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A Step-by-Step Derivation of the Standard

New Keynesian Phillips Curve1

by

Jens Sebastian Sienknecht (sebastian.sienknecht@gmail.com)

Introduction

This document intends to

• show as many derivation steps as possible towards the standard New Keynesian

Phillips curve,

• give students a useful guideline for more complicated versions of this equation,

and

• use the logical steps of the well-known textbook of Galí (2015).

Basic assumptions

• There is a continuum of firms indexed by i ∈ [0, 1]. Each firm produces a differentiated good Yt (i) using a firm-specific amount of labor Nt (i).

• Technology is identical across firms and upward shifts of the production function

are exogenously determined by At :

Yt (i) = At Nt (i)1−α

,

α ∈ [0, 1) .

(1)

• The demand (or household consumption) for the good produced by firm i can be

derived from a consumption bundle optimization as

−

Pt (i)

Ct , > 1.

(2)

Ct (i) =

Pt

• Aggregate consumption Ct and the price index Pt are CES functions that will be

defined below (see equations (43) and (69)).

1

Following Galí (2015): Monetary Policy, Inflation, and the Business Cycle. Princeton University

Press.

1

• Given that firms operate under monopolistic competition, there are positive (nominal) profits D(i) in the production sector. These profits flow to the households,

who are assumed to be shareholders.

Calvo pricing

• After setting its price, a firm can re-adjust in the upcoming period only with a

certain probability.

• In the continuum of firms: the reset probability becomes the fraction of firms that

can re-adjust in the next period.

• Formally, the reset probability is given by 1 − θ, θ ∈ [0, 1].

• The probability of no re-setting in the next period (or the fraction of firms not

re-setting) is θ.

• Firms allowed to reset their price take into account that they probably won’t be

able to do so in the following periods.

• Formally, the probability that a price chosen at period t prevails until t + k is θk .

⇒ A firm setting its price maximizes (expected) discounted future profits until it can

re-optimize again.

Stochastic discount factor

• Profits are discounted by the stochastic discount factor Λ.

• Assumption: complete asset markets.

• Λt,t+k denotes how the household values a payoff at period t + k, given its knowledge up to time t. This provides the link to the Euler (or asset pricing) equation.

• Definition of the stochastic discount factor:

k

Λt,t+k = β Uc,t+k /Uc,t = β

k

Ct+k

Ct

−σ

.

(3)

• β = (1 + ρ)−1 represents (deterministic) discounting, which is determined by the

time preference rate ρ ∈ [0, 1].

• The stochastic part is determined by the marginal utilities of consumption Uc,t+k

and Uc,t . This term makes Λt,t+k a stochastic variable.

−σ

Ct+k

,

• A CRRA household utility specification with separable terms leads to Ct

σ ≥ 1.

2

⇒ Λt,t+k is the stochastic discount factor for real profits that flow to the household

(the owner of intermediate firms).

⇒ Suppose that Uc,t ↑ and therefore Λt,t+k ↓. This means that real profits that flow

at t are now valued higher by the household, relative to profits that arrive at

period t + k. If the household were to receive these profits at t + k instead of t,

they would be valued lower in terms of utility. The household has become more

impatient to receive firm profits at period t.

The profits of firm i are given by D(i)

• Reminder: D(i) represents nominal dividends of firm i paid to the households.

• D(i) is defined as

D(i) = P (i)Y (i) − C (Y (i)) .

(4)

• C (Y (i)) is nominal cost as a function of firm’s output Y (i).

• At time t, the firm chooses the optimal price Pt∗ (i). Given this price, profits at

t + k are

Dt+k (i) = Pt∗ (i)Yt+k|t (i) − Ct+k Yt+k|t (i) .

(5)

• Yt+k|t (i) denotes output in period t + k, given the chosen price Pt∗ (i).

• Ct+k Yt+k|t (i) denotes nominal costs, given (via output) the chosen price Pt∗ (i).

• Assume now that firms that can choose their price at time t face the same profit

maximization problem. Consequently, the firm index i can be dropped:

Dt+k = Pt∗ Yt+k|t − Ct+k Yt+k|t .

