5. CONCLUSION
2.1 Optimal Interest Rate
The efficiency monetary policy is calculated by considering the inflation variation and the optimal output variation. Optimal output and inflation variation occurring in the interest rate results in the minimum los function. Rodenbush and Svensson (Walsh, 2003:508-512 stated that the optimal monetary policy was given by the equation of output and inflation as follows.
t t t t t
t
t a y a y a i E u
y 1 1 2 2 3( 1 1 ) (1)
t t t
t y
1 (2) From the equation (2) inflation expectation may be made into equation: Et1 t Etyt1 and inserted into the output equation, it is abstained:
1 1 3
1 2 1
1 ( )
t t t t t t t
t a y a y a i E y u
y
1 3
3 1 2 1
1
) (
t t t t ut
a i a y a y a
(3)
If :
3 3 1 2 1
1
) ( a
i a y a y
a t t t t
t
so output and inflation of period t+1 is:
1
1
t
t u
y (4)
1
1
t t t t
t v
(5)
Where: vt1 ut1 t1
The value function from the loss function is:
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( ) ( )
2
min Et 1 yt21 t21 V t 1 L
t
(6)
Minimization of lost function with equation obstacle (4)and (5) is:
( ) ( )
2 ) 1 2 (
min Et 1 t ut 1 2 t t vt 1 2 V t t vt 1 L
t
(7)
The First Order Condition is:
0 ) ( )
(2 tt EtV t1 (8) from envelope theorem, it is obtained:
) ( 1
t t EtV t
L (9)
By multiplying the equation (8) by, and being added to the equation (9), so it results in: V(t)t. By withdraw the time one period and making expectation, EtV(t1) can be eliminated from the equation (8) becoming:
0 )
(2 ttEtt1 (10) or:
2 1
2
t t t
t E
(11)
Optimal condition occurs in t Bt. Optimal B value is:
0 )
( 2
2
B B (12) Considering
3 3 1 2 1
1
) ( a
i a y a y
a t t t t
t
, this optimal
monetary policy is:
1 3 2 3
1 3
3 ) 1
1 (
t t t
t y
a y a a a a
a
i B
(13)
Conference Papers 2.2 Efficient Monetary Policy
Efficient monetary policy is a policy that minimizes loss as a side effect the implementation of a monetary policy. Monetary policy efficiency may be seen by comparing the actual monetary policy with the monetary policy efficiency frontier. The monetary policy efficiency frontier is reduced from the minimization of loss function for discretion monetary policy.
Loss function contains output and inflation variation like the equation (14). While is level of tolerance of the monetary policy maker, towards the output gap variation for keeping stable inflation. In other words, may be required as a preference level of monetary policymakers towards the output gap variation.
Several economists gave the optimal value of . McCallum (2000), Nelson (2000), and Jensen (2002) gave the value of 0.25 for , while Roberts gave the value of 0.3 for (Walsh, 2003:517-533).
1
2
2 )
( )
1 ( ˆ) (
i
i t i t i
t y
E
L (14)
The result of equation minimization (14) with many combinations of output and inflation variation add certain level of will from a tradeoff line between output and inflation variation as shown in Figure 2. The smaller the value , the closer the tradeoff line to the point of origin (point 0). The monetary policy efficiency frontier for discretion policy will tangent the tradeoff line of output and inflation variation.
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Figure 2. Monetary Policy Efficiency Frontier
Figure 2 illustrates monetary policy efficiency frontier. Every point at the curve represents output and inflation variation for every value of . The level of social marginal cost of output and inflation variation is at the efficiency frontier which is also an indifferent curve. Clarida, Gali, and Gertler (1999) suggest: if shock has no correlation, the optimal policy will occur when the central bank preference equals the public preference or ˆ . But if shock has correlation, the optimal policy made is ˆ.
Inflation Variation
Output Variation Source: Walsh (2003)
Tradeoff of output and inflation variation
Monetary Policy Efficiency Frontier
0
Conference Papers
Figure 3. Monetary Policy Efficiency
The efficiency monetary policy may be measure from the distance of actual point of the output and inflation variation towards the monetary policy efficiency frontier. The closer the point observed with the efficiency frontier, the more efficient the monetary policy. In Figure 3, the degree of the efficiency of monetary policy can be measured from the distance of observed point A with the curve of the monetary policy efficiency frontier (Ceccehtti et. al., 2006:412; Mishkin dan Schmidt Hebbel, 2007:52).
Cecchetti et. al. (2006) calculated a change of the monetary policy efficiency in an almost similar way. Cecchetti focused on the change of the monetary policy efficiency through a graphic method and mathematic calculation of loss function as a function of output and inflation variation by weighted the preference of and (1 – ). The loss function can be written as follows:
1 0
), Var(
) 1 ( ) (
Var
y
Loss
(15) where: λ = parameter of central bank preference towards inflation variation; Var(•) = inflation deviation squared from its target or
Source : Ceccehtti et.al. (2006:412); Mishkin dan Schmidt Hebbel (2007:52)
Observed point A
Inflation variation
Output gap variation q*
0
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output deviation from potential output; π = inflation; and y represent the output.
Based on the loss function formed, the measurement of the macro economic performan-ce can be formulated at the period of i (Pi) given by the equation:
) Var(
) 1 ( ) (
Var i i
i y
P (16) The optimal macro economic performance (Si) is the economic performance resulting from the minimum Pi, with Si :
*
* (1 )Var( )
) (
Var i i
i y
S (17)
where var(π)* and var(y)* are inflation variance and output variance in the optimal condition. If ∆S = S2 – S1 has a negative value, it indicates that the performance of economy decreases.
To determine var (π)* andvar (y)*, we can see a homothetic shift of the original frontier curve outward tangent the performance point as shown in Figure 4. The optimal variance is the intersection point between original frontier and the line drawn from the original point to the performance point.
.Figure 4. Monetary Policy Efficiency Frontier (Original Frontier) and Performance Point
Optional Variance [var()*,var(y)*]
Performance Point [var(),var(y)]
Original frontier
Variance of output
Variance of inflation Original
0
Conference Papers The efficiency of the monetary policy is calculated from
the distance of the actual performance toward the optimal performance. The inefficency level for every period i is:
] ) Var(
) Var(
)[
1 ( ] ) ( Var ) ( Var
[
i i i i
i y y
E (18)
Then, the variable of monetary policy efficiency ∆E is calculated, based on the proportion toward ∆P that is:
|
| P
Q E
(19)
Furthermore, the average inefficiency between two periods wil be calculated by assumed that central bank chooses the interest rate minimizing the loss function that is minimizing deviation squared from the inflation and output of the average target. The average is formulated as follows:
] ) (
) 1 ( ) (
[ )
(L E * 2 y y 2
E i i (20)
Where π* and y* are the inflation target and the output target for every t taken from the average value, andtis the inflation level for every t taken from the average value.
3. MEASUREMENT FOR EFFICIENCY OF