CONCEPT OF PRODUCTION FUNCTION
The production function relates the output of a firm to the amount of inputs, typically capital and labor
In a general mathematical form, a production function can be expressed as:
Q = f(X1,X2,X3,...,Xn)
Q = quantity of output
X1,X2,X3,...,Xn = quantities of factor inputs (such as capital, labour, land or raw materials). This general form does not encompass joint production; that is a production process that has multiple co-products or outputs
COBB-DOUGLAS PRODUCTION FUNCTION
A standard production function which is applied to describe much output two inputs into a production process make. It is used commonly in both macro and micro examples.
For capital K, labor input L, and constants a, b, and c, the Cobb-Douglas production function is:
f(k,n) = bkanc
If a+c=1 this production function has constant returns to scale. (Equivalently, in mathematical language, it would then be linearly homogenous.) This is a standard case and one often writes (1-a) in place of c. Log-linearization simplifies the function, meaning just that taking logs of both sides of a Cobb-Douglass function gives one better separation of the components.
In the Cobb-Douglass function the elasticity of substitution between capital and labor is 1 for all values of capital and labor
STAGES IN PRODUCTION FUNCTION
To simplify the interpretation of a production function, it is common to divide its range into 3 stages. In Stage 1 (from the origin to point B) the variable input is being used with increasing output per unit, the latter reaching a maximum at point B (since the average physical product is at its maximum at that point). Because the output per unit of the variable input is improving throughout stage 1, a price-taking firm will always operate beyond this stage.
In Stage 2, output increases at a decreasing rate, and the average and marginal physical product are declining. However the average product of fixed inputs (not shown) is still rising, because output is rising while fixed input usage is constant. In this stage, the employment of additional variable inputs increases the output per unit of fixed input but decreases the output per unit of the variable input. The optimum input/output combination for the price-taking firm will be in stage 2, although a firm facing a downward-sloped demand curve might find it most profitable to operate in Stage 1. In Stage 3, too much variable input is being used relative to the available fixed inputs: variable inputs are over-utilized in the sense that their presence on the margin obstructs the production process rather than enhancing it. The output per unit of both the fixed and the variable input declines throughout this stage. At the boundary between stage 2 and stage 3, the highest possible output is being obtained from the fixed input