I
The Basic New Keynesian Model
January 11th 2012
Lecture notes by
Drago Bergholt, Norwegian Business School [email protected]
II
Contents
1. Introduction ... 1
1.1 Prologue ... 1
1.2 The New Keynesian model – Key features ... 1
2. Households ... 3
2.1 Setup ... 3
2.2 Optimal consumption vector and the aggregate price index ... 4
2.3 Optimal allocation of consumption and labor ... 6
3. Firms ... 11
3.1 Aggregate inflation ... 11
3.2 Optimal price setting ... 12
3.3 Log-linearization ... 13
4. Equilibrium ... 18
4.1 Market clearing ... 18
4.2 The New Keynesian Phillips curve and the Dynamic IS equation ... 20
5. Equilibrium determinacy ... 26
5.1 System representation ... 26
5.2 Blanchard and Kahn conditions ... 27
6. Shocks ... 30
6.1 Effects of a monetary policy shock ... 30
6.2 Effects of a technology shock ... 33
7. Distortions to the efficient allocation... 36
7.1 The efficient steady state ... 36
7.2 Distortions caused by market power ... 37
7.3 Distortions caused by sticky prices ... 38
7.4 Monetary policy solutions to equilibrium distortions ... 39
8. The welfare loss function ... 44
8.1 Introduction ... 44
8.2 The simplest case – A welfare loss function when real rigidities are absent ... 44
8.3 Introduction of cost push shocks ... 51
8.4 A welfare loss function when real rigidities are present ... 53
9. Welfare based evaluation of monetary policy ... 58
9.1 Introduction ... 58
9.2 An efficient steady state under discretion ... 58
9.3 An efficient steady state under commitment ... 62
9.4 A distorted steady state under discretion ... 66
III
9.5 A distorted steady state under commitment ... 68
10. Wage rigidities ... 69
10.1 Introduction ... 69
10.2 Firms ... 69
10.3 Households ... 70
10.4 Inflation equations and the Dynamic IS equation ... 79
10.5 System representation and equilibrium determinacy... 82
10.6 Shocks ... 83
10.7 Monetary policy design with sticky wages... 85
11. A small, open economy model ... 93
11.1 Introduction ... 93
11.2 Households ... 93
11.3 Terms of trade, domestic inflation and CPI inflation ... 98
11.4 The real exchange rate ... 99
11.5 International risk sharing ... 100
11.6 Uncovered interest rate parity ... 101
11.7 Firms and technologies ... 102
11.8 Equilibrium – Aggregate demand and output ... 103
11.9 Equilibrium – The trade balance ... 109
11.10 Equilibrium – The supply side: Marginal cost and inflation dynamics ... 109
11.11 The New Keynesian Phillips curve and the Dynamic IS equation ... 111
11.12 Equilibrium determinacy ... 113
11.13 Equilibrium dynamics ... 116
11.14 Optimal monetary policy in the small open economy ... 118
11.15 Welfare losses ... 123
References ... 127
Appendix ... 128
A. Dynare codes – A monetary policy shock with sticky prices ... 128
B. Dynare codes – A technology shock with sticky prices ... 129
C. Dynare codes – A monetary policy shock with sticky prices and wages ... 131
1
1. Introduction
1.1 Prologue
These lecture notes take the reader through a basic New Keynesian model with utility maximizing households, profit maximizing firms and a welfare maximizing central bank. I follow Gali’s (2008) book as closely as possible. The notes were born during my participation at a couple of PhD courses in monetary policy, taught by Antti Ripatti (Bank of Finland) and Krisztina Molnar (Bank of Norway), respectively. Both courses built on the excellent book by Gali. The aim of the notes is to provide the reader with all relevant calculations which are left out of the book. In addition, the notes also go through equilibrium determinacy conditions in more detail, following benchmark articles such as Blanchard and Kahn () and Bullard and Mitra (2002). Chapters 2, 3 and 4 characterize the basic New Keynesian model. I first analyze households, then firms. Results are combined to establish general equilibrium. I derive a dynamic IS equation and a New
Keynesian Phillips curve. Determinacy and shocks are discussed in chapters 5 and 6. I perform some welfare analysis of monetary policy in chapters 7, 8 and 9. Chapter 10 augments the basic model with sticky wages in addition to sticky prices, following Erceg et al. (2000). Finally, the small open economy model established by Gali and Monacelli (2005) is derived in chapter 11.
