In 2022, the U.S. is running a trade deficit of $1,000 billion. The expected surplus in 2024 is $200 billion. Assuming the interest rate is a constant
3
%
, calculate the trade balance in 2023 such that the present discounted value of the sum of (expected) trade deficits and surpluses (from 2022 to 2024) is zero.

Use a spreadsheet for your calculations and round your answer to the nearest $ billion.

Note: Enter a trade surplus is a positive number and the trade deficit as a negative number. It is easiest to discount the imbalances in 2023 and 2024 back to 2022 (i.e., calculate the present value in 2022 dollars).

Question

In 2022, the U.S. is running a trade deficit of $1,000 billion. The expected surplus in 2024 is $200 billion. Assuming the interest rate is a constant 
3
%
, calculate the trade balance in 2023 such that the present discounted value of the sum of (expected) trade deficits and surpluses (from 2022 to 2024) is zero.

Use a spreadsheet for your calculations and round your answer to the nearest $ billion.

Note: Enter a trade surplus is a positive number and the trade deficit as a negative number. It is easiest to discount the imbalances in 2023 and 2024 back to 2022 (i.e., calculate the present value in 2022 dollars).

BlackTom AI · Accepted Answer

Present discounted value (PDV) of the three-year sequence must equal zero. Using 2022 as the discounting date, the equation is:
-1000 + TB2023/(1.03) + 200/(1.03^2) = 0.

Relevant concepts:
- PDV: future amounts divided by (1+r)^n to express them in present terms.
- A trade surplus is positive; a deficit is negative (2022 is -1000, 2024 is +200).
- r = 3% = 0.03.

Step-by-step solution:
1) Compute the 2024 term discounted back two years:
(1.03)^2 = 1.0609.
200 / 1.0609 = 188.519 (rounded to three decimals).

2) Substitute into the PDV equation:
-1000 + TB2023/1.03 + 188.519 = 0.

3) Isolate TB2023/1.03:
TB2023/1.03 = 1000 - 188.519 = 811.481.

4) Solve for TB2023:
TB2023 = 1.03 × 811.481 = 835.82543.

5) Round to the nearest $ billion:
835.82543 ≈ 836.

Why this answer is correct:
Discounting the 2023 and 2024 balances to 2022 dollars and equating their sum with the 2022 deficit enforces the PDV condition. The algebra above directly follows from that condition and yields TB2023 ≈ 835.825, which rounds to 836 (a surplus, positive as required).

Common mistakes (why other answers would be wrong):
- Forgetting to discount 2024 by (1.03)^2 or 2023 by 1.03 produces incorrect numbers.
- Using r = 3 instead of 0.03 or mis-signing the 2022 deficit would flip the result's sign.
- Rounding too early could slightly change the third-digit but not the rounded billion value.

Final confirmation: The computed 2023 trade balance is +836 (billion), matching the provided answer.

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