A fully-coherent simple model of the central bank with portfolio choice by households Model PC.
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Transcript of A fully-coherent simple model of the central bank with portfolio choice by households Model PC.
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