Chapter 9, Problem 4NP

a)

To determine

To describe: The general equilibrium levels of the real wage, employment and output is to be calculated.

Answer to Problem 4NP

The real wage = $w=9$

The employment = N = 100

Output = $Y=950$

Explanation of Solution

The following equation will represent the production of economy −$Y=A\left(5N-0.0025N^2\right)$

Where,

A = productivity

And the labor equation for the marginal production −$MPN=5A-0.005AN$

When A = 2, then the labor supply curve will be given as −$NS=55+10\left(1-t\right)$

Where,

NS = supply of labor

w = real wage

The tax on the wage income is given as −

t = 0.5

The equations to calculate the desired consumption and desired investment can be given as −$C^d=300+0.8\left(Y-T\right)-200r$

  $I^d=258.5-250r$

If the government is purchasing, G = 50 then taxes are $T=20+0.25Y$ . And based on this data the equation for the money demand will be given as −$\frac{M^d}{P}=0.5Y-250\left(r+{\pi }^c\right)$

And the expected rate of inflation can be given as −${\pi }^c=0.02$

Now, nominal money supply = 9150

The following equation is used to calculate the labor demand, ND −$w=MPN=5A-0.005A\times ND$

Now, put the above equation in labor supply equation. Then, $ND=NS=N$

Given that −

A = 2 and t = 0.5

Then, $NS=55+10\left(1-t\right)w$

Or, $NS=55+10\left(1-t\right)\left(5A-\left(0.005A\right)ND\right)$

Or, $NS=55+10\left(1-t\right)\left(5\times 2-\left(0.005\times 2\right)N\right)$

  $N=55+5\left(10-0.01N\right)$

Now, solve the equation for the value N −$N=55+50-0.005N$

  $\begin{array}{l}1.05N=105\\ N=100\end{array}$

The equilibrium value of employment = N = 100

Now, put the above value in labor demand or labor supply equation. Then, the equation is −$NS=55+10\left(1-t\right)w$

Or, $100=55+10\left(1-0.5\right)w$

Or, $w=9$ = equilibrium value of real wage.

Now, put N=100 in production function to calculate equilibrium output −$Y=A\left(5N-0.0025N^2\right)$

  $Y=2\times \left(5\times 100-0.0025{\left(100\right)}^2\right)$

  $Y=950$

Economics Concept Introduction

Introduction: The real interest rate can be defined as the interest of an investor, saver or lender which is received after the inflation. The price level can be defined as the average of current prices which are present in entire spectrum of goods and services.

b)

To determine

To describe: The general equilibrium values of the real interest rate, consumption and investment is to be described.

Answer to Problem 4NP

The real interest rate = r = 0.05

The consumption = C = 844

The investment = I = 246

Explanation of Solution

The equation for the goods market which clear real interest market rate by using the desired saving equation −$S^d=Y-C^d-G$

Put the above equation in desired consumption by using government purchases, G and taxes, T −$S^d=Y-C^d-G$

  $S^d=Y-\left[300+0.8\left(Y-T\right)-200r\right]-G$

Or, $S^d=Y-\left[300+0.8\left(0.75Y-20\right)-200r\right]-G$

Simplify the above equation −$S^d=Y-\left[300+0.4Y-16-200r\right]-G$

  $S^d=0.6Y-284+200r-G$

To calculate the value of r, then $S^d=I^d$

Then, $0.6Y-284+200r-G=258.5-250r$

Or, $r=\frac{542.5+G}{450}-\frac{0.6Y}{450}$

To calculate the full employment output (Y), put G=50 in Equ −$\begin{array}{l}r=\frac{542.5+50}{450}-\frac{0.6\times 950}{450}\\ r=0.05\end{array}$

Now, the real interest rate = r = 0.05

Now, put the calculated values in desired consumption equation −$\begin{array}{l}{C}^d=300+0.8\left(Y-T\right)-200r\\ C^d=300+0.8\left(Y-\left[20+0.25Y\right]\right)-200r\end{array}$

  $C^d=300+0.8\left(950-\left[20+0.25\times 950\right]\right)-200\times 0.05$

  $C^d=290+554=844$

Equilibrium value of consumption = 844

To calculate the value $I^d$ , put the value of real interest rate in desired investment equation −$\begin{array}{l}{I}^d=258.5-250r\\ I^d=258.5-250\times 0.05=246\end{array}$

The equilibrium value of investment = 246

Economics Concept Introduction

Introduction: The real interest rate can be defined as the interest of an investor, saver or lender which is received after the inflation. The price level can be defined as the average of current prices which are present in entire spectrum of goods and services.

c)

To determine

To describe: The general equilibrium value of the price levels is to be calculated.

