Perceiving change can be a funny thing. Why is it we don’t notice change when it is happening right before our eyes? Every time grandma comes to visit, she gushes over how big the kids have grown, but you hardly notice while seeing them day after day. You likely don’t notice the slow leak in a cars tire, or a minor change in temperature of just a few degrees. Change is hard to detect, unless you have a manner of comparison. How do you know when something is materially different? How much must change before we even realize there is even a change afoot? Well, it’s called the Just Noticeable Difference (JND) and it has many implications in our financial lives.
Weber’s Law and the Just Noticeable Difference
In 1834, psychophysicist E. H. Weber formulated one of the most fundamental insights into human perception. Weber determined how to quantify the perception of change in a given stimulus. He found there is a relationship between the quantity or size of something and how much more must be added to it for us to be able to perceive that an increase has taken place. The perceived change is called the just noticeable difference and this insight became coined as Weber’s Law.
According to Weber the change in a stimulus that will be just noticeable is a constant ratio of the original stimulus. Weber’s Law looks like this:
where I is the original intensity of the stimulation, ΔI is the addition to it required for the change to be perceived (the JND), and k is a constant. If you graph it out (like the example graph below), you will notice there is a linear relationship where as the intensity increase, so does the JND.
Interestingly, experiments have determined different threshold constants that exist for different sensations for most people. For light intensity the threshold has been found to be 8%. That is, the light intensity must increase by 8% before the average person will notice the change in intensity. For determining a change in weight the change must be at least 2% heavier or lighter than the original weight for most people to sense the difference.
So how does Weber’s Law work in practice? Let’s look at a thought experiment he put together. If you hold an object weighing 20 pounds in your outstretched hand you can perceive how much it weighs. But how much would you have to add to your hand for you to notice that the weight is heavier? (for those of you who dislike math, please bear with me for a moment)
You can rearrange the formula, so the known variables are on the same side of the equation to get ΔI=I*k. This alternative says the same thing, but in a different way. The change in intensity needed to just notice a difference (JND) is proportional (k) to the original intensity (I). This implies that low intensity stimuli (e.g. a dim light or a light weight) require very small changes in their intensity in order to be just noticeably different, but that high intensity stimuli (e.g. a bright light or a heavy weight) require a much larger change in their intensity to be just noticeably different.
Using the above example we take I=20 pounds (weight stimulus) and multiply by 0.2 (known weight constant of 2%) to determine the JND for this particular weight. In this case the JND for a change from 20 pounds is 4 pounds. This means you might not notice the change in weight if you only add 2.5 lb. However, you would notice if you added a 5 lbs.
Now, if you start with a 100 lb. weight, it would take as much as 20 pounds to be added to the original weight for you to notice the difference. In other words, the just noticeable difference (JND) changes depending on what the starting quantity is. The larger the starting point, the larger the change must be for it to be noticed. Still not clear? Khan Academy has a nice 9-minute video explaining Weber’s Law. You can watch it here. So, what does Weber’s Law have to do with our finances?
Weber’s Law and Spending Decisions
Weber’s Law has been used across all different senses as well as an explanation for a consumer’s sensitivity to pricing. It can be applied to pricing by identifying the point at which a price change is ‘noticed’ by the consumer sufficiently to change how they think and act. In effect, this means that when the price is low, a small change in price is seen as significant, while higher prices may vary quite significantly before we change our preference. Depending on how large of a price you start with will determine your sensitivity to price changes. This phenomenon can be a real problem for our bank account.
We often see small price increases across our products and services and don’t think much of them. Look at Netflix; they continue to raise their prices year-to-year but do it in such a way that retains their customers and doesn’t cause people to drop their service. This is because the perception of the actual price change isn’t big enough to change most people’s behavior.
We often make poor spending choices because we are sensitive to the cost of the wrong things. When I go grocery shopping, I am always comparison shopping between brands for like items. I am very sensitive to price differences between similar items (Cheerios vs Toasted Oats) and will generally opt for the store brand item over the name brand simply because of the cheaper price (assuming items are qualitatively similar).
You might think the same frugal habits translate across all types of goods, but you would be wrong. Most people are not nearly so cost conscious when it comes to buying large items. This past year in Colorado we were hit with two terrible hail storms a week apart. As a result, my stalwart 14-year-old pickup truck was totaled. I wasn’t planning on being in the car buying business, but mother nature had other plans. My recent car buying experience serves as a good example of Weber’s Law in action.
I narrowed my search down to two similar vehicles priced a few thousand dollars apart. This difference in price didn’t deter me from considering the more expensive model. In fact, I bought it! Afterwards, the irony struck me. Here I was willing to modify my consumption preferences when it came to grocery shopping and consistently opt for the cheaper option, but when it came to the difference in price of a few thousand dollars with a major car purchase, I was less price sensitive.
