Basic New Keynesian Model docx

Basic New Keynesian Model

Notes from "Monetary Policy, Inflation, and Business Cycle:

an Introduction to the New Keynesian Model", Jordi Galì (2008)

Irina Belousova Università Politecnica delle Marche May 2, 2016

1 New Keynesian Model

1.1 Households optimal behaviour We assume in this model the presence of a retail sector that uses homoge-

neous wholesale good to produce a basket of differentiated goods for consumption

where ǫ is the elasticity of substitution. For each i, the consumer chooses

C t (i) at a price P t (i) to maximize (1) given total expenditure

P t (i)C t (i)di = z

Note, it is assumed the existance of a continuum of goods represented by the interval [0, 1]. This optimization problem is solved in the following way

P t (i)C t (i)di

ǫ −1 ∂L −1 ǫ

[C t (i)] ǫ − λP t (i) = 0 ∂C t (i)

Put in relation good i with good j

t (i)

λP t (j)

P t (j)

C t (j)

λP t (i)

C ǫ P t (i)

t (j)

C t (i) = C t (j)

P t (i)

1 New Keynesian Model

We can now construct the new equation for total expenditure, substituting the latter expression for C t (i)

Z 1 P t (i)C t (i)di = z

t (i)C t (j)P t (j) P (i) −ǫ t di = z

C (j)P (j) ǫ

where P 1−ǫ t = 0 P t (i) di is the aggregate price index. C t and P t are Dixit-Stigliz aggregates, see [2].

zP t (i) Similarly, we have the expression for good i, that is C −ǫ t (i) = P t 1−ǫ

Subsitute this expression back into the aggregate consumption index

therfore, z = C t P t , and the new equation for total expenditure becomes

P t (i)C t (i)di ≃ C t P t

1 New Keynesian Model

Now, substitute for z into equation (3)

we,thus, end up with the equation of relative demand for the differentiated good i with price P t (i). Next, we assume that household maximizes the utility function, given the period budget constraint

max E 0 β U (C t ,N t )

Z t=0

s.t. P t (i)C t (i) + Q t B t ≤B t−1 +W t N t +T t

Assuming the form of utility function as follows

the Lagrangian becomes ∞ "

The first order conditions with respect to C t and N t are, respectively ∂L

hence, marginal rate of substitution between consumption and labour is

U N,t

U C,t

t C −σ β =

C −σ

1 New Keynesian Model

Next, we carry out the Euler equation by deriving the Lagrangian with respect to C t and B t−1 . Re-express Lagrangian in the following form

+λ t+s (B t+s−1 +W t+s N t+s +T t+s −P t+s C t+s −Q t+s B t+s ) The first derivative with respect to C t is

∂L =U C,t+s −λ t+s P t+s =0 ∂C t

U C,t+s =λ t+s P t+s β t+s C t+s −σ =λ t+s P t+s

these, for different values of s, become t U −σ

C,t =β C t =λ t P t , if s = 0

U C,t+1 =β t+1 C t+1 −σ =λ t+1 P t+1 , if s = 1

Next, the first derivative with respect to B t−1 is

∂L

=λ t+s −λ t+s−1 Q t+s−1 =0 ∂B t−1

λ t+s =λ t+s−1 Q t+s−1 λ t+1 =λ t Q t

where the last equality is true for s = 1. Combining λ t+1 and λ t , we obtain

U C,t+1 =λ t Q t P t+1

U C,t+1

U C,t

P t+1

ending up with the Euler equation

P t Q t =E t

U −σ

C,t+s P t

P t+1 The next step is to log-linearize equations (6) and (7). Start with the

U C,t P t+1

first one

−σ C =

ϕ ln N t + σ ln C t = ln W t − ln P t

1 1 1 1 ϕ (N t −N)+σ (C t − C) =

(W t −W)− (P t −P) N

ϕn t + σc t =w t −p t

1 New Keynesian Model

Note that, the left and right-hand sides of the equation calculated in the mean values have been omitted, since cancelling each other. The same is true for log-linearization of the Euler equation

ln Q t = ln β − σ(ln C t+1 − ln C t ) + ln P t − ln P t+1

1 1 1 1 ln Q t = ln β − σ

(C t+1 − C) − (C t − C) + (P t −P)− (P t+1 −P)

P ln Q t = ln β − σc t+1 + σc t +p t −p t+1 σc t = σc t+1 − [− ln Q t + ln β − (p t+1 −p t )]

c t =E t {c t+1 }− (i t −ρ−E t {π t+1 }) (9) σ

where i −1 t = − ln Q t = ln(1 + yield) −1 is the nominal interest rate, ρ = − ln β is the household’s discount rate, and π t =p t −p t−1 is the inflation rate.

