Given a production function q= AL^aK^ß where L and K are quantities of labor and capital used in producing q and A. i. Derive the elasticity of substitution of the production function.?

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Answer

Derivation of the Elasticity of Substitution

The elasticity of substitution measures the rate at which capital can be substituted for labor in the production process without affecting the level of output. It is given by the formula:

$$ \sigma = -\frac{d\ln(\frac{L}{K})}{d\ln(\frac{MPL}{MPK})} $$

where MPL and MPK are the marginal products of labor and capital, respectively.

Step 1: Calculate the Marginal Products of Labor and Capital

The marginal product of labor (MPL) is given by the derivative of the production function with respect to labor:

$$ MPL = \frac{\partial q}{\partial L} = aAL^{a-1}K^{\beta} $$

Similarly, the marginal product of capital (MPK) is given by the derivative of the production function with respect to capital:

$$ MPK = \frac{\partial q}{\partial K} = \beta AL^{a}K^{\beta-1} $$

Step 2: Calculate the Ratio of Marginal Products

Next, we calculate the ratio of the marginal products of labor and capital:

$$ \frac{MPL}{MPK} = \frac{aAL^{a-1}K^{\beta}}{\beta AL^{a}K^{\beta-1}} = \frac{a}{\beta}\cdot\frac{K}{L} $$

Step 3: Compute the Elasticity of Substitution

Now, we substitute the ratio of marginal products into the formula for the elasticity of substitution:

$$ \sigma = -\frac{d\ln(\frac{L}{K})}{d\ln(\frac{a}{\beta}\cdot\frac{K}{L})} = -\frac{d\ln(\frac{L}{K})}{d\ln(\frac{a}{\beta}) - d\ln(\frac{K}{L})} = -\frac{1}{1} = -1 $$

Therefore, the elasticity of substitution for the production function q= AL^aK^ß is -1, indicating that labor and capital are perfect substitutes in the production process.

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