Question

A savings account has a balance of​ $1. Savings Plan A will add​ $1,000 to the account each​ month, and Plan B will double the amount each month.

Which plan is better in the short run? The long run?

Answer 1

To compare Savings Plan A and Plan B, we can evaluate how much money each plan will accumulate over time.

**Attributes of the Plans:**
- Starting balance: $1
- Plan A: Adds $1,000 each month.
- Plan B: Doubles the balance each month.

### Short Run Analysis (First Few Months)

1. **Plan A:**
- Month 0: $1 (initial balance)
- Month 1: $1 + $1,000 = $1,001
- Month 2: $1,001 + $1,000 = $2,001
- Month 3: $2,001 + $1,000 = $3,001
- Month 4: $3,001 + $1,000 = $4,001

2. **Plan B:**
- Month 0: $1 (initial balance)
- Month 1: $1 × 2 = $2
- Month 2: $2 × 2 = $4
- Month 3: $4 × 2 = $8
- Month 4: $8 × 2 = $16

### Summary of Short Run Results
- After 1 month:
- Plan A = $1,001
- Plan B = $2

- After 2 months:
- Plan A = $2,001
- Plan B = $4

- After 3 months:
- Plan A = $3,001
- Plan B = $8

- After 4 months:
- Plan A = $4,001
- Plan B = $16

**In the short run, Plan A is significantly better** as it accumulates a much larger balance compared to Plan B.

### Long Run Analysis (Exponential Growth)

As time goes on, Plan B will begin to outpace Plan A due to the nature of exponential growth.

1. **Plan A** grows linearly, adding $1,000 every month.
- After \( n \) months:
\[ \text{Balance of Plan A} = 1 + 1,000n \]

2. **Plan B** grows exponentially.
- After \( n \) months, the amount will be:
\[ \text{Balance of Plan B} = 1 \times 2^n \]

### Comparison in the Long Run
- As \( n \) increases, the balance of Plan B (which is \( 2^n \)) will grow much faster than linear growth from Plan A (which is \( 1 + 1,000n \)).

**Mathematically:**
- When \( n \) becomes large:
- For example, at \( n = 10 \):
- Plan A: \( 1 + 1,000 \times 10 = 10,001 \)
- Plan B: \( 1 \times 2^{10} = 1,024 \)

- But as \( n \) gets very high, Plan B will surpass Plan A:
- At \( n = 20 \):
- Plan A: \( 1 + 1,000 \times 20 = 20,001 \)
- Plan B: \( 1 \times 2^{20} = 1,048,576 \)

### Conclusion
- **Short Run:** Plan A is better.
- **Long Run:** Plan B will be vastly superior due to exponential growth.

Thus, while Plan A offers immediate benefits, Plan B is the better choice over an extended timeframe.