A concave relationship is one where the dependent variable rises with the explanatory variable but at a diminishing rate, so the curve bends downward and eventually flattens. Marginal effects shrink as the input grows, distinguishing it from a linear or convex (accelerating) response.
How it works
Formally, a function is concave when its second derivative is non-positive: each additional unit of input yields a smaller increment of output. Graphically the line increases monotonically but bows toward the horizontal — rising then flattening — as opposed to convexity, where the slope steepens.
Why it matters now
In 2025-26 central-bank reaction-function research, a concave link between policy effort and uncertainty implies diminishing returns to hawkishness: beyond a threshold, additional tightening or signalling buys progressively less stabilisation, complicating forward guidance and terminal-rate calibration.
Example
If the MPC's stabilising response to economic uncertainty rises sharply at low uncertainty but plateaus at high uncertainty, the response curve is concave: moving from low to moderate uncertainty might add 50bp of implied tightening, while moving from moderate to extreme adds only 10bp — the marginal policy reaction fades even as uncertainty climbs.
Frequently asked
- What is a concave relationship?
- A concave relationship is one where output rises with the input but at a shrinking rate, so the curve bends downward and flattens. Its second derivative is non-positive: each additional unit of input adds less than the previous one. This contrasts with linear responses (constant slope) and convex ones, where the slope steepens as the input grows.
- How does a concave relationship differ from a convex one?
- A concave relationship has diminishing marginal effects (second derivative ≤ 0, curve flattening), while a convex relationship has increasing marginal effects (second derivative ≥ 0, curve steepening). Concavity means the curve lies above its chords; convexity
Glossary · convexity
Convexity is the curvature in the relationship between an asset's price and an underlying variable such as yield — the second-order term beyond linear (delta or duration) sensitivity. Positive convexity means gains accelerate and losses decelerate as the variable moves, an asymmetry that rewards holders when volatility rises.
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- Why does concavity matter for central-bank reaction functions?
- Concavity matters because it implies diminishing returns to monetary effort: beyond a threshold, extra tightening or signalling buys progressively less stabilisation. A concave link between policy response and uncertainty means moving from low to moderate uncertainty triggers a large reaction, but moving from moderate to extreme adds little — complicating terminal-rate calibration and forward guidance
Glossary · forward guidance
Forward guidance is a central bank's communication about the likely future path of policy rates, used to shape market expectations and steer financial conditions today. It comes in calendar-based (date-contingent), state-based (outcome-contingent on inflation or unemployment thresholds), or qualitative open-ended forms.
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- What is an example of a concave function in economics?
- Diminishing marginal utility is the classic concave example: each extra unit of consumption raises utility, but by less than the prior unit. Production functions with diminishing returns to a single input, log utility, and the short-run Phillips curve at high activity levels are also concave. The shared feature is a positive but shrinking slope.
- How do you identify concavity in a dataset or model?
- Concavity is identified when fitted marginal effects decline as the explanatory variable rises, or when a second derivative estimate is consistently negative. Analysts test it with quadratic or spline regressions, interaction terms, or by inspecting whether the curve lies above the straight line joining its endpoints. A significant negative squared term signals concave curvature.