Dynare codes are provided in the appendix. A few words about notation: Variables in levels are denoted with capital letters, logged variables with small letters. Percentage deviations are denoted with small letters with a hat. Let us illustrate by an example: The percentage deviation in from is presented by a first-order Taylor expansion:
̂ ( )
1.2 The New Keynesian model – Key features
So, what kind of features do the New Keynesian models possess? The most important are:
Dynamic, stochastic, general equilibrium (DSGE) modeling: Agents’ behavior today affects future environments. Agents know this and behave accordingly. Still, uncertainty arises because at least some processes in the economy are exposed to exogenous shocks. General equilibrium, in the sense that it incorporates all markets in the economy, is provided.
Monopolistic competition: Prices are set by private economic agents in order to maximize their objectives, as opposed to being determined by an anonymous Walrasian auctioneer seeking to clear all competitive markets at once.
Nominal rigidities: At least some firms are subject to constraints on the frequency with which they can adjust prices of the goods and services they sell. Alternatively, firms may face some
2
costs of adjusting those prices. The same kind of friction applies to workers in the presence of sticky wages.
Short run non-neutrality of monetary policy: As a consequence of nominal rigidities, changes in short term nominal interest rates are not matched by one-for-one changes in expected inflation, thus leading to variations in real interest rates. The latter brings about changes in real quantities. In the long run, however, all prices and wages adjust, and the economy reverts back to its natural equilibrium.
While the first bullet point is a common feature in most modern macroeconomic models, including those in the RBC literature, the last three are special ingredients in New Keynesian models. Now it is time to present the basic model.
3
2. Households
2.1 Setup
We will study households and the implications of market power first. Consider an economy consisting of many identically, infinitely-lived households, with measure normalized to one. The representative household has an instantaneous (and time separable) money-in-utility function of the form:
( ) (2.1)
The consumption level is denoted , is labor, and is real money holdings. One can think of as a composite of many goods. We make the following assumptions about preferences:1
, , , , , To simplify the analysis, we also assume that the marginal utility of one specific element in the utility function is independent of the level of other elements, i.e. that . A representative household maximizes lifetime utility, and discounts the future proportionally by a factor :
{∑ ( )} (2.2)
The consumption index is the sum of consumption of all goods , and there exists a continuum of goods represented by the interval [ ]:
(∫
) (2.3)
Note that utility is a nested function of , where is increasing in and ( ) is increasing in . Thus, utility is increasing in . The CES-aggregator given in (2.3) is an assumption about preferences. Given this assumption, goods become imperfect substitutes, a feature which equips firms with market power.2 Households’ maximization problem is subject to a one-period budget constraint:
∫ (2.4)
In this setup, is the number of bonds purchased last period, each yielding a payoff of one, and is the price per bond bought today.
1 The expression represents (
)
( ) throughout the text.
2 Equation (2.3) also nests free competition as a special case. In particular, taking the limit as approaches infinity, (2.3) becomes ∫ .
4
2.2 Optimal consumption vector and the aggregate price index
The household’s decision problem can be dealt with in two stages. First, for any given level of consumption expenditures, it will be optimal to purchase the consumption vector that maximizes total consumption .3 Second, given this optimal bundle of consumption goods, the household must choose the utility maximizing combination of consumption, labor and money. Let us find the optimal consumption vector first. For a given level of consumption expenditures, say
∫ , the consumption maximization problem is given by:
(∫
)
s.t. (2.5)
∫
This problem can be used to derive an aggregate price index in addition to the optimal consumption vector. Let us solve the problem:
(∫
) (∫ ) FOC:
: (∫
)
⇒ (∫
) [(∫
) ]
⇒
The equality must hold for all goods, so the relationship between two different goods must be:
(
)
⇒ (
) (2.6)
Insert (2.6) into the constraint and solve for : ∫ ∫ (
) ∫
3 Alternatively, one can find the consumption vector that minimizes total consumption expenditures for a given level of consumption. The two problems are equivalent and give identical results.
