## Returns to Scale Estimation

In most instances, returns to scale can be easily estimated. For example, assume that all inputs in the unspecified production function Q = f(X, Y, Z) are increased by using the constant factor k, where k = 1.01 for a 1 percent increase, k = 1.02 for a 2 percent increase, and so on. Then, the production is

where h is the proportional increase in Q resulting from a k-fold increase in each input factor. From Equation 7.20, it is evident that the following relationships hold:

• If h > k, then the percentage change in Q is greater than the percentage change in the inputs, eQ > 1, and the production function exhibits increasing returns to scale.

• If h = k, then the percentage change in Q equals the percentage change in the inputs, eQ = 1, and the production function exhibits constant returns to scale.

• If h < k, then the percentage change in Q is less than the percentage change in the inputs, eQ < 1, and the production function exhibits decreasing returns to scale.

For certain production functions, called homogeneous production functions, when each input factor is multiplied by a constant k, the constant can be completely factored out of the production function expression. Following a k-fold increase in all inputs, the production function takes the form hQ = knf(X,Y,Z). The exponent n provides the key to returns-to-scale estimation. If n = 1, then h = k and the function exhibits constant returns to scale. If n > 1, then h > k, indicating increasing returns to scale, whereas n < 1 indicates h < k and decreasing returns to scale. In all other instances, the easiest means for determining the nature of returns to scale is through numerical example.

To illustrate, consider the production function Q = 2X + 3Y + 1.5Z. Returns to scale can be determined by learning how an arbitrary, say 2 percent, increase in all inputs affects output. If, initially, X = 100, Y = 200, and Z = 200, output is found to be

Qj = 2(100) + 3(200) + 1.5(200) = 200 + 600 + 300 = 1,100 units

Increasing all inputs by 2 percent (letting k = 1.02) leads to the input quantities X = 102, Y = 204, and Z = 204, and

Q2 = 2(102) + 3(204) + 1.5(204) = 204 + 612 + 306 = 1,122 units

Because a 2 percent increase in all inputs has led to a 2 percent increase in output (1.02 = 1,122/1,100), this production system exhibits constant returns to scale. 