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ECON 401

Advanced Macroeconomics

Topic 2

New Keynesian Macroeconomics

Fabio Ghironi

University of Washington

NEW KEYNESIAN THEORY: MONOPOLY PRICING

RELEVANT MARKET STRUCTURE(S)?

Introduction

 Real business cycle (RBC)/neoclassical theory  All (goods) prices are determined in perfect competition  In both consumption-leisure and consumption-savings dimensions  Critical assumption: no firm is a price setter  no firm has any market

power

 New Keynesian theory  Starting point: firms do wield (at least some) market power  Critical assumption: firms do set their (nominal) prices  Purposeful setting/re-setting of (nominal) prices may entail costs of

some sort  “Menu costs,” but soon interpret more broadly  Central issue in macro: how do “costs of adjusting prices” (“sticky prices”)

affect monetary policy insights and recommendations?

 Upcoming analysis  Step 1: Develop theory in which firms are purposeful price setters, not

price takers  Step 2: Superimpose on the theory some “costs” of setting/re-setting

nominal prices  Step 3: Study optimal monetary policy

MONOPOLISTIC COMPETITION

Basics of New Keynesian Theory

 Monopolistically-competitive view of goods markets the foundation of NK theory  An intermediate market structure between pure perfect competition

and pure monopoly

 Framework  Allows for purposeful price setting by firms  Retains some competitive features of pure supply-and-demand theory  Assumes that goods are imperfect substitutes

 The foundation/essence of market power  In contrast to the perfect substitutability of goods in theory of pure perfect

competition

 Markup  The ratio of a firm’s unit sales price to its marginal cost of production  A key concept in the theory of monopoly/monopolistic competition  A key measurable (sort of…) empirical concept

 Perfect competition: markup = _____

NK MODEL OVERVIEW

Basics of New Keynesian Theory

 Monopolistic competition in goods markets the underlying market structure  Operationalize by dividing goods markets into two “sectors”

 Retail firms  Each sells a perfectly-substitutable “retail good” in a perfectly-

competitive market  Purchase differentiated “wholesale goods” in monopolistically-

competitive markets

 Wholesale firms  An “infinite” number of them  Each produces a “wholesale good” imperfectly substitutable with any

other “wholesale good”   each wholesale firm is a price setter  “Wholesale goods” sold to retail firms

 Conceptual separation allows for separate consideration of  Price setting at the microeconomic level  Determination of market outcomes at the macroconomic level

RETAIL FIRMS

Basics of New Keynesian Theory

 A representative retail firm  Operates a “production function” that “bundles together” wholesale goods

into retail goods  Inputs: wholesale goods ONLY, no labor or other inputs required  Example: a retail store that produces no goods of its own

 Dixit-Stiglitz aggregator function/”production function”  Workhorse building block of NK theory

 Parameter ε measures curvature  Elasticity of substitution between any pair of differentiated wholesale

goods is ε/(ε-1)  ε also the critical determinant of profit-maximizing markup  Restriction for NK model to make any sense: ε > 1  Setting ε = ____ recovers perfect competition (i.e., RBC, not NK)

1 1/

0t it y y di

     

Output of the retail good

(Also a basic building block of theory of international trade)

An infinity (continuum) or differentiated wholesale goods

See all this soon

RETAIL FIRMS

Basics of New Keynesian Theory

 A continuous infinity of wholesale goods  A metaphor for “many varieties of goods”  Easier to deal with mathematically than discrete infinity (tools of

calculus can be applied!)  And normalize to continuum [0,1] (could also say, i.e., [0,2], etc…)

 Schematic structure of goods markets

 Representative retail firm’s profit function

0t t it it P y P y di

Nominal price of the retail good Nominal price of wholesale good i

Substitute Dixit-Stiglitz production function

1 11/

0 0t it it it P y di P y di

      

RETAIL FIRMS

Analysis

 Representative retail firm’s profit-maximization problem

 FOC with respect to yjt (for any j)

 …after several rearrangements

  0..

1 11/

0 0 max

it i t it it it

y P y di P y di

 

 

      Chooses profit-maximizing quantity of input of each wholesale good. Focus analysis on any arbitrary wholesale good – call it yjt.