5
⇒
∫
Insert the result above into and evaluate the result for :
(∫
) [∫ (
∫ )
]
[
∫
(∫ )
]
[(∫ )
]
(∫ )
Define as the expenditure needed to purchase a unit-level of , that is | . Using this definition we can solve the above equation for :
(∫ ) (2.7)
Thus, equation (2.7) can conveniently be defined as an aggregate price index. We will use it throughout the notes. To find the optimal consumption vector, insert (2.6) into the expenditures level equation. Then, insert (2.7) and solve for consumption of good :
∫ ∫ (
)
∫
⇒ [(∫ ) ]
( )
⇒ ( ) (2.8)
Insert (2.8) into (2.3) and rearrange:
(∫
) (∫ [( )
]
)
(∫ [ ] )
(∫ ) [(∫ ) ]
⇒
⇒ ∫ (2.9)
Finally, we get the demand function for good by inserting (2.9) into (2.8):
( ) (2.10)
6
Equation (2.10) is the solution to (2.5), the first stage of a representative household’s decision problem. Once the household knows prices and has decided on , it also knows how much to consume of each good. The next step is to decide .
2.3 Optimal allocation of consumption and labor
The problem in the second stage is established by using (2.2), (2.4) and (2.9):
{∑ ( )}
s.t. (2.11)
Problems such as the one above are most often solved by using either Kuhn-Tucker conditions or by dynamic programming. The results should be the same, of course. I will now show both of these methods. First, the Kuhn-Tucker approach starts by setting up the Lagrangian. Let us go through the steps:
∑ { ( ) ( )} (2.12) FOC:
: (2.13)
: (2.14)
: (2.15)
: (2.16)
From (2.16):
(2.17)
From (2.13):
{
} ( ) {
}
⇒ ( ) {
} (2.18)
From (2.14) and (2.13):
(2.19)
From (2.15) and (2.13):
⇒
(2.20)
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Equations (2.18), (2.19) and (2.20) determine the intertemporal consumption allocation (the Euler equation), the labor-leisure choice and the money demand, respectively. Together, those
equations determine the rational, forward-looking household’s allocation decisions. An alternative approach to derive (2.18)-(2.20) from (2.11) is to use dynamic programming. Point of departure is the observation that the structure of the household’s optimization problem in period is identical to the one in period , , etc. To see this, we first define total financial wealth at the beginning of period as:
Second, rewrite the budget constraint to:
⇒ ( )
Third, assume that the budget constraint holds with equality and solve for :
( ( ) ) (2.21)
Fourth, recast (2.11) into a Bellman equation where is treated as the state variable and as the control variable:
( ) { ( ) ( )} (2.22) Equation (2.22) captures the core idea of dynamic programming, as it already defines a necessary condition any solution to (2.11) has to fulfill. The Bellman equation basically states that the highest obtainable value of the decision problem in period , ( ), is given by the control which maximizies the sum of current period utility and the discounted value of the decision problem next period. The Euler equation for this problem states that the marginal cost of allocating more wealth today is equal to the marginal benefit of allocating more wealth tomorrow. It is written as:
( )
( )
When we plug (2.21) into (2.1), this optimality condition becomes:
( ) (2.23)
The envelope theorem for the problem states that the marginal change in the value function today from a change in total wealth must be equal to the marginal change in today’s utility. This optimality condition is written as:
( ) ( )
When we plug (2.21) into (2.1), the envelope theorem yields:
8
( ) (2.24)
Iterate (2.24) one period forward:
( )
(2.25)
Insert (2.25) into (2.23) and we get the following consumption Euler equation:
(2.26)
Further, we characterize the remaining optimality conditions using (2.21) and (2.1):
: (2.27)
: ( ) (2.28)
From (2.26):
(2.29)
From (2.27):
(2.30)
From (2.28):
(2.31)
Equations (2.29)-(2.31) determine the intertemporal consumption allocation (the Euler equation), the labor-leisure choice and the money demand, respectively. Notice that they are identical to (2.18)-(2.20), highlighting the fact that the household’s optimization problem should have the same solutions regardless of solution method. To proceed we need to specify utility. As an example, consider the following per-period utility function:4
( ) (
)
(2.32)
The marginal utilities of consumption, labor and money become:
( )
The Euler equation given by (2.18) or (2.29) writes:
{( )
} (2.33)
4 Gali (2008) excludes real money balances from the utility function, but instead imposes an ad-hoc log-linearized money demand given by , where is the interest rate elasticity in the money demand equation.
We will see soon that this is equivalent to setting in (2.32).
