Some Characteristics of the New Keynesian Models Found in the Derivation Process of the New Keynesian Phillips Curve

Some Characteristics of the New Keynesian Models Found in the Derivation Process of the New Keynesian Phillips Curve

ニュー・ケインジアン・フィリップス曲線の導出過程に見られるニュー・ケイ ンジアン・モデルの特質

NISHII Kentaro

Abstract: The Keynesian theories in the mainstream in 1960’s were strongly criticised by Lucas in “Econometric Policy Evaluation: A Critique (1976)” for lack of ‘micro foundation.’ He emphasised that the policy analysis models should be dynamic and presume economic agents who behaved in expectation of the future.

After his critique, ‘new’ Keynesians introduced those micro foundations into their New Keynesian (NK) models. To discuss some characteristics of the NK models, this paper takes the new Keynesian Philips curve (NKPC) as an example of the basic NK models and shows the derivation process of the NKPC in Section two. Then, it compares the NKPC to the traditional Phillips curve in Section three, and finally, it shows how the NK models have made progress as a policy analysis tool with the dynamic stochastic general equilibrium (DSGE) model introduced in Section four.

Keywords: new Keynesian Phillips curve, Lucas Critique, micro foundation, DSGE model

要約：1960年代に主流であったケインジアンによる経済理論は、1976年のいわゆる

『ルーカス批判』によって、ミクロ的基礎付けを持たない点で強い批判を受けた。ル ーカスは、政策分析モデルは動学的であるべきで、将来への期待に基づいて行動する 経済主体を仮定すべきだと強調した。ルーカス批判以降、ニュー・ケインジアン達は 指摘されたミクロ的基礎付けを自らのモデルに取り入れ、ニュー・ケインジアン(NK) モデルを確立していった。本論文では、NKモデルの特質を論ずるために、第二章で ニュー・ケインジアン・フィリップス曲線(NKPC)を例にとり、その導出過程を示し た。さらに第三章では、NKPCとそれまでのフィリップス曲線を比較、最後に第四章 では動学的確率的一般均衡(DSGE)モデルを紹介しつつ、政策分析ツールとしての

NK

モデルがどのように発展してきたかを示した。

キーワード: ニュー・ケインジアン・フィリップス曲線、ルーカス批判、ミクロ的基 礎付け、動学的確率的一般均衡モデル

1. Introduction

In the 1960’s, Keynesian theories represented by John Hicks’ IS-LM model were in the

prime of time as macroeconomics policy analysis tools. However, in the 1970’s, they

suddenly started to decay. This decay was caused by Lucas (1976), who won the Nobel

Prize in economics in 1995. He concludes in his “Econometric Policy Evaluation:

A Critique” as follows:

“given that the structure of an econometric model consists of optimal decision rules of economic agents, and that optimal decision rules vary systematically with changes in the structure of series relevant to the decision maker, it follows that any change in policy will systematically alter the structure of econometric models.”

In other words, he points out that, for example, the IS-LM model is static and not appropriate to forecast effects of policies because parameters in the IS-LM model could be changed with implementation of the policies. Therefore, for a more precise policy analysis, the models should be dynamic and presume economic agents who behave in expectation of the future. That is why “microeconomic foundations”, which are solving optimizing problems in other words, were introduced into the Keynesian models.

To show a clear-cut example of progress from the traditional Keynesian theories towards the New Keynesian theories, this paper examines the steps involved in deriving the standard new Keynesian Philips curve (NKPC) in Section two, and then focuses on the differences between the NKPC and the traditional Philips curve in Section three.

Additionally, Section four introduces how the New Keynesian model further progressed and has been adopted in the dynamic stochastic general equilibrium (DSGE) model as one of most commonly-used policy analysis tools today, especially among central banks.

2. Steps towards the New Keynesian Philips Curve

2.1. Assumptions

Following Walsh (2003), we assume the following:

1. The model is based on optimizing behavior.

2. There is imperfect competition in goods market implying product differentiation.

3. Imperfect competition in labor market implies labor heterogeneity leading to wages heterogeneity.

4. Prices are set by monopolistically competitive firms (Calvo type price stickiness).

5. Monetary policy is based on interest rates rather than demand for money.

6. Endogenous variation in capital is ignored because these new Keynesian models

are short-run stabilization policy models.