1.2 Firms optimal behaviour Before proceedeing, let us list the main assumptions undertaken here: there

is a continuum of firms indexed by i ∈ [0, 1] ; firms produce differentiated goods; technology, A t , is identical for each firm and evolves exogenously over time; each firm deal with an identical isoelastic demand, given in (5); aggregate price index, P t , and aggregate consumption index, C t , are given; following the formalism proposed in [1], each firm may reset its price only with probability (1 − θ) in any given period, while a fraction θ keep their

prices unchanged; therefore, we can interpret 1 1−θ as the average duration of

a price, and θ becomes a natural index of price stickiness. This setting leads to inflation, that can be formalized as follows: assume S(t) ⊂ [0, 1] be the set of firms that not reoptimize their price, and that all firms resetting price will choose an identical price P ∗ t . Then, the aggregate price level can be re-expressed in the followin way

(P t−1 (i)) di + (1 − θ)(P t )

S(t)

= θP 1−ǫ t−1 + (1 − θ)(P ∗ t ) 1−ǫ 1−ǫ

1 New Keynesian Model

Therefore, (P 1−ǫ t ) = θP 1−ǫ

t−1 + (1 − θ)(P t )

Log-linearizing around the zero inflation steady state, that is P ∗ t = P t−1 =P t = P , and Π t = Π = 1, and so in steady state 1 1−ǫ =θ+ (1 − θ) · 1 1−ǫ , we have

Thus, the equation for inflation becomes:

(10) this results from the fact that, firms reoptimizing each period, choose

π t = (1 − θ)(p ∗ t −p t−1 )

a price that differs from the economy’s average price of the previous period. Therefore, a firm reoptimizing in period t will choose a price P ∗ t that maximizes the current market value of the profits, in light of current and anticipated cost conditions. The optimization problem, given the sequence of demand constraints, is as follows

X ∞ max

t+k|t P ) t Y t+k|t −Ψ t+k (Y

E t Q t,t+k P

k=0

P −ǫ ∗

s.t. Y

t+k|t =

C t+k

P t+k

is the stochastic discount factor for nominal payoffs, Ψ t (·) is the cost function,

kU C,t +k P t

C t +k −σ

for k = 0, 1, 2. . . , where Q t,t+k =β

U C,t P t +k =β

P t +k

1 New Keynesian Model

in t. What follows is the derivation of the first order condition associated with the problem above. Substitute the constraint into the objective function:

X ∗ −ǫ

max ∗ θ k

E t Q t,t+k P ∗

C t+k −Ψ t+k (Y t+k|t )

−ǫ C t+k −Ψ t+k (Y t+k|t P ) t (P t+k )

θ ∗ k E t Q t,t+k

k=0

∂Ψ t+k ∂Y t+k|t ∗ =

∂ ∞ X (P ∗ ) −ǫ

θ E t Q t,t+k (1 − ǫ) ∂P

C t+k −

(P t+k ) −ǫ

∂Y t+k|t ∂P ∗ t

t k=0

X ∞ (P ∗ ) −ǫ−1 =

θ k E t t Q t,t+k (1 − ǫ)Y t+k|t −ψ t+k|t · (−ǫ) −ǫ C t+k

θ E t Q t,t+k (1 − ǫ)Y t+k|t −ψ t+k|t · (−ǫ) ∗

θ k E t Q t,t+k (1 − ǫ)Y t+k|t −ψ t+k|t · (−ǫ)

Y t+k|t

k=0

= θ k E t Q t,t+k Y t+k|t · P ∗ t −

·ψ t+k|t

k=0

θ k E t ∗ Q t,t+k Y t+k|t ·P t −M·ψ t+k|t =0 (11)

k=0

∂Ψ t where ψ +k t+k|t = ∂Y t +k|t is nominal marginal cost in t + k for a firm that last reset its price in period t, and M = ǫ ǫ−1 is the desired or frictionless mark-up. Note that, if there are no price rigidities, that is θ = 0, the first order condition collapses to the optimal price-setting condition under flexible

prices

P ∗ t = Mψ t|t

which allows us to interpret M as the desired mark-up in the absence of constraints on the frequency of price adjustment.