(6)

• The firm is uncertain about future possibilities to adjust. Therefore, it has to form

(rational) expectations:

θk Et {Dt+k } = θk Et Pt∗ Yt+k|t − Ct+k Yt+k|t .

(7)

• Real profits are obtained by dividing with the aggregate price level:

Dt+k

k

.

= θk Et (1/Pt+k ) Pt∗ Yt+k|t − Ct+k Yt+k|t

θ Et

Pt+k

(8)

• Multiply with the stochastic discount factor to obtain discounted real profits:

Dt+k

k

= θk Et Λt,t+k (1/Pt+k ) Pt∗ Yt+k|t − Ct+k Yt+k|t

.

(9)

θ Et Λt,t+k

Pt+k

3

• The problem of the infinitely-lived firm is to maximize its expected sum of discounted profits (9), subject to the following version of the demand function (2)

at Pt∗ :

∗ −

Pt

Yt+k|t =

Ct+k

for

k = 0, 1, 2, ...

(10)

Pt+k

Profit maximization in real terms2

∞

∞

X

X

Dt+k

k

max

=

max

θ

E

Λ

θk Et Λt,t+k (1/Pt+k ) Pt∗ Yt+k|t − Ct+k Yt+k|t

t

t,t+k

∗

∗

Pt

Pt

Pt+k

k=0

k=0

(11)

subject to the demand function

∗ −

Pt

Yt+k|t =

Ct+k

for

k = 0, 1, 2, ...

(12)

Pt+k

Observations in (11) and (12) before the first-order derivative with respect to Pt∗ :

• In Pt∗ Yt+k|t : Yt+k|t depends on Pt∗ : take derivatives using the product rule.

• In Ct+k Yt+k|t : take the derivative of Ct+k Yt+k|t with respect to Yt+k|t (outer

derivative) times the derivative of Yt+k|t with respect to Pt∗ (chain rule).

First-order derivative of profits:

∞

X

0 !

0

0

θk Et Λt,t+k (1/Pt+k ) Yt+k|t + Pt∗ Yt+k|t

− Ct+k

Yt+k|t Yt+k|t

= 0.

(13)

k=0

0

• The derivative of total nominal costs with respect to output Ct+k

Yt+k|t represents nominal marginal costs.

First-order derivative of demand:

∗ − −1

∗ −

∂Yt+k|t

Pt

Pt

1

1

0

= −

Ct+k

= −

Ct+k ∗ .

Yt+k|t =

∗

∂Pt

Pt+k

Pt+k

P

P

| t+k {z

} t

(14)

=Yt+k|t

0

0

Insert Yt+k|t

= − Yt+k|t (Pt∗ )−1 and denote marginal costs as Ψt+k|t ≡ Ct+k

Yt+k|t :

∞

X

θk Et Λt,t+k (1/Pt+k ) Yt+k|t − Yt+k|t + Ψt+k|t Yt+k|t (Pt∗ )−1 = 0.

k=0

2

See Galí (2015), p. 56.

4

(15)

−1

Multiply with −1 Yt+k|t

Pt∗ :

∞

X

θk Et Λt,t+k (1/Pt+k ) −1 Pt∗ − Pt∗ + Ψt+k|t = 0.

(16)

k=0

Use the factorization −1 Pt∗ − Pt∗ = ( −1 − 1) Pt∗ = − (1 − −1 ) Pt∗ :

∞

X

θk Et Λt,t+k (1/Pt+k ) − 1 − −1 Pt∗ + Ψt+k|t = 0.

(17)

k=0

−1

Multiply with − (1 − −1 ) :

∞

X

o

n

−1

Ψt+k|t

= 0.

θk Et Λt,t+k (1/Pt+k ) Pt∗ − 1 − −1

(18)

k=0

Denoting the frictionless markup as M = (1 − −1 )

∞

X

−1

= / ( − 1):

θk Et Λt,t+k (1/Pt+k ) Pt∗ − MΨt+k|t = 0.

(19)

k=0

This is equation (10) in Galí (2015), p. 56.

Log-linearization around the steady state.

Perfect foresight zero-inflation steady state version of (19):

∞

X

θk Λ (1/P ) (P ∗ − MΨ) = 0.