9
The labor-leisure choice given by (2.19) or (2.30) writes:
(2.34) The money demand equation given by (2.20) or (2.31) becomes:
( )
⇒ ( )
⇒ ( ) (2.35)
Finally, it is convenient to log-linearize (2.33)-(2.35). We denote small letter variables as the log of large letter variables. With respect to the Euler equation, define the following:
Using this, (2.33) can be rewritten to:
[ ( ( ) )] ( ) ( ) It is clear from the equation above that in steady state where . Thus, a first-order Taylor expansion of the Euler equation around steady state yields:
( ) [ ( ) ( ) ( ) ( )]
⇒ ( ) ( )
⇒
⇒ ( ) (2.36)
The linearized version of the labor supply equation (2.34) is:
⇒ (2.37)
Finally, let us linearize the money demand equation given by (2.35):
[ ( ) ]
⇒ ( )
10
⇒ [ ( ) ( )( )] [ ( ) ] ( )
⇒ [ ( ) ] ( )
If we discard the constant term and assume an income elasticity of one, where this assumption implies that , the money demand equation can be written as (2.38), where ( ) :
(2.38)
This ends the analysis of households in the New Keynesian model. We now turn to firms.
11
3. Firms
3.1 Aggregate inflation Assume Cobb-Douglas technology:5
(3.1)
Here, is the output produced by firm in period , is the economy-wide technology level and is the labor force used by the firm. One key ingredient in the New Keynesian model is price rigidity. When firms set their prices, they can do so freely. However, they do not know a priori when the next opportunity to price change emerges. The probability of being unable to change the price in any given period is . Thus, this is the fraction of all firms that is stuck with the price they had last period while the remaining firms reset their prices. The aggregate price dynamics (inflation) in period can be calculated as follows, where is the aggregate price level, is the optimal price set by firms who are able to reoptimize in that period, and ( ) [ ] represent the set of firms not reoptimizing their posted price:
[∫ ( ) ( ) ] [ ( ) ]
⇒ ( ) ([ ( ) ]
)
( ) (
)
⇒ ( ) (
) (3.2)
The aggregate gross inflation is defined as
. Steady state is defined by zero inflation, implying that and . Linearizing (3.2) around steady state yields:6 ( ) [ ( ) ( ) ] ( )( ) ( ) ( )
⇒ ( ) ( )( )( )
⇒ ( )( ) (3.3)
Equation (3.3) makes it clear that inflation results from the fact that firms reoptimizing in any given period choose a price that differs from the economy’s average price in the previous period.
5 The capital stock is treated as fixed and investment is set to zero in the short run. These two specifications follow McCallum and Nelson (1999), who argued that capital do not play a major role in most monetary policy and business cycle analyses.
6 Remember the first-order Taylor expansion: ( ) ( ) ∑ ( ) where is the vector of variables one wants to linearize around. Using this as a point of departure, it is often convenient to define a new variable ̂ as the log deviation in from : ̂
. This implies that ̂ , and the Taylor expansion can be rewritten to a formula for log-linearization via Taylor series expansion: ( ) ( ) ∑ ̂ .
12
Hence, in order to understand inflation over time one needs to analyze the factors underlying firms’ price setting decisions.
3.2 Optimal price setting
Basically, when firms are faced with the problem of setting optimal price today, they must take into consideration that this price often determine profit in the future as well, as the probability of being stuck with today’s price periods ahead is . Thus, a firm who reoptimizes in period will choose the price that maximizes current market value of the profits generated while that price remains effective. The stochastic discount factor for nominal payoffs in period is
, which is given by:7
( )
(3.4)
The representative firm’s maximization problem is thus given by:
{∑ [ ( | | ( | ))]}
s.t. (3.5)
| (
)
Let us spend a couple of seconds on the problem given in (3.5). | is the output in period for a firm that last set its price in period . | ( | ) is the total cost in period as a function of this output. The nominal, undiscounted profit in period is thus
| | ( | ). The firm’s problem is subject to a sequence of demand
constraints as given by (2.10), and market clearing in period implies that the firm produces
| (
) . The problem can be rewritten to an unconstrained one by inserting the constraint into the profit function. We also insert for the discount factor. This gives us:
{∑ [ ( )
( (
) | ((
) ))]
} (3.6)
Let us find the optimal price : ∑ [ ( )
( (
) | ((
) ))]
FOC:
