# Constant returns to scale

A process displays constant returns to scale when increasing all inputs by a factor of s leads to an increase in output by the same factor. In economics, this is in terms of the production function of the process — for instance, that of a business, company, or economy as a whole. Mathematically, if the function is $F(\vec{x})$, where $\vec{x} \in \bold{R}^{n}$ is a vector of inputs, constant returns to scale are characterized by $F(s\vec{x}) = sF(\vec{x})$, where s > 0.

A function displaying constant returns to scale is homogeneous of degree one, and by Euler's Theorem can be written as:

$F(\vec{x}) = x_1\frac{\partial F}{\partial x_1} + x_2\frac{\partial F}{\partial x_2} + ... + x_n\frac{\partial F}{\partial x_n}$

Since for a competitive market, the payment to each factor of production (xi) is $\frac{\partial F}{\partial x_i}$, this implies that that the total payments to all factors of production exhaust output (F), and the market has zero profit.

## Examples of functions displaying constant returns to scale

• Linear demand or supply: $F(\vec{x}) = \vec{a}\vec{x}$
• Cobb-Douglas production function with $\alpha \in [0,1]$: Y = ALαK1 − α
The variables have the interpretation of Y as output, L as labour, K as capital, and A is multifactor productivity. In general, empirical studies have shown that the United States economy has approximately α = 0.33.