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Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

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BuyFindarrow_forward

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 money

If 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=bank reserves, D=deposits of 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 aRi±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=0.2DC0.2D=0

Other equations are,

R=0.1DR0.1D=0

And

H=R+CHRC=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:

[1010120100.200010.10011010]

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 R25R2, R310R3.

[1010120100.200010.10011010][10101205010001010011010]

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

Perform the operations R2R25R1, R4R4+R1

[10101205010001010011010][101012000606000101000111120]

In the next step,

Perform the operation R216R2

[101012000606000101000111120][101012000101000101000111120]

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

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