An economy is currently subject to a negative demand shock
𝑎
¯
=
−
0.01
. You know that
𝑏
¯
=
1
4
and
𝑟
¯
=
3.5%. Compute the real interest rate
𝑅
𝑡
that is consistent with
𝑌
~
=
0.01
.

Round your answer to the nearest tenth of a percent.

Note: You should convert interest rates to decimal values before using them in calculations.

Question

An economy is currently subject to a negative demand shock 
𝑎
¯
=
−
0.01
. You know that 
𝑏
¯
=
1
4
 and 
𝑟
¯
=
 3.5%. Compute the real interest rate 
𝑅
𝑡
 that is consistent with 
𝑌
~
=
0.01
.

Round your answer to the nearest tenth of a percent.

Note: You should convert interest rates to decimal values before using them in calculations.

BlackTom AI · Accepted Answer

We start by restating the given components of the problem to ensure we understand the setup. The economy faces a negative demand shock with an autonomous effect a_bar = -0.01, a parameter b_bar = 1/4, and a given baseline real interest rate r_bar = 3.5%. We are asked to find the real interest rate R_t that is consistent with an output gap Ŷ = 0.01. The problem notes that rates should be converted to decimal values for calculations.

A common linear relationship used in this kind of IS-type or elasticity-based framework expresses the output gap as a function of the real interest rate deviation from its baseline, adjusted by a sensitivity parameter. A plausible specification is:
Ŷ = a_bar + b_bar*(R_t − r_bar)
This says: the output gap is driven by an autonomous shock a_bar plus the tightening/loosening effect of moving the real rate away from its baseline, scaled by b_bar.

With this equation, solving for R_t requires algebra:
Ŷ = a_bar + b_bar*(R_t − r_bar)
Rearrange: Ŷ − a_bar = b_bar*(R_t − r_bar)
Divide: (Ŷ − a_bar)/b_bar = R_t − r_bar
Thus R_t = r_bar + (Ŷ − a_bar)/b_bar
Plug in the numbers (convert to decimal where needed):
- Ŷ = 0.01, a_bar = −0.01, so Ŷ − a_bar = 0.01 − (−0.01) = 0.02
- b_bar = 1/4 = 0.25
- (Ŷ − a_bar)/b_bar = 0.02 / 0.25 = 0.08
- r_bar = 3.5% = 0.035 in decimal terms
Then R_t = 0.035 + 0.08 = 0.115, which as a percent is 11.5%

Thus, under the assumed linear relationship, the real interest rate consistent with Ŷ = 0.01 is 11.5% (rounded to the nearest tenth of a percent: 11.5%).

If you were using a slightly different specification for how Ŷ relates to R_t (for example, Ŷ = a_bar + b_bar*R_t without subtracting r_bar, or using a different normalization for r_bar), you would obtain a different numeric result. The key steps are to (1) identify the model relation between Ŷ, R_t, a_bar, b_bar, and r_bar, (2) isolate R_t, and (3) substitute the given values, converting percentages to decimals when performing the arithmetic.

Note: In this explanation, I followed a common linear-demand/IS-type structure. If your course or problem uses another dynamic or a different form for the relationship, you would adjust the algebra accordingly, but the general approach—substitute the knowns into the model equation and solve for R_t—remains the same.

FA25 ECON 302 002 Homework #8 (IS Curve)

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