**[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?

**[A:]** Thanks for your question!

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

**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.****Q=.5KL**Again we put in our multipliers and create our new production function.Q’ = .5(K*m)*(L*m) = .5*K*L*m

^{2}= Q * m^{2}Since m > 1, then m

^{2}> m. Our new production has increased by more than*m*, so we have**increasing returns to scale**.**Q=K**Again we put in our multipliers and create our new production function.^{0.3}L^{0.2}Q’ = (K*m)

^{0.3}(L*m)^{0.2}= K^{0.3}L^{0.2}m^{0.5}= Q* m^{0.5}Since m > 1, then m

^{0.5}< 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.

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