Question

Recall the formula that states that \( SV=1 / P,\) where \(V\) is the value of the dollar and \(P\) is the price level. If the price level falls from 1 to \(0.75,\) what will happen to the value of the dollar? a. It will rise by a third \((33.3\) percent). b. It will rise by a quarter \((25\) percent). c. It will fall by a quarter \((-25\) percent). d. It will fall by a third \((-33.3\) percent).

Short answer

a. The value of the dollar will rise by a third (33.3%).

Step-by-step solution

Understanding the Formula

The given formula is \( V = \frac{1}{P} \), which expresses the value of the dollar \( V \) in terms of the price level \( P \). This formula indicates that the value of the dollar is the reciprocal of the price level.

Determine Initial and Final Values

Initially, the price level \( P \) is 1. Using the formula, we find the initial value of the dollar: \( V_{initial} = \frac{1}{1} = 1 \). When the price level falls to 0.75, the new value of the dollar becomes \( V_{final} = \frac{1}{0.75} \).

Calculate the Final Value

Calculate \( V_{final} \) by evaluating \( \frac{1}{0.75} \): \( \frac{1}{0.75} = \frac{4}{3} \approx 1.3333 \). Thus, the final value of the dollar is approximately 1.3333.

Determine the Percentage Change in Value

To determine the percentage change, use the formula: \[ \text{Percentage Change} = \left( \frac{V_{final} - V_{initial}}{V_{initial}} \right) \times 100\% \]. Substitute the values: \( \left( \frac{1.3333 - 1}{1} \right) \times 100\% = 33.33\% \).

Conclusion

The value of the dollar rises by approximately 33.3 percent when the price level falls from 1 to 0.75. Therefore, the correct answer is option (a): The value of the dollar will rise by a third \((33.3\%\)).

Key concepts

Price Level

The price level is an essential concept in economics. It refers to the average of current prices across the entire spectrum of goods and services produced in the economy. When price levels change, they impact the purchasing power of money. This is because if prices increase, each unit of currency buys fewer goods and services, and vice versa.

In the context of the exercise, the price level decreased from 1 to 0.75. This implies that prices have dropped, consequently increasing the value of money. Such shifts in price level are significant as they reflect inflationary or deflationary trends, affecting economic decisions at the personal, business, and governmental levels. Understanding how the price level interacts with other economic indicators is crucial for comprehending broader economic trends.

Percentage Change

Percentage change is a mathematical concept that represents the degree of change over time. It is particularly useful in understanding how economic variables evolve. In our exercise, the percentage change is used to determine how much the value of the dollar increased as a result of a fall in the price level.

To calculate the percentage change, we employ the formula:

• \[\text{Percentage Change} = \left( \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \right) \times 100\% \]

This formula provides a clear understanding of the relative change between the two values. In the exercise, this concept helps us see how the decrease in the price level caused the value of the dollar to rise by 33.3%. This measure is critical for economic analysis, providing a simple yet powerful tool to compare changes over time.

Economics Formula

Economics often relies on mathematical formulas to simplify and model the relationships between different economic variables. In the given exercise, the formula \( V = \frac{1}{P} \) is used to show the relationship between the value of the dollar \( V \) and the price level \( P \).

This formula signifies a reciprocal relationship where the value of the dollar is inversely proportional to the price level – as one increases, the other decreases. Such formulas help in visualizing and predicting economic outcomes based on different scenarios. Familiarity with common economic formulas enhances one's ability to critically assess financial information and make well-informed decisions.

Reciprocal Relationship

A reciprocal relationship means that two variables are inversely related. When one variable increases, the other decreases, and vice versa. In the formula \( V = \frac{1}{P} \), this kind of relationship is showcased. It explains how a fall in the price level results in a rise in the value of the dollar.

To break it down further:

• If \( P \) increases, then \( V \) decreases, meaning it takes more dollars to purchase the same amount of goods.
• Conversely, if \( P \) decreases, as seen in the exercise, then \( V \) increases, meaning the dollar can purchase more goods.

This negative or inverse correlation is crucial in economics. It underscores how variables interact under different economic conditions, allowing policymakers and economists to understand and predict how changes in one aspect of the economy might impact other areas.