Chapter 50: Money, Monetary Standards, Velocity, and Inflation. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Where does extra money go?

When the money stock rises 25 percent, how much of it shows up in prices?

The equation of exchange is an identity. Extra money raises prices one for one only if velocity and output stay put; a fall in velocity or a rise in output absorbs part or all of it.

$Equation, written in LaTeX: P_0=\frac{1{,}000\times4}{2{,}000}=2.$

$Equation, written in LaTeX: P_1=\frac{1{,}250\times4}{2{,}000}=2.5,$

$Equation, written in LaTeX: P_1=\frac{1{,}250\times4}{2{,}200}\approx2.2727,$

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M is the money stock, V velocity (how often a unit of money is spent per period), Y real output and P the price index, linked by MV = PY. The economy starts at M0 = 1,000, V0 = 4 and Y0 = 2,000.

Predict first. Money rises 25 percent to 1,250 while velocity falls to 3.2 and output stays 2,000. What happens to the price index?

Your prediction It rises It stays at 2 It falls

Money stock after issue

Velocity after issue

Real output

P0 = M0 V0 / Y0: 2.0000
P1 = M1 V1 / Y1: 2.5000
Change in P: 25.00%
Change in money: 25.00%

P0 = 1,000 x 4 / 2,000 = 2.0000. After the change, P1 = 1,250 x 4.0 / 2,000 = 2.5000, so the price index rises 25.00 percent. Money changed 25.00 percent; the price change differs from it whenever velocity or output moves.

Worked steps

P0 = 1,000 x 4 / 2,000 = 4,000 / 2,000 = 2.0000
P1 = 1,250 x 4.0 / 2,000 = 5,000 / 2,000 = 2.5000
Change = (2.500000 - 2.0000) / 2.0000 = 25.00%

Use the idea

Before forecasting inflation from money growth, state what you assume about velocity and real output, because the forecast is conditional on both.

Where the conclusion applies

The identity holds by definition; the causal reading needs velocity and output to be independent of the money change, which the chapter treats as a conditional benchmark.

Check your understanding: With M1 = 1,250, V = 4 and output 2,200, what is the percent change in P?

P1 = 1,250 x 4 / 2,200 = 2.2727, and (2.2727 - 2) / 2 = 13.64 percent.

Chapter 50 source: section "Quantity Theory of Money".

Demonstration 2 of 4

Falling prices raise real debt

When money and velocity contract together, what happens to a fixed nominal debt?

Debt contracts are fixed in money. When the price index falls, each unit owed buys more goods, so the borrower must give up more real resources to repay, even though nothing in the contract changed.

$Equation, written in LaTeX: P_0=\frac{M_0V_0}{Y_0}=\frac{500\times4}{1{,}000}=2.$

$Equation, written in LaTeX: P_1=\frac{450\times3.8}{980}\approx1.7449.$

$Equation, written in LaTeX: M^*=\frac{P_0Y_1}{V_1}=\frac{2\times980}{3.8}\approx515.79.$

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The economy starts at M0 = 500, V0 = 4 and Y0 = 1,000, so P0 = 2. After a banking disruption output is 980. A borrower owes 200 nominal units; its real debt is 200 / P in consumption baskets. M* is the money stock that would restore P = 2 at the new velocity and output.

Predict first. Money falls 10 percent to 450 but velocity holds at 4. Does the borrower's real debt still rise?

Your prediction Yes, it still rises No, it falls

Money after the disruption

Velocity after the disruption

Figure: Falling prices raise real debt. Left: the price index moves from 2 to 1.7449. Right: the real value of 200 owed moves from 100.00 to 114.62 baskets. — Money after the disruption: 450, Velocity after the disruption: 3.8
Constructed example: the chapter's hypothetical economy (M0 500, V0 4, Y0 1,000; after the shock 450, 3.8 and 980; debt of 200); money stocks of 475 and 500 and velocities of 3.6, 4.0 and 4.2 are added for comparison.

