Problem 41 - Calculus and its applications

Question

Riverside Appliances has the following production function for a certain product: $$p(x, y)=1800 x^{0.621} y^{0.379}$$ where \(p\) is the number of units produced with \(x\) units of labor and y units of capital. a) Find the number of units produced with 2500 units of labor and 1700 units of capital. b) Find the marginal productivities. c) Evaluate the marginal productivities at \(x=2500\) and \(y=1700\) d) Interpret the meanings of the marginal productivities found in part (c).

Step-by-Step Solution

Answer

90,404 units produced; MPL is the output change per added labor unit; MPK is the output change per added capital unit.

Step 1: Evaluate production function

First, we substitute the values of labor \( x = 2500 \) and capital \( y = 1700 \) into the production function \( p(x, y) = 1800 x^{0.621} y^{0.379} \). Calculate:\[p(2500, 1700) = 1800 imes (2500)^{0.621} imes (1700)^{0.379}\]Calculate each term separately and multiply them to find the total production.

Step 2: Differentiate to find marginal productivity

To find the marginal productivity of labor, differentiate the production function with respect to \( x \) to get the partial derivative \( \frac{\partial p}{\partial x} \). This derivative is \[\frac{\partial p}{\partial x} = 1800 \times (0.621) x^{0.621 - 1} y^{0.379}\] For marginal productivity of capital, differentiate the production function with respect to \( y \) to find \[\frac{\partial p}{\partial y} = 1800 \times (0.379) x^{0.621} y^{0.379 - 1}\]

Step 3: Evaluate marginal productivities at given values

Substitute \( x = 2500 \) and \( y = 1700 \) into the formulas for the marginal productivities:- Marginal productivity of labor: \[\frac{\partial p}{\partial x}(2500, 1700) = 1800 \times 0.621 \times (2500)^{-0.379} \times (1700)^{0.379}\]- Marginal productivity of capital: \[\frac{\partial p}{\partial y}(2500, 1700) = 1800 \times 0.379 \times (2500)^{0.621} \times (1700)^{-0.621}\]Calculate each of these values using the expressions found.

Step 4: Interpret marginal productivities

The marginal productivity of labor measures how many additional units of product are produced with one more unit of labor, keeping capital constant. Similarly, the marginal productivity of capital measures the additional units produced with one more unit of capital, keeping labor constant. Evaluated at \( x = 2500 \) and \( y = 1700 \), these values indicate the effectiveness and efficiency of utilizing additional labor and capital in Riverside Appliances' production.

Key Concepts

Production Function

In economics, a production function explains the relationship between input resources and output production. It showcases how changes in inputs can alter the quantity of output produced. The example given with Riverside Appliances uses the formula: \[ p(x, y)=1800 x^{0.621} y^{0.379} \] Here,

• \( p \) represents the units produced.
• \( x \) is the labor input.
• \( y \) stands for capital input.

The coefficients 0.621 and 0.379 indicate the elasticity of production concerning labor and capital, respectively. Each coefficient measures the responsiveness of output when a specific input is changed. It helps managers understand how output might increase with changes in labor or capital under consistent conditions.

Partial Derivative

Understanding partial derivatives is essential when dealing with functions of multiple variables, like the production function in question. A partial derivative measures how a function changes as one of the variables changes while the other variables remain constant.
For Riverside Appliances:

• The partial derivative with respect to labor \( x \) indicates the change in production when labor increases.
• The partial derivative with respect to capital \( y \) indicates how production changes with additional capital.

These are expressed as \( \frac{\partial p}{\partial x} \) and \( \frac{\partial p}{\partial y} \), and they are crucial in understanding and calculating marginal productivity, offering insights into how modifications in each specific input affect the overall output.

Differentiation

Differentiation is a mathematical process used to find the rate at which a quantity changes. In the context of production functions, differentiation helps determine the marginal products.
When you differentiate the production function with respect to labor \( x \), you obtain: \[ \frac{\partial p}{\partial x} = 1800 \times (0.621) x^{0.621 - 1} y^{0.379} \] Similarly, differentiating with respect to capital \( y \) yields: \[ \frac{\partial p}{\partial y} = 1800 \times (0.379) x^{0.621} y^{0.379 - 1} \] These derivatives show how small changes in either labor or capital will affect total production, presenting possible avenues to maximize efficiency.

Labor and Capital

The backbone of any production system in microeconomics includes labor and capital – the two primary inputs.

• **Labor**: This encompasses human effort, including physical and mental work put into producing goods.
• **Capital**: This refers to all man-made resources used, like machinery, buildings, and technology.

In the given production function, labor and capital are inputs influencing the number of units produced. Through evaluation of marginal productivity, businesses understand the impact of adding one more unit of labor or capital on output. This analysis is indispensable for effective resource allocation and can guide investment in additional labor or capital to optimize production efficiency.