The reasoning is because the size of the starting point. The difference between $3.99 and $2.99, while monetarily small, is a quarter of the total purchase price. Alternately, The difference between $25,000 and $27,000, while monetarily larger, still only represents 7% of the purchase price. Understanding how we can sabotage ourselves because our JND for price sensitivity on larger versus smaller goods can help us become savvier shoppers and consequently better savers too.
Weber’s Law and Debt
Weber’s Law has other implications with our finances than just price sensitivity and shopping. It has an impact on the psychology of indebtedness. Our ability to accrue and payoff debt is subject to the same linear principles found with weightlifting. If you start with a small amount of debt, each purchase you put on your credit card is noticed and likely elicits a negative emotional response. However, if you’ve already accrued a large amount of debt, you may not even notice additional purchases added along the way.
Dave Ramsey’s debt snowball is a good example of Weber’s Law in action. The reason the debt snowball works, is because it takes advantage of our perceptions of change. By now you probably can see why. A change in a small starting point is perceived greater than an equal change in a larger starting point.
If you start paying off a large debt first, you are less likely to maintain the momentum required to be debt free because the initial stimuli (in this case debt) is large, it will take a much larger decrease in the balance for you to notice that you are making any kind of headway. Focusing on a small debt first has the opposite effect.
Assume you have budgeted $250 a month towards debt repayment. You have two debts. The first is a consumer credit card with a balance of $10,000, the second is a medical bill of $1000. That same $250 payment represents either 1/4 or 1/40th of the total balance. You can guess which one you will feel progress and a sense of accomplishment sooner.
If you find yourself in debt, follow Dave Ramsey’s debt reduction strategy. We can make a logical argument to tackle the higher percentage first, however, Weber’s Law would tell us Dave is on to something. Start small and build momentum, you’ll be glad you did.
Weber’s Law and Investing
Hopefully by now you understand how small numbers matter. Much like our sensitivity to pricing, we have a predilection to not notice small numbers while making investment decisions. That’s because when we are trying to determine what funds to invest in we are usually dealing with large sums of money (or the hope they will grow into large sums over time), and as such, our starting point is large.
When it comes to retirement, we are often talking about the need to make our investments grow into hundreds of thousands of dollars. Because this “number” is so large we often don’t bat an eye at expense ratios and tax consequences of the funds we choose. When comparing similar funds, a tenth of a percentage point in fees doesn’t seem like a significant factor. After all, we all think we’re going achieve gains of 10-12% right? Unfortunately, even small reductions in our return cause by expenses, fees, and taxes will have a materially large effect on our overall return. (see my post on why your investment returns are lower than you think).
Every fund has an expense ratio which measures operating expenses relative to the total value of the fund. Operating expenses consist of the fund manager’s fee, marketing costs (also known as a 12b-1 fee), custodial services, record keeping, taxes, legal expenses, accounting fees, etc. Adding all these fees together can seriously influence your overall returns.
Here is an example found over at Vanguard how investment costs might not seem like a big deal, but they add up, compounding along with your investment returns. In other words, you don’t just lose the tiny amount of fees you pay, but you also lose all the growth that money might have had for years into the future.
Imagine you have $100,000 invested. If the account earned 6% a year for the next 25 years and had no costs or fees, you’d end up with about $430,000. If, on the other hand, you paid 2% a year in costs, after 25 years you’d only have about $260,000. That small little 2% you paid in fees every year would wipe out almost 40% of your final account value. Be wary of your errors in perception caused by Weber’s Law. Small numbers matter when we are compounding their effect year after year
Weber’s Law and Savings
Saving enough can be tough. We know this is a problem because the average savings rate in the U.S. sits at a paltry 6 percent. Fortuantley, just as much as Weber’s Law works against us in our spending and investing, it can actually help trick us into saving. There is a great book by David Bach called the Automatic Millionaire. The premise of the book is straight out of Weber’s Law.
A quick synopsis; while Bach was teaching finance at a local adult education program, he met a man who completely changed his perception of how to save. The man had saved over a million dollars to include owning two houses out right, all on a salary of $53,946. How did he do it? He set up automatic savings deductions straight from his paycheck and essentially paid himself first. He didn’t live some super frugal life, but he was diligent month in and month out.
He also didn’t start with a huge savings goal. Had he done so, and aggressively set aside too much of his total earnings, he likely would have abandoned his strategy. Rather, he focused on automatically investing just 10 percent of his paycheck every month. This amount was below his just noticeable difference threshold. By being insensitive to the amount of money that went into savings he was able to sustain the savings habit for decades. The best quote comes from his wife, who was an equal partner in their savings strategy.
“You can’t spend what you can’t see!”