Wednesday, October 25, 2017

A tale of two limits

The Trump administration is considering a reduction in the yearly 401(k) contribution limit from the current $18,000 to $2400.
Edit: Trump has stated 401(k) limit reductions are not going to be a part of the plan.
Comment: Trump says a lot of things, so whatever.

The goal is offset lost tax revenue due to the nominal lowering of taxes (though without knowing the exact details, we can't be sure who really is hurt or helped and by how much). To understand this a little better, we have to start with what a 401(k) account is.

A 401(k) is a long-term, incentivized, savings account. Money placed into it by an individual is not taxed now. Generally speaking, contributed money is invested (bonds, stocks, mutual funds, T-bills, ... ) and grows. The money and all gains are also locked away until age 59.5 (modulo penalties and very specific circumstances that don't contribute to this discussion). Once a person starts taking money out, it is considered like normal income and is subject to tax. As such, it is a tax deferral and not a full tax shelter. Any reference to tax sheltering is a mischaracterization. This is important because it's easy to try and cast it as an example of people using loopholes.

Because the money is not taxed now and the gains are not directly taxed, it provides a meaningful incentive for people to save long-term. In addition to being good for you, it's good if people have enough money in retirement to take care of themselves: they exit the workforce and make space for younger people, they aren't a drag on social safety nets or their kids, they get have a better quality of life as physical abilities diminish, etc. In short, everyone being self-sufficient for the long-term is good for us all.

It might be tempting to think that any incentivized savings is great, and that's a reasonably true statement. However, $2400 per year vs $18,000 per year is materially different because of _math_. Career lengths, incomes, and market rates of return vary, but we can use a couple canonical examples to illustrate the point.

Suppose someone starts working at age 18 and spends 45 years in the workforce. They contribute $2400 per year to their 401(k). If we assume a market-normal-ish return of 7% per year, they would reach age 63 with about $686k. That sounds like a nice amount, but consider the effects of inflation. At about 2% per year, we should expect our money to have devalued by about a factor of 2.5; that nest egg is worth about 280k in today's money. We have to think about these numbers in today's money because that's the only way to have a fair assessment of what its spending power is. If we assume someone has to live off it for 20 years, we're looking at 14-25k per year depending on how conservative or optimistic we are about consistent growth of the funds during retirement. The lower end of that range is right around the poverty line. Repeating the same math with the current cap yields about $5.1M, or a hair over $2M in today's dollars. If that needs to last 20 years, even a conservative approach will yield about $100k a year.

We can re-run the math for a variety of scenarios. Supposing a more modest 5% return, the Trump proposal would yield $157k in today's money. Even a most optimistic projection from there leaves the owner at poverty levels of income, while the current limit yields $1.1M in today's money and a median ~50k income in conservative projections. Or, if a person saves for 35 years, the range falls to $63-135k; yielding only a fraction of poverty-level income.

It's fair and important to point out that not everyone can take full advantage of a maximal 401(k) contributions. A median person making $45-50k per year is unlikely to have $18k to spare from that. However, they may be able to spare $5000. Being able to do that doubles their savings over time. The chart below shows accumulated savings in current dollars (assuming 2% inflation) at $2400 and $18000 per year savings rates. Green cells show a guaranteed 20 years at median income, red cells show 20 years guaranteed income at less than the poverty threshold ($12,084 per year). Savings scales linearly with contribution. For example, if you wanted to calculate the savings from $5000 per year over 35 years at 6%, just find the cell below under 2400 and multiply by 5000/2400 (and the answer would be $278,602).

2400 18000
Years\Percent 5 6 7 5 6 7
5 12011 12254 12501 90085 91902 93755
10 24764 25951 27202 185729 194631 204017
15 38480 41507 44811 288597 311299 336082
20 53406 59414 66213 400544 445602 496598
25 69819 80260 92525 523641 601949 693940
30 88029 104750 125158 660221 785623 938683
35 108390 133729 165893 812928 1002968 1244201
40 131302 168217 216991 984763 1261624 1627431
45 157220 209440 281312 1179153 1570801 2109842


There are many what-ifs in here, and clearly the current 401(k) limits don't guarantee a healthy retirement savings (due to inability to save, poor investment strategies, etc). But, a $2400 per year limit guarantees the savings is AT MOST barely yielding poverty-level disbursements at retirement age. When people have shown over and over that they are not good at saving, taking away a major savings-incentive vehicle seems not just imprudent but downright negligent.

