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 <math>F(\vec{x})</math>, where <math>\vec{x} \in \bold{R}^{n}</math> is a vector of inputs, constant returns to scale are characterized by <math>F(s\vec{x}) = sF(\vec{x})</math>, where <math> s > 0 </math>.

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

<math> 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}</math>

Since for a competitive market, the payment to each factor of production (<math>x_i</math>) is <math>\frac{\partial F}{\partial x_i}</math>, 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: <math> F(\vec{x}) = \vec{a}\vec{x}</math>

• Cobb-Douglas production function with <math>\alpha \in [0,1]</math>: <math>Y = AL^{\alpha}K^{1-\alpha}</math>

  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 <math>\alpha = 0.33</math>.