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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\vec{x})} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} ) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\vec{x}) = \vec{a}\vec{x}}

• Cobb-Douglas production function with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in [0,1]} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0.33} .