Problem 27 - Calculus: One and Several Variables

Question

Suppose that only \(90 \%\) of the outstanding currency is recirculated into the economy: then \(90 \%\) of that is spent, and so on. Under this hypothesis, what is the long-term economic value of a dollar?

Step-by-Step Solution

Answer

The long-term economic value of a dollar is \(10 dollars\). Under the hypothesis, a dollar could eventually circulate up to 10 dollars worth of economic value.

Step 1: Identify the first term, common ratio, and their substituents

Here, the first term \( a = 1 \) (the initial dollar) and the common ratio \( r = 0.9 \) (each successive term is 90% of the prior term).

Step 2: Substitute the values in the formula

We use the formula for the sum of an infinite geometric series, \( S = \frac{a}{1 - r} \), where \( S \) is the sum we seek, and substitute the values for \( a \) and \( r \) identified in the previous step,

Step 3: Evaluate the expression

After substituting \( a = 1 \) and \( r = 0.9 \) into the formula, we evaluate the expression to get the long-term economic value of a dollar.

Step 4: Simplify the expression

Simplify the expression and give the final answer.

Key Concepts

Understanding Long-term Economic Value

When we talk about the long-term economic value in relation to an infinite geometric series, we're looking at how much a currency or resource is worth when used or invested over time, given certain conditions. In the context of the exercise, this involves understanding the behavior of money that keeps getting spent and re-spent in the economy.

So what does it mean for a dollar when 90% of it is always put back into the system? It implies that the dollar doesn’t just stop at its initial face value. Over an indefinite period, this cycle of recirculation increases the overall value of that dollar.

• This idea showcases compounding effects in an economic system.
• It provides insight into how spending can drive economic growth.

Calculating the sum of an infinite series helps us predict this long-term value, providing a clear snapshot of the ripple effects of continuous spending.

Decoding the Common Ratio

In infinite geometric series, the common ratio is the factor by which each term is multiplied to get the next term in the series. Here, our common ratio is 0.9, or 90%, which reflects the proportion of the money that continues to circulate in the economy each step of the way.

Understanding the common ratio is crucial because it determines how quickly the terms of our series decrease. A common ratio below 1 means that each term will be smaller than the preceding one, aligning with the concept of diminishing value over time.

• A ratio closer to 1 means slower decay of values.
• A ratio much smaller than 1 would show a faster decrease.

The smaller the common ratio, the quicker the series' overall sum converges to its final value. In economic terms, it gives insight into how quickly resources can deplete without sustained replenishment.

Calculating the Sum of the Series

To find the long-term value of the currency scenario given in the exercise, we use the sum of an infinite geometric series formula: \[ S = \frac{a}{1 - r} \].

Here, \( a = 1 \) (our initial dollar) and \( r = 0.9 \). Plug these into the formula:\[ S = \frac{1}{1 - 0.9} = \frac{1}{0.1} = 10 \].

This calculation tells us that the long-term economic value of a single dollar, under these conditions, is $10.

• The formula captures how constant recirculation amplifies the dollar's impact on the economy.
• It's a powerful tool for understanding economic activity over time.

This result highlights the potential sustained value created through continual reinvestment or spending within an economic framework.