econ 4325 monetary policy lecture 3 - universitetet i oslo€¦ · 3.solve for the equilibrium real...

ECON 4325Monetary Policy

Lecture 3

Martin Blomhoff Holm

Outline

1. Brief review of lecture 2.

2. The firm problem without sticky prices.

3. Monetary policy without sticky prices.

Holm Monetary Policy, Lecture 3 1 / 26

Part I: Brief review of lecture 2.


Let’s check what you remember from last time.


Part II: Firm problem without sticky prices.


The firm problem

Assume that we have many firms denoted by i who produce according to

Yi ,t = AtN1−αi ,t

where Yi ,t is production by firm i , At is aggregate TFP, Ni ,t is labor infirm i , and α ∈ [0, 1) is the curvature of the production function.

We further assume that aggregate TFP follows an AR(1) in logs:

log(At) = ρa log(At−1) + εAt

at = ρaat−1 + εAt


The firm problem

Let firm i maximize profits

maxNi,t

PtYi ,t −WtNi ,t

maxNi,t

PtAtN1−αi ,t −WtNi ,t

The first order condition is

Wt

Pt= (1− α)AtN

−αi ,t

i.e. all firms choose the same level of labor.

As a result, we can aggregate all firms to one representative firm. Fromnow on, we therefore remove the dependence on i .


The first order condition

Wt

Pt= (1− α)AtN

−αt

What does it mean? Let us take logs on both sides and solve in terms oflabor demand.

wt − pt = log(1− α) + at − αntor rearranged

nt = − 1

α(wt − pt − at − log(1− α))

Takeaways:

I If wages increase, firms reduce labor demand by 1α of the percentage

reduction in wages.

I 1/α is the elasticity of labor demand to wage changes.

I α = 0: firms adjust labor infinitely much in response to wage changes.

I α→ 1 firms adjust one-for-one for any increase in wages.


EquilibriumWe can now summarize the complete model by the following equations:

Household side:

C−σt = β(1 + it)Et

{C−σt+1

(Pt

Pt+1

)}Wt

Pt= Cσt N

φt

Firm side:

Yt = AtN1−αt

Wt

Pt= (1− α)AtN

−αt

Market Clearing:

Yt = Ct (supply = demand in goods market)

Nt = Nt (supply = demand in labor market)


Let’s log-linearize I: Consumption-Euler

1 = Et

{β(1 + it)

(Ct

Ct+1

)σ Pt

Pt+1

}Step 1: Take exponential and logs of RHS

1 = Et exp {it − ρ− σ(ct+1 − ct)− πt+1}

Step 2: Taylor expansion of RHS (step on blackboard)

Et{1 + (it − i)− σ(c̃t+1 − c̃t)− (πt+1 − π)}

Step 3: Note that i = π + ρ in steady state such that and LHS = RHS

1 = Et{1 + it − ρ− σ(ct+1 − ct)− πt+1}

Step 4: Rearrange to obtain

ct = Et{ct+1} −1

σ(it − Et{πt+1} − ρ)


Let’s log-linearize II: Labor supply

Wt

Pt= Cσt N

φt

ωt = σct + φnt


Let’s log-linearize III: Production

Yt = AtN1−αt

yt = at + (1− α)nt


Let’s log-linearize IV: Labor demand

Wt

Pt= (1− α)AtN

−αt

ωt = log(1− α) + at − αnt


Part III: Monetary policy without sticky prices.


Equilibrium Definition

1. Households maximize the discounted stream of utility

2. Firms maximize profits

3. Market clearing: Yt = Ct and NDt = NS

t


The Model Equations

ωt = σct + φnt (1)

ct = Etct+1 −1

σ(it − Etπt+1 − ρ) (2)

yt = at + (1− α)nt (3)

ωt = yt − nt + log(1− α) (4)

yt = ct (5)


Five Steps to Solve the Model

1. Combine (1) and (4) to solve for nt .

2. Use the solution for nt to solve for yt(= ct) using (3).

3. Solve for the equilibrium real interest rate using the Euler-equation(2). Remember that rt = it − Etπt+1.

4. Use the solution for yt in step 2 to find nt from the solution in step 1.

5. Use either (1) or (4) to solve for wt .


DO IT!


The Solution

ct = yt

yt =1 + φ

σ(1− α) + φ+ αat +

(1− α) log(1− α)

σ(1− α) + φ+ α= ψyaat + ξy

rt = ρ+ σψyaEt{∆at+1}

nt =1− σ

σ(1− α) + φ+ αat +

log(1− α)

σ(1− α) + φ+ α= ψnaat + ξn

ωt =σ + φ

σ(1− α) + φ+ αat +

[σ(1− α) + φ] log(1− α)

σ(1− α) + φ+ α= ψωaat + ξω


Interpretations


Result: The Classical Dichotomy

I All real variables (real wage, output, consumption, and labor hours)are independent of monetary policy: monetary policy neutrality

I Technology is the only driving force of real variables

I Nominal variables (inflation, nominal interest rate) are not uniquelydetermined

I We need one more equation to specify how monetary policy isconducted


Equilibrium behavior of nominal variables I

I The real interest rate is entirely determined by the real side of theeconomy. The nominal interest rate and expected inflation is thendetermined by the Fisher equation

it = Etπt+1 + rt

I Since we have one equation with two unknowns. The inflation andnominal interest rate is indeterminate. Give me any path of prices andI can specify the right path of nominal interest rate, and opposite.


Equilibrium behavior of nominal variables II

I Example: The Taylor rule:

it = ρ+ φππt

With this equation, we also know the path of πt and it .

I However, the model has two predictions that are in sharp contrast toempirical evidence:

1. Prices respond instantaneously to any shock.2. Since rt is fixed, a higher nominal interest rate will result in higher

expected inflation.


Recall for example from lecture 1

Romer-Romer 2004Holm Monetary Policy, Lecture 3 24 / 26

What is needed?

I For nominal interest rate changes to have to real effects, we need tointroduce a friction.

I In the canonical New-Keynesian model and in this class, we introducesticky prices and monopolistic competition.

I But remember! There are many ways to non-neutralityI Wage rigidities, information frictions ...


Next week

I The firm problem with sticky prices.


econ 4325 monetary policy lecture 3 - universitetet i oslo€¦ · 3.solve for the equilibrium real...

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