The economy consists of households, firms, and central bank. Households maximise the utility, firms maximise profits and central bank controls the nominal interest rate.

2.2. Households

The preferences of the representative household are given by U =  ^



 





 



 





 





 

 

 





 

0 

1 1 1

1 1

i 1

i t b

i t

i t i

i t t

N P

M b E C

 



  ^{ } ₍₁₎

showing that representative household maximizes the expected present discounted value of utility, where C _t denotes composite consumption good of differentiated products,

t

t P

M / is real money balances, N _t is the time devoted to market employment, and

leisure is 1  N _t (as N _t + l= 1), respectively.

Dixit and Stiglitz preferences (1977) defined C _t as

1 1 0

1 



 

 



   ^







dj c

C

_{t jt}

, (2) where  is the price elasticity of demand and greater than 1. Here, a higher  implies less monopoly. Then, the household faces two stage decision problems.

First stage: the household minimises cost of buying C _t . The decision problem is to find

min 

0¹

p

_jt

c

_jt

dj

cjt

(3) subject to

t

jt

dj C

c  



 





^{ }



0^{1 1}

1









, (4)

where p _jt is the price of good j . The first order condition for good j is

0

1 1 1

0 1

 



 



  

^

^

^{ }

^







_t



_{jt jt}

jt

c dj c

p _, (5)

where  _t is the Lagrangian multiplier.

Solving for  _t ,

_t

p

_jt

dj    P

_t

 

 

^{ }



¹

1 1

0

1

. (6) Rearranging equation (5) for C jt we get the product demand,

t t jt

jt

C

P c p





 

 



  (7)

Second stage: the household maximises utility with respect to consumption C _t , labor supply N _t , money M _t , and bond holdings B _t .

Household’s decision problem is to find

Max U =  ^

















 





 





 

 

 





 



0

1 1 1

1 1

1

i

i t b

i t

i t i

i t t

N P

M b E C

 



  ^{ } (8)

subject to the budget constraint

 

_t

t t t t

t t t

t t t t t t

P i B P

N M P W P B P

C M    

 



 



 



 



 





^1

1

_1 ^1

, (9)

where B _t is holding of one-period bonds that pay a nominal rate of interest i _t .  _t is real profits and W is nominal wages.

Solving the maximization problem yields the following first order condition:







^











 



 



1

1 1

) 1

(

_t

t t t t

t

C

P E P i

C , (10)

which is the Euler equation, showing that value of consumption in this term is equal to

the value of discounted consumption in the next term. The household optimises in such

a way to smooth consumption over time.

Solving the next first order condition yields

t t t

b

t t

i i C

P M

 

 

 











1



. (11) This equation shows that the marginal rate of substitution (MRS) between real money balances and consumption is equal to the opportunity cost of holding money. The decision of whether to consume or to keep a money balance is determined by interest rate i. If the interest rate goes up, the household foregoes some consumption today for holding money.

Solving the last first order condition yields

t t t

t

P W C

N

_^



 _. (12) This equation shows that the MRS between leisure and consumption is equal to the real wages. The decision of whether to consume and enjoy life or to exploit real wages by working harder is determined by the real wages themselves.

2.3. Firms

The production function of the firm, which we assume to be linear (constant returns to scale (CRS)), is a function of labour input N _jt indicating employment in a particular

industry j and an aggregate productivity disturbance Z _t : 1 ) (

, 

 _{t jt t}

jt Z N E Z

c . (13) Profit-maximizing firms face the following two stage decision problems:

First, firms choose cost minimization inputs, that is, they minimize W t N jt subject to producing c _jt  Z _t N _jt . This problem can be written in real terms as

) (

min

_{t t jt t jt}

t t

N

N c Z N

P W

t



 



 



  , (14)

where the Lagrangian multiplier  _t in this cost minimization problem is the firm’s real marginal cost. Solving the first order condition, we obtain

j





t

t t t

t

Z

W P

  , where Z _t = MPL jt . (15) We assume that the CRS production function and the firms employ the same labour.

Therefore, the marginal cost for all the firms is the same.

