Problem 49 - College Algebra and Calculus: An Applied Approach

Question

The earnings per share \(z\) (in dollars) for Starbucks Corporation from 1998 through 2006 can be modeled by \(z=0.106 x-0.036 y-0.005\), where \(x\) is sales (in billions of dollars) and \(y\) is the shareholder's equity (in billions of dollars). (Source: Starbucks Corporation) (a) Find the earnings per share when \(x=8\) and \(y=5\). (b) Which of the two variables in this model has the greater influence on the earnings per share? Explain.

Step-by-Step Solution

Answer

The earnings per share when \(x=8\) and \(y=5\) is calculated in Step 1. And in Step 2, it is concluded that sales (\(x\)) has a greater influence on the earnings per share than shareholder's equity (\(y\)), because the coefficient of \(x\) in the equation is larger than that of \(y\).

Step 1: Calculating Earnings Per Share

Substituting the given values of \(x\) and \(y\) into the model \(z = 0.106x - 0.036y - 0.005\), where \(x = 8\) and \(y = 5\), we get \(z = 0.106*8 - 0.036*5 - 0.005\). Solving this gives us the value of \(z\).

Step 2: Comparing the influence of x and y on z

To determine which variable, \(x\) (sales) or \(y\) (shareholder's equity), has a greater influence on the earnings per share (\(z\)), we need to compare the absolute values of their coefficients in the equation. The absolute coefficients give us the rate of change in \(z\) per billion dollars change in \(x\) or \(y\). This can be seen from the equation: for each billion dollars increase in \(x\), \(z\) increases by \(0.106\) and for each billion dollars increase in \(y\), \(z\) decreases by \(0.036\). Since \(0.106 > 0.036\), \(x\) has a greater influence on the earnings per share.

Key Concepts

Understanding Earnings Per Share

Earnings per share (EPS) is a crucial indicator in assessing a company's profitability. It represents the amount of profit attributed to each share of common stock. EPS is often used by investors to gauge the financial health of a company. In the context of Starbucks, the EPS can be calculated using the formula:\[ z = 0.106x - 0.036y - 0.005 \]Here, the variables are as follows:

• \(x\): Sales in billions of dollars
• \(y\): Shareholder's equity in billions of dollars

To find the earnings per share, we simply substitute the values of \(x\) and \(y\) into the equation. For instance, if sales \(x = 8\) billion dollars and shareholder's equity \(y = 5\) billion dollars, substituting these into the model gives:\[ z = 0.106 \times 8 - 0.036 \times 5 - 0.005 \]Solving this equation yields the EPS for Starbucks under these conditions. This tells us how each factor contributes to the company's profitability per stock share.

Influence of Variables in the EPS Model

The equation \( z = 0.106x - 0.036y - 0.005 \) helps us understand not just the earnings per share, but also the impact of different variables on EPS. Here, the coefficients of \(x\) and \(y\) indicate their influence. The coefficient \(0.106\) next to \(x\) implies that for every billion-dollar increase in sales, EPS increases by 0.106. Conversely, the coefficient \(-0.036\) for \(y\) means that for every billion-dollar increase in shareholder's equity, EPS decreases by 0.036.
These coefficients provide insights into the factors driving company value:

• An increase in sales ( \(x\)) tends to enhance shareholder value.
• Increasing shareholder's equity ( \(y\)) slightly reduces the EPS value according to the model.

In practical terms, this means boosting sales may have a more pronounced effect on profitability per share compared to raising equity.

Coefficient Comparison and Its Importance

In algebraic modeling, coefficient comparison is a fundamental technique to comprehend the relative impact of variables. In the Starbucks EPS model:\[ z = 0.106x - 0.036y - 0.005 \]The two coefficients 0.106 and -0.036 offer clear insights. The coefficient 0.106 attached to sales (\(x\)) is larger than -0.036 attached to shareholder's equity (\(y\)). This comparison showcases that sales have a more substantial influence on enhancing earnings per share than equity reductions.
Here's why this matters:

• Businesses can prioritize increasing sales as a strategy to boost their EPS more effectively.
• Understanding these comparisons aids in financial decision-making and strategic planning.

This comparison helps to create more targeted and efficient business strategies, leading to optimized profitability for shareholders.