1 jt

jt t t

P y y

  

    

DEMAND FUNCTION FOR GOOD j

WHOLESALE FIRMS

Basics of New Keynesian Theory

 Focus on the activities of an arbitrary wholesale firm j  Symmetry: assume that every wholesale firm makes decisions analogously

 Consistent with the representative-agent approach  So can speak of a “representative” wholesale firm

 Assume zero fixed costs of production

 Operates a constant-returns-to-scale (CRS) production technology in order to produce its unique, differentiated output  CRS: if all inputs are scaled up by the factor x, total output is scaled up

by the factor x  Implementation of theory requires specifying neither the factors of

production (i.e., labor, capital, etc) nor a production function (f(.))

 Marginal cost of production  = average cost of production  is invariant to the quantity produced

 i.e., mc is NOT a function mc(quantity)

Together, these imply a simple description of production

WHOLESALE FIRMS

Analysis

 Representative wholesale firm’s profit-maximization problem

 The sole choice object is Pjt  Compute FOC!

max jt

jt jt t jt jt P

P y Pmc y

mc is NOT a function of quantity produced – CRS assumption.

FC = 0  mc = ac

Conversion of production costs into nominal terms requires factor Pt, NOT Pjt. Because costs are not denominated in the firm’s own prices.

Total revenue depends on firm’s production and its own product price.

1 1 max

jt jt jt t t jt tP

t t

P P P y Pmc y

P P

       

       

Substitute in demand function for wholesale good j.

The critical point of analysis of monopoly: the firm understands and internalizes the effect of its price on the quantity that it sells.

WHOLESALE FIRMS

Analysis

 Representative wholesale firm’s profit-maximization problem

 FOC with respect to Pjt (lengthy algebra…)

.  …after several rearrangements

 Define relative price as

1 1 max

jt jt jt t t jt tP

t t

P P P y Pmc y

P P

       

       

2 1 2 1 1 1 1 11 0

1 1jt t t jt t t t P P y P P mc y

       

 

      

 

jt jt

P mc

P 

Optimal relative price of wholesale firm j is a markup ε over marginal cost of production.

KEY PRICING RESULT OF DIXIT- STIGLITZ THEORY.

jt jt

P p

P  jt jtp mc

In which case can express pricing rule as

/jt jt jtp mc  Or as optimal markup rule

THE DIXIT-STIGLITZ FRAMEWORK

Conclusion

 Key prediction of basic Dixit-Stiglitz theory

 Firms aim to keep their prices at a constant markup over marginal cost

 Empirical relevance of DS constant markup prediction?  Not very in the short run…  …but maybe in the long run

 Markups generally observed to be countercyclical (with respect to GDP)  During expansions, markups decline; during recessions, markups rise  (detrended, business cycle frequencies)

 DS framework has long been the main starting point for pricing theories; recent incorporation into studying  Customer switching effects  Brand loyalty  Search costs

jt jt

p mc

  

NEXT: The Dixit-Stiglitz framework as the foundation of New Keynesian sticky-price models.

NEW KEYNESIAN THEORY: THE MODERN STICKY-PRICE MODEL

RELEVANT MARKET STRUCTURE(S)?

Introduction

power

some sort  “Menu costs,” but soon interpret more broadly  Central issue in macro: how do “costs of adjusting prices” (“sticky prices”)

affect monetary policy insights and recommendations?

 Upcoming analysis  Step 1: Develop theory in which firms are purposeful price setters, not

price takers  Step 2: Superimpose on the theory some “costs” of setting/re-setting

nominal prices  Step 3: Study optimal monetary policy

NOW

MENU COSTS

Basics of New Keynesian Theory

 Do firms incur “costs” in the very act of setting/re-setting nominal prices?  If so, what is the nature and prevalence of these costs?  A central issue in price theory

 Menu cost – any and all costs incurred directly due to the price (re-)setting process  Independent of any physical production costs – i.e., NOT a cost captured by

standard “production functions”

 Two common views of nature of menu costs  Fixed menu cost: total menu cost is independent of the magnitude of the price

change being considered  Example: cost of printing new prices on restaurant menus is probably independent

of what the new prices are

 Convex menu cost: total menu cost is convex and increasing in the magnitude of the price change being considered

 Example: if “menu cost” includes “cost of angering customers,” “managerial time,” etc., convexity assumption may be more appropriate

 Both fixed and convex are likely aspects of menu costs  Formal theoretical NK model typically focuses only on convex menu costs

Anderson and Simester;

Zbracki et al papers

MODELING CONVEX MENU COSTS

Basics of New Keynesian Theory

 Introduce menu costs at level of wholesale firms  Because they actually (re-)set prices!  What does it mean for a firm that is not a price-setter to incur costs of

setting prices?...

