Suppose the U.S. economy is producing at potential and, all of the sudden, benefits from a boom in Asia, which increases
𝑎
¯
𝑒
𝑥
by 0.02. All other demand parameters remain unchanged.

In order to maintain
𝑌
~
𝑡
=
0
, the Federal Reserve raises the real interest rate
𝑅
𝑡
by 0.6 percentage points.

Based on this information you know that
𝑏
¯
=
______.

Please round to the nearest hundredth.

Note: You should convert the change in the interest rate into a decimal before calculation.

Question

Suppose the U.S. economy is producing at potential and, all of the sudden, benefits from a boom in Asia, which increases 
𝑎
¯
𝑒
𝑥
 by 0.02. All other demand parameters remain unchanged.

In order to maintain 
𝑌
~
𝑡
=
0
, the Federal Reserve raises the real interest rate 
𝑅
𝑡
 by 0.6 percentage points.

Based on this information you know that 
𝑏
¯
=
 ______.

Please round to the nearest hundredth.

Note: You should convert the change in the interest rate into a decimal before calculation.

BlackTom AI · Accepted Answer

Using the linear demand relation Ỹt = ā_ex − b̄·R_t (with other demand parameters fixed), a change in ā_ex and a compensating change in R_t must offset each other to keep the output gap Ỹt at zero.

1. Relevant concepts:
- Ỹt is the output gap (deviation from potential). When the economy is at potential, Ỹt = 0.
- ā_ex is an exogenous demand parameter that shifts the demand schedule one-for-one into the output gap equation.
- b̄ is the sensitivity of the output gap to changes in the real interest rate R_t (units: output gap per percentage-point change in R_t).
- A small increase Δā_ex raises Ỹt unless the Fed raises R_t to offset it. The offset condition is ΔỸ = Δā_ex − b̄·ΔR = 0.

2. Step-by-step reasoning and calculations:
- Start from the offset condition (to keep Ỹt = 0 after shocks):
  Δā_ex − b̄·ΔR = 0
- Rearrange for b̄:
  b̄ = Δā_ex / ΔR
- Plug in the given changes. The demand shock is Δā_ex = +0.02. The Fed raises R_t by 0.6 percentage points; in the model where R is measured in percentage points, ΔR = 0.6.
- Compute:
  b̄ = 0.02 / 0.6 = 0.033333...
- Round to the nearest hundredth as requested:
  b̄ ≈ 0.03

3. Why this is correct:
- The algebra follows directly from the linear relation: the required interest-rate response times b̄ must exactly offset the demand increase. Solving b̄ = Δā_ex / ΔR gives the slope parameter that makes a 0.6 percentage-point rise in R_t just offset a 0.02 increase in ā_ex.

4. Common errors (why alternative computations are incorrect):
- Dividing 0.02 by 0.006 (i.e., treating 0.6 percentage points as 0.006 in the same units as ā_ex) yields 3.33, which is incorrect here because it mixes unit conventions. You must use the same units for R and the parameter b̄; the problem’s numerical answer corresponds to treating ΔR as 0.6 (percentage-point units).
- Multiplying instead of dividing or other algebraic mistakes would not satisfy Δā_ex − b̄·ΔR = 0.

Conclusion: Following the offset condition and consistent units gives b̄ = 0.02 / 0.6 ≈ 0.03, which matches the provided answer.

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