Do Bigger US Banks Run Leaner? Exact Elasticity Accounting for the Efficiency Ratio of 4,280 FDIC-Insured Institutions, June 2026

The efficiency ratio, non-interest expense divided by net interest income plus non-interest income, is the operating-cost measure that banks and supervisors watch most. I use FDIC Call Report data for all 4,313 insured institutions on 30 June 2026 (4,280 after removing loss-making outliers). Two exact identities organise the analysis. The aggregate ratio, 56.24%, is the revenue-weighted mean of bank ratios, so it is dominated by the largest banks, while the median bank's ratio is 62.45%. And the least-squares slope of the log ratio on log assets is exactly the cost elasticity minus the revenue elasticity. Non-interest expense rises with assets with elasticity 0.915 and revenue with elasticity 0.961, so the ratio falls by 0.046 log points per log point of assets, about 10% for each tenfold rise in size; with controls for specialisation and state the fall is 0.067. The gain is concentrated among small banks. A piecewise fit gives slopes of minus 0.056 below USD 1 billion and minus 0.040 from USD 1 to 10 billion, but only minus 0.016 (standard error 0.024) from USD 10 to 100 billion and plus 0.011 (0.025) above USD 100 billion. Median ratios fall from 70.8% for banks under USD 100 million to 54.7% for those above USD 250 billion. Quantile regressions show the size effect is twice as strong for the least efficient banks (minus 3.65 points per log point at the 90th percentile) as for the most efficient (minus 1.91 at the 10th). Median return on assets peaks at 1.26% for USD 1 to 10 billion banks.