Increasing, Decreasing and Constant Returns to Scale

How to identify increasing, decreasing and constant returns to scale

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The term "returns to scale" relates to how well a business or company is producing. It tries to pinpoint increased production in relation to factors that contribute to that production over a period of time. 

Most production functions include both labor and capital as factors. So how can you tell if that function is increasing returns to scale, decreasing returns to scale, or if the returns are constant or unchanging to scale?

These three definitions look at what happens when you increase all inputs by a multiplier

For illustrative purposes, we'll call the multiplier m. Suppose our inputs are capital or 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.

About Multipliers 

The multiplier must always be positive and greater than 1 because the goal here is to look at what happens when we increase production. An m of 1.1 indicates that we've increased our inputs by .1 or 10 percent. An m of 3 indicates that we've tripled the amount of inputs we use.

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 whatever your professor requires. 

Three Examples of Economic Scale

  1. Q = 2K + 3L. We will increase both K and L by m and create a new production function Q’. Then 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 I replaced (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. So we have constant returns to scale.

  1. Q=.5KL Again we put in our multipliers and create our 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.

  2. Q=K0.3L0.2 Again we put in our multipliers and create our 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 constant returns to scale, this way is the fastest and easiest. By using the m multiplier and simple algebra, we can answer our economic scale questions.

Remember that even though people often think about returns to scale and economies of scale as interchangeable, they are importantly different. Returns to scale only consider production efficiency while economies of scale explicitly consider cost.