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| ====== Cobb-Douglas Production Function ====== | ====== Cobb-Douglas Production Function ====== | ||
| - | + | The Cobb-Douglas Production Function is a particular form of the [[:en:production-function|]]. | |
| - | The Cobb-Douglas Production Function is a particular form of the [[Production Function]]. | + | |
| It takes the following form: | It takes the following form: | ||
| - | Q(L,K) = A L<sup>β</sup>K<sup>α</sup> | + | Q(L,K) = A L<sup>β</sup> K<sup>α</sup> |
| - | * L:labor | + | * L:labor |
| * K:capital | * K:capital | ||
| * Q:output | * Q:output | ||
| Line 15: | Line 14: | ||
| * 0<β<1 | * 0<β<1 | ||
| - | The main characteristics of the Cobb-Douglas production function are: | + | ======= Properties of Cobb Douglas Production Function ======== |
| - | - The [[marginal product]] is positive and decreasing. | + | - The [[:en:marginal-product|marginal product]] is positive and decreasing. |
| - | - [[Output elasticity]] is constant, equal to α for L or β for K. | + | - [[:en:cobb-douglas-output-elasticity|Output elasticity]] is constant, equal to α for L or β for K. |
| - | - [[Return to scale]] are α+β | + | - [[:en:return-to-scale|Return to scale]] are α+β |
| Plot of a Cobb-Douglas production function: | Plot of a Cobb-Douglas production function: | ||
| - | {{:en:cobb-douglas-ew.jpg?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 ===== | ||
| Line 29: | Line 28: | ||
| This behavior is usually seen in a lot of real world examples. To find the marginal product of a production factor, we derivate the total output with respect that factor. For example, to find the marginal product of capital: | This behavior is usually seen in a lot of real world examples. To find the marginal product of a production factor, we derivate the total output with respect that factor. For example, to find the marginal product of capital: | ||
| - | ∂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. |
| Graphically: | Graphically: | ||
| - | {{:es:cobb-douglas-marginal.png?nolink|}} | + | {{:es:cobb-douglas-marginal.png?nolink&}} |
| ===== Output elasticity is constant ===== | ===== Output elasticity is constant ===== | ||
| - | Output elasticity is defined as the percent change in output, when there is a percent change in one production factor. | + | Output elasticity is defined as the percent change in output, when there is a percent change in one production factor. |
| - | In the case of the Cobb-Douglas production function, output elasticity is constant. Output elasticity of labor is β and output elasticity of capital is α. | + | In the case of the Cobb-Douglas production function, output elasticity is constant. Output elasticity of labor is β and output elasticity of capital is α. |
| ==== Proof ==== | ==== Proof ==== | ||
| - | By definition, output elasticity is: | + | By definition, output elasticity is: |
| - | (∂Q/Q) / (∂L/L) = | + | (∂Q/Q) / (∂L/L) = |
| - | = (∂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 ===== | ||
| - | Returns to scale measure the proportional change in output, given a proportional change in the quantity of every factor of production. | + | Returns to scale measure the proportional change in output, given a proportional change in the quantity of every factor of production. |
| ==== Proof ==== | ==== Proof ==== | ||
| - | |||
| 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> | ||
| + | |||
| + | = c<sup>β</sup> c<sup>α</sup> AL<sup>β</sup> K<sup>α</sup> | ||
| - | = Ac<sup>β</sup>L<sup>β</sup>c<sup>α</sup>K<sup>α</sup> | + | = c<sup>β+α</sup> Q |
| - | = c<sup>β</sup>c<sup>α</sup>AL<sup>β</sup>K<sup>α</sup> | + | This means that if we increase every production factor by c, the output level will increase in c<sup>β+α</sup> . |
| - | = c<sup>β+α</sup>Q | + | If β+α = 1, the output will increase in c. In this case, the Cobb-Douglas production function has constant return to scale. |
| - | 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 proportional increase in output will be lower than the proportional increase in production factors. In this case, the Cobb-Douglas production function has decreasing returns 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. | + | If β+α > 1, the proportional increase in output will be higher than the proportional increase in production factors. In this case, the production function has increasing returns to scale. |
| - | If β+α < 1, the proportional increase in output will be lower than the proportional increase in production factors. In this case, the Cobb-Douglas production function has decreasing returns to scale. | + | [[:en:cobb-douglas-output-elasticity|]] |
| - | If β+α > 1, the proportional increase in output will be higher than the proportional increase in production factors. In this case, the production function has increasing returns to scale. | ||