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Chapter 8: Problem 44

Cocks \(^{6}\) determined a Cobb-Douglas production function for Eli Lilly and Company. He found that \(f(L, K)=A L^{1.31939}K^{2.09583},\) where \(A\) is a positive constant, \(f\) is a measure of physical output, \(L\) is a measure of labor input, and \(K\) is a measure of physical capital input. Explain in words the meaning of the two exponents.

Short Answer

Expert verified
The exponents 1.31939 and 2.09583 indicate the output elasticity of labor and capital, suggesting increasing returns to scale.

Step by step solution

01

Understanding Cobb-Douglas Production Function

The Cobb-Douglas production function is an economic model representing the relationship between inputs (labor and capital) and the output they produce. It is commonly expressed as: \(f(L, K) = A L^\alpha K^\beta\). In this function, \(\alpha\) and \(\beta\) are the output elasticities of labor and capital respectively.
02

Interpreting Exponent of Labor (L)

In the given function, \(f(L, K)=A L^{1.31939}K^{2.09583}\), the exponent of labor \(L\) is 1.31939. This represents the output elasticity of labor, indicating that a 1% increase in labor input \(L\) will yield approximately a 1.31939% increase in the output \(f\), holding capital \(K\) constant.
03

Interpreting Exponent of Capital (K)

Similarly, the exponent of capital \(K\) is 2.09583. This suggests that a 1% increase in capital input \(K\) will produce around a 2.09583% increase in the output \(f\), assuming that labor \(L\) is held constant.
04

Analyzing Returns to Scale

The sum of the exponents (1.31939 + 2.09583 = 3.41522) indicates increasing returns to scale. This means that if both labor \(L\) and capital \(K\) are increased by 1%, the output \(f\) increases by more than 1%, precisely by 3.41522%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Output Elasticity
Output elasticity in the Cobb-Douglas production function highlights the responsiveness of output to a proportional change in inputs. The model helps us see how changes in labor or capital influence the total production.
  • The exponent \( \alpha = 1.31939 \) indicates how much output changes with labor. If you increase labor by 1%, the output grows by about 1.31939%.
  • Meanwhile, \( \beta = 2.09583 \) refers to capital's impact. A 1% hike in capital raises the output by roughly 2.09583%.
In essence, these exponents tell us that both inputs significantly affect production, with capital having a more noticeable impact compared to labor. Each increment in labor or capital will change output according to these elasticity measures, assuming the other variable remains constant.
Returns to Scale
Returns to scale in production functions describe what happens to output when all inputs are increased by the same proportion. This concept is crucial in understanding how efficient a production process is.
  • In the function with exponents \(1.31939\) and \(2.09583\), the sum \( 3.41522 \) shows the type of returns to scale.
  • Because the sum exceeds 1, it's labeled as increasing returns to scale. It implies efficiency advantages where doubling inputs results in more than double the output.
This property is particularly important when businesses plan expansions or consider scaling their operations. By assessing returns to scale, companies can predict better outcomes from increasing their labor force and capital investments.
Economic Model
An economic model like the Cobb-Douglas production function is essential for simplifying and explaining complex economic processes and relationships. It offers a structured way to quantify how inputs such as labor and capital can impact output.
  • The model is mathematical in nature, providing a concrete formula through which relationships are predicted and measured.
  • It is instrumental in making informed decisions, setting policies, and understanding economic dynamics in production scenarios.
Overall, its explanatory power lets companies and economists anticipate changes in output, craft strategies, and analyze the production efficiency of a system with greater precision.

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