A production function satisfies decreasing returns to scale when a proportional increase in all inputs results in an increase in output by a smaller proportion. This concept applies in the long run when all factors of production are scaled up simultaneously.

Mathematically, if the inputs $x_1$ and $x_2$ are increased by a factor $t$ (where $t > 1$), the production function exhibits decreasing returns to scale if the new output is less than $t$ times the original output. This is expressed as $f(tx_1, tx_2) < t.f(x_1, x_2)$.

For example, if a firm doubles all inputs (increasing them by a factor of 2), but the resulting output is less than doubled, the production function displays decreasing returns to scale. In the context of a Cobb-Douglas production function ($q = x_1^\alpha x_2^\beta$), this occurs when the sum of the exponents is less than one ($\alpha + \beta < 1$).