 Wholesale firm j incurs real menu cost of nominal price adjustment

 REAL cost of price adjustment – denominated in goods  Parameter Ψ > 0 governs “importance” of menu costs

 Ψ = 0 means no menu cost, which recovers basic Dixit-Stiglitz framework

 Convex: the larger the percentage deviation of Pjt from Pjt-1, the larger the menu cost  Implication: a disincentive to adjusting prices “too quickly”

 Question: are downward adjustments just as costly as upward adjustments?  Intuition: “no” Anderson and Simester evidence: “maybe?...”

1 2

P P





    

 

RETAIL FIRMS

Analysis

 Representative retail firm’s profit-maximization problem

 FOC with respect to yjt (for any j)

 …after several rearrangements

 IDENTICAL TO BASIC (FLEXIBLE-PRICE) DIXIT-STIGLITZ FRAMEWORK!

  0..

1 11/

0 0 max

it i t it it it

y P y di P y di

 

 

      Chooses profit-maximizing quantity of input of each wholesale good. Focus analysis on any arbitrary wholesale good – call it yjt.

1 jt

jt t t

P y y

  

    

DEMAND FUNCTION FOR GOOD j

WHOLESALE FIRMS

Basics of New Keynesian Theory

 Focus on the activities of an arbitrary wholesale firm j  Symmetry: assume that every wholesale firm makes decisions analogously

 Consistent with the representative-agent approach  So can speak of “the” wholesale firm

 Assume zero fixed costs of production

by the factor x  Implementation of theory requires specifying neither the factors of

production (i.e., labor, capital, etc) nor a production function (f(.))

 Marginal cost of production  = average cost of production  is invariant to the quantity produced

 i.e., mc is NOT a function mc(quantity)

 AND ALSO INCUR QUADRATIC MENU COSTS

Together, these imply a simple description of production

1 2

P P





    

 

The basis for “sticky” or “sluggish” nominal price adjustment

WHOLESALE FIRMS

Analysis

 Representative wholesale firm’s period-t profit function

 Presence of menu cost makes wholesale firm’s profit-maximization problem a DYNAMIC one  Because any nominal price chosen in a given period has consequences

for profits in the subsequent period through menu costs  Firm pricing problem is forward-looking

 Dynamic (two-period) profit function

ax 1m 2jt

jt jt jt t jt jt

j P t

P P y Pmc y P

P 



      

 

mc is NOT a function of quantity produced – CRS assumption. FC = 0  mc = ac OF PRODUCTION!

Conversion of production costs into nominal terms requires factor Pt, NOT Pjt. Because costs are not denominated in the firm’s own prices.

Total revenue depends on firm’s production and its own product price.

Period-t menu costs, conversion of which into nominal terms requires factor Pt, NOT Pjt. Because costs are not denominated in the firm’s own prices.

2 2

1 1 1 1

max 1 1

1 2 2jt

jt jt jt jt t jt jt t jt jt t t j

t P t t

jt jt

P P P y Pmc y P P y P mc y P

P P   

     

 



                    

  

Discount factor requires inflation adjustment.

And background assumption: no agency problem.

WHOLESALE FIRMS

Analysis

 Representative wholesale firm’s profit-maximization problem

2 2

1 1 1 1 1 1 1

1 1

max 1 2 1 1 2jt

jt jt jt jt t jt jt t jt jt t t jt tP

jt t jt

P P P y Pmc y P P y P mc y P

P P   

 

       

                        

Substitute in demand function for wholesale good j in both period t and t+1 (and t+2, t+3, t+4, …)

The critical point of analysis of monopoly: the firm understands and internalizes the effect of its price on the quantity that it sells.

2 1 1

1 1 1

2 1 1

1 1 1 1 1 1 1 1 1

max 1 2

1 1 2

jt t t jt t

jt jt jt jt

jt jt

tP jt jt jt

jt jt jt jt t

jt t t t t

y Pmc y P P P P

P P P P

P P P

y P mc y

P P P P

   



  

 

  

    

      

                  

       

                             In period t, firm chooses Pjt.

