# Consider the following total cost function of a firm: C =50 +100q - 6q(TO SECOND POWER) + 0.5q...

## Question:

Consider the following total cost function of a firm:

C =50 +100q - 6q(TO SECOND POWER) + 0.5q (to third power)

Where, C is total cost and q is the level of output.

Determine the returns to scale for the following production function: Q=5K(to second power) +2LK + L (to second power)

## Returns to scale

Returns to scale refers to the changes in the output if the inputs were changed by some factor. For example, if you double the inputs and the output more than doubles, its called increasing returns to scale; if the output exactly doubles, its called constant returns to scale; and if the output less than double,s its called decreasing returns to scale.

## Answer and Explanation:

Given:

{eq}Q(L,K) = 5K^2 + 2LK + L^2 {/eq}

The easiest way to find returns to scale for any production function is to increase the amount of input by some positive constant. Therefore:

{eq}Q(tL,tK) = 5(tK)^2 + 2(tL)(tK) + (tL)^2 \\Q(tL,tK) = t^2*5K^2 + t^2*2LK + t^2*L^2 \\Q(tL,tK) = t^2*Q(L,K) {/eq}

This means that when you increase your input by t, the output will increase by {eq}t^2 {/eq}. Therefore, this represents increasing returns to scale. Note that the cost function is irrelevant in determining the returns to scale.

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Chapter 3 / Lesson 71