**Problem:** Find the elasticity of scale and the elasticity of substitution for the CES production function $ f(x_1, x_2) = (x_1^\rho + x_2^\rho)^{\frac{1}{\rho}} $, where $ 0 \neq \rho < 1 $.---**Explanation:**This mathematical problem relates to the analysis of the CES (Constant Elasticity of Substitution) production function. The function is defined as:\[ f(x_1, x_2) = (x_1^\rho + x_2^\rho)^{\frac{1}{\rho}} \]Here, $ \rho $ is a parameter that influences the substitutability between the inputs $ x_1 $ and $ x_2 $. The condition $ 0 \neq \rho < 1 $ indicates the range for $ \rho $ within which the analysis is conducted.**Key Concepts:**1. **Elasticity of Scale:** This measures how output changes with a proportional change in all inputs. For the CES production function, it helps determine if returns to scale are increasing, constant, or decreasing.2. **Elasticity of Substitution:** This quantifies the ease with which one input can be substituted for another, maintaining the same level of output. In the context of the CES function, it is often related to the parameter $ \rho $.There are no graphs or diagrams in the image, only a textual representation of a mathematical concept. Understanding these concepts is essential for analyzing how firms can optimize production and manage input costs effectively.

Let the production function be: Q = x1ρ+x2ρ1ρElasticity of scale can be calculated by:μ=∂Q∂x1*x1Q+∂Q∂x2*x2Q∂Q∂x1=x1ρ-1(x1ρ+x2ρ)1ρ-1 ∂Q∂x2=x2ρ-1(x1ρ+x2ρ)1ρ-1 μ=x1ρ-1(x1ρ+x2ρ)1ρ-1*x1x1ρ+x2ρ1ρ+x2ρ-1(x1ρ+x2ρ)1ρ-1*x2x1ρ+x2ρ1ρμ=x1ρx1ρ+x2ρ+x2ρx1ρ+x2ρμ=x1ρ+x2ρx1ρ+x2ρμ=1 Elasticity of scale = 1Elasticity of substitution:MPx1=∂Q∂x1MPx1=x1ρ-1(x1ρ+x2ρ)1ρ-1 MPx2=x2ρ-1(x1ρ+x2ρ)1ρ-1MRTS = MPx1MPx2MRTS = x1ρ-1(x1ρ+x2ρ)1ρ-1 x2ρ-1(x1ρ+x2ρ)1ρ-1MRTS = x1ρ-1x2ρ-1MRTS = x1x2ρ-1Elasticity of substitution = (x1/x2)ρ-1

Answered: *Find the elasticity of scale and the…

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*Find the elasticity of scale and the elasticity of substitution for the CES production function f(x1,#2) = (x{ + a£)+,where 0 # p < 1.

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

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Problem 1QTC

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Question

**Problem:** Find the elasticity of scale and the elasticity of substitution for the CES production function $ f(x_1, x_2) = (x_1^\rho + x_2^\rho)^{\frac{1}{\rho}} $, where $ 0 \neq \rho < 1 $.

---

**Explanation:**

This mathematical problem relates to the analysis of the CES (Constant Elasticity of Substitution) production function. The function is defined as:

\[ f(x_1, x_2) = (x_1^\rho + x_2^\rho)^{\frac{1}{\rho}} \]

Here, $ \rho $ is a parameter that influences the substitutability between the inputs $ x_1 $ and $ x_2 $. The condition $ 0 \neq \rho < 1 $ indicates the range for $ \rho $ within which the analysis is conducted.

**Key Concepts:**

1. **Elasticity of Scale:** This measures how output changes with a proportional change in all inputs. For the CES production function, it helps determine if returns to scale are increasing, constant, or decreasing.

2. **Elasticity of Substitution:** This quantifies the ease with which one input can be substituted for another, maintaining the same level of output. In the context of the CES function, it is often related to the parameter $ \rho $.

There are no graphs or diagrams in the image, only a textual representation of a mathematical concept. Understanding these concepts is essential for analyzing how firms can optimize production and manage input costs effectively.

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