Chapter 15, Problem 8PA

(a)

To determine

The equations for dynamic aggregative demand (DAD) curve and dynamic aggregative supply (DAS) curve with respect to random shocks on expected inflation.

Explanation of Solution

The equation for the expected inflation with respect to random shocks in period $\text{t}-1$  is as follows.

${\text{E}}_{\text{t}}{}_{–1}{\text{π}}_{\text{t}}={\text{π}}_{\text{t}}{}_{–1}+{\text{η}}_{\text{t}}{}_{–1}$ (1)

Here, ${\text{η}}_{\text{t}}{}_{–1}$ is the random shock, which is zero. If it deviates from zero due to some events beyond past inflation, it will change the expected inflation. Therefore, the equation for expected inflation can be represented as follows:

${\text{E}}_{\text{t}}{\text{π}}_{\text{t}+1}={\text{π}}_{\text{t}}+{\text{η}}_{\text{t}}$ (2)

The equation for the dynamic aggregative supply (DAS) curve with the presence of random shocks can be derived as follows.

Consider the Philips curve equation in the dynamic AD-AS model.

The equation for Philips curve:

${\text{π}}_{\text{t}}={\text{E}}_{\text{t}-\text{1}}{\text{π}}_{\text{t}}+\varphi ({\text{Y}}_{\text{t}}-{\overline{\text{Y}}}_t)+{\upsilon }_t$ (3)

Now, substitute Equation (1) in Equation (3) to get the equation for DAS curve.

${\text{π}}_{\text{t}}={\text{π}}_{\text{t}}{}_{–1}+{\text{η}}_{\text{t}}{}_{–1}.+\varphi ({\text{Y}}_{\text{t}}-{\overline{\text{Y}}}_t)+{\upsilon }_t$ (4)

Therefore, the Philips curve becomes ${\text{π}}_{\text{t}}={\text{π}}_{\text{t}}{}_{–1}+{\text{η}}_{\text{t}}{}_{–1}.+\varphi ({\text{Y}}_{\text{t}}-{\overline{\text{Y}}}_t)+{\upsilon }_t$.

The equation for DAD curve can be derived by using the following equations in the dynamic AD-AS model:

The equation for the demand for goods and services:

${\text{Y}}_{\text{t}}={\overline{\text{Y}}}_{\text{t}}-{\text{α(r}}_{\text{t}}-\text{ρ})+{\text{ε}}_{\text{t}}$ (5)

The fisher equation:

${\text{r}}_t={\text{i}}_{\text{t}}-{\text{E}}_{\text{t}}{\text{π}}_{\text{t}+\text{1}}$ (6)

Consider Equation (5).

${\text{Y}}_{\text{t}}={\overline{\text{Y}}}_{\text{t}}-\alpha ({\text{r}}_{\text{t}}-\text{ρ})+{\text{ε}}_{\text{t}}$

Substitute Equation (6) into Equation (5).

${\text{Y}}_{\text{t}}={\overline{\text{Y}}}_{\text{t}}-\text{α}({\text{i}}_{\text{t}}-{\text{E}}_{\text{t}}{\text{π}}_{t+1}-\text{ρ})+{\text{ε}}_{\text{t}}$

Now, substitute Equation (2) instead of ${\text{E}}_{\text{t}}{\text{π}}_{t+1}$ in this equation.

${\text{Y}}_{\text{t}}={\overline{\text{Y}}}_{\text{t}}-\text{α}({\text{i}}_{\text{t}}-{\text{π}}_{\text{t}}-{\text{η}}_{\text{t}}-\text{ρ})+{\text{ε}}_{\text{t}}$ (7)

Consider the monetary policy rule in the dynamic AD-AS model.

The equation for the monetary-policy rule.

${\text{i}}_{\text{t}}={\text{π}}_{\text{t}}+\text{ρ}+{\text{θ}}_{\text{π}}{\text{(π}}_{\text{t}}-{\text{π}}_{\text{t}}^{\text{*}})+{\text{θ}}_{\text{Y}}({\text{Y}}_{\text{t}}-{\overline{\text{Y}}}_{\text{t}})$ (8)

Substitute Equation (8) in Equation (7).

${\text{Y}}_{\text{t}}={\overline{\text{Y}}}_{\text{t}}-\text{α}({\text{π}}_{\text{t}}+\text{ρ}+{\text{θ}}_{\text{π}}{\text{(π}}_{\text{t}}-{\text{π}}_{\text{t}}^{\text{*}})+{\text{θ}}_{\text{Y}}{\text{(Y}}_t-{\overline{\text{Y}}}_t)-{\text{π}}_{\text{t}}-{\text{η}}_{\text{t}}-\text{ρ})+{\text{ε}}_{\text{t}}$

Now, rearrange the above equation to get the equation for DAD curve.