7 Go back to the Euler equation of the consumers to get the intuition.
13
∑ [ ( )
(( ) (
)
| (
)
)]
∑ [ (( ) | | | (
)
)]
∑ [ | (( ) | )]
⇒ ∑ [ | ( | )]
⇒ ∑ ( | ) ∑ ( | | ) Next, we insert for and | and solve for the optimal price :
∑ ( ( )
(
)
)
∑ ( ( )
(
)
| )
⇒ ∑ ( ) ∑ ( | )
⇒ ∑ ( ) ∑ ( |
)
⇒ ∑ ∑ |
(3.7)
Divide both sides by to get the optimal real price as a weighted average of future real marginal costs:8
∑ (
) |
∑ ( )
(3.8) Notice that in the case with flexible prices, i.e. when , (3.6) collapses to a one period
problem and (3.7) becomes:
|
|
| (3.9)
Thus, (3.9) gives the desired or frictionless markup.
3.3 Log-linearization
8 Note that the real marginal cost in period is denoted | |
.
14
The next step is to log-linearize (3.7) around the steady state. In a zero inflation steady state, we must have that:
| |
| | |
The last three identities follow from the zero inflation definition and from market clearing.
Before log-linearizing it is convenient to divide both sides of (3.7) by :
∑ |
∑
⇒ ∑
∑ |
(3.10)
A first-order Taylor expansion of the LHS of (3.10):9
9 This is just a simple first-order Taylor expansion. The first term is the LHS of (3.10) in steady state. The four last terms contain the first derivatives with respect to , , and respectively, all evaluated in steady state.
15
∑
∑ ( )
∑ ( )
∑ ( ) ( )
∑ ( )
( )
∑
∑ ( )
∑ ( )
∑ ( )( )
∑ ( )( )
∑ [ ( ) ( ) ( )( )
( )( )]
∑ [ ( )( )
( )( )]
A first-order Taylor expansion of the RHS of (3.10):10
10 The first term is the RHS of (3.10) in steady state. The four last terms contain the first derivatives with respect to
, , and | respectively, all evaluated in steady state.
16
∑
∑ ( )
∑ ( )
∑ ( ) ( )
∑ ( | )
∑
∑ ( )
∑ ( )
∑ ( )( )
∑ ( | )
∑ [ ( ) ( )
( )( ) ( | )]
∑ [ ( ) ( )( )
( | )]
Finally we equate LHS with RHS and solve for :
∑ [ ( )( ) ( )( )]
∑ [ ( ) ( )( )
( | )]
17
⇒ ∑ ( ) ∑ ( | )
⇒ ∑ [( | ) ]
⇒ ( ) ∑ [( | ) ]
⇒ ( ) ∑ [ | ] (3.11)
We see from (3.11) that firms will set a price that corresponds to the desired markup,11 given by , over a weighted average of their current and expected nominal marginal costs, with the weights being proportional to the probability of the price remaining effective at each horizon
.
11 Because , we have that ( ( ) ) .
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4. Equilibrium
4.1 Market clearing
Market clearing in the goods market implies:
(4.1)
Let aggregate output be defined as:
(∫
) (4.2)
Insert (4.1) into (4.2), and then (2.10) into (4.2), to get the aggregate market clearing condition:
(∫
) {∫ [( )
]
}
(∫
)
( ∫ ) (∫ )
[(∫ ) ]
⇒
Finally, taking logs on both sides yields:
(4.3)
Equation (4.3) is the aggregate market clearing condition. Insert (4.3) into (2.36) and get:
( ) (4.4)
Market clearing in the labor market:
∫ (4.5)
From (3.1) we see that ( ) . Insert this into (4.5) as well as the goods market clearing condition (4.3) and the consumption demand (2.10):
∫ ( ) ∫ ( ) ∫ ((
)
)
( ) ∫ ( ) (4.6) Next we take the log of (4.6):
( ) [∫ ( ) ]
19
⇒ ( ) ( ) [∫ ( ) ] ( ) (4.7)
I will now show that because ∫ ( ) up to a first-order approximation around , but first I must show that ∫ . Recall the consumer price index
(∫ ) . Rearranging gives:
(∫ ( ) ) (∫ ( )( ) )
⇒ ∫ ( )( ) (4.8)
A second order approximation of (4.8) gives us:
∫ [ ( )( ) ( ) ( ) ]
( ) ∫ ( ) ( )
∫ ( )
( ) ( ) ∫ ( )
∫ ( )
⇒ ( ) ( ) ∫ ( ) ∫ ( )
⇒ ∫ ∫ ( ) (4.9)
