The term "returns to scale" refers to how well a business or company is producing its products. It tries to pinpoint increased production in relation to factors that contribute to production over a period of time.
Most production functions include both labor and capital as factors. How can you tell if a function is increasing returns to scale, decreasing returns to scale, or having no effect on returns to scale? The three definitions below explain what happens when you increase all production inputs by a multiplier.
For illustrative purposes, we'll call the multiplier m. Suppose our inputs are capital and labor, and we double each of these (m = 2). We want to know if our output will more than double, less than double, or exactly double. This leads to the following definitions:
- Increasing Returns to Scale: When our inputs are increased by m, our output increases by more than m.
- Constant Returns to Scale: When our inputs are increased by m, our output increases by exactly m.
- Decreasing Returns to Scale: When our inputs are increased by m, our output increases by less than m.
The multiplier must always be positive and greater than one because our goal is to look at what happens when we increase production. An m of 1.1 indicates that we've increased our inputs by 0.10 or 10 percent. An m of 3 indicates that we've tripled the inputs.
Three Examples of Economic Scale
Now let's look at a few production functions and see if we have increasing, decreasing, or constant returns to scale. Some textbooks use Q for quantity in the production function, and others use Y for output. These differences don't change the analysis, so use whichever your professor requires.
- Q = 2K + 3L: To determine the returns to scale, we will begin by increasing both K and L by m. Then we will create a new production function Q'. We will compare Q' to Q.Q' = 2(K*m) + 3(L*m) = 2*K*m + 3*L*m = m(2*K + 3*L) = m*Q
- After factoring, we can replace (2*K + 3*L) with Q, as we were given that from the start. Since Q' = m*Q we note that by increasing all of our inputs by the multiplier m we've increased production by exactly m. As a result, we have constant returns to scale.
- Q=.5KL: Again, we increase both K and L by m and create a new production function. Q' = .5(K*m)*(L*m) = .5*K*L*m2 = Q * m2
- Since m > 1, then m2 > m. Our new production has increased by more than m, so we have increasing returns to scale.
- Q=K0.3L0.2:Again, we increase both K and L by m and create a new production function. Q' = (K*m)0.3(L*m)0.2 = K0.3L0.2m0.5 = Q* m0.5
- Because m > 1, then m0.5 < m, our new production has increased by less than m, so we have decreasing returns to scale.
Although there are other ways to determine whether a production function is increasing returns to scale, decreasing returns to scale, or generating constant returns to scale, this way is the fastest and easiest. By using the m multiplier and simple algebra, we can quickly solve economic scale questions.
Remember that even though people often think about returns to scale and economies of scale as interchangeable, they are different. Returns to scale only consider production efficiency, while economies of scale explicitly consider cost.