economics 311 money and income chapter 4-the demand for money. department of economics college of...

Economics 311Money and Income

Chapter 4-The Demand for Money.

Department of Economics

College of Business and Economics

California State University-Northridge

Professor Kenneth Ng

Friday, April 21, 2023

Introduction

Examine 3rd major decision made by household.

Want to explain the level of cash balances a households holds (money demand). Can think of “cash balances” in a simplistic fashion

as the amount of currency a household holds.

Two-Step process of explanation. First, examine the problem graphically. Second, examine the problem mathematically.

Simple Numeric Example

Consider a person who: Earns $12,000 of income a year and has his income

direct deposited into his checking account. Is paid monthly. Spends $12,000 evenly throughout the year Visits the bank once every month once a month and

withdraws just enough to get him to support his spending until his next planned visit to the bank.

What do the level of his cash balances look like over time Examine problem graphically.

Time (Years)

CashBalances

12

1

$1,000

Income = $12,000

Transaction Interval=1/12

•Note what is on the axes.

•Time measured in years.

•How will the amount of cash balances held by the individual change over time?

•How much must be withdrawn at the beginning of the month to support his spending for the month?

•$1000

•What will happen to his cash balances as the month progresses?

Cash balances over time.

Time (Years)

CashBalances

12

1

$1,000

Cash balances over time.•Cash balances will follow the saw-tooth pattern depicted below.

•What is his transaction interval?

•Transaction interval=time measured in years between visits to the bank to replenish cash balances.

•What are his average cash balances?

•$500

•Notice this is half the amount he withdraws.

•What is velocity?

•Velocity=number of times on average each dollar is used to purchase goods and services in a year.

•The household is using an average cash balance of $500 to purchase $12,000 in goods and services so:

•Velocity=12,000/500=24

What happens if he goes to the bank more frequently?

Time (Years)

CashBalances

12

1

$1,000

24

1

24

3

12

2

$500

•Suppose the person decreases his transaction interval from 1/12th to 1/24th?

•What will happen to the number of trips made to the bank annually?

•Increase from 12 to 24.

•If the household is making twice as many trips but annual spending is unchanged, what happens to the amount of cash withdrawn at each visit to the bank?

•Decreases from $1000 to $500

•What will his pattern of cash balances over time look like?

What happens if he goes to the bank more frequently?

Time (Years)

CashBalances

12

1

$1,000

Income = $12,000

Transaction Interval=1/24

24

1

24

3

12

2

$500

•Cash balances will follow the saw-tooth pattern depicted below.

•What is his transaction interval?

•Transaction interval=time measured in years between visits to the bank to replenish cash balances.

•1/24th of a year

•What are his average cash balances?

•$250

•Notice this is half the amount he withdraws.

•What is velocity?

•Velocity=number of times on average each dollar is used to purchase goods and services in a year.

•The household is using an average cash balance of $250 to purchase $12,000 in goods and services so:

•Velocity=12,000/250=48

Time (Years)

Money in Bank

12

1

$1,000

Time (Years)

CashBalances

12

1

$500

The economic forces underlying the demand for money

The bottom graph shows the amount of money the household keeps in the bank earning interest with a transaction interval of 1/24th of a year.

For the first half of each month, the household has $500 in the bank.

For the second half of each month the household has $0 in the bank.

The household is maintaining an average bank balance of $250.

Time (Years)

Money in Bank

12

1

$1,000

Time (Years)

CashBalances

12

1

$500

The economic forces underlying the demand for money•What are average bank balances when the household chooses a transaction interval of 1/2th?

•0$

•What is the benefit of choosing a smaller transaction interval and making more trips to the bank?

•More interest earned.

•Each dollar held as cash balances means foregone interest.

•What is the benefit of choosing a smaller transaction interval and making more trips to the bank?

•Resources used making trips.

•What transaction interval is optimal for the household?

•The one which minimizes the total cost of holding cash balances or optimally trades off the cost of trips and interest foregone.

$1000

The Demand for Money

The demand for money (cash balances) is determined by the transaction interval (T) chosen by the household.

The optimal transaction interval is determined by the interplay of two costs. Interest foregone from holding cash balances. Cost of making a trip to the bank to get cash

balances. Can solve for the optimal transaction interval

mathematically.

