c) Consider a more conservative central banker whose loss function is represented by L = ½ (yt − k)² + ½¼½ (π₁) ² - Apparently, the preferred inflation rate would read Eb 1+6 What can you say about the relationship between 1 and b? Is în larger or smaller than b? Provide an explanation. π£ = ô(k − y) - - d) Given the above output function and y = 0, such that Yt = πt − π£ + &, the inflation rate and the loss-function in c), it can be shown that the loss function can be expressed as a function of L (6) − } { x² (8² + 6) + (6 +6³ 7 B [2]} = b+ô² E (1+6³) [e²] Assuming that I represents the social welfare function, the socially optimal level of conservatism ô can be obtained from the first-order condition of L(1) with respect to ĥ. Show from this condition whether ☎ is larger or smaller than b.

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Chapter1: Making Economics Decisions
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Give answer for c and D only

Exercise 1
Consider the following scenario.
Aggregate output y at date t is specified as follows,
Yt = Tt – T +ỹ
where the trend level of output is represented by y.
The central bank's loss function is captured by
L
2
(y: – k)² +(T)
2
Where b> 0, and where k > 0 is a given parameter reflecting the normal level of
aggregate production or the target level of output. The central bank takes the public's
expectations as given. The society loss function is the same as the central bank loss
function.
Notation: y;: output/production; T: inflation rate; n: expected inflation rate; L:
loss function; b, k: constant parameters; t: time index.
Transcribed Image Text:Exercise 1 Consider the following scenario. Aggregate output y at date t is specified as follows, Yt = Tt – T +ỹ where the trend level of output is represented by y. The central bank's loss function is captured by L 2 (y: – k)² +(T) 2 Where b> 0, and where k > 0 is a given parameter reflecting the normal level of aggregate production or the target level of output. The central bank takes the public's expectations as given. The society loss function is the same as the central bank loss function. Notation: y;: output/production; T: inflation rate; n: expected inflation rate; L: loss function; b, k: constant parameters; t: time index.
Exercise 2
Consider a modified aggregate supply function which takes account for the emergence of
random business cycle shocks (&t) with E&t] =O in the sense that
Yt = Tt – T + ÿ +&t
The loss function is the same as in exercise 1:
(yt – k)? +(72)?
Notation: Et: random shock; E[e,]: expected value of €t; b: constant parameter; all
other variables see Exercise 1.
Having considered the scenario above complete the following tasks:
a) Derive the central bank's preferred inflation rate and explain.
b) The result in a) reads as
b(k – 9) – H5
ɛb
Explain the economic intuition of this result.
c) Consider a more conservative central banker whose loss
function is represented by
L = (4 – k)² + } (Tt)²
Apparently, the preferred inflation rate would read
6(k – 9) –
||
1+ô
What can you say about the relationship between ô and b? Is i
larger or smaller than b? Provide an explanation.
d) Given the above output function and y
Yt = Tt – T +€4, the inflation rate and the loss-function in c), it
can be shown that the loss function can be expressed as a
:0, such that
||
function of b
z (1) – }{* (* » +) + 1}
():
3),
2 (6² + ô) + E [=*]}
(1+8')
Assuming that L represents the social welfare function, the
socially optimal level of conservatism b can be obtained from the
first-order condition of L(ô) with respect to 6. Show from this
condition whether b is larger or smaller than b.
Transcribed Image Text:Exercise 2 Consider a modified aggregate supply function which takes account for the emergence of random business cycle shocks (&t) with E&t] =O in the sense that Yt = Tt – T + ÿ +&t The loss function is the same as in exercise 1: (yt – k)? +(72)? Notation: Et: random shock; E[e,]: expected value of €t; b: constant parameter; all other variables see Exercise 1. Having considered the scenario above complete the following tasks: a) Derive the central bank's preferred inflation rate and explain. b) The result in a) reads as b(k – 9) – H5 ɛb Explain the economic intuition of this result. c) Consider a more conservative central banker whose loss function is represented by L = (4 – k)² + } (Tt)² Apparently, the preferred inflation rate would read 6(k – 9) – || 1+ô What can you say about the relationship between ô and b? Is i larger or smaller than b? Provide an explanation. d) Given the above output function and y Yt = Tt – T +€4, the inflation rate and the loss-function in c), it can be shown that the loss function can be expressed as a :0, such that || function of b z (1) – }{* (* » +) + 1} (): 3), 2 (6² + ô) + E [=*]} (1+8') Assuming that L represents the social welfare function, the socially optimal level of conservatism b can be obtained from the first-order condition of L(ô) with respect to 6. Show from this condition whether b is larger or smaller than b.
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