P1: 1.7449
Price change: -12.76%
Real debt before: 100.00
Real debt after: 114.62
Money that restores P = 2: 515.79
Extra money needed: 65.79

P1 = 450 x 3.8 / 980 = 1,710 / 980 = 1.7449, so the price index falls 12.76 percent. The borrower's real debt rises, from 200 / 2 = 100.00 to 200 / 1.744898 = 114.62 baskets. Holding velocity at 3.8 and output at 980, the money stock that keeps P at 2 is 2 x 980 / 3.8 = 515.79, 65.79 more than 450.

Worked steps

P0 = 500 x 4 / 1,000 = 2.0000
P1 = 450 x 3.8 / 980 = 1.7449
Change = (1.744898 - 2) / 2 = -12.76%
Real debt = 200 / 1.744898 = 114.62 (was 100.00)
M* = 2 x 980 / 3.8 = 515.79

Use the idea

When money or spending contracts, track real debt burdens alongside prices, because falling prices transfer resources from debtors to creditors.

Where the conclusion applies

The equation of exchange with output fixed at 980; M* is an accounting counterfactual that holds velocity and output at their new levels, not a policy promise.

Check your understanding: With M1 = 450, V1 = 3.8 and Y1 = 980, how much extra money restores P = 2?

M* = 2 x 980 / 3.8 = 515.79, so 515.79 - 450 = 65.79 units.

Chapter 50 source: section "Deflation and Monetary Contraction".

Demonstration 3 of 4

How often to visit the bank

How many transfers a year minimize the cost of holding transaction cash?

More transfers cost more in fees but cut the average cash balance and so the interest given up. The total cost curve is lowest where the two margins balance, at n* = sqrt(iT / (2b)); with whole transfers, compare the neighbouring integers.

$Equation, written in LaTeX: n^*=\sqrt{\frac{0.04\times24{,}000}{2\times6}}=\sqrt{80}\approx8.944.$

$Equation, written in LaTeX: n^*=\sqrt{\frac{0.09\times24{,}000}{12}}=\sqrt{180}\approx13.416,$

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A business spends T = 24,000 dollars a year evenly. Each transfer from its interest-bearing account costs b dollars; i is the interest rate given up on cash. With n transfers the average cash balance is T / (2n) and the annual cost is C(n) = b n + i T / (2n).

Predict first. If the interest rate more than doubles from 4 to 9 percent, does the optimal number of transfers more than double?

Your prediction Yes, more than double No, it rises by less

Interest rate

Cost per transfer (dollars)

Figure: How often to visit the bank. Annual cost of cash management against transfers per year. The total cost curve bottoms out at n* = 8.944; the best whole number is 9 at $107.33. — Interest rate: 4%, Cost per transfer (dollars): $6
Constructed example: the chapter's hypothetical business (24,000 dollars a year, 6 dollars a transfer, interest 4 and 9 percent); an interest rate of 2 percent and transfer costs of 3 and 12 dollars are added for comparison.

n* (continuous): 8.944
Best whole n: 9
C(best n): $107.33
Average balance at best n: $1,333.33
Average balance at n*: $1,341.64
Cost at the whole numbers either side of n*: C(8) = $108.00 and C(9) = $107.33

n* = sqrt(0.04 x 24,000 / (2 x 6)) = sqrt(80.00) = 8.944. Comparing the neighbouring whole numbers: C(8) = 6 x 8 + 0.04 x 24,000 / 16 = 48 + 60.00 = 108.00; C(9) = 6 x 9 + 0.04 x 24,000 / 18 = 54 + 53.33 = 107.33, so 9 transfers is the least-cost whole number. The average cash balance at n* is $1,341.64.

Worked steps

n* = sqrt(0.04 x 24,000 / 12) = sqrt(80.00) = 8.944
C(8) = 48 + 60.00 = $108.00
C(9) = 54 + 53.33 = $107.33
Average balance at n*: sqrt(6 x 24,000 / (2 x 0.04)) = $1,341.64

Use the idea

Set cash-transfer frequency with the square-root rule, then round by comparing the costs of the two nearest whole numbers.

