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LeoMessi

  • 2 years ago

What type of returns to scale does the following production function exhibit?

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  1. LeoMessi
    • 2 years ago
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    \[q = 10K^{2/3} + L^{1/2}\] K = # machines, L = # labor units.

  2. dpflan
    • 2 years ago
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    OK, there are (i.) decreasing returns, (ii.) increasing returns, and (iii.) constant returns. Your function is one of K and L so \[q=f(K,L)\]. if we increase K and L by 2, is the quantity produced greater than, less than, or equal to 2 * the production function?

  3. dpflan
    • 2 years ago
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    \[2*F(K,L) versus F(2*K, 2*L)\]

  4. LeoMessi
    • 2 years ago
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    \[F(2*K, 2*L) = 10*(2K)^{2/3} + (2*L)^{1/2} = 10 * 2^{2/3} * K + 2^{1/2} * L\] and \[2*F(K,L) = 2 *(10 * K^{2/3} + L^{1/2}\]

  5. dpflan
    • 2 years ago
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    right, to easily compare the 2 factor out a 2 in \[F(2*K, 2*L)\] \[(2) * (10 * 2^{-1/3} * K + 2^{-1/2} * L)\] given the exponents, that is obviously less than \[2*(10*2^{2/3}*K + 2^{1/2}*L)\]

  6. LeoMessi
    • 2 years ago
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    yep, so decreasing returns to scale

  7. LeoMessi
    • 2 years ago
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    right on, so it seems safe to assume that when the exponents are less than 1, decreasing returns are most likely, and when the exponents are 1 then constant, and if greater than 1 than increasing

  8. LeoMessi
    • 2 years ago
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    cool stuff, thanks man

  9. dpflan
    • 2 years ago
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    yeah, I think you get the idea, try out some more problems

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