So what does the proposal gain? We can't compute this exactly because we need access to a lot of savings figures and the details of the tax proposal, but we can make some estimates. For someone putting away a full $18,000 per year, the cap would mean they need to take $15,600 of their income instead of deferring it. For a high-income individual, that would be taxed at 33-39%. I think some people in the 25% tax bracket could still contribute the max, but it's unlikely people in lower brackets would be able to afford this. The benefit to the federal government runs about $4000-$6000 per such person per year. The number of such income earners is maybe the top 20%, and again this assumes everyone who can afford to does contribute the max. Multiplying the middle of the range by about 20% of the 160M tax filers yields about $160B. It's meaningful money, no doubt. Due to the imprecision and optimism of the estimations, I'd recast that number in the $100-150B range. It could be less, it could be more, but probably not by a factor of 2 off either end. Sadly, I think the majority of people are unable to or don't save more than $2400 a year into a retirement account (so the tax revenue benefit would be zero under the new plan).

The current tax proposal has been estimated to cost $1.5T over 10 years, so this 401(k) clawback actually seems to address a significant portion of the lost money, but at what expense? In addition to eroding the likelihood that individuals save for retirement, it pulls an accounting trick on us: the fed is collecting tax on current disbursements from 401(k) that have balances seeded with much larger limits AND will not front-load collection of taxes from future moneys that can't go into 401(k) anymore. As existing retirement assets run out (are disbursed), those taxes reduce drastically (because now they are fed by the new-model small 401(k) accounts) and we're left only with the new-model taxes. Essentially, the change gives us an in-between period where taxes get to double-dip and _look_ a lot more impressive than they will be for the long-term. It's literally borrowing from the future.

Wednesday, October 11, 2017

Hungry eyes

According to a recent article, there's an interesting wrinkle in the world of driverless cars: Apparently all the computing required to run one uses 2-4kW. As an additional quirk, this usage needs to happen more or less all the time, no matter how slow a car is going. In a typical gas-operated scenario, mileage is largely independent of speed (up to a point where air resistance starts going up a lot), but is heavily affected by starts and stops. Electrics improve upon this by sitting in pure standby while idle and recapturing some of the energy used for acceleration during braking.

The dynamic of a constantly operating energy sink alters mileage computations significantly, especially if it's using 2-4kW (about 3-5hp). I'll use the Nissan Leaf as an example (30 kW-hr battery, 100 mile range). Other electrics fare similarly enough for this analysis in consumption per mile.

We can calculate the loss of range off a full charge due to the computing. I'll use the following shorthand in computations:
S: average speed of travel (in mph)
D: distance traveled (in mi)
U: power utilized by computing (in kW)
C: battery capacity (in kW-hr)
T: time traveled (in hr)
R: maximum range


Full charge = energy used for miles + energy used over time
C = CD/R + UT

And the time spent traveling is:
T = D/S

C = CD/R + UD/S
C = D(C/R + U/S)
D = C / (C/R + U/S)

A semi-urban commute might happen at an average of 20-35mph. This seems counterintuitive because most speed limits are 35 or higher, but consider how often you have to stop at a light, or traffic is backed up. I'll use what I consider common examples, but your mileage may vary.

at 35mph, range is reduced to 72-84 miles.
at 20mph, range is reduced to 60-75 miles.
at 10mph, range is reduced to 43-60 miles.

Here's a table showing what fraction of nominal range is maintained at different speeds and power utilizations:

U1 U2 U3 U4 U5 U6 U7
Speed 4 3 2 1 0.5 0.3 0.1
75 0.840 0.875 0.913 0.955 0.977 0.986 0.995
70 0.831 0.867 0.908 0.952 0.975 0.985 0.995
65 0.820 0.859 0.901 0.948 0.973 0.984 0.995
60 0.808 0.849 0.894 0.944 0.971 0.982 0.994
55 0.794 0.837 0.885 0.939 0.969 0.981 0.994
50 0.778 0.824 0.875 0.933 0.966 0.979 0.993
45 0.759 0.808 0.863 0.927 0.962 0.977 0.992
40 0.737 0.789 0.849 0.918 0.957 0.974 0.991
35 0.710 0.766 0.831 0.908 0.952 0.970 0.990
30 0.678 0.737 0.808 0.894 0.944 0.966 0.988
25 0.637 0.700 0.778 0.875 0.933 0.959 0.986
20 0.584 0.651 0.737 0.849 0.918 0.949 0.982
15 0.513 0.584 0.678 0.808 0.894 0.933 0.977
10 0.412 0.483 0.584 0.737 0.849 0.903 0.966
5 0.260 0.318 0.412 0.584 0.737 0.824 0.933


Depending on the speed of a commute, the computer can realistically impose a 20% or even higher overhead on range. I've highlighted the chart to show classes of ranges. I've somewhat arbitrarily decided that more than a 20% loss is red, 10-20% loss is orange, 5-10% is yellow, etc. The green category I chose at 2% or less loss because that amounts to less than a mile per gallon at the mileage achieved by current hybrid cars, though full electrics are generally rated around double that (~100mpg equivalent).