Answer to Problem 4NP

The equilibrium value of the price level = P =20

Explanation of Solution

Equate the money demand equation to money supply for the calculation of value r. Then the equation can be represented as −$\frac{M^d}{P}=\frac{M}{P}$ ……………..Equ (1)

Also given that −${\pi }^e=0.02$

The above equation can also be represented as −$\begin{array}{l}0.5Y-250(r+{\pi }^c)=\frac{9150}{P}\\ 0.5Y-\frac{9150}{P}=250\left(r+0.02\right)\end{array}$

  $0.5Y-\frac{9150}{P}=250r+5$

Continue to solve for r −$\frac{0.5Y}{250}-\frac{9150}{250P}-\frac{5}{250}=r$

  $0.002Y-\frac{36.6}{P}-0.02=r$

Put the following values in above Equ −

Y = 950

r = 0.05$0.002\times 950-\frac{36.6}{P}-0.02=0.05$

  $1.83=\frac{36.6}{P}$

Or, $P=\frac{36.6}{1.83}=20$

Equilibrium value of price level = P =20

Economics Concept Introduction

Introduction: The real interest rate can be defined as the interest of an investor, saver or lender which is received after the inflation. The price level can be defined as the average of current prices which are present in entire spectrum of goods and services.

d)

To determine

To describe: The general equilibrium values of the real wage, employment, output, the real interest rate, consumption, investment and price level are to be described.

Answer to Problem 4NP

The real interest rate = 0.10

Consumption = C = 834

Investment = I= 233.5

Price level = P = 20.56

The values of real wage, employment and output will not change.

Explanation of Solution

Now, the government purchase, G = 72.5

The IS and LM equation will be used to calculate the real wage, employment, output, real interest rate and price level −

Given that −

G = 72.5

Y = 950

Now calculate value of r −$r=\frac{542.5+G}{450}-\frac{0.6Y}{450}$

Put the given values in above Equ −$r=\frac{542.5+72.5}{450}-\frac{0.6\times 950}{450}$

  $r=0.10$

Equilibrium value of real interest rate = 0.10

Now, put the full employment output and real-interest in desired consumption equation −$C^d=300+0.8\left(Y-T\right)-200r$

Or, $C^d=300+0.8\left(Y-\left[20+0.5Y\right]\right)-200r$

Put the given values in above Equ −$C^d=300+0.8\left(950-\left[20+0.5\times 950\right]\right)-200\times 0.10$

  $C^d=300+0.8\times \left(950-257.5\right)-20$

  $C^d=280+554=834$

The equilibrium value of consumption = C = 834

Put the real interest value in desired investment equation −$\begin{array}{l}{I}^d=258.5-250r\\ I^d=258.5-250\times 0.10\end{array}$

  $I^d=233.5$

The equilibrium value of investment = 233.5

To solve for P, put the given values in LM equation −

Given that −

Y = 950

r = 0.10$0.002Y-\frac{36.6}{P}-0.02=r$

Put the given values in above Equ −$0.002\times 950-\frac{36.6}{P}-0.02=0.10$

  $1.78=\frac{36.6}{P}$

Continue to solve for P −$P=\frac{36.6}{1.78}=20.56$

The equilibrium value of price level = 20.56

The equilibrium value of real wage, employment and output will not be affected by the increased government purchase.

Economics Concept Introduction

Introduction:

The real interest rate can be defined as the interest of an investor, saver or lender which is received after the inflation. The price level can be defined as the average of current prices which are present in entire spectrum of goods and services.