So FOC with respect to Pjt…

WHOLESALE FIRMS

Analysis

 Representative wholesale firm’s profit-maximization problem

 FOC with respect to Pjt

 If ψ = 0, collapses to

 Existence of menu costs (ψ > 0) complicates pricing rule

2 1 1

1 1 1

2 1 1

1 1 1 1 1 1 1 1 1

max 1 2

1 1 2

jt t t jt t

jt jt jt jt

jt jt

tP jt jt jt

jt jt jt jt t

jt t t t t

y Pmc y P P P P

P P P P

P P P

y P mc y

P P P P

   



  

 

  

    

      

                  

       

                            

2 1 1 111 1

2 1

1 1

1 11 1 1 0

1 1 1 jt jtt t

t t t t t jt

jt jt jt

jt jt jtjt t

P PP P P y P mc y

P P

P P P PP P

  

  

  

   

 

  



                    

jt t

P mc

P  Exactly the flexible-price Dixit-Stiglitz pricing rule

SYMMETRIC EQUILIBRIUM

Analysis

 Now drop the distinction between “retail goods” and “wholesale goods”  Suppose “goods” are all identical

 A macro perspective  The “representative good”…  …since macro analysis is most concerned with aggregates

 Impose symmetry by now dropping j indexes – i.e., now suppose Pjt = Pt

2 1 2 1 1 1 11 1 1 1

1 1 1

1 1 1 0

1 1 1 t t t t t

t t t t t t t t t t t t t

P P P P P P P y P P mc y

P P P P P

        

  

       

  

            

     

…and use definition of inflation Pt/Pt-1 = 1 + πt

  1 1 1 1 1 1

1 1 1 1 0

1 1 t t t t t

t t t t t t t t

P P P P P mc y

P P P P P 

   

  

  

            

    

Several terms combine… = 1 = 1

NEW KEYNESIAN PHILLIPS CURVE

Analysis

 The New Keynesian Phillips Curve (NKPC)

 Links period-t inflation to period-t marginal costs of production and period- (t+1) inflation

 “Classical” Phillips Curve  A link between period-t inflation and one component of period-t

marginal costs of production (employment)  No “forward-looking” elements in it

 Forward-looking pricing/inflation behavior the key idea articulated by NKPC  Pricing decisions are inherently dynamic

 NKPC the cornerstone idea in New Keynesian theory

 Here derived from Rotemberg framework…can derive off alternative theories

  1 1 1

1 (1 ) (1 ) 0 1 t t t t t t

mc y       

      

ECON 401

Advanced Macroeconomics

Alternative New Keynesian Foundation

Fabio Ghironi

University of Washington

Introduction

• When we studied the monopolistic competition foundation of the New Keynesian model that Sanjay Chugh introduces in his textbook and slides, I mentioned that we could have

set up the model without introducing the separation between wholesalers and retailers

by just assuming that the representative consumer consumers a bundle of differentiated

final products produced by firms with monopoly power that operate under monopolistic

competition.

• These notes/slides show you how to do that.

A Consumption Bundle

• Suppose that instead of the homogeneous retail good produced by Chugh’s perfectly competitive retailers, households in our economy consume a bundle of differentiated

products that combines these products according to:

Ct =

(∫ 1 0

ct(j) θ−1 θ dj

) θ θ−1

where θ > 1 is the elasticity of substitution between differentiated goods in the bundle, and

ct(j) denoted consumption of product j, which is produced under monopolistic competition

by firm j.

• The consumer has period utility U(Ct) (or U(Ct,1 − Nt) if we want to have endogenous labor supply in the model) and the dynamics of Ct are determined by an intertemporal

maximization problem as usual.

• We are interested in the expressions for the CPI of this economy (Pt) and for how the consumer allocates the Ct determined by the appropriate Euler equation to the individual

differentiated ct(j)’s.

The Welfare-Consistent CPI

• The welfare-based consumer price index of this economy is obtained as solution to the problem of finding the minimum amount of spending needed to purchase one unit of the

bundle Ct.

• Formally,

Pt ≡ min ct(j)

∫ 1 0

pt(j)ct(j)dj subject to Ct = 1,

where pt(j) is the price of good j, or, substituting the expression for Ct,

Pt = min ct(j)

∫ 1 0

pt(j)ct(j)dj subject to

(∫ 1 0

ct(j) θ−1 θ dj

) θ θ−1

= 1.

