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# Increasing, Decreasing, and Constant Returns to Scale

## Increasing, Decreasing, and Constant Returns to Scale

[Q:] I was wondering if you could help me. If I have a production function that has both labor and capital as factors, how can I tell if it is increasing returns to scale, constant returns to scale, or decreasing returns to scale?

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

Our multiplier must always be positive, and greater than 1, since we want to look at what happens when we increase production. An m of 1.1 indicates that we've increased our inputs by 10% and an m of 3 indicates that we've tripled the amount of inputs we use. Now we will look at a few production functions and see if we have increasing, decreasing, or constant returns to scale. Note that some textbooks use Q for quantity in the production function, and others use Y for output. It does not change this analysis any, so use whatever your professor uses.

### 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.

2. 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.

3. 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

Since 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 for determining whether or not 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.

If you have a question on economies of scale, macroeconomics, or any other economics topic you'd like answered please use the feedback form.

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