Question

Consider the economy described in Applied Problem 23.

a. Derive expressions for the MP curve and the AD

curve.

b. Assume that p = 2. What are the real interest rate

and the equilibrium level of output?

c. Suppose government spending increases to $4 trillion.

What happens to equilibrium output?

d. If the Fed wants to keep output constant, then what

monetary policy change should it make?

Short answer

A. The Aggregate Demand equation is $Y=16.4-1.6\pi$

B. Hence, the aggregate output is $\$13.2$trillion when inflation rate is given as $2\%$in the economy.

C. Hence, the aggregate output increase from $\$13.2$trillion to $\$15.2$trillion when government expenditure increased by $\$0.5$trillion.

D. The real interest rate is$4.25\backslash \%.$

Step-by-step solution

:Concept Introduction 

Monetary policy is used by the central bank to control the liquidity of plutocrat from frugality to bring the frugality to stablecondition.This is done through operation of interest rate and the plutocrat force. Objects of a financial policy are high employment and affair stability, profitable growth, stability of fiscal requests, interest rate stability, and stability in foreign exchange calls. Aggregate demand is the complete claim of wares and graces in an frugality. Aggregate claim is the sum of consumption, blockade, administration charge and net exports.
affectedness is the overall increase in general price position of an frugality- affectedness causes increase in the price of goods and services which decreases exports and fall in aggregate importunity.

Explanation of Solution

A.

Given,

$\overline{\mathrm{C}}$is $\$3.25$trillion.

I is $\$1.3$trillion.

$\overline{\mathrm{G}}$is $\$3.5$trillion.

$\overline{T}$is $\$3$trillion.

$\overline{\mathrm{NX}}$is -$1 trillion.

$\overline{f}$is $1.$

$\mathrm{mpc}$is $0.75.$

d is$0.3.$

xis 0.1.

$\overline{\mathbf{r}}$is $1$.

$\lambda$is $1.$

Monetary Policy curve equation is:

$r=\overline{r}+\lambda \pi \left(1\right)$

Aggregate Demand curve equation is:

$\mathrm{Y}=[\overline{\mathrm{C}}+\overline{\mathrm{I}}-\mathrm{d}\overline{f}+\overline{\mathrm{G}}+\overline{\mathrm{NX}}-mpc\times \overline{\mathrm{T}}]\times \frac{1}{1-mpc}-\frac{\mathrm{d}x\mathrm{s}}{1-mp\mathrm{c}}\times (\overline{\mathrm{r}}+\lambda \pi )\left(2\right)$

Where,

- $Y$is the Aggregate demand.

-$\overline{\text{Cis autonomous consumption.}}$

- lis the investment expenditure.

- $d$is investment responsiveness to real interest rate.

- $\overline{f}$is financial frictions.

- G is Government expenditure.

-$\overline{\mathrm{NX}}$ is net export.

- mpc is marginal propensity to consume.

- T is the tax.

- x is net export responsiveness to real interest rate.

- r is autonomous component of real interest rate.

- $\lambda$is the responsiveness of real interest rate to inflation rate.

- $\pi$is the inflation rate.

Substitute the values of $\mathrm{r}$and $\lambda$in the equation (1),

$r=1+\pi$

Hence, the Monetary Policy curve equation is $r=1+\pi$.

Substitute the values in the equation (2),

$$\begin{gathered} Y=[3.25+1.3-0.3\times 1+3.5-1-0.75\times 3]\times \frac{1}{1-0.73}-\frac{0.340.1}{1-0.75}\times (1+\pi ) \\ \mathrm{Y}=[4.5]\times \frac{1}{0.25}-\frac{0.4}{0.25}\times (1+\pi ) \\ \mathrm{Y}=18-1.6\times (1+\pi ) \\ \mathrm{Y}=16.4-1.6\pi \end{gathered}$$

Hence, the Aggregate Demand equation is role="math" localid="1648507358113" width="113" height="20">Y=16.4-1.6π.

:Explanation of Solution 

B.

Given,

$\overline{\mathrm{C}}$is $\$3.25$trillion.

$\overline{I}$is$\$1.3$trillion.

$\overline{\mathrm{G}}$is $\$3.5$trillion.

$\overline{T}$is $\$3$trillion.

$\overline{\mathrm{NX}}$is - 1 trillion.

$\overline{f}$is 1.

$\mathrm{mpc}$is $0.75$.

$d$is $0.3$.

xis$0.1.$

ris$1.$

$\lambda$is $1.$

$\pi$is$2.$

Monetary policy curve equation:

$\mathrm{r}=\mathrm{r}+\lambda \pi$

Aggregate Demand curve equation is:

$\mathrm{Y}=[\overline{\mathrm{C}}+\overline{\mathrm{I}}-\mathrm{d}\overline{f}+\overline{\mathrm{G}}+\overline{\mathrm{NX}}-mpc\times \overline{\mathrm{T}}]\times \frac{1}{1-mpc}-\frac{\mathrm{d}+x}{1-mpc}\times (\overline{\mathrm{r}}+\lambda \pi )$

Where,

- $Y$is the Aggregate demand.

-$\overline{Cisautonomousconsumption}$

- lis the investment expenditure.

- $d$is investment responsiveness to real interest rate.

- $\overline{f}$is financial frictions.

- Gis Government expenditure.

- $\overline{\mathrm{NX}}$is net export.

- mpc is marginal propensity to consume.

- Tis the tax.

- xis net export responsiveness to real interest rate.

- $\overline{n}s$autonomous component of real interest rate.

