Question

Recall the formula that states that \(S V=1 / P,\) 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. It will rise by a third (33.3 percent).

Step-by-step solution

Understand the Formula

The formula given is \( S V = \frac{1}{P} \), which implies that the value of the dollar \( V \) is equal to the inverse of the price level \( P \). So, \( V = \frac{1}{P} \).

Initial Value of the Dollar

Initially, the price level \( P \) is 1. Therefore, the initial value of the dollar is \( V_1 = \frac{1}{1} = 1 \).

New Value of the Dollar

The price level falls to 0.75. Therefore, the new value of the dollar is \( V_2 = \frac{1}{0.75} \). Simplifying this gives \( V_2 = \frac{4}{3} \approx 1.3333 \).

Calculate the Change in Dollar Value

The change in the value of the dollar is \( V_2 - V_1 = 1.3333 - 1 = 0.3333 \).

Determine the Percentage Increase

To find the percentage increase, use the formula: \( \text{Percentage Increase} = \frac{\text{Change}}{\text{Original Value}} \times 100\% = \frac{0.3333}{1} \times 100\% = 33.3\% \).

Key concepts

Price Level

The price level is a useful concept that represents the average of the current prices for goods and services in an economy. It provides a snapshot of how much we need to spend to buy everyday items. When prices increase, we say there's inflation, and when they decrease, it's called deflation. A change in the price level reflects how economic forces either push prices up or down.
To put it simply:

• A higher price level means you have to pay more for items, reducing purchasing power.
• A lower price level means goods are cheaper, enhancing purchasing power.

In our example, the price level drops from 1 to 0.75, meaning that the economy's goods overall become less expensive. This drop effectively increases the value received from spending a dollar since each dollar now buys more goods than before.

Percentage Increase

Percentage increase is a way to express the change in value as a proportion of the original amount. It helps us compare changes relative to the starting point, and it's a common tool in economics to describe how much something has grown from an initial figure.
Here's how we calculate it:
1. Find the change: Subtract the original value from the new value.
2. Divide the change by the original value.
3. Multiply the result by 100 to convert it into a percentage.

In the exercise, the value of the dollar increases from 1 to 1.3333 due to the decrease in the price level. The calculation for percentage increase is:\[\text{Percentage Increase} = \frac{1.3333 - 1}{1} \times 100\% = 33.3\%\]This indicates that the value of the dollar increased by 33.3%.

Inverse Relationship

An inverse relationship refers to a connection between two variables in which as one variable increases, the other decreases. This concept is crucial in understanding the formula given: \[ V = \frac{1}{P} \] In this equation:

• \(V\), the value of the dollar, increases when \(P\), the price level, decreases.
• Conversely, if the price level \(P\) goes up, \(V\) goes down.

The inverse relationship in our case study shows how a lower price level enhances the purchasing power of the dollar. It highlights that you could get more value out of a dollar when prices are low, showcasing this negative correlation in a straightforward, mathematical way.