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 ${\vec {x}}\in {\mathbf {R}}^{n}$ 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 ($x_{i}$) 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 $\alpha \in [0,1]$: $Y=AL^{\alpha }K^{1-\alpha }$

  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 $\alpha =0.33$.