Second, the firms choose the optimal price p _jt to maximize their profit. Following the Calvo price stickiness model, we assume that in each period a fraction of firms denoted by  keep their prices fixed and the remaining 1   adjust their prices optimally. Thus,  determines the degree of nominal rigidity. The larger  is, the higher nominal rigidity will be. We also assume that the probability of price stickiness is constant because the probability that firms adjust their prices is time-dependent.

Prices are set to maximise not only the value of current profit but also the expected discounted value of future real profit because, in future periods, they may not be able to adjust the prices. So, in fixing the price, the firms also consider their future marginal cost.



^

   









 



   

 



 

0 , i

i jt i t i jt i t

jt i t i i

t

c c

P

E  p  , (16)

where  i _, t  i is the discount factor given by



^

_

^



 

 



t i i t

C C . (16) is a function of subjective discount factor and also the marginal substitution of consumption between two periods. Putting the value of c jt in equation (7) into (16), the objective function can be written as

 =

_{t i}

i t i

jt i t i

t jt i t i i

t

C

P p P

E

^

p

_

















_ ^





 





 

 



 

 

 



 

0

1

,







 . (17)

Since they are essentially identical, all the firms that adjust in period t will set the same price p _t ^ , thus allowing us to remove subscript j . Meanwhile, the other firms will keep their prices fixed. The first order condition for the optimal choice of p _t ^ is

 





p

jt

  1 0

1

0

,

  

 



 



 



 



 



   

 



 



_









  







 ^ti

i t

t

i t

i t i t

t i

t i i

t

C

P p p P

E p









(18)

Using the facts that p

_j

 p

_t^*

and  _{i , t  i} =



^

_

^





 





t i i t

C C , equation (18) can be written as











 







 



 

 

 





 

 





 

 





 

 

 





0

1 1

0 1

1

i

t i t i t i i t

i

t i t i t i t i i t

t t

P C P E

P C P

E P

p





















 _. (19)

This is the price ratio of firms which do adjust their price ( 1   ) . 2.4. Flexible Prices and Price Setting

If all firms are able to adjust their prices in each period, then   0 and prices are flexible. Then, equation (19) can be simplified to

t t t

t

P

p  



 _

 

 





 

 

 





^

1 , (20) where

 1

 

  is the mark-up.

Real price is equal to mark-up of marginal cost. In a symmetric Nash equilibrium, where optimal price of each firm is equal to aggregate money supply, we get that marginal cost is equal to the inverse of mark-up:  _t  1 /  . From the definition of real marginal cost,



^t

t

t

Z

P

W  , (21)

which means real wage is equal to the inverse of mark-up times the marginal product of labour.

In a perfect competition model, real wage is equal to marginal cost, but here, because of monopolistic competition, we have marginal product of labour times the inverse of the mark-up. However, the real wage also needs the marginal rate of substitution between leisure and consumption to be consistent with household optimization. From equation (12), this condition implies that







 ^

^



t t t t

t

C N Z P

W . (22)

Let x ˆ _t f be the log linearization of X _t around its steady state X. Log linearizing (22) around the steady state yields  n ˆ _t ^f   c ˆ _t ^f  z ˆ _t . Log linearizing the production function (13), y ˆ _t ^f  n ˆ _t ^f  z ˆ _t . Because output is equal to consumption in equilibrium, y ˆ _t ^f  c ˆ _t ^f . Combining these conditions, the flexible-price equilibrium output y ˆ _t ^f can be written as

t f

t

z

y 1 ˆ

ˆ  

 







 





 , (23) which is a standard equilibrium output equation in a model of flexible price and monopolistic competition. This means that the only thing that affects output is elasticity.

Since the mark-up is constant, it drops out in log linearization.

Setting   0 , the model is reduced to the neoclassical model. Also, we have output inefficiency because the price is greater than the marginal cost.