Where the conclusion applies

Even spending through the year, a fixed fee per transfer and a constant interest rate. Real firms face uncertain payments, which add a precautionary buffer.

Check your understanding: At i = 0.09, what are n* and the optimal average balance?

n* = sqrt(0.09 x 24,000 / 12) = sqrt(180) = 13.416; the average balance is sqrt(6 x 24,000 / 0.18) = $894.43.

Chapter 50 source: section "Baumol-Tobin transactions demand for money".

Demonstration 4 of 4

The inflation tax on existing money

How much purchasing power does a one-time money issue take from existing holders?

New money dilutes existing balances. Once prices adjust, the old holdings buy less, and under the one-time proportional assumptions the real value lost by holders equals the real value gained by the issuer.

$Equation, written in LaTeX: \frac{10{,}000}{1}-\frac{10{,}000}{1.10}=10{,}000-9{,}090.91=909.09.$

$Equation, written in LaTeX: 0.10(\frac{10{,}000}{1.10})=909.09.$

$Equation, written in LaTeX: s=\frac{1{,}000}{1.10}=909.09.$

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The public holds existing money with P0 = 1. The issuer adds new units; prices rise in proportion, so P1 = (holdings + issue) / holdings. Exact erosion is holdings / P0 - holdings / P1; seigniorage s is the issue valued at P1.

Predict first. Is the shortcut "inflation rate times beginning balances" too high or too low?

Your prediction Too high Too low Exact

New money issued

Existing public holdings

Figure: The inflation tax on existing money. Left: existing holdings fall in real value from 10,000 to 9,090.91, an erosion of 909.09. Right: exact erosion 909.09, shortcut 1,000.00, seigniorage 909.09. — New money issued: 1,000, Existing public holdings: 10,000
Constructed example: the chapter's hypothetical issue (holdings 10,000, P0 = 1, issue 1,000, P1 = 1.10); issues of 500 and 2,000 and holdings of 5,000 and 20,000 are added, with prices assumed proportional to money as in the book's case.

P1: 1.1000
Inflation: 10.00%
Exact erosion: 909.09
Shortcut (rate x beginning balances): 1,000.00
Shortcut error: 90.91
Seigniorage: 909.09

With prices proportional to money, P1 = (10,000 + 1,000) / 10,000 = 1.1000, inflation of 10.00 percent. Exact erosion is 10,000 - 10,000 / 1.1000 = 10,000 - 9,090.91 = 909.09. The shortcut 0.10 x 10,000 = 1,000.00 is too high by 90.91. The issuer's real gain is 1,000 / 1.1000 = 909.09, equal to the holders' loss.

Worked steps

P1 = (10,000 + 1,000) / 10,000 = 1.1000
Exact erosion = 10,000 - 10,000 / 1.1000 = 10,000 - 9,090.91 = 909.09
Same via final balances: 0.10 x 9,090.91 = 909.09
Shortcut = 0.10 x 10,000 = 1,000.00, too high by 90.91
Seigniorage s = 1,000 / 1.1000 = 909.09

Use the idea

Measure an inflation tax at post-adjustment prices; the beginning-balance shortcut overstates it, and the more so the bigger the issue.

Where the conclusion applies

A one-time issue, prices proportional to money, fixed output and desired balances, non-interest money and no production cost. The equality of loss and gain fails when any of these change.

Check your understanding: If 2,000 units are issued on 10,000 with proportional prices, what is the exact erosion?

P1 = 1.2, so 10,000 - 10,000 / 1.2 = 10,000 - 8,333.33 = 1,666.67, against the 2,000 shortcut.

Chapter 50 source: section "Seigniorage / Inflation Tax".

Where does extra money go?

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

Falling prices raise real debt

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

How often to visit the bank

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

The inflation tax on existing money

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies