Now suppose that
𝑏
¯
=
1
5
. Moreover, you want to "move" your economy closer to potential output and you are targeting
𝑌
~
=
0
%
. Compute the real interest rate
𝑅
𝑡
(in percent) that moves the economy to this target short-run output, assuming everything except
𝑏
¯
is the same as in the previous question.

Question

Now suppose that 
𝑏
¯
=
1
5
. Moreover, you want to "move" your economy closer to potential output and you are targeting 
𝑌
~
=
0
%
. Compute the real interest rate 
𝑅
𝑡
 (in percent) that moves the economy to this target short-run output, assuming everything except 
𝑏
¯
 is the same as in the previous question.

BlackTom AI · Accepted Answer

We start by restating what the question provides and what it asks for, so the setup is clear. The given pieces are: the average or steady-state parameter b̄ is 1/5 = 0.2, and the target short-run output is Y~ = 0%. Everything else is the same as in the previous question, and we are asked to compute the real interest rate R_t (in percent) that moves the economy to the target short-run output.

Because the full context (the exact form of the IS/monetary policy relationship from the prior question) isn’t reproduced here, we proceed by describing the standard mechanism that links the real interest rate to the output gap via the sensitivity parameter b̄.

- Conceptual idea: In a typical simplified macro model with an IS-like relationship, the output gap (the deviation of actual output from potential) responds to the real interest rate. The parameter b̄ captures how strongly the output responds to changes in the real rate: a higher b̄ means a given change in R_t moves output more. If the target output gap is zero (Y~ = 0%), then we want the actual output to equal potential in the short run, which requires setting the real rate so that the IS condition is satisfied at that point.

- General relation (conceptual): ΔR_t is the change in the real rate needed to achieve a target output gap. A common linearized form is something like: OutputGap = - b̄ × ΔR_t, where ΔR_t is the deviation of the real rate from some baseline or natural rate, and OutputGap is the percentage deviation of actual output from potential. If the target OutputGap is 0%, then the required ΔR_t is zero only if the current output gap is already zero in the baseline; otherwise, the needed ΔR_t would be ΔR_t = - (CurrentOutputGap) / b̄. In other words, you adjust R_t by an amount inversely proportional to b̄ to close the gap.

- Applying the given numbers: we have b̄ = 0.2. The target is Y~ = 0%, implying the desired output gap is 0. Without the explicit value for the current output gap or the baseline real rate from the previous question, we cannot compute an exact numeric ΔR_t from first principles here. The calculation would require the current short-run output deviation and the baseline (or natural) real rate from the prior question.

- Why the numeric result 3.1% (if it appears in the context) would make sense: if the current output gap is positive (economy above potential) or negative (below potential) in the prior setup, plugging those values into ΔR_t = - (CurrentOutputGap) / b̄ with b̄ = 0.2 would yield ΔR_t = -5 × (CurrentOutputGap) in percentage points. A resulting real rate of around 3.1% would reflect a modest tightening or loosening depending on the sign and magnitude of the gap, scaled by the 0.2 sensitivity. But the exact numeric value depends on the prior question’s baseline rate and the precise current output gap.

- Important caveat: since the problem statement says “everything else is the same as in the previous question” and provides only b̄ and Y~, you would typically need at least the current output gap (or the baseline R_t and the current deviation in Y) to compute a numeric R_t. In practice, educators often supply those prior values or the implied gap from the previous part so that you can carry the calculation forward. If you only know b̄ and the target gap, you cannot determine a numeric R_t without the starting point.

- Summary of reasoning steps you would follow once the missing pieces are available: (1) identify the current short-run output gap from the prior question, (2) apply the linear relation ΔR_t = - (CurrentOutputGap) / b̄ with b̄ = 0.2, (3) add ΔR_t to the baseline real rate from the prior question to obtain the required R_t, and (4) convert to a percentage if the rate is in decimal form.

If you can share the specific current output gap and the baseline real rate (or the prior question’s data), I can compute the exact R_t value step by step for you. The provided answer of 3.1% would then be verified or derived directly from those numbers.

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