(20)

k=0

Noting from the stochastic discount factor that Λ = β k :

∞

X

(βθ)k (1/P ) (P ∗ − MΨ) = 0.

(21)

k=0

Note that the sum does not apply to time-dependent variables anymore, such that

∞

X

k

0

(βθ) = (βθ) +

k=0

∞

X

k+1

(βθ)

= 1 + βθ

k=0

∞

X

k

(βθ) →

k=0

Therefore, the (convergent) sum is just a constant.

∞

X

k=0

∞

P

(βθ)k =

1

.

1 − βθ

(22)

(βθ)k (1/P ) can be cancelled out

k=0

from (21) to obtain

P ∗ = MΨ.

5

(23)

This is the same relationship under fully flexible prices (θ = 0):3

Pt∗ = MΨt|t .

(24)

An approximation of the first-order condition (19) involves approximating the expression in curly brackets:

Λt,t+k (1/Pt+k ) Pt∗ − MΨt+k|t .

(25)

For a first-order approximation in logarithms define

˜ t+k|k = log Λt+k|k

λ

,

pt+k = log (Pt+k )

µ = log(M)

, ψt+k|t = log(Ψt+k|t ).

,

p∗t = log (Pt∗ )

Noting that for example Pt∗ = exp {p∗t }, substitute

Λt,t+k (1/Pt+k ) Pt∗ − MΨt+k|t

o

n

n

o

˜ t+k|k − pt+k exp {p∗ } − exp µ + ψt+k|t

.

= exp λ

t

First-order Taylor approximation around the non-stochastic steady state:

n

n

o

o

˜ t+k|k − pt+k exp {p∗ } − exp µ + ψt+k|t

exp λ

t

o

n

˜ − p (exp {p∗ } − exp {µ + ψ})

≈ exp λ

∂ {. . . }

∂ {. . . } ˜

∂ {. . . } ∗

˜

+

λt+k|k − λ +

(pt+k − p) +

(pt − p∗ )

∗

˜

∂p

∂p

∂λ

∂ {. . . }

∂ {. . . }

+

(µ − µ) +

ψt+k|t − ψ .

∂µ

∂ψ

(26)

(27)

(28)

Note that the steady state P ∗ = MΨ implies that exp {p∗ } − exp {µ + ψ} = 0 and

therefore:

n

o

˜

exp λ − p (exp {p∗ } − exp {µ + ψ}) = 0.

(29)

Moreover, exp {p∗ } − exp {µ + ψ} = 0 turns the following derivatives equal to zero:

∂ {. . . }

∂ {. . . }

= 0.

=

˜

∂p

∂λ

4

(30)

3

To derive the price setting under fully flexible prices use (19) and write the term that results from

∞

P

k = 0 outside the sum: θ0 Λt,t (1/Pt ) Pt∗ − MΨt|t +

θk Et Λt,t+k (1/Pt+k ) Pt∗ − MΨt+k|t = 0.

k=1

Notice that setting θ = 0 has different implications for the first and second terms. In general,

it is true

that 00 ≡ 1 and 01 = 02 = · · · = 0. The expression remaining is Λt,t (1/Pt ) Pt∗ − MΨt|t = 0, which

is equivalent to (24).

n ˜ ∗

o

∂ eλ−p ep −eµ+ψ

∗

˜

4

For example,

= 1 · eλ−p ep − eµ+ψ = PΛ (P ∗ − MΨ) = 0.

˜

∂λ

6

The approximation breaks down to

Λt,t+k (1/Pt+k ) Pt∗ − MΨt+k|t

∂ {. . . } ∗

∂ {. . . }

ψt+k|t − ψ .

≈

(pt − p∗ ) +

∗

∂p

∂ψ

The remaining steady-state derivatives are

n

o

n

o

∂ {. . . }

∂ {. . . }

∗

˜

˜

= − exp λ − p + µ + ψ .

= exp λ − p + p

,

∂p∗

∂ψ

Rewrite the first steady-state derivative using P ∗ = P (see (75)):

n

o

∂ {. . . }

ΛP ∗

∗

˜

= Λ = βk.

