Part 1If piπ denotes the rate of inflation and ii denotes the nominal rate of interest, then the amount a borrower repays in a year on a one-dollar loan is [input] ▼ empty selection and the inflation-adjusted purchasing power of the originally borrowed dollar is [input] ▼ i plus pii+π 1 plus pi1+π 1 minus pi1−π empty selection .Part 2The inflation-adjusted purchasing power of the originally borrowed dollar is subtracted from the amount a borrower repays in a year on a one-dollar loan.The result is the ____________. A. inflation-adjusted cost of the loan. B. real price of the loan. C. real interest rate. D. all of the above. E. A and B only.

Question

Part 1If piπ denotes the rate of inflation and ii denotes the nominal rate of​ interest, then the amount a borrower repays in a year on a​ one-dollar loan is [input] ▼  empty selection and the​ inflation-adjusted purchasing power of the originally borrowed dollar is [input] ▼  i plus pii+π 1 plus pi1+π 1 minus pi1−π empty selection .Part 2The​ inflation-adjusted purchasing power of the originally borrowed dollar is subtracted from the amount a borrower repays in a year on a​ one-dollar loan.The result is the​ ____________. A. ​inflation-adjusted cost of the loan. B. real price of the loan. C. real interest rate. D. all of the above. E. A and B only.

BlackTom AI · Accepted Answer

The nominal repayment on a one-dollar loan equals 1 + i, and the original dollar’s purchasing power one year later (adjusted for inflation π) equals 1/(1+π) measured in year‑1 dollars.

1) Relevant concepts
- Nominal repayment: money actually repaid in dollars = 1 + i.
- Inflation adjustment: to compare amounts across time, convert to the same price-level units. A dollar today is worth 1/(1+π) of a dollar one year from now.
- Real (inflation‑adjusted) cost/price: the repayment amount measured in constant (inflation‑adjusted) dollars minus the inflation‑adjusted principal.
- Real interest rate: a percentage rate r such that (1 + r) = (1 + i)/(1 + π).

2) Step-by-step reasoning and calculations
- Nominal amount repaid in one year on a $1 loan: 1 + i.
- Inflation‑adjusted purchasing power of the originally borrowed $1 in year‑1 dollars: 1/(1+π).
- Subtract the inflation‑adjusted principal from the nominal repayment to get the inflation‑adjusted amount the borrower actually pays:
  (1 + i) − 1/(1 + π).

Show intermediate algebra comparing this to the real interest rate:
- Express the subtraction over a common denominator:
  (1 + i) − 1/(1 + π) = [(1 + i)(1 + π) − 1] / (1 + π)
                      = [1 + i + π + iπ − 1] / (1 + π)
                      = (i + π + iπ) / (1 + π).
- The real interest rate r is (1 + i)/(1 + π) − 1 = (i − π) / (1 + π), which is different from the expression above. The subtraction yields an inflation‑adjusted dollar amount (a real cost), not the percentage real rate.

3) Why the correct answer is E (A and B only)
- A (inflation‑adjusted cost of the loan) is correct because the subtraction produces the real dollar cost paid by the borrower.
- B (real price of the loan) is also correct because “real price” is another way to refer to the inflation‑adjusted amount the borrower repays net of the inflation‑adjusted principal.
- C (real interest rate) is incorrect because the real interest rate is a rate ((1+i)/(1+π) − 1), not the dollar amount produced by the subtraction.
- D (all of the above) is incorrect because C is wrong.

Conclusion: The subtraction yields the inflation‑adjusted cost of the loan and the real price of the loan, so the correct choice is E. A and B only.

类似问题

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