Define the variables.

rinterest of RateNominalR

Inflation of Rate

ateInterest R Realr

sCommoditie of nConsumptio AnnualC

LevelPriceP

Bank toTrip ofCost Nominal

(years) Intervalent ReplenishmT

Formulas

2

1

P Money HoldingofCost Total

2

P

mRoregoneInterest F

2

HoldingsCash Average Nominal

1

P Cost nTransactio Real

TCR

T

TCR

Tcp

m

T

Real Transaction Costs

T

1

P Cost nTransactio Real

Real transaction costs are equal to the real cost of a single trip times the number of trips.

If the transaction interval is 1/12th how many trips will be made per year.

Answer: 12.

Nominal Average Cash Holdings

Nominal average cash holdings are equal to total spending times the transaction interval .

PC=total spending

PCT=amount withdrawn at each visit to the bank.

Dividing the amount withdrawn by 2 gives average cash holdings.

2

HoldingsCash Average Nominal

Tcp

m

Interest Foregone by Holding Cash Balances

Interest forgone is the interest rate times average cash balances.

Simplifying yields the formula.

2

2

m

p

mRoregoneInterest F

TCR

Tcp

p

R

pR

Total Cost of Holding Money

The total cost of holding money (cash balances) can now be expressed with the formula above.

The total cost of holding money has two components.

The cost of trips to the bank (first term of equation).

Interest forgone (second term of equation).

2

1

P Money HoldingofCost Total

TCR

T

2

1

PTC

TCR

T

Can express the household's problem as the following--Want to choose the transaction interval (T) that minimizes the total cost of holding cash balances. Algebraically, can solve the problem:

Differentiating TC with respect to T yields:

22 RC

TPT

TC

Setting equal to 0 and solving for T yields:

Intervaln Transactio Optimal2

22

20

*

2

2

2

RCPT

RCPT

TP

RC

RCT

P

Can now substitute T* to determine optimal cash balances:

cT2

1

P

m

Substituting T*for T:

R

c

P

RCP

c

2

2

2P

m*

Finally can solve for Velocity by substituting for m/p:

PRcV

Rc

P

cV

Pm

cc

2

2

m

PV

*

*

This allows us to express general formulas for Money Demand and Velocity:

cP

rVV

cP

rLP

m

,,,

,,,

Can also display the forces at work graphically:

$

Transaction Interval

What happens to the number of trips to bank per year as the transaction interval increases?

Fewer Trips-------

Can also display the forces at work graphically:

$

Transaction Interval

What happens to the cost of making trips (transaction costs) as the transaction interval increases?

Using simple algebra, as T gets bigger, the fraction gets smaller.

Fewer Trips-------

T

1

P

Can also display the forces at work graphically:

$

Transaction Interval

Fewer Trips-------

T

1

P

2

TCR

What happens to amount of interest foregone as the transaction interval increases?

Again using simple algebra, as T increases the fraction increases.

Can also display the forces at work graphically:

$

Transaction Interval

The total cost of cash balances is the sum of the vertical heights of the interest foregone and the transaction costs curve.

Fewer Trips-------

T

1

P

2

TCR

2

1

P

TCR

T

T1

The Optimal Transaction Interval (T*)

$

Transaction Interval

Fewer Trips-------

T

1

P

2

TCR 2

1

P

TCR

T

T*T1

The optimal transaction interval is the one which minimizes the total cost of holding cash balances.

Simple geometry indicates that this occurs at T*.

Can you explain in words why the household will be better off moving from T1 to T*?

The Optimal Transaction Interval (T*)

$

Transaction Interval

Fewer Trips-------

T

1

P

2

TCR 2

1

P

TCR

T

T*T1

Moving from T1 to T* what is happening to the number of trips to the bank and transaction costs?

Fewer trips and less transaction costs.

Moving from T1 to T* what is happening to the interest foregone by holding cash balances?

Fewer trips means more money must be withdrawn at each visit to the bank so bank balances are lower and interest foregone is greater.

Moving from T1 to T* will make the household better off (lower the cost of cash balances) if the extra interest foregone is less than the transaction costs saved from making fewer trips.

