+p)]x A2015 - W2016.

Note that [(1+i) - (1+p)] = .03, which is whey you see 30,000 in asset income in the retirement account balance report.

The [(1+i) - (1+p)] is the real return in 2015 dollars converted into 2016 dollars, i.e., it's (1+p) x [(1+i)/(1+p) - 1],

where (1+i)/(1+p) - 1 is the model's real rate of return.

best, Larry

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