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| ====== Elasticity of Production ====== | ====== Elasticity of Production ====== | ||
| - | The elasticity of production, also called **output elasticity**, is the percentaje change in the production of a good by a firm, divided the percentaje change in an input used for the production of that good, for example, labor or capital. | + | The elasticity of production, also called **output elasticity**, is the percentaje change in the production of a good by a firm, divided the percentage change in an input used for the production of that good, for example, labor or capital. |
| - | The elasticity of production shows the **responsiveness** of the output when there is a change in one input. It is defined as de proportional change in the product, divided the proportional change in the quantity of an input. | + | The elasticity of production shows the **responsiveness** of the output when there is a change in one input. |
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| + | It is defined as de proportional change in the product, divided the proportional change in the quantity of an input. | ||
| **For example**, if a factory employs 10 people, and produces 100 chairs per day. If the number of people employed in the factory increases to 12, that is, a 20% increase, and the number of chairs produced per day increases to 110 (that is, a 10% increase), the elasticity of production is: | **For example**, if a factory employs 10 people, and produces 100 chairs per day. If the number of people employed in the factory increases to 12, that is, a 20% increase, and the number of chairs produced per day increases to 110 (that is, a 10% increase), the elasticity of production is: | ||
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| **ΔQ/Q / ΔL/L** = 10/100 / 2/10 = 0.1 / 0.2 = 0.5 | **ΔQ/Q / ΔL/L** = 10/100 / 2/10 = 0.1 / 0.2 = 0.5 | ||
| - | If the production function contains only one input, the elasticity of production measures the degree of returns to scale. In this case: | + | If the production function contains only one input, the elasticity of production measures the degree of [[https://www.econowiki.com/doku.php?id=en:returns-to-scale|returns to scale]]. In this case: |
| - | - if the elasticity of production is 1, the production has constant return to scale, at that point. | + | |
| - | - if the elasticity of production is greater than one, the production has increasing returns to scale at that point. | + | |
| - | - if the elasticity of production is less than one, the production has decreasing returns to scale at that point. | + | |
| - | # Using a production function | + | - if the elasticity of production is 1, the production has **constant returns to scale**, at that point. |
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| + | - if the elasticity of production is greater than one, the production has **increasing returns to scale** at that point. | ||
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| + | - if the elasticity of production is less than one, the production has **decreasing returns to scale** at that point. | ||
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| + | ===== Using a production function ===== | ||
| If a production function, for example: Q=f(K,L), is used to calculate the input, and the function is diferentiable, the elasticity of production can be calculated using derivatives: | If a production function, for example: Q=f(K,L), is used to calculate the input, and the function is diferentiable, the elasticity of production can be calculated using derivatives: | ||
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| That is, the marginal product of capital, divided the average product of capital. | That is, the marginal product of capital, divided the average product of capital. | ||
| - | ## Example: Elasticity of Production of a Cobb Douglas Production Function | + | ==== Example: Elasticity of Production of a Cobb Douglas Production Function ==== |
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| + | The [[:en:cobb-douglas-production-function|Cobb-Douglas production function]] is a function that is used a lot in economics. The form of a Cobb-Douglas production function is: | ||
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| + | Q(L,K) = A L<sup>β</sup> K<sup>α</sup> | ||
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| + | To calculate the [[:en:cobb-douglas-output-elasticity|elasticity of production of the Cobb-Douglas production function]], with respect to K, we must find the proportional change in the production, divided the proportional change in K: | ||
| - | The Cobb-Douglas production function is a function that is used a lot in economics. The form of a Cobb-Douglas production function is: | + | \begin{equation} \frac {\frac{\partial Q}{Q}} {\frac{\partial K}{K}} = \frac {\frac{\partial Q}{\partial K}} {\frac{Q}{K}} = \frac { α A L ^{β} K^{α-1} }{ \frac {A L^β K^α}{K} } \end{equation} |
| - | Q(L,K) = A L<sup>β</sup> K<sup>α</sup> | + | \begin{equation} = \frac { α A L ^{β} K^{α} K^{-1} }{ \frac {A L^β K^α}{K} } \end{equation} |
| - | To calculate the elasticity of production of the Cobb-Douglas production function, with respect to K, we must find the proportional change in the production, divided the proportional change in K: | + | \begin{equation} = \frac { \frac {α A L ^{β} K^{α}}{K} }{ \frac {A L^β K^α}{K} } \end{equation} |
| - | (∂Q/Q) / (∂K/K) = (∂Q/∂K) / (Q/K) = = [ Aβ L<sup>(β-1)</sup> K<sup>α</sup> ] / [ A L<sup>β</sup> K<sup>α</sup> / L ] | + | \begin{equation} = \frac { \frac {α Q}{K} }{ \frac {Q}{K} } \end{equation} |
| - | \begin{equation} | + | \begin{equation} = α \end{equation} |
| - | \frac {\frac{\partial Q}{Q}} {\frac{\partial K}{K}} | + | |
| - | \end{equation} | + | |
| - | \begin{equation} | + | [[production-function-example]] |
| - | \frac {\frac{\partial Q}{\partial K}} {\frac{Q}{K}} | + | |
| - | \end{equation} | + | |