Clearly with the current tech this would be a major hit, especially for commuter situations (where the vast majority of driving, especially of shorter range electrics). I've focused on electrics for the moment because it's easier to do the math without converting between gas and electric consumptions. That said, if we use the baseline that current EVs are rated at 100mpg equivalent, it means they are about 2-3 times as efficient at producing miles per unit energy. Perhaps ironically, this means gasoline engines take a proportionally smaller hit from the self-driving computers since the energy used to power the computers is proportionally less than that used by the engine itself. Here's a similar table of a typical hybrid (assume 50mpg):

U1 U2 U3 U4 U5 U6 U7
Speed 4 3 2 1 0.5 0.3 0.1
75 0.918 0.938 0.957 0.978 0.989 0.993 0.998
70 0.913 0.933 0.955 0.977 0.988 0.993 0.998
65 0.907 0.929 0.951 0.975 0.987 0.992 0.997
60 0.900 0.923 0.947 0.973 0.986 0.992 0.997
55 0.892 0.917 0.943 0.971 0.985 0.991 0.997
50 0.882 0.909 0.938 0.968 0.984 0.990 0.997
45 0.871 0.900 0.931 0.964 0.982 0.989 0.996
40 0.857 0.889 0.923 0.960 0.980 0.988 0.996
35 0.840 0.875 0.913 0.955 0.977 0.986 0.995
30 0.818 0.857 0.900 0.947 0.973 0.984 0.994
25 0.789 0.833 0.882 0.938 0.968 0.980 0.993
20 0.750 0.800 0.857 0.923 0.960 0.976 0.992
15 0.692 0.750 0.818 0.900 0.947 0.968 0.989
10 0.600 0.667 0.750 0.857 0.923 0.952 0.984
5 0.429 0.500 0.600 0.750 0.857 0.909 0.968


Also, ironically, as locomotive efficiency rises, the proportional drag of an always-on system rises. The same table for a 200mpg takes some pretty big hits even for a 100W system:

U1 U2 U3 U4 U5 U6 U7
Speed 4 3 2 1 0.5 0.3 0.1
75 0.738 0.789 0.849 0.918 0.957 0.974 0.991
70 0.724 0.778 0.840 0.913 0.955 0.972 0.991
65 0.709 0.765 0.830 0.907 0.951 0.970 0.990
60 0.692 0.750 0.818 0.900 0.947 0.968 0.989
55 0.673 0.733 0.805 0.892 0.943 0.965 0.988
50 0.652 0.714 0.789 0.882 0.938 0.962 0.987
45 0.628 0.692 0.771 0.871 0.931 0.957 0.985
40 0.600 0.667 0.750 0.857 0.923 0.952 0.984
35 0.568 0.636 0.724 0.840 0.913 0.946 0.981
30 0.529 0.600 0.692 0.818 0.900 0.938 0.978
25 0.484 0.556 0.652 0.789 0.882 0.926 0.974
20 0.429 0.500 0.600 0.750 0.857 0.909 0.968
15 0.360 0.429 0.529 0.692 0.818 0.882 0.957
10 0.273 0.333 0.429 0.600 0.750 0.833 0.938
5 0.158 0.200 0.273 0.429 0.600 0.714 0.882


However, it's likely that by the time cars are twice as efficient much will have changed about our traffic. It might coincide with entire fleets of driverless cars that can coordinate in real-time and completely avoid traffic jams. Slow speeds may completely disappear.


In the nearer term, the question is whether this 2-4kW consumption is likely to drop significantly. The current systems are, as far as I know, running as software on relatively commodity compute devices. This is necessary because the technology is nascent. Over time, many routines can move to FPGAs or ASICs, which tend to come with gigantic power savings. This white paper suggests an FPGA has nearly an order of magnitude savings over GPUs. There will also be fabrication improvements that drive efficiency per operation downwards; it's not unrealistic to combine those two factors along with other efficiencies and get down to the 100W range in the next 5-10 years. At 100W, it's no worse than a stereo, and its highly optimized driving can probably save that much power compared to a human-operated gas pedal.