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en:cobb-douglas-production-function [2018/03/16 17:46] federico |
en:cobb-douglas-production-function [2019/04/06 08:26] federico |
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| Plot of a Cobb-Douglas production function: | Plot of a Cobb-Douglas production function: | ||
| - | {{:en:cobb-douglas-ew.jpg?nolink&}}{{ :en:cobb-douglas-1.svg.png?nolink&}} | + | {{:en:cobb-douglas-ew.jpg?nolink&|cobb-douglas-ew.jpg}}{{ :en:cobb-douglas-1.svg.png?nolink&}} |
| ===== The marginal product is positive and decreasing ===== | ===== The marginal product is positive and decreasing ===== | ||
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| ∂Q / ∂K = | ∂Q / ∂K = | ||
| - | = α * (A L<sup>β</sup> ) K<sup>(α-1)</sup> | + | = α * (A L<sup>β</sup> ) K<sup>(α-1)</sup> |
| Given that α is positive and lower than 1, the marginal product is positive and decreasing. | Given that α is positive and lower than 1, the marginal product is positive and decreasing. | ||
| Line 52: | Line 52: | ||
| = (∂Q/∂L) / (Q/L) That is the marginal product of labor divided the medium product of labor. | = (∂Q/∂L) / (Q/L) That is the marginal product of labor divided the medium product of labor. | ||
| - | = [ Aβ L<sup>β-1</sup> ) K<sup>α</sup> ] / [ A L<sup>β</sup> K<sup>α</sup> / L ] | + | = [ Aβ L<sup>β-1</sup> ) K<sup>α</sup> ] / [ A L<sup>β</sup> K<sup>α</sup> / L ] |
| - | Given that 1/L is L<sup>-1</sup> , AL<sup>β</sup> K<sup>α</sup> /L is AL<sup>β-1</sup> K<sup>α</sup> . It follows that output elasticity is: | + | Given that 1/L is L<sup>-1</sup> , AL<sup>β</sup> K<sup>α</sup> /L is AL<sup>β-1</sup> K<sup>α</sup> . It follows that output elasticity is: |
| - | = AβL<sup>β-1</sup> K<sup>α</sup> / AL<sup>β-1</sup> K<sup>α</sup> | + | = AβL<sup>β-1</sup> K<sup>α</sup> / AL<sup>β-1</sup> K<sup>α</sup> |
| The only difference between the numerator and the denominator is β; then: | The only difference between the numerator and the denominator is β; then: | ||
| - | Output elasticity = AβL<sup>β-1</sup> K<sup>α</sup> / AL<sup>β-1</sup> K<sup>α</sup> = β | + | Output elasticity = AβL<sup>β-1</sup> K<sup>α</sup> / AL<sup>β-1</sup> K<sup>α</sup> = β |
| ===== Return to scale are α+β ===== | ===== Return to scale are α+β ===== | ||
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| If we increase every factor in a given constant c, the new output level will be: | If we increase every factor in a given constant c, the new output level will be: | ||
| - | Q<sup>'</sup> = A(cL)<sup>β</sup> (cK)<sup>α</sup> | + | Q<sup>'</sup> = A(cL)<sup>β</sup> (cK)<sup>α</sup> |
| - | = Ac<sup>β</sup> L<sup>β</sup> c<sup>α</sup> K<sup>α</sup> | + | = Ac<sup>β</sup> L<sup>β</sup> c<sup>α</sup> K<sup>α</sup> |
| - | = c<sup>β</sup> c<sup>α</sup> AL<sup>β</sup> K<sup>α</sup> | + | = c<sup>β</sup> c<sup>α</sup> AL<sup>β</sup> K<sup>α</sup> |
| - | = c<sup>β+α</sup> Q | + | = c<sup>β+α</sup> Q |
| - | This means that if we increase every production factor by c, the output level will increase in c<sup>β+α</sup> . | + | This means that if we increase every production factor by c, the output level will increase in c<sup>β+α</sup> . |
| If β+α = 1, the output will increase in c. In this case, the Cobb-Douglas production function has constant return to scale. | If β+α = 1, the output will increase in c. In this case, the Cobb-Douglas production function has constant return to scale. | ||