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en:cobb-douglas-output-elasticity [2016/01/02 14:54] federico |
en:cobb-douglas-output-elasticity [2017/08/31 14:36] (current) federico |
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| ====== Cobb-Douglas Output Elasticity ====== | ====== Cobb-Douglas Output Elasticity ====== | ||
| - | The Cobb-Douglas Output Elasticity is constant and equal to α or β. | + | The [[en:cobb-douglas-production-function|Cobb-Douglas]] [[en:elasticity-of-production|Output Elasticity]] is constant and equal to α or β. |
| - | If the Cobb-Douglas production function is Q(L,K) = A L<sup>β</sup>K<sup>α</sup>, the output elasticity with respect to labor (L) is α and the output elasticity with respect to capital (K) is β. | + | If the Cobb-Douglas production function is **Q(L,K) = A L<sup>β</sup>K<sup>α</sup>**, the output elasticity with respect to labor (L) is β and the output elasticity with respect to capital (K) is α. |
| To prove this, we must take into account the definition of output elasticity: it is the porcentual change in output in respond to a porcentual change in levels of either labor or capital. | To prove this, we must take into account the definition of output elasticity: it is the porcentual change in output in respond to a porcentual change in levels of either labor or capital. | ||
| If we want to calculate the output elasticity with respect to labor, we must use the following equation: | If we want to calculate the output elasticity with respect to labor, we must use the following equation: | ||
| + | <code> | ||
| (∂Q/Q) / (∂L/L) | (∂Q/Q) / (∂L/L) | ||
| + | </code> | ||
| This is equal to: | This is equal to: | ||
| + | <code> | ||
| (∂Q/∂L) / (Q/L) | (∂Q/∂L) / (Q/L) | ||
| + | </code> | ||
| Now we have the marginal product divided by the average product. Applying the Cobb-Douglas production function: | Now we have the marginal product divided by the average product. Applying the Cobb-Douglas production function: | ||
| = [ 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 ] | ||
| - | |||
| = [ Aβ L<sup>(β-1)</sup> K<sup>α</sup> ] / [ A L<sup>(β-1)</sup> K<sup>α</sup> ] | = [ Aβ L<sup>(β-1)</sup> K<sup>α</sup> ] / [ A L<sup>(β-1)</sup> K<sup>α</sup> ] | ||
| - | |||
| = β | = β | ||
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| It is noteworthy that both output elasticities are constant and greater than 0 and smaller than 1. | It is noteworthy that both output elasticities are constant and greater than 0 and smaller than 1. | ||
| + | |||
| + | See Also: | ||
| + | [[Elasticity of Production]] | ||