A fully-coherent simple model of the central bank with portfolio

choice by householdsModel PC

Table 4.1 : Balance sheet of Model PC

1. Households 2. Production 3. Government 4. Central Bank

Money + H H 0

Bills +Bhh B + Bcbcb 0

Balance (networth)

V + V 0

0 0 0 0

Table 4.2: Transactions flow matrix of Model PC

1. Households 2. Production 3. Government 4. Central bank

Current Capital

Consumption C + C 0

Governmentexpenditures

+G G 0

Income = GDP + Y Y 0

Interest payments + r-1-1.Bhh-1-1 r-1-1.B-1-1 + r-1-1.Bcbcb-1-1 0

Central bankprofits

+ r-1-1.Bcbcb-1-1 r-1-1.Bcbcb-1-1 0

Taxes T + T 0

Change in money H + H 0

Change in bills Bhh + B Bcbcb 0

0 0 0 0 0 0

The PC model: national accounting equations

(4.1) Y C + G(4.2) YD Y - T + r-1.Bh-1

(4.3) T = .(Y + r-1.Bh-1)

Portfolio decisions, based on expected wealth and values:The Brainard-Tobin formula amended

(4.7E) Bd/Ve = 0 + 1.r - 2.(YDe/Ve)(4.6E) Hd/Ve = (1 - 0) - 1.r + 2.(YDe/Ve)(4.13) Hd = Ve - Bd

(4.14) Ve V-1 + (YDe - C)

The consumption function with propensities to consume out of income and out of wealth (the so-called Modigliani

consumption function)is equivalent to a wealth adjustement mechanism with a target wealth to disposable income ratio.

The assumption of stable stock-flow norms (Godley and Cripps 1982) is derived from the assumption of relatively

stable propensities to consume

Vh = 2.( 3.YD - Vh-1) Vh = 2.( 3.Vh

T - Vh-1)

where 3 = (1 - 1)/ 2.The wealth to income ratio is: V/YD = 3

Realized and expected values

(4.4) V V-1 + (YD - C) (4.5) C = 1.YDe

+ 2.V-1 0 < 1 , 2 < 1(4.15) Bh = Bd

(4.6) Hh = V- Bh

(4.16A) YDe = YD-1

(4.16) YDe = YD.(1 + Ra)

The government, the central bank, and the hidden equation

(4.8) Bs Bs - Bs-1 (G + r-1.Bs-1) - (T + r-1.Bcb-1)(4.9) Hs Hs - Hs-1 Bcb

(4.10) Bcb = Bs - Bh

(4.11) r = r

The hidden equation(4.12) Hh = H s

Chart 4.1a: The stock of Hd & Hh over time with random fluctuations in disposable income

(4.17) Hh - Hd = YD - YDe

Chart 4.1b: Changes of Hd & Hh over time (1st differences)With random fluctuations in disposable income

Chart 4.2: Bills (Bh) & Cash (Hh) held by householdsafter an increase in the interest rate on bills (in 1960)

Chart 4.3: Y, YD and V after an increase in the interest rate (in 1960)When propensities to consume are constants

Chart 4.4: Y, YD and V after an increase in the propensity to consume (a1) in 1960

Chart 4.5: Bills (Bh) & Cash (Hh) held by households after an increase of the propensity to consume (a1) in 1960:

Chart 4.6: Y, YD V and C after a rise in the interest rate (in 1960)which affects the propensity to consume a1 and hence the implicit target wealth to income ratio.

(4.30) 1 = 10 .r 1

Alternative closuresIt is possible to have an alternative closure, a

neoclassical one, by assuming the following changes

• Replace the equation:• Bcb = Bs – Bh

• Delete the interest rate equation (where r was a constant)

• Set Bcb as a constant (through open market operations)

• Add the equation• Bh = Bs – Bcb

• We now have two equations that set Bh. The rate of interest must become a price-clearing variable (in the portfolio equation)

Variations on the PC model:Adding long-term bonds

• It is easy to add long-term bonds to the PC model, with their possible capital gains or losses

• Both the short and the long rates can be made exogenous, if the Treasury accepts to see wide fluctuations in the composition of its liabilities.

• Or the long rate of interest can be made endogenous, either because the Treasury changes long rates when the share of bonds in national debt diverges from a band; or because the monetary authorities let the prices on long-term bonds fluctuate freely, keeping still the amount or the share of bonds in total debt.

Share of bondsbeing detained by the public:Bonds/(bonds+bills)

55 %

45 %Acceptable range

Adding-up constraints in portfolio choice

• Brainard-Tobin have emphasized the vertical adding-up constraints

• Godley has emphasized the horizontal adding-up constraints

• B. Friedman has advocated the symmetry constraints

• With symmetry and vertical constraints, horizontal constraints are necessarily fulfilled.

d

d

hd

hd bL

nce m

b

bL

nce

re

M 1

M 2

B

B L p

Vrr

rr

rE R r

V Y D

.

/ ( )

1 0

2 0

3 0

4 0

11 1 2 1 3 1 4

2 1 2 2 2 3 2 4

3 1 3 2 3 3 3 4

4 1 4 2 4 3 4 4

1 5

2 5

3 5

4 5

1

(ADUP.1) 10 + 20 + 30 + 40 = 1(ADUP.2) 15 + 25 + 35 + 45 = 0

(ADUP.3) 11 + 21 + 31 + 41 = 0(ADUP.4) 12 + 22 + 32 + 42 = 0(ADUP.5) 13 + 23 + 33 + 43 = 0(ADUP.6) 14 + 24 + 34 + 44 = 0

(ADUP.7) 11 = -(+ 12 + 13 + 14)(ADUP.8) 22 = -(+ 21 + 23 + 24)(ADUP.9) 33 = -(+ 31 + 32 + 34)(ADUP.10) 44 = -(+ 41 + 42 + 43)

Symmetry constraints

• (ADUP.7) λ12 = λ21

• (ADUP.8) λ13 = λ31

• (ADUP.9) λ23 = λ32

• (ADUP.10) λ14 = λ41

• (ADUP.11) λ24 = λ42

• (ADUP.12) λ34 = λ43