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8. Barro – Gordon – Model
min L s. t. N Nn c(- e ),
min b ( *^ )^2 ( N c( e) N *)^2
n
2 goals of monetary policy:
- Minimize deviations of inflation from its optimal rate π *
- Try to achieve efficient employment, N* > Nn
Static Phillips curve: N = Nn + c ( π – π e),
Loss function L = b ( π – π )^2 + (N – N)^2
2
b c
b c N Nn c e
Response of the central bank on given expectations
Barro – Gordon – Model
2
b c
b c N N c E
e e n
e * ( N * N n ) b
c
Rational expectations: π e^ = E( π )
c
discretionary solution
discretionary solution is inefficient!
Inflation bias
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Barro – Gordon – Model
Response of central bank to inflation expectations π ( π e) without commitment
N
π
Nn
π *
π for π e= π *
N*
Iso-loss curves
Phillips curve for π e= π *
Barro – Gordon Model
Equilibrium with rational expectations:
N
Nn
π *
π equilibrium
N*
Phillips curve for π e^ = π equilibrium
Iso-loss curve
equilibrium
Discretionary solution =
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Barro – Gordon – Model
For avoiding the inflation bias and stabilizing the economy in an efficient way,
- the CB must aim at closing the output gap, instead of trying to achieve efficiency, N* = N (^) n ,
With L = b ( π – π *)^2 + (N – N (^) n)^2 and π e^ = π *. we get L = b E( π – E( π ))^2 + E(N – E(N))^2 = b Var( π ) + Var(N).
- the CB must be independent from short-run political goals of the government. Otherwise, there is always an incentive to increase employment in the short run, N* > Nn.
Stabilizing Demand and Supply Shocks
Stochastic economy:
- Liquidity shocks can be absorbed by monetary
policy.
- Shocks of commodity demand: if CB stabilizes
inflation, output and employment are stabilized as well.
- Supply shocks affect productivity. They are bound to
have real effects.
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8.1 Stabilizing supply shocks
What is the optimal response of monetary policy to
shocks in productivity?
production function y = a N + θ
Productivity shock θ : E( θ ) = 0 Var( θ ) = σ^2
Labor demand N = Nn + c ( π – w + θ )
Wages w = π e
All variables may be interpreted as growth rates.
Phillips curve
a c 1
1
Stabilizing supply shocks
N
π
Nn
π e
Phillips curve in face of supply shocks
N = N (^) n + c ( π – π e^ + θ )
1/c
Phillips curve für θ = 0
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Stabilizing supply shocks
Solving the trade-off between stabilizing inflation and
employment.
Min b ( π – π *)^2 + (N – Nn)^2
= Min b ( π – π *)^2 + ( c ( π – π e^ + θ ) )^2
Since N*=N (^) n, the inflation bias is zero, so that π e^ = π *.
=> 2 b ( π – π *) + 2 c^2 ( π – π * + θ ) = 0
=> (b+c^2 ) ( π – π *) = – c^2 θ
2
c
b c
Stabilizing supply shocks
L
π
Min b ( π – π *)^2 + (N – N (^) n)^2
N = N (^) n + c ( π – π e^ + θ )
Phillips curve for θ = 0
employment fluctuations
fluctuations of inflation
1/c
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8.2 Rules versus Discretion
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What are optimal rules in the face of demand and supply
shocks?
production function y = a N + θ
Productivity shock θ : E( θ ) = 0 Var( θ ) =
Labor demand N = Nn + c ( π – w + θ ),
Wages w = π e
Demand side (quantity theory) μ + η = π + y
Demand shock η : E( η ) = 0 Var( η ) =
Phillips curve
2
Loss function L = b ( π – π )^2 + (N – N)^2
a
c
1
1
Rules versus Discretion
Rule 1: constant rate of inflation, π = π *
=> π e^ = π * => N = N (^) n + c θ
E ( N N ^ )^2 b ( )^2 E ( Nn N * c )^2
( N n N *^ )^2 2 ( Nn N *) cE ( ) E (^ c )^2
( *^ )^222 N (^) n N c
Rules versus Discretion
Comparing welfare loss for rule 1 versus rule 2
2
2
N n N b b Nn N
2 2
2 2
( 1 ( 1 ) )
b b
2
2 2
2
1 ( 1 )
b
b
( 1 )^2
b
E(loss for constant μ) > E(loss for constant π)
Constant money supply growth leads to higher expected costs than constant inflation, if (i) the weight on pricestability is sufficiently large, or (ii) the variance of demand shocks is large compared to the variance of supply shocks.