Can express the three variables, the transaction interval, real cash balances, and velocity as

functions.

cP

rVV

cP

rLP

m

cP

rTT

,,,

,,,

,,,*

Using these three formulas, you should be able to explain what happens to the three variables and why when there is a change in the world.

The sign under each variables shows what will happen to the value of the function when that variable increases in value.

Analysis (1)

cP

rVV

cP

rLP

m

cP

rTT

,,,

,,,

,,,*

What would happen to the transaction interval, the number of trips to the bank, the demand for money, and velocity if interest rates rose?

T* will decrease.

Cash balances will decrease.

Velocity will increase.

Can you explain why?

Effect of an increase in interest rates.

$

Transaction Interval

Fewer Trips-------

T

1

P

2

TCR

2

1

P

TCR

T

T*

Will the increase in interest rates effect transaction costs?

No—no shift in red line.

Will the increase in interest rates effect interest foregone?

Yes—Blue line will shift up.

When we vertically sum the transaction cost and interest foregone curves to get the total cost curve, the total cost will have shifted up and to the left.

The optimal transaction interval (T*) will now be at a smaller value.

Does this make sense?

T*

Analysis (2)

cP

rVV

cP

rLP

m

cP

rTT

,,,

,,,

,,,*

What would happen to the transaction interval, the number of trips to the bank, the demand for money, and velocity if there was a technological change in the banking industry such as the introduction of the ATM?

T will decrease.

Cash balances will decrease.

Velocity will increase

Effect of a technological change.

$

Transaction Interval

Fewer Trips-------

T

1

P

2

TCR

2

1

P

TCR

T

T*

Will the increase in the cost of a trip to the bank affect interest foregone?

No—Blue line will not shift.

Will the decrease in the cost of a trip to the bank affect transaction costs?

Yes—red line will shift down.

When we vertically sum the transaction cost and interest foregone curves to get the total cost curve, the total cost will have shifted down and to the left.

The optimal transaction interval (T*) will now be at a smaller value.

Does this make sense?

T*

Analysis (3)

cP

rVV

cP

rLP

m

cP

rTT

,,,

,,,

,,,*

What would happen to the transaction interval, the number of trips to the bank, the demand for money, and velocity if inflation were expected to be higher in the future?

T would decrease.

Real cash balances would decrease.

Velocity would increase.

Fisher’s Equation

inflation expected

ateInterest R Realr

ateInterest R Nominal

)(

R

rR

Fisher’s equation (named after economist Irving Fisher says that the nominal rate of inflation equals the real rate plus the expected rate of inflation.

Note the role of expectations.

Later we will find out that inflation is determined by changes in the money supply which are controlled by the FED.

Effect of an increase in expected inflation.

$

Transaction Interval

Fewer Trips-------

T

1

P

2

TCR

2

1

P

TCR

T

T*

Will the increase in expected inflation effect transaction costs?

No—no shift in red line.

Will the increase in expected inflation effect interest foregone?

Yes—if expected inflation is higher, household’s expected the real value of their cash balances to be degraded, i.e. the cost of holding cash balances increases.

Yes—Blue line will shift up.

When we vertically sum the transaction cost and interest foregone curves to get the total cost curve, the total cost will have shifted up and to the left.

The optimal transaction interval (T*) will now be at a smaller value.

Does this make sense?

T*

Analysis (4)

cP

rVV

cP

rLP

m

cP

rTT

,,,

,,,

,,,*

What would happen to the transaction interval, the number of trips to the bank, the demand for money, and velocity if consumption were to increase?

T would decrease.

Real cash balances would increase.

Velocity would increase.

How can the transaction interval decrease while cash balances increase?

Effect of an increase in consumption

If c (real income) were to increase, according to the formulas cash balances would increase, velocity would increase and would decrease.

Mystery is how can cash balances increase at the same time velocity is increasing?

Answer: As C increases, the cost of holding cash balances increases. Household responds by increasing the number of trips but not by increasing the number of trips to enough keep V constant.

PC T PC/T M/P V

12,000 1/12 1000 500 24

24,000 1/12 2000 1000 24

24,000 1/24 1000 500 48

24,000 1/20 1200 600 40