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Recall that a Cobb-Douglas production function has the form P = cL(^alpha) K(^beta) with c,...

Question:

Recall that a Cobb-Douglas production function has the form {eq}P = cL^\alpha K^\beta{/eq} where {eq}c, \alpha, \beta > 0{/eq}.

Economists talk about increasing returns to scale if doubling {eq}L{/eq} and {eq}K{/eq} more than doubles {eq}P{/eq}, constant returns to scale if doubling {eq}L{/eq} and {eq}K{/eq} exactly doubles {eq}P{/eq}, and decreasing returns to scale if doubling {eq}L{/eq} and {eq}K{/eq} less than doubles {eq}P{/eq}.

What conditions in the sum {eq}\alpha + \beta{/eq} lead to:

(a) constant returns to scale?

(b) decreasing returns to scale?

Fully justify your answers mathematically.

Returns to Scale:

Returns to scale in economics measure the changes in output with respect to changes in inputs. In particular:

  • If the quantity of all inputs is increased by a constant proportion, and the output increases by the same proportion, production is said to have constant returns to scale.
  • If the output increases by less than the same proportion, then production is said to have decreasing returns to scale.
  • If the output increases by more than the same proportion, then production is said to have increasing returns to scale.

Answer and Explanation: 1

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Consider a case where both K and L were doubled; the new output would be {eq}c(2L)^\alpha (2K)^\beta = 2^{\alpha+\beta}cL^\alpha K^\beta =...

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Returns to Scale in Economics: Definition & Examples

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Chapter 3 / Lesson 71
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Understand the meaning of returns to scale in economics. Learn about increasing returns to scale, constant returns to scale and decreasing returns to scale.


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