Suppose That The Economy Responds To The Real Interest

Sheet 1suppose That The Economy Responds To The Real Interest Rate Acc

Suppose that the economy responds to the real interest rate according to the following equation: Yt = Y – Y (it-1 – 0.025). Potential GDP is $25 trillion; expectations are adaptive; the natural rate of unemployment is 5.5%; the 2010 inflation rate was 3.5%. The Phillips curve is: Ï€t = Ï€te + 0.75 (Yt-1 – Y)/Y. Based on this information, answer the following questions:

Paper For Above instruction

This paper aims to analyze various macroeconomic indicators and their interrelations based on the provided model and data. It will elucidate the relationships between output, inflation, and interest rates, and project future economic conditions, including potential policy implications. The analysis integrates the Phillips curve, the dynamic response of GDP to the real interest rate, and the adaptive expectations framework, facilitating a comprehensive understanding of economic trajectories from 2010 through 2014.

Question 51) Suppose the unemployment rate in 2011 was indicated on blackboard. What is GDP in points?

To determine the GDP in 2011, we need to link the unemployment rate to output via the Phillips curve. The natural rate of unemployment (u*) is 5.5%. Using Okun's law and the Phillips curve, the deviation of unemployment from its natural rate relates to output gap. Assuming the unemployment rate (u) in 2011 is known from the blackboard, the output gap can be estimated as:

  • u - u ≈ − (Yt - Y) / (some proportionality constant, often around 2)

In the absence of the exact unemployment rate, but knowing the Phillips curve:

Ï€t = Ï€te + 0.75 (Yt-1 – Y) / Y

To solve for GDP in 2011 (Y2011), further data on the unemployment rate or inflation deviation is required. Since these are not provided, a common approach is to assume that the unemployment rate in 2011 corresponds to a certain output gap. If, for example, the unemployment rate was slightly below natural rate, indicating an output above potential, approximate GDP points can be inferred accordingly. Without specific unemployment data, an exact numerical answer cannot be provided; rather, the procedure involves translating the unemployment deviation into an output gap, then converting that into points of GDP relative to potential GDP ($25 trillion).

Question 52) Suppose output in 2010 was indicated on blackboard. Calculate the inflation rate in points.

Given the 2010 inflation rate π2010 = 3.5%, and the Phillips curve:

Ï€t = Ï€te + 0.75 (Yt-1 – Y)/Y

Expectations are adaptive, implying πte for 2010 equals previous inflation plus some adjustment, possibly tied to core inflation expectations. Assuming adaptive expectations update gradually, πte in 2010 approximately equals the previous rate of 3.5%, and the actual inflation aligns accordingly.

The output level in 2010 is given; thus, the inflation in 2010 can be cross-validated with the Phillips curve. If output in 2010 was at potential, then (Y2010 – Y)/Y ≈ 0, leading to:

Ï€2010 ≈ Ï€te. Thus, inflation in 2010 was approximately 3.5%, matching the provided data. Therefore, the inflation rate in points in 2010 is 3.5%.

Question 53) What will the 2012 output be?

To forecast 2012 output, we use the economy's response equation:

Yt = Y – Y (it-1 – 0.025)

The real interest rate (it-1) influences output; assuming the interest rate remains constant or follows a predictable policy path, the equation can be applied recursively. Without specific interest rate data, a baseline assumption might be that the interest rate in 2012 (i2) is similar to the previous period.

Suppose the interest rate in 2011 was at a certain value, then the 2012 output can be computed as:

Y2012 = 25 – 25 * (i2011 – 0.025)

If, for example, interest rates held steady at 2%, then:

Y2012 = 25 – 25 (0.02 – 0.025) = 25 – 25 (–0.005) = 25 + 0.125 = 25.125 trillion.

Therefore, assuming interest rates are at 2%, the 2012 GDP would be approximately $25.125 trillion, slightly above potential GDP.

Question 54) What will the 2012 and 2013 inflation rates be?

Inflation in each year is given by the Phillips curve:

Ï€t = Ï€te + 0.75 (Yt-1 – Y)/Y

Assuming inflation expectations are adaptive and have been anchored at 3.5% in 2010, and the output gaps are known from previous calculations, we can estimate inflation:

For 2012:

Ï€2012 = 3.5% + 0.75 * (Y2011 – 25)/25

If Y2011 was, for example, 25.125 trillion (from earlier assumptions), then:

Y2011 – 25 = 0.125 trillion, so:

Ï€2012 = 3.5% + 0.75 (0.125 / 25) = 3.5% + 0.75 0.005 = 3.5% + 0.00375 = 3.50375%

Similarly, for 2013, using the projected Y2012:

Ï€2013 = 3.50375% + 0.75 * (Y2012 – 25)/25

If Y2012 is 25.125 trillion, then:

Ï€2013 ≈ 3.50375% + 0.75 * 0.005 = 3.50375% + 0.00375 ≈ 3.5075%

Thus, inflation rates in 2012 and 2013 would be approximately 3.5% and 3.51%, respectively, assuming stable interest rates and modest output deviations.

Question 55) If the federal reserve wants to bring inflation to 2% in 2014 what will output need to be in 2013?

To reduce inflation to 2% by 2014, the Fed must influence the output gap in 2013. According to the Phillips curve:

Ï€2014 = Ï€e + 0.75 (Y2013 – Y) / Y*

Assuming the expectation πe is anchored at 2%, we solve for the necessary output level in 2013:

2% = 2% + 0.75 * (Y2013 – 25) / 25

Subtract 2% from both sides:

0 = 0.75 * (Y2013 – 25) / 25

This implies Y2013 must be exactly at potential GDP (Y*), which is $25 trillion, to keep inflation at target 2%.

Question 56) The fed wants inflation to be 1% in 2013. In question 55 you calculated the 2013 output level to get there. Now calculate the 2012 interest rate necessary to achieve that.

To lower inflation to 1% in 2013, the Fed must induce a larger negative output gap. First, determine the output level that achieves 1% inflation:

1% = Ï€e + 0.75 (Y2013 – 25)/25

Assuming expectations πe have been adjusted downward to 1% (or anchored at 1%), then:

1% = 1% + 0.75 * (Y2013 – 25)/25

This simplifies to:

0 = 0.75 * (Y2013 – 25)/25

Again, Y2013 must be at potential GDP (Y* = 25 trillion) to target 1% inflation. To achieve this, the interest rate in 2012 needs to be set such that the output is at potential. Rearranging the core equation:

Y2013 = 25 – Y* (i2012 – 0.025)

Given that Y2013 is targeted at potential GDP, the interest rate i2012 should be at the level that aligns output with potential. Therefore:

i2012 = 0.025

which corresponds to the neutral real interest rate aligned with the Federal Reserve’s policy to maintain both the inflation target and output at potential. Slight deviations from this rate would create output gaps necessary for infla

tion adjustments, but for the purpose of setting interest rates to target 1% inflation in 2013, the interest rate should be approximately 2.5%.

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