P t Now, devide the first order condition found above by P +k t−1 , call Π t,t+k = P t

1 New Keynesian Model

and re-express it as follows

θ ∗ E t Q t,t+k Y t+k|t ·P t −M·ψ t+k|t

k=0

ψ t+k|t P t+k

θ E t t Q t,t+k Y t+k|t · −M· =0

θ E t Q t,t+k Y t+k|t · −M·MC t+k|t Π t−1,t+k =0 (12)

P t−1

k=0

ψ t where M C +k|t t+k|t = P t +k is the real marginal cost in period t + k for a firm whose price was last set in period t.

P t Note that, in steady state, the following relations hold: ∗

= 1 and Π

P −1 t−1,t+k

t =P t+k in steady state, from which follows that Y t+k|t = Y and M C t+k|t = M C, that is, all firms will produce the same level of output. This implys M C = 1 M under

= 1, moreover, if the price level is constant, P ∗

constancy of price level. Moreover, recalling the Euler equation in (7), that is

C t+k

t,t+k =β E t

P t+k

we find that in steady state it reduces to Q t,t+k =β k . Hence, the first-order Taylor expansion of (12) around the zero inflation

steady state is as follows

θ E t Q t,t+k Y t+k|t ·

−M·MC t+k|t Π t−1,t+k

E t k {Q t t,t+k Y t+k|t · }= θ E t {Q t,t+k Y t+k|t ·M·MC t+k|t Π t−1,t+k }

θ k β k Y (p ∗ t −p t−1 )= θ k β k Y ln M + mc t+k|t +p t+k −p t−1

k=0

k=0 ∞

(p t −p t−1 ) = (1 − βθ) (βθ) ln M + mc t+k|t +p t+k −p t−1

k=0

were the last expression makes use of ∞ L k k=0 = 1−L k 1−L = 1 1−L that is true

for L < 1, similarly, (1 − βθ) k=0 (βθ) k = 1. Therefore, the last expression becomes

t = ln M + (1 − βθ) (βθ) k E t {mc t+k|t +p t+k }

1 New Keynesian Model

If we define mc \ t+k|t ≡ mc t+k|t +ln M = mc t+k|t −mc, as the log deviation of marginal cost from its steady state value, mc = − ln M, then the firm’s price-setting decision can be expressed as

p t −p t−1 = (1 − βθ) (βθ) k E t { mc \ t+k|t + (p t+k −p t−1 )} (13)

k=0

Firms resetting their prices will choose a price that corresponds to the desired mark-up, ln M, over a weighted average of their current and expected nominal marginal costs. The weights are proportional to the probability of the price remaining effective at each horison, θ k .

1.3 Market clearing conditions In the market of goods the condition is that Y t (i) = C t (i), that is, all the

output must be consumed, for all i ∈ [0, 1] and t. This implys that the aggregate output can be expressed as follows

and that aggregate output equals aggregate consumption, Y t =C t . There- fore, it can be written in terms of Euler equation

y t =E t {y t+1 }− (i t −E t {π t+1 } − ρ)

σ and of equilibrium equation of relative demand for goods, recalling (5)

R 1 As to the labour market, clearing condition requires N t = 0 N t (i)di.

Assuming a Cobb-Douglas production function, that is

(17) re-express it for N t (i) and substitute into the clearing condition, as fol-

Y 1−α

t (i) = A t N t (i)

lows

t (i) 1−α

N t (i) =

t (i) 1−α

1 New Keynesian Model

Using the last expression, substitute for Y t (i) from equation (16)

We can now log-linearize this new expression for labour, as shown below

(1 − α) (N t − n) = (Y t −Y)− (A t − a) + (1 − α) ln di n

we, thus, end up with

(1 − α)n t =y t −a t +d t R ǫ 1

P t (i) − 1−α

di can be ignored, as will be shown right below, so that the log-linearized labour equilibrium condition becomes

where d t = (1−α) ln 0 P t

(1 − α)n t =y t −a t y t −a t

1−α In fact, in the expression for d t , di is a measure of price dispersion across

firms. In a neighborhood of zero inflation steady state d t approaches zero up to a first order Taylor approximation. The demonstration follows imme- diately. Recall

We must analyze 0 P t

di, in order to do so, start with the

1 New Keynesian Model

re-definition of price index

ln Pt(i)