$\begin{array}{c}{\text{Y}}_t={\overline{\text{Y}}}_{\text{t}}-{\text{α(π}}_{\text{t}}+\text{ρ}+{\text{θ}}_{\text{π}}{\text{(π}}_{\text{t}}-{\text{π}}_{\text{t}}^{\text{*}})+{\text{θ}}_{\text{Y}}{\text{(Y}}_{\text{t}}-{\overline{\text{Y}}}_{\text{t}})-{\text{π}}_{\text{t}}-{\text{η}}_{\text{t}}-\text{ρ})+{\text{ε}}_{\text{t}}\\ {\text{Y}}_t={\overline{\text{Y}}}_{\text{t}}-\text{α}\left({\text{θ}}_{\text{π}}{\text{(π}}_{\text{t}}-{\text{π}}_{\text{t}}^{\text{*}})+{\text{θ}}_{\text{Y}}({\text{Y}}_{\text{t}}-{\overline{\text{Y}}}_{\text{t}})-{\text{η}}_{\text{t}})\right)+{\text{ε}}_{\text{t}}\\ \left(1+{\text{αθ}}_{\text{Y}}\right){\text{Y}}_{\text{t}}=\left(1+{\text{αθ}}_{\text{Y}}\right){\overline{\text{Y}}}_{\text{t}}-{\text{αθ}}_{\text{π}}({\text{π}}_{\text{t}}-{\text{π}}_{\text{t}}^{\text{*}})+{\text{αη}}_{\text{t}}+{\text{ε}}_{\text{t}}\\ {\text{Y}}_{\text{t}}={\overline{\text{Y}}}_{\text{t}}-\left(\frac{{\text{αθ}}_{\text{π}}}{\text{1}+{\text{αθ}}_{\text{Y}}}\right)({\text{π}}_{\text{t}}-{\text{π}}_{\text{t}}^{\text{*}})+\left(\frac{\text{α}}{1+{\text{αθ}}_{\text{Y}}}\right){\text{η}}_{\text{t}}+\left(\frac{1}{1+{\text{αθ}}_{\text{Y}}}\right){\text{ε}}_{\text{t}}\end{array}$

Therefore, the equation for the DAD curve can be represented as follows:

${\text{Y}}_{\text{t}}={\overline{\text{Y}}}_{\text{t}}-\left(\frac{{\text{αθ}}_{\text{π}}}{1+{\text{αθ}}_{\text{Y}}}\right)({\text{π}}_{\text{t}}-{\text{π}}_{\text{t}}^{\text{*}})+\left(\frac{\text{α}}{\text{1}+{\text{αθ}}_{\text{Y}}}\right){\text{η}}_{\text{t}}+\left(\frac{1}{1+{\text{αθ}}_{\text{Y}}}\right){\text{ε}}_{\text{t}}$

Economics Concept Introduction

Dynamic aggregative demand curve: The dynamic aggregative demand curve indicates the negative association between output and inflation which determines the economy’s short-run equilibrium.

Dynamic aggregative supply curve: The dynamic aggregative supply curve indicates positive association between output and inflation which determines the economy’s short-run equilibrium.

 (b)

To determine

Explain the effect of random shock if it is greater than zero on DAD, DAS, output level, inflation rate, nominal interest rate, and real interest rate.

Explanation of Solution

The DAD curve shows the relationship between output and inflation. Therefore, the value of random shock is greater than zero, which causes to shift the DAD curve right ward. This rightward shift of DAD curve increases the rate of inflation and output level. In such situation, according to the monetary policy rule, the central bank increases the nominal interest rate and real interest rate. As a result, people expect a higher inflation in coming year; thus, they will increase the purchases today. On the other hand, the presence of random shock does not immediately make any changes on DAS curve because the DAS curve also depends on the lagged value of the shock parameter.

Economics Concept Introduction

Dynamic aggregative demand curve: The dynamic aggregative demand curve indicates the negative association between output and inflation which determines the economy’s short-run equilibrium.

Dynamic aggregative supply curve: The dynamic aggregative supply curve indicates positive association between output and inflation which determines the economy’s short-run equilibrium.

 (c)

To determine

The changes in DAD, DAS, output, inflation, nominal, and real interest rate in period $\text{t}+1$.

Explanation of Solution

As described in part (a), in period $\text{t}+1$, the expected inflation with random shock can be represented as follows:

${\text{E}}_{\text{t}}{}_{+1}{\text{π}}_{\text{t}}={\text{π}}_{\text{t}}{}_{+1}+{\text{η}}_{\text{t}}{}_{+1}$

Here, ${\text{η}}_{\text{t}}{}_{+1}$ is the random shock, which is zero. Therefore, in period $\text{t}+1$, the DAD curve shifts back to its original position; whereas the DAS curve depends on the variable ${\text{η}}_{\text{t}}$, the positive random shock, which causes to shift the DAS curve leftward. Hence, a leftward shift of DAS curve reduces the level of output and increases the value of inflation rate. A higher inflation with lower output lead to the situation called stagflation.

Economics Concept Introduction

Stagflation: The stagflation is an economic condition where slow economic growth is accompanied by rising inflation.

 (d)

To determine

The changes in subsequent periods.

Explanation of Solution

As described in part (c), in period $\text{t}+1$, the positive random shock which causes to shift the DAS curve leftward, which reduces the level of output and increases the value of inflation rate. However, in following periods, eventually, the lower level of output reduces the inflation, thereby the expected inflation rate. As a result, the DAS curve moves back to its original position, but it does not mean that the DAS curve immediately moves back to its original position because the rise in short-run inflation (described in part (c)) causes to increase the expected inflation above its long-run value.

Economics Concept Introduction

Dynamic aggregative supply curve: The dynamic aggregative supply curve indicates one of the two relationships between output and inflation which determines the economy’s short-run equilibrium.

 (e)

To determine

The statement that inflation scares self- fulfilling.

Explanation of Solution

As described in part (b), if the value of random shock is greater than zero, it causes to shift the DAD curve rightward. This rightward shift of DAD curve increases the rate of inflation and output level. This means when people expect a higher inflation, such type of random shocks act in a way that it actually does. Therefore, the given analysis illustrates that the inflation scares can be self-fulfilling.