TA Session Compilation PDF

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TA Session - Compilation

June 6, 2020

1

Inflation dynamics

The inflation equation in the New Keynesian model is given by πt = βE t { πt+1 } − λ(µt − µ),

(1)

where πt ≡ pt − pt−1 is the inflation rate, µt is the average price markup and µ is the price markup under flexible prices. 1. Explain in words but as rigorously as possible the assumptions underlying (1). Answer: The assumption underling equation (1): • Monopolistic competition: Firms are price setters - and not price takers; • Nominal rigidity: Firms are subject to price adjustment costs, i.e., firms are allowed to reset their prices only with some probability. Therefore in the present model, inflation results from aggregate consequences of purposeful price-setting decisions by firms, which adjust their prices in light of current and anticipated cost conditions. 2. What is the intuition for the negative relation between ”the markup gap” and inflation?

1

Answer: Solving (1) forward, inflation is expressed as the discounted sum of current and expected future deviations of average markups from their desired levels:

∞

π t = −λ ∑ β k E t { µt+ κ − µ}. κ =0

Thus, inflation will be positive when firms expect average markups to be below their desired level µ, for in that case firms that have the opportunity to rest prices will, on average, choose a price above the economy’s average price level in order to realign their markup closer to its desired level. 3. Derive, starting from (1), the New Keynesian Phillips curve πt = βE t { πt+1 } + κy˜t where y˜t ≡ y t − ytn is the log deviation between output and natural output. Explain the assumptions made in each step. What is the ”divine coincidence” and under what conditions does it hold? Answer: The average price markup can be expressed as µt = pt − |{z} price

ψt |{z}

marginal cost

= −(wt − pt ) + ( a t − αnt + log(1 − α)) = −(σy t + φnt ) + ( a t − αnt + log(1 − α))     1+α φ+α yt + a t + log(1 − α). =− σ+ 1−α 1−α

Under flexible prices we get     φ+α n 1+α µ=− σ+ a t + log(1 − α). yt + 1−α 1−α Combining the last two relations we get   φ+α (y t − ytn) µˆ t = − σ + 1−α | {z }

y˜ t : output gap

2

Finally, we can substitute the markup gapµˆt in (1), which yields πt = βE t { πt+1 } + κ y˜ t , φ +α

where κ ≡ λ(σ + 1− α ).

The divine coincidence refers to the property of New Keynesian models that there is no trade-off between the stabilization of inflation and the stabilization of the welfare-relevant output gap (the gap between actual output and efficient output) for central banks. To see this let rewrite the NKPC as

∞

πt = κ ∑ β k E t {y˜t+κ }.

(2)

κ =0

If the central bank succeeds in stabilizing fully the output gape we get that the output gap is y˜t = 0

∀t ∈ [0, ∞),

πt = 0

∀t ∈ [0, ∞),

implying that

which means that stabilizing the output gap is sufficient to stabilize inflation. 4. Derive, starting from the NKPC, an equation relating inflation to the deviation of output from steady state yˆ t ≡ y t − y, on y. Determine the behavior of inflation when the central bank chooses to stabilize fullyyˆt , assuming that yˆtn ≡ y nt − y follows an AR(1) process with autoregressive coefficient ρy . Why is that policy sub-optimal?. Answer: The NKPC can be rearranged as πt = βE t { πt+1 } + κyˆt − κ yˆtn.

3

Iterating forward we get ∞

πt = κ

∑ β j Et { yˆ t+ j } − κ j= 0

=κ

∞

∞

j= 0

j= 0

j

j

∑ β j Et { yˆ t+ j } − κ ∑ β j Et {ρy yˆtn + ∑ ε t+i } i

∞

∑ β j Et { yˆ t+ j }−κ ˆytn ∑( βρ y ) j . j= 0

|

∑ β j Et { yˆtn+ j } j= 0

∞

=κ

∞

j= 0

{z

}

|

= 0 by stabilization

{z

1 1 − βρ y

}

Rearranging everything we get the following equation for inflation πt = −

κ yˆn . 1 − βρ y t

This policy is sub-optimal because it does not focus on the welfare-relevant measure yˆtn. This is so since we know that a certain amount of fluctuations are indeed efficient - as we saw in the classical monetary model.

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