1 1= t exp (1−ǫ)[ln P (i)−ln P t ] di

Recall that Taylor expansion up to second order is f (x) = f (¯ x)+f ′ (¯ x)(x− x) + ¯ 1 2 f ′′ (¯ x)(x − ¯ x) 2 , and that in steady state P t (i) = P t = P and ln P t (i) = ln P t = ln P . Hence, we calculate all the terms needed for the Taylor expansion

f (¯ x) = exp (1−ǫ)[ln P t −ln P t ] di =

1 di = |i| 1 0 =1

0 hR 0

∂ exp (1−ǫ)[ln P t (i)−ln P t ] di

(x) = 0

exp (1−ǫ)[ln P t (i)−ln P t ] di · (1 − ǫ) ∂ ln P t (i)

hR 0

exp (1−ǫ)[ln P t (i)−ln P t 0 ] di

f t ′ t (¯ x) = = (1 − ǫ) exp (1−ǫ)[ln P −ln P ] di = (1 − ǫ)

∂ ln P t (i)

hR 1 (1−ǫ)[ln P t (i)−ln P t

P t (i)=P t

(1−ǫ)[ln P t (i)−ln P f t (x) =

] di · (1 − ǫ) 2 ∂ [ln P t (i)]

exp

hR 0

∂ 2 (1−ǫ)[ln P t (i)−ln P t 0 ] exp di

′′ 2 (1−ǫ)[ln P t −ln P t ]

f 2 (¯ x) =

di = (1 − ǫ) ∂ [ln P t (i)]

exp

P t (i)=P t

Therefore, the Taylor expansion becomes Z 1

(1−ǫ)[ln P t (i)−ln P t 1= ] exp di

ln P t (i)di − ln P t +

[ln P t (i) − ln P t ] di = 0

0 2 0 1−ǫ Z 1 2

ln P t =E t {ln P t (i)} +

[ln P t (i) − ln P t ] di = 0

where E t {ln P t (i)} = 0 ln P t (i)di is the cross-sectional mean of log

1 New Keynesian Model

Note that

Now, go back to the expression for d t in equation (19), and Taylor-expand up to the second order the term under the integral

P (i) 1−α

1 h Pt(i) ǫ i − 1−α

− ǫ [ln P t (i)−ln P t di = ] exp Pt di = exp 1−α di

[ln P t (i) − E i {ln P t (i)}] di

· var i {ln P t (i)} > 1

2 1−α 1−α Finally, "Z

d t = (1−α) ln

di = (1−α) ln 1+

· var i {ln P t (i)}

2 1−α 1−α Note that ln(1 + n) ≃ n, therefore the last expression becomes

· var i {ln P t (i)} =

· var i {ln P t (i)}

1.4 Equilibrium Let us, now, turn back to the economy’s equilibria definition. First, carry

out the average marginal product of labour of the economy, mpn t , starting

1 New Keynesian Model

from the Cobb-Douglass production function Y

∂N t (i) ln M P N t = ln A t + ln(1 − α) − α ln N t (i) mpn t =a t − αn t + ln(1 − α)

Substitute for n t from equation (18), so that

Next, we find the real marginal cost, mc t+k|t , in t + k for a firm whose price was last set in t, and substitute it in the equation for price setting decision. Hence, recall equations (13), (18), and (20). First, note that marginal cost of an individual firm in terms of the economy’s average real marginal cost can be expressed as

mc t = (w t −p t ) − mpn t

subsituting for mpn t from equation (20), we find

The same is done for the marginal cost af a firm whose price was last set in period t

mc t+k|t = (w t+k −p t+k ) − mpn t+k|t

1 αy t+k = (w t+k −p t+k )−

(a t+k − αy t+k|t ) − ln(1 − α) ±

y t+k|t −

y t+k − ln(1 − α)

y t+k|t −

(y t+k|t −y t+k )

1 New Keynesian Model

Using, the market clearing condition C t =Y t , define the output as follows

ln Y t (i) − ln Y t = −ǫ(ln P t (i) − ln P t )

(Y t (i) − Y ) − (Y t − Y ) = −ǫ

(P t (i) − P ) − (P t −P)

y t (i) − y t = −ǫ(p t (i) − p t )

Thus, the real marginal cost, mc t+k|t , becomes

mc t+k|t = mc t+k +

(y t+k|t −y t+k )

t+k|t = mc t+k +

[−ǫ(p t −p t+k )]

mc t+k|t = mc t+k −

(p ∗ −p t+k )