2.5. The Standard NKPC (with Calvo price stickiness (1983)) Transforming equation (6), which is the consumer price index, to

 

 

 

^



p dj

p

_t ¹ _jt

0

1 1

, (24) where prices are sticky (   0 ) , since a fraction of ( 1   ) of firms adjusts prices at time t and the rest  keep prices as in the previous period, the above equation becomes





  ^  _ ^

    ¹  ₁ ¹

1 ( 1 )( ) ( )

)

( p _t p _t p _t . (25)

This equation shows the average price of the fraction ( 1   ) of firms which set their price in period t and the average price of the remaining fraction  of firms who set their price in the earlier period t -1.

Using (19) and the log linearization of equation (25) around a zero inflation steady-state, we obtain an expression for an aggregate inflation of the form

t t

t i

t

E k ~ m c ˆ

1



  

_

 , (26) where





 )( 1 ) 1

~  (  

k .

This equation is the NKPC. Alternatively, in terms of the output gap, we can rewrite the above equation as

t t t i

t   E  _{ 1}  k x ˆ

 . (27) The empirical problem of implementing this model is that data on real marginal cost is not observed (Whelan 2007). Data normally contains information that affects the average cost, such as wages, but not the real marginal cost. Therefore NKPC, which uses a measure of output gap, is used instead of the marginal cost.

3. Difference between the NKPC and the Traditional Philips Curve

Examining the derivation process of the standard NKPC, we obtain differences between the NKPC and the traditional Philips curve as follows:

1. In the NKPC, the inflation process is forward-looking and current inflation is a function of expected future inflation. Meanwhile, the traditional Phillips curve, which is often written as

y  y ˆ =  (    ^e ) ,

is not a function of future inflation. In other words, it is only a function of expectation at

time t. The traditional Philips curve can be termed as derived from a static process

containing the inflation surprise (    ^e ) , whereas the NKPC is a dynamic form of

forward-looking expectation. While setting its price, the firm will also be concerned

with inflation in the future, because it may be difficult to adjust its price for several

periods due to, for example, the menu cost.

2. The NKPC implies that the real marginal cost (MC) is the most important variable for the inflation process. In NKPC, inflation depends on real MC, though it is possible to relate real MC to an output gap measure, whereas the traditional Philips curve expresses inflation as a function of the output gap or unemployment rate relative to its natural rate.

3. The third difference between the NKPC and the Philips curve is that only the NKPC is derived from a model of optimizing behaviour of price setters where the traditional Philips curve was never derived (Sergeant (1982)). The strong point of NKPC, being based on such micro foundations, is that it provides a clear picture of how the structural parameters  and  affect k ~ , which is the response of inflation to real MC. A higher  implies that the firms care more about the future expected profits, and this leads to the decline of k ~ . Inflation is less sensitive to current MC and thus, the response of inflation to marginal cost is slow, whereas a higher  (increased price rigidity) reduces k ~ . Lower is the response of inflation to marginal cost if nominal rigidity is increased.

4. Conclusions and Further Progress of New Keynesian Models

While traditional Keynesians were held out of the centre of macroeconomics after the Lucas Critique (1976), neoclassical economics recovered lost grounds. Among their outstanding performances, the real business cycle (RBC) model by Kydland and Prescott (1982), who won the Nobel Prize in economics in 2004, is the foundation of many useful policy analysis tools represented by the dynamic stochastic general equilibrium (DSGE) model.

However, disadvantages of the RBC model were also pointed out as it was widely

used by policy makers. For example, this model is based upon an unrealistic assumption

of no market imperfection. Since the 1990’s, new Keynesians have attempted to take

market imperfections, such as price and wage stickiness and adjustment costs of

investment, into original frameworks of the RBC model. Consequently, the DSGE

model has significantly progressed. Moreover, it has even enabled both neoclassical and

new Keynesians to present their arguments by using the same policy analysis tool in this

century.

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Dixit, A.K.and Stiglitz J. E. (1977): “Monopolistic Competition and Optimum Product Diversity,” The American Economic Review, 67(3), 297-308.

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動学的一般均衡モデルによる財政政策の分析

（財団法人三菱経

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モデルによるマクロ実証分析の方法

（公益財団法人三菱

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Roberts, J.M., (1995), “New Keynesian Economics and the Phillips Curve”, Journal of Money, Credit and Banking, Vol.27, No.4, Part 1, 975 – 984.

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http://www.karlwhelan.com/Teaching/msc07.html

Received on January 7, 2014.