• The Lagrangian for this problem is:

L =

∫ 1 0

pt(j)ct(j)dj + λt

[ 1−

(∫ 1 0

ct(j) θ−1 θ dj

) θ θ−1 ] ,

where λt is the Lagrange multiplier.

The Welfare-Consistent CPI, Continued

• Taking the derivative of this expression with respect to ct(j), setting it equal to zero, using(∫ 1 0 ct(j)

θ−1 θ dj ) θ

θ−1 = Ct, and rearranging yields:

ct(j) =

( pt(j)

λt

)−θ Ct,

or:

ct(j) =

( pt(j)

λt

)−θ once we recall the constraint Ct = 1.

• Now substitute the expression ct(j) = (pt(j)/λt)−θ into (∫ 1

0 ct(j)

θ−1 θ dj ) θ

θ−1 = 1.

• Simple algebra then allows you to obtain:

λt =

(∫ 1 0

pt(j) 1−θdj

) 1 1−θ

The Welfare-Consistent CPI, Continued

• Therefore,

ct(j) =

( pt(j)

λt

)−θ = pt (j)

−θ (∫ 1

pt(j) 1−θdj

) θ 1−θ

• Recall that Pt is defined as the value of spending ( ∫ 1 0 pt(j)ct(j)dj) such that spending needed

for Ct = 1 is minimized, i.e., such that ct(j) obeys this expression.

– As usual in this class, the shapes of objective function and constraint are such that it is

not necessary to verify second-order conditions.

• Hence,

Pt =

∫ 1 0

pt(j) 1−θ (∫ 1

pt(j) 1−θdj

) θ 1−θ

dj =

(∫ 1 0

pt(j) 1−θdj

) 1 1−θ

• Note that this expression for the price index implies the usual property that Pt = pt in the symmetric equilibrium of the model.

Differentiated Good Demand

• The optimal demand for each individual differentiated good j is found by solving the problem:

max ct(j)

Ct subject to...

∫ 1 0

pt(j)ct(j)dj = St,

where St is an exogenously imposed amount of spending.

– Note that this does not mean finding a different Ct from the one implied by the

intertemporal utility maximization problem.

– Ct remains determined by the relevant Euler equation.

– The solution to the problem we are studying now will tell us how best to allocate Ct across the individual differentiated goods.

Differentiated Good Demand, Continued

• Substituting the expression for Ct in the maximization problem above yields:

max ct(j)

(∫ 1 0

ct(j) θ−1 θ dj

) θ θ−1

subject to...

∫ 1 0

pt(j)ct(j)dj = St,

and the Lagrangian for this problem is

L =

(∫ 1 0

ct(j) θ−1 θ dj

) θ θ−1

+ φt

( St −

∫ 1 0

pt(j)ct(j)dj

) .

• Taking the derivative of the Lagrangian with respect to ct(j), setting it equal to zero, using(∫ 1 0 ct(j)

θ−1 θ dj ) θ

θ−1 = Ct, and rearranging yields:

ct(j) = (φtpt(j)) −θ Ct.

• Substituting this into Ct = (∫ 1

0 ct(j)

θ−1 θ dj ) θ

θ−1 yields an equation that can be solved for φt to

obtain:

φt =

(∫ 1 0

pt(j) 1−θdj

)− 1 1−θ

• But comparing this to the expression for Pt obtained above immediately implies that φt = P

−1 t .

Differentiated Good Demand, Continued

• Hence,

ct(j) =

( pt(j)

)−θ Ct.

• Note that this demand function has the same expression as the demand of a differentiated wholesale good by the retailer in Sanjay Chugh’s version of the New Keynesian framework.

• Hence, everything else follows identically.

• In particular, producer j’s profit maximization problem will yield: pt (j)

Pt =

θ −1 mct(j).

• Identical production functions and symmetry across producers will imply mct(j) = mct, pt (j) = pt, and the price index expression will imply Pt = pt, as we already noted.

• The demand expression for ct(j) in the symmetric equilibrium will in turn imply ct = Ct, very much like we had Yt (output of the retail bundle) = yt (output of each wholesale good) in

Chugh’s framework.

• All the properties of the framework, under flexible or sticky prices, follow identically.