Chapter 4.3, Problem 37E

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Chapter
Section

### Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

# Econometrics (from the GRE Economics Test) This and the next exercise are based on the following simplified model of the determination of the money stock: M = C + D C = 0.2 D R = 0.1 D H = R + C where M = Money Stock C = Currency in circulation R = Bank Reserves D = Deposits of the public H = High-powered moneyIf the money stock were $120 billion, what would bank reserves have to be? To determine To calculate: The bank reserves if the money stock were$120 billion such that

M=C+DC=0.2DR=0.1DH=R+C

Where M is money stock, C is currency in regulation, R=bank reserves, D=deposits of public, H is high powered money.

Explanation

Given Information:

The money stock is $120 billion. It is given that M=C+DC=0.2DR=0.1DH=R+C Where M is money stock, C is currency in regulation, R=bankĀ reserves, D=depositsĀ ofĀ public, H is high powered money. Formula Used: Elementary row operations Type 1: Replacing the row Ri by aRi, where a is a nonzero number. Type 2: Replacing the row Ri by aRiĀ±bRj, where a is a nonzero number. Gauss Jordan reduction method: Step 1: First clear the fractions or decimals if any, using operations of type 1. Step 2: Select the first nonzero element of the first row as pivot. Step 3: Use the pivot to clear its column using operations of type 2. Step 4: Select the first nonzero element in the second row a pivot and clear its column. Step 5: Turn all the selected pivot elements into a 1 using operations of type 1. Calculation: Consider the provided information, The money stock is$120 billion.

So,

M=\$120 billion

Consider the provided equations,

M=C+DC=0.2DR=0.1DH=R+C

So,

M=C+DC+D=120

Also,

C=0.2DCā0.2D=0

Other equations are,

R=0.1DRā0.1D=0

And

H=R+CHāRāC=0

The augmented matrix for the given system of equations is set up with the unknowns in order: C, R, D, H is shown below:

[101012010ā0.20001ā0.100ā1ā1010]

Apply Gauss Jordan reduction method to get the solution of the given system of equation.

Begin with simplifying rows 2 and 3

Perform the operations R2ā5R2, R3ā10R3.

[101012010ā0.20001ā0.100ā1ā1010]ā[101012050ā100010ā100ā1ā1010]

Pivot the first nonzero element of the first row and clear its column.

Perform the operations R2āR2ā5R1, R4āR4+R1

[101012050ā100010ā100ā1ā1010]ā[101012000ā60ā600010ā1000ā111120]

In the next step,

Perform the operation R2ā16R2

[101012000ā60ā600010ā1000ā111120]ā[101012000ā10ā100010ā1000ā111120]

Next pivot the first nonzero element of the second row and clear its column

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