The Marriner S. Eccles Federal Reserve Board Building in Washington. (Andrew Harrer/Bloomberg News)
Jared Bernstein, a former chief economist to Vice President Joe Biden, is a senior fellow at the Center on Budget and Policy Priorities and author of 'The Reconnection Agenda: Reuniting Growth and Prosperity'.

Those of us following key economic variables right now — unemployment, inflation, wages, debt, interest rates — may soon want to heed the warning of mathematician Jordan Ellenberg: “Thinking nonlinearly is crucial, because not all curves are lines.” (If we ever want to find the road back to Factville, Ellenberg’s book “How Not to be Wrong” provides the map.)

What is nonlinearity, why does it matter, and how does it show up in the economy?

Suppose you’re shopping and a mango is $1. (Honk if you love mangoes as much as I do.) If two mangoes cost $2, three cost $3, and so on, we’re into a linear, one-to-one relationship. If, however, the grocer really wants to move the mangoes, she might sell 3 for $2.50, breaking the linear pattern and introducing a nonlinearity.

In economics, nonlinearities often arise when dormant relationships reactivate.

Unemployment and inflation (or wages), or budget deficits and interest rates, have long belied the textbook correlations that are supposed to exist between them. Tight labor markets, so the story goes, generate higher labor costs, which bleed into higher prices. But that correlation has been low for years, leading many to declare its death.

Research by economist Joe Gagnon shows that the death knell may be premature. The reason is — you guessed it — a nonlinearity in the relationship between employment and inflation. As his figures below reveal, at low inflation, the correlation is nowhere to be seen. (In his data, a higher “employment gap” means a tighter labor market.) At higher inflation, it comes back to life.

Source: Joe Gagnon (Jared Bernstein/Joe Gagnon)

What about the relationship between lower unemployment and faster wage growth? This one looms large in my work, as I’ve long argued that tighter labor markets provide workers more of the bargaining clout they so sorely lack. That correlation is in the data, but there’s a clear nonlinearity: Going from, say, 7 percent to 6 percent unemployment does less for wage growth than going from 4 percent to 3 percent.

These scatterplots show yearly, nominal wage growth for middle-wage workers at high unemployment (greater than 5.5 percent) and low unemployment (less than 4.5 percent). Again, the figure on the left shows a pretty random scatter of dots. But once the job market tightens, the dots start to line up in the expected order.

Nonlinearities in wage growth at high and low unemployment (Jared Bernstein/The Washington Post)

Finally, recent data shows something we haven’t seen in a long time in advanced economies: Interest rates have been climbing. The figure below shows rates on government bonds at various maturities, which all started rising around 2017.

Source: Federal Reserve data (Jared Bernstein/Jared Bernstein)

This is, of course, largely the work of the Federal Reserve. After holding the benchmark rate it controls at zero for years during and after the last recession, it started slowly raising the rate in 2016. But another factor in play is our highly unusual fiscal stance, wherein we’re stimulating a near-full-capacity economy with hundreds of billions of deficit spending and tax cuts. Here again, another long-dormant relationship could turn out to be nonlinear: Rates have been unresponsive to growth, low unemployment and budget deficits for years. That could be changing.

There could also be nonlinear interactions in play: Inflation behaves as Gagnon suggests, leading bond investors to insist on higher premiums against inflation, a mechanism by which the inflation nonlinearity maps onto the interest rate nonlinearity.

We don’t have great explanations for nonlinear economics, but I suspect it’s pretty simple. Consider an employer who resists raising pay, even as the jobless rate falls because she can still find the workers she needs without bumping up her wage offer. (The absence of union power is another big problem here.) But at some point, because the job market is so tight, she can no longer resist. She either raises wages or leaves potential profits on the table.

A similar dynamic may prevail for pricing power. Because American shoppers are price-sensitive and inflation expectations are “well-anchored” (meaning we expect inflation to stay around the Fed’s 2 percent target), producers all along the supply chain resist passing price hikes down the chain until their profit margins start taking haircuts they can’t live with. Then the dam breaks, and boom! A nonlinearity is born.

The policy implication of nonlinearities is also simple: Do not assume constant, unchanging correlations. Policymakers can’t fall asleep at the assumed-to-be-linear switch. Neither should they overreact, freaking out when expected relationships reappear. The fact that inflation, interest rates and wage growth are crawling off the mat does not signal a fire that must be extinguished with aggressive rate hikes. To the contrary, they are signs of a healthy economy.

After all, nonlinearities are a lot more interesting then linearities. I don’t know about you, but I’m really glad that not all curves are lines.

Correction: The mislabeled lines in the interest rate figure have been fixed. Thanks to BDub Miami in comments.