From equation (4.9) it is also clear that ∫ up to a first-order approximation. Next, let us do a second order approximation of ∫ ( ) :
∫ ( )
∫ [ ( )]
∫
( ) ∫ [ (
) ( ) ]
∫( ) (
) ∫( )
∫ (
) ∫( ) Now, insert (4.9) and get:
20
∫ ( )
(∫
∫ ( ) )
∫
( ) ∫( )
( )
( )∫ ( ) (
) ∫( ) [ ( )
( ) ( ) ] ∫ ( ) ( )( )
( ) ∫ ( ) ( )
( ) ∫ ( )
(4.10) From (4.10) we conclude that up to a first-order approximation, ∫ ( ) . This implies that:
( ) [∫ ( ) ] ( ) Thus, (4.7) can be rewritten to:
( ) (4.11)
4.2 The New Keynesian Phillips curve and the Dynamic IS equation
Next, an expression for individual firms’ marginal cost as a function of the economy’s average real marginal cost is derived. The latter is derived in (4.12), where we insert from (4.11):12
12 The nominal marginal cost by using labor is the wage . The nominal marginal gain is the income increase, that is the price times the marginal increase in production by adding a little more labor. Thus, the real marginal cost is the nominal cost relative to the nominal gain, i.e. . Linearizing gives . It follows from the average production function that marginal productivity is ( ) . Thus, ( ) .
21
( ) ( )
⇒ ( ) (4.12)
Similarly, a firm’s real marginal cost in period is:
| | | ( ) (4.13) Now, the market clearing condition and the demand schedule (2.10) imply that firm output is
| ( |
) , which in linearized terms gives | ( | ) .13 Use this as well as (4.13) and (4.12) to get:
|
[ |
( )]
[
( )]
( | ) [ ( ) ( ) ]
( )
⇒ | ( ) (4.14)
Notice that the last term in (4.14) disappears if there is constant returns to scale, i.e. if . Then | , which implies that the marginal real cost is independent of the production level; it is common across all firms. We shall now derive an expression for inflation.
The point of departure is (3.11), which we rewrite to:
( ) ∑ [ | ] Insert (4.14):
( ) ∑ [
( ) ]
( ) ∑ ( ̂
)
⇒ ( ) ∑ ( ̂ )
⇒ ( ) ∑ ( ̂ )
13 Note that | .
22
Define and subtract ( ) on both sides.:
( ) ∑ ( ̂ )
( ) ∑ ( ̂ )
( ) ∑ ̂
( ) ∑ ( )
( ) ∑ ̂
( ) [ ( ) ( ) ( ) ]
( ) ∑ ̂
( ) [ ( ) ( ) ] ( ) ∑ ̂
[ ( ) ( ) ] [ ( ) ( ) ] ( ) ∑ ̂
[ ]
( ) ∑ ̂
∑
If we take out the terms of each summation operator, the equation can be written more compactly as a difference equation:
( ) ∑ ̂
∑
( ) ̂
[( ) ∑ ̂
∑
] ( ) ̂ ( ) ( ) ̂ Next we insert (3.3) and solve for inflation ( )( ):
( ) ( ) ̂ ( )( )
23
⇒ ( ) ( ) ( ) ̂
⇒ ( ) ( ) ̂
⇒ ( )( ) ( ) ( ) ( )( ) ̂
⇒ ̂ (4.15)
Equation (4.15) expresses inflation as the sum of (discounted) expected inflation and real marginal costs, and we have defined ( )( ) ( )( )
to ease the notation. It is clear from (4.15) that inflation is strictly decreasing in price stickiness , in the measure of decreasing returns , and in the demand elasticity . An alternative presentation of inflation is found by solving (4.15) forward:
{ [ ( ̂ ) ̂ ] ̂ } ̂ ∑ ̂
Equivalently, and defining the average markup in the economy as – , we see that inflation will be high when firms expect average markups to be below their steady state or desired level – . In that case firms that have the opportunity to reset prices will choose a price above the
economy’s average price level in order to realign their markup closer to its desired level. Thus, in the present model, inflation results from the aggregate consequences of purposeful price-setting decisions by firms, which adjust their prices in light of current and anticipated cost conditions.