= exp λ − p + p = exp log

∂p∗

P

Rewrite the second steady-state derivative using P ∗ = MΨ:

o

n

∂ {. . . }

ΛMΨ

˜

= − exp λ − p + µ + ψ = − exp log

∂ψ

P

∗

ΛP

= − exp log

= −Λ = −β k .

P

The approximation of (25) then reads

Λt,t+k (1/Pt+k ) Pt∗ − MΨt+k|t

≈ β k (p∗t − p∗ ) − β k ψt+k|t − ψ .

(31)

(32)

(33)

(34)

(35)

The first-order condition (19) now becomes

∞

X

θk Et β k (p∗t − p∗ ) − β k ψt+k|t − ψ = 0.

(36)

k=0

Factorize β k and get

∞

X

(βθ)k Et p∗t − p∗ − ψt+k|t + ψ = 0.

(37)

k=0

Pull the sum inside and separate sum-independent terms:

(p∗t − p∗ + ψ)

∞

X

(βθ)k −

k=0

∞

X

(βθ)k Et ψt+k|t = 0.

(38)

k=0

Compute the geometric sum (see 22):

∞

(p∗t − p∗ + ψ)

X

1

−

(βθ)k Et ψt+k|t = 0.

1 − βθ k=0

7

(39)

Multiply with (1 − βθ):

p∗t

∗

− p + ψ − (1 − βθ)

∞

X

(βθ)k Et ψt+k|t = 0.

(40)

k=0

Solve for the optimal reset price:

p∗t = p∗ − ψ + (1 − βθ)

∞

X

(βθ)k Et ψt+k|t .

(41)

k=0

Noting that at the steady state p∗ = µ + ψ:

p∗t

= µ + (1 − βθ)

∞

X

(βθ)k Et ψt+k|t .

(42)

k=0

• ψt+k|t = log Ψt+k|t is the log (nominal) marginal cost.

• µ = log M is the desired (frictionless) gross markup.

• To the extent that prices are sticky (θ > 0), firms set prices in a forward-looking

way.5

• The chosen price corresponds to the desired markup over a weighted average of

current and expected marginal costs.

• The weights are proportional to the probability of the price remaining effective

at each horizon θk , times the cumulatice discount factor β k .

Equilibrium in the goods market

Yt (i) = Ct (i).

(43)

• Each product variety is in equilibrium of supply and demand.

R

−1

−1

1

, it follows that Yt = Ct . As• Given the CES aggregator Yt = 0 Yt (i) di

sumption: there are no additional demand components (government expenditures,

etc.)

Equilibrium in the labor market

Z

1

Nt =

Nt (i)di.

0

• Nt is the aggregate amount of labor (hours) supplied by the household.

5

θ = 0: a monopolistic firm sets its price in a standard way, namely p∗t = µ + ψt|t .

8

(44)

•

R1

Nt (i)di is the aggregate amount of labor hours demanded across the (differen0

tiated) firms sector.

Aggregation

Linking aggregate labor to aggregate production:

• Note that there is a discrepancy between individual output Yt (i) and aggregate

output Yt due to the discrepancy between individual prices Pt (i) and Pt (see (2)).

• This, in turn, implies that the production function of the single firm i may not be

simply imposed at the aggregate level (i.e. by simply dropping the firm index i).

Solving the production function (1) of a single firm i for labor gives

Nt (i) =

Given the demand function Ct (i) =

the goods market, it follows that

Pt (i)

Pt

Yt (i) =

Yt (i)

At

−

1

1−α

.

(45)

Ct and the market clearing condition in

Pt (i)

Pt

−

Yt .

(46)

Insert individual labor demand from the production function (45) into the aggregate

labor (44):

1

Z 1

Yt (i) 1−α

Nt =

di.

(47)

At

0

Insert the demand function (46) for Yt (i) and get

Z

Nt =

0

1

Pt (i)

Pt

−

At

1

1−α

Yt

di.

(48)

Factorize aggregate output and the technology variable (they are not indexed with i):

Nt =

Yt

At

1

1−α

Z

1

0

Pt (i)

Pt

− 1−α

di.

Take logs:

1

log Nt =

(log Yt − log At ) + log

1−α

9

Z

0

1

Pt (i)

Pt

(49)

− 1−α

di.

(50)

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