- $\lambda$is the responsiveness of real interest rate to inflation rate.

- $\pi$is the inflation rate.

The Monetary Policy curve equation is $r=1+\pi$.

The Aggregate Demand equation is $Y=16.4-1.6\pi$.

Substitute, value of$\pi$to calculate real interest rate and aggregate output.

Real interest rate$=1+2$

$=3\%$

Hence, the real interest rate is $3\%$when inflation rate is given as $2\%$in the economy.

Aggregate output$=16.4-1.6\times 2$

$=\$13.2$

Hence, the aggregate output is $\$13.2$trillion when inflation rate is given as $2\%$in the economy.

:Explanation of Solution 

C.

Given,

$\overline{\mathrm{C}}$is $\$3.25$trillion.

$\overline{\text{I}}$is $\$1.3$trillion.

$\overline{\mathrm{G}}$is $\$4$trillion.

$\overline{\mathrm{T}}$is $\$3$trillion.

$\overline{\mathrm{NX}}$is$-\$1$trillion.

$\overline{f}$is $1$.

mpc is $0.75$.

$dis0.3.$

$$\begin{gathered} xis0.1. \\ Tis1. \end{gathered}$$

$\lambda$is 1 .

$\pi$is 2 .

Monetary policy curve is:

$r=\overline{r}+\lambda \pi$

Aggregate demand curve equation is:

$\mathrm{Y}=[\overline{\mathrm{C}}+\overline{\mathrm{I}}-\mathrm{d}\overline{f}+\overline{\mathrm{G}}+\overline{\mathrm{NX}}-mpc\times \overline{\mathrm{T}}]\times \frac{1}{1-mpc}-\frac{\mathrm{d}sx}{1-mpc}\times (\mathrm{r}+\lambda \pi )$

Where,

-$Y$is the Aggregate demand.

-$.\overline{Cisautonomousconsumption}$

- lis the investment expenditure.

- d is investment responsiveness to real interest rate.

- $\overline{f}$is financial frictions.

-$\overline{\text{Gis Government expenditure.}}$

- $\overline{\mathrm{NX}}$is net export.

- mpc is marginal propensity to consume.

- Tis the tax.

- xis net export responsiveness to real interest rate.

- ris autonomous component of real interest rate.

- $\lambda$is the responsiveness of real interest rate to inflation rate.

- $\pi$is the inflation rate.

Substitute the given values in equation (1) to determine change in aggregate output due to change in government expenditure,

$$\begin{gathered} Y=[3.25+1.3-0.3\times 1+4-1-0.75\times 3]\times \frac{1}{1-0.75}-\frac{0.3+0.1}{1-0.75}\times (1+2) \\ \mathrm{Y}=\left[5\right]\times \frac{1}{0.25}-\frac{0.0}{0.25}\times (1+2) \\ \mathrm{Y}=20-1.6\times (1+2) \\ \mathrm{Y}=\$15.2 \end{gathered}$$

Hence, the aggregate output increase from $\$13.2$trillion to $\$15.2$trillion when government expenditure increased by $\$0.5$trillion.

:Explanation of Solution

D .

In order to keep output constant at$\$13.2$the Fed will have to increase the real interest rate. The Fed can control the real interest rate through the autonomous component of real interest rate $\overline{r}$.

Given,

$\overline{\mathrm{C}}$is $\$3.25$trillion.

$\overline{I}$is $\$1.3$trillion.

$\overline{G}$is $\$4$trillion.

$\overline{\mathrm{T}}$is $\$3$trillion.

$\overline{\mathrm{NX}}$is $-\$1$trillion.

$\overline{f}$is 1 .

$\mathrm{mpc}$is $0.75$.

d is 0.3.

x is 0.1.

Monetary Policy curve equation

$r=\overline{r}+\lambda \pi$

Aggregate Demand curve equation is as,

$\mathrm{Y}=[\overline{\mathrm{C}}+\overline{\mathrm{I}}-\mathrm{d}\overline{f}+\overline{\mathrm{G}}+\overline{\mathrm{NX}}-mpc\times \overline{\mathrm{T}}]\times \frac{1}{1-mpc}-\frac{\mathrm{d}\pm x}{1-mpc}\times (\overline{\mathrm{r}}+\lambda \pi )$

Where,

- $Y$is the Aggregate demand.

-$\overline{Cisautonomousconsumption.}$

- Tis the investment expenditure.

- $d$is investment responsiveness to real interest rate.

- $\overline{f}$is financial frictions.

- $\overline{G}$is Government expenditure.

- $\overline{\mathrm{NX}}$is net export.

- mpc is marginal propensity to consume.

- Tis the tax.

- xis net export responsiveness to real interest rate.

- ٓ̄s autonomous component of real interest rate.

- $\lambda$is the responsiveness of real interest rate to inflation rate.

-$\pi$is the inflation rate.

Let the aggregate output is $\$13.2$trillion, now to keep aggregate output $\$13.2$trillion calculate the real interest rate:

$$\begin{gathered} 13.2=[3.25+1.3-0.3\times 1+4-1-0.75\times 3]\times \frac{1}{1-0.75}-\frac{0.3+0.1}{1-0.75}\times \left(\mathrm{r}\right) \\ 13.2=\left[5\right]\times \frac{1}{0.25}-\frac{0.4}{0.25}\times \left(\mathrm{r}\right) \\ 13.2=20-1.6\times \left(\mathrm{r}\right) \\ \mathrm{r}=4.25\% \end{gathered}$$

Hence, the real interest rate is$4.25\%.$