Next, a relation is derived between the economy’s real marginal cost and a measure of aggregate economic activity. We have derived earlier that ( ) . Insert this into ( ) , and use that ( ) :
( ) ( ) [ ( ) ] ( ) [ ( )]
Insert for and get to:
( ) ( ) (4.16)
In the case with flexible prices we know from before that . Define natural output level as the equilibrium level under full price flexibility. In this case (4.16) can be rewritten to:
( ) ( ) (4.17)
Solve (4.17) for natural output:
( )
( )
⇒ ( ) [ ( )] ( ) ( )[ ( )]
( )
24
⇒ (4.18)
If we subtract (4.17) from (4.16) we get a measure of the real marginal cost gap ̂ as a function of the output gap from natural output, denoted ̃ :
̂
[ ( )
( )]
[ ( )
( )]
( )
( )
⇒ ̂ ( ) ̃ (4.19)
Finally, the New Keynesian Phillips curve is established by inserting (4.19) into (4.15):
̂ ( ) ̃
⇒ ̃ (4.20)
The New Keynesian Phillips curve (NKPC) is one of the key building blocks of the New
Keynesian model, and the parameter is defined by ( ) . The second key equation is the dynamic IS equation. If we use the definition of the real interest rate, , equation (4.4) becomes ( ). In a similar vein, the natural output is given as a function of the natural interest rate:
( ) (4.21)
Subtracting (4.21) from (4.4) gives the output gap from the natural output, i.e. the dynamic IS equation (DIS):
̃ [ ( )] [ ( )]
⇒ ̃ ̃ ( ) (4.22)
Equations (4.20) and (4.22) together with an equilibrium process for the natural rate , which in general will depend on all exogenous forces in the model, constitute the non-policy block of the basic New Keynesian model. That block has a simple recursive structure: The NKPC determines inflation given a path for the output gap, whereas the DIS equation determines the output gap given a path for the exogenous natural rate and the actual real rate. To see the latter, assume the transversality condition . Then one can solve (4.22) forward to yield:
̃ ∑ ( ) (4.23)
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Equation (4.23) emphasizes the fact that the output gap is proportional to the sum of current and anticipated deviations between the real interest rate and its natural counterpart. To gain further insight into the natural interest rate, note first that (4.4) implies ( ).
Second, note that the first difference of (4.18) gives . Now, solve (4.22) for and use these two observations to yield an expression for the natural real rate. From (4.22):
( ̃ ̃ ) [ ( ) ( )]
[ ]
[ ( ) ]
⇒ (4.24)
Thus, the natural real rate is a function of households’ discount rate and expected technological progress. In some cases it is convenient to work with deviations in the natural real rate from the discount rate, which we define as:
̂ (4.25)
Note that if one turns off technology shocks, the real rate becomes the discount rate. Once a process for the technological progress is specified, one can identify the real interest rate path in (4.24). In order to close the model, we supplement (4.20) and (4.22) with one or more equations determining how the nominal interest rate evolves over time, i.e. with a description of how monetary policy is conducted. Observe from (4.23) that the equilibrium path of real variables cannot be determined independently of monetary policy when prices are sticky. The output gap is directly determined by the real interest rate gap, which is directly determined by the nominal interest rate set by central banks. This important feature of the New Keynesian Model is in contrast to classical models where monetary policy is neutral.
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5. Equilibrium determinacy
5.1 System representation
Throughout this and the next sections I will look at a specific monetary policy rule, more specifically an interest rate rule. I assume the central bank follows a rule of the form:
̃ (5.1)
Standard reasoning implies that and are non-negative, which we assume from now on.
The first task when analyzing monetary rules is to check whether the specified policy yields a unique and stable equilibrium. While doing this, it is convenient to work with a reduced form representation of (4.20) and (4.22) who takes into account the policy rule under consideration.
We first derive a forward looking version of the dynamic IS equation. Insert (5.1) into (4.22), and then (4.20) into (4.22). Solve the resulting equation for ̃ :
̃ ̃ ( ) ̃ ( ̃ ) ̃ [ ( ̃ ) ̃ ̂ ]
̃
̃ ̂
⇒ ̃ ̃ ̂
⇒ ̃ [ ̃ ( ) ( ̂ )] (5.2) Equation (5.2) shows the current output gap as a function of expected output gap, expected inflation, and shocks. We next achieve a similar representation of current inflation. Insert (5.2) into (4.20) and get:
̃
{
[ ̃ ( ) ( ̂ )]}
̃ ( ) ( )
( ̂ )
̃ ( )
( ̂ )
⇒ { ̃ [ ( )] ( ̂ )} (5.3)
