Suppose that inflation is 2.64% per year (rather than 1.85% as noted above), and there is no change in Amy's income growth and the return on her savings. What would Amy's first deposit (in nominal terms) have to be in order to achieve the same real value in her retirement savings account balance when she retires?

To solve this problem, we need to adjust Amy's first deposit to ensure that she achieves the same real value in her retirement savings account balance, despite the increase in the inflation rate from 1.85% to 2.64%. Here’s the approach:

### Step 1: **Identify the Given Variables**

- **Original inflation rate (i₁):** 1.85%
- **New inflation rate (i₂):** 2.64%
- **Original first deposit (D₁):** This is unknown, but we'll denote it as \( D_1 \).
- **New first deposit (D₂):** This is what we need to find.
- **Real return rate (r):** The rate of return on her savings minus the inflation rate.

### Step 2: **Understand the Impact of Inflation**
  
The real value of Amy's savings at retirement should be the same under both inflation rates. To find the equivalent nominal first deposit, we'll consider the following:

- If \( D_1 \) was the initial deposit with inflation \( i_1 \), and the same real value is to be maintained with a new deposit \( D_2 \) under inflation \( i_2 \), then \( D_2 \) needs to compensate for the higher inflation.

### Step 3: **Calculate the Real Interest Rate**

The real interest rate \( r \) is calculated using the formula:

\[
r = \frac{1 + \text{nominal interest rate}}{1 + \text{inflation rate}} - 1
\]

Since the real rate is the same in both scenarios (as Amy's income growth and return on savings don't change), we equate the future value of the first deposit under both scenarios:

\[
D_1 \times (1 + r)^n = D_2 \times (1 + r')^n
\]

Where:
- \( r \) is the real rate with the original inflation.
- \( r' \) is the real rate with the new inflation.
- \( n \) is the number of years until retirement.

### Step 4: **Equating and Solving for \( D_2 \)**

Since the real return rate \( r \) remains the same, we have:

\[
D_2 = D_1 \times \frac{1 + i_1}{1 + i_2}
\]

Plugging in the values:

\[
D_2 = D_1 \times \frac{1 + 0.0185}{1 + 0.0264}
\]

\[
D_2 = D_1 \times \frac{1.0185}{1.0264}
\]

\[
D_2 = D_1 \times 0.9923
\]

### Step 5: **Interpretation**

Amy's new first deposit \( D_2 \) should be approximately 99.23% of her original first deposit \( D_1 \). In other words, because inflation is higher, the nominal amount she needs to save initially should decrease slightly to maintain the same real value at retirement.

### Final Expression:

\[
D_2 \approx 0.9923 \times D_1
\]

If you know the value of \( D_1 \), you can calculate the exact nominal first deposit \( D_2 \).

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### Related Questions:
1. How does changing inflation affect the purchasing power of savings?
2. What factors determine the real interest rate?
3. How can Amy ensure her savings keep up with inflation?
4. What happens if Amy’s income growth outpaces inflation?
5. How can Amy adjust her savings strategy in a fluctuating economy?

### Tip:
Always consider the impact of inflation on long-term savings to ensure your retirement goals are met in real terms.

Learn how to adjust Amy's first deposit in nominal terms to maintain the same real value in her retirement savings account, despite an increase in inflation from 1.85% to 2.64%. Understand the impact of inflation and real interest rates in financial planning. Suitable for high school students studying economics or mathematics.

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