Problem 50 - Calculus an Applied Approach

Question

Shareholder's Equity The shareholder's equity z (in billions of dollars) for Wal-Mart Corporation from 2000 to 2006 can be modeled by z = \(0.205 x-0.073 y-0.728\), where \(x\) is net sales (in billions of dollars) and y is the total assets (in billions of dollars). (a) Find the shareholder's equity when \(x=300\) and \(y=130\). (b) Which of the two variables in this model has the greater influence on shareholder's equity? Explain.

Step-by-Step Solution

Answer

For part (a), the shareholder's equity is approximately 51.282 billion dollars. For part (b), the net sales have a greater influence on the shareholder's equity than the total assets.

Step 1: Calculate Shareholder's Equity

Substitute the given values into the equation \(z = $0.205x - 0.073y - 0.728\). So when \(x=300\) and \(y=130\), the equation becomes: \(z = 0.205*300 - 0.073*130 - 0.728\)

Step 2: Carry out the calculation

By performing these operations, we get \(z = 61.5 - 9.49 - 0.728 = 51.282\) billion dollars.

Step 3: Identify the influence of the variables

Identify the greater influence on equity by comparing the coefficients of \(x\) and \(y\). Here, the coefficient of \(x\) is 0.205 while that of \(y\) is -0.073. The variable with the larger absolute coefficient value has a greater impact on shareholder's equity.

Step 4: Evaluate the influence

Given that 0.205 is greater than -0.073 in absolute terms, it can be concluded that net sales (x) has a greater influence than total assets (y) on the shareholder's equity. This is because a small change in net sales will have a larger effect on the shareholder's equity than the same change in total assets. Remember that the negative sign associated with \(y\) indicates that an increase in total assets decreases shareholder's equity, and vice versa.

Key Concepts

Shareholder's Equity

Shareholder's equity is a key financial metric that represents the amount of money that would be returned to shareholders if all the company's assets were liquidated and all its debts were paid off. It is often viewed as a measure of a company's financial health, reflecting the owner's value in the company. Calculating shareholder’s equity can be done using several models and formulas, one of which includes net sales and total assets, as seen in the equation:

\[ z = 0.205x - 0.073y - 0.728 \]
Here:

• \(x\) represents net sales (in billions),
• \(y\) stands for total assets (in billions).

In this equation, the influence of net sales and total assets on shareholder's equity is quantified by their respective coefficients. By calculating shareholder's equity using specific values of net sales and total assets, a company can understand its current financial standing and what portion belongs to shareholders. This understanding is crucial for both internal management and external investors in making informed decisions.

Mathematical Modeling

Mathematical modeling involves creating equations and formulas to represent real-world situations, providing a way to predict and analyze outcomes. In financial contexts, mathematical models can estimate various financial health indicators such as shareholder's equity. Understanding these models allows businesses to simulate different scenarios and make better strategic decisions.

In the model provided:

• The equation \( z = 0.205x - 0.073y - 0.728 \) models the relationship between net sales, total assets, and shareholder's equity.
• This linear equation indicates how changes in net sales (\(x\)) and total assets (\(y\)) affect shareholder’s equity (\(z\)).

The coefficients determine the degree of influence each variable has inside the model. Positive and negative coefficients reveal whether an increase in that particular variable will increase or decrease the outcome, respectively. By substituting specific values into the model, businesses can conduct scenario analyses to see how changes in sales or assets can alter shareholder equity.

Net Sales Influence Analysis

Net sales play a pivotal role in a company's financial outcomes and are crucial in determining shareholder's equity. In the given model, \(x = 300\) represents the net sales in billions of dollars, contributing significantly to the final calculation of shareholder’s equity.

Key points to note about net sales influencing shareholder's equity include:

• The coefficient of net sales (\(0.205\)) is higher in absolute value compared to the total assets' coefficient (\(-0.073\)), indicating that changes in net sales have a more significant impact.
• If net sales increase by one billion dollars, shareholder's equity increases by approximately $205 million.
• Since this coefficient is positive, an increase in net sales leads to an increase in shareholder's equity, reflecting healthy and profitable business operations.

Conversely, a reduction in net sales implies a decrease in shareholder's equity, highlighting the importance for the company to maintain or improve sales figures to sustain or grow its equity. Thus, analyzing the effect of net sales allows businesses to prioritize sales strategies that safeguard and accelerate equity growth, benefiting shareholders.