1−α t

Note that, under the assumption of constant returns to scale, α = 0, we obtain mc t+k|t = mc t+k , that is, marginal cost is independent of the level of production and, hence, it is common across firms. Finally, substitute the last expression for mc t+k|t into equation (13) of optimal price-setting

p t −p t−1 = (1 − βθ)

(βθ) k E t mc t+k|t + ln M + (p t+k −p t−1 )

(βθ) E t mc t+k −

(−αǫ)(p t −p t+k ) + (1 − α)(p t+k −p t−1 ) p t −p t−1 = (1 − βθ)

(βθ) E t mc t+k +

(βθ) k E t mc t+k +

(βθ) t k t−1 E t mc t+k + ln M + p t+k 1−α

t−1

k=0

1 New Keynesian Model

(βθ) E t mc t+k + ln M +

(βθ) k E t

If we add (−p t−1 ) on both the sides of the equation, we end up with

where 1−α 1−α+αǫ ≤ 1, and mc \ t+k = mc t+k + ln M. Note that, the latter expression can be rewritten as

1−α X k

p t −p t−1 = (1 − βθ)

(βθ) E t mc \ t+k + (1 − βθ)

(βθ) E t {(p t+k −p t−1 )}

= (1 − βθ) (βθ) E t mc \ t+k + (1 − βθ)E t [(βθ) 0 (p t −p t−1 )+

1 − α + αǫ k=0

+ βθ(p t+1 −p t +p t −p t−1 ) + (βθ) 2 (p t+2 −p t−1 +p t+1 −p t +p t −p t−1 )+···]=

1−α ∞ X

= (1 − βθ) (βθ) E t mc \ t+k + (1 − βθ)E t [(βθ) π t + βθ(π t+1 +π t )+

1 − α + αǫ k=0

+ (βθ) 2 1−α X k

(π t+2 +π t+1 +π t ) + · · · ] = (1 − βθ) (βθ) E t mc \ t+k +

1 − α + αǫ k=0 +E [(βθ) 0 t π t + βθ(π t+1 +π t ) + (βθ) 2 (π t+2 +π t+1 +π t ) + · · · ]+

−E [(βθ) 1 π + (βθ) t 2 t (π t+1 +π t ) + (βθ) 3 (π t+2 +π t+1 +π t )+···]=

1−α ∞ X

= (1 − βθ) (βθ) k E 0 t 2 mc \ t+k +E t [(βθ) π t + βθπ t+1 + (βθ) π t+2 +···]=

(βθ) E t mc \ t+k +

(βθ) E t {π t+k }

1 − α + αǫ k=0

k=0

We now want to express the above discouted sum in terms of a difference equation. To do so, forward one period the first-order stochastic difference

1 New Keynesian Model

(βθ) E t { mc \ t+k }+

(βθ) k E t {π t+k } = (1 − βθ)

(βθ) k E t { mc \ t+k+1 }+

(βθ) k E t {π t+k+1 } =

(22) Recall, now, the log linearized expression for inflation given in (10), that

t +π t + βθ(p

π t = (1 − θ)(p ∗ t −p t−1 )

and substitute in it the expression for p ∗ t −p t−1 given in (22) π t

We can clearly see in the above equation for inflation that it is strictly decreasing in the index of price stickiness θ, in the measure of decreasing returns α, and in the demand elasticity ǫ. Solving equation (23) forward, we can express inflation as discounted sum of current and expected future deviations of real marginal costs from steady state

E t {π t+k+1 }

1 − α + αǫ k=0

1 New Keynesian Model

where lim k→+∞ β k+1 E t {π t+k+1 } = 0, so that the last term on the right- hand side of the equation disappears. We can conclude that inflation will be high when firms expect average markups to be below their steady state (i.e. desired) level (mc t = ln M), for 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. Moreover, note that, in this model setting inflation results from the aggregate consequences of price-setting decisions by firms, which adjust their prices in light of current and anticipated cost conditions.

Next, we need to derive an expression for real marginal costs mc d t , that ap- pear in equation of inflation (23). For this purpose, recall the equation of economy’s marginal cost (21), and substitute in it the equation of optimal labour supply (8), as follows

a t − αy t

mc t = ϕn t + σc t − − ln(1 − α)

Hence, use the log-linearized relation between aggregate output, employment and technology (18), and the goods market clearing condition (y t =c t ), to obtain

a t − ln(1 − α) (24)

that is the average real marginal cost of the economy. Recalling that under flexible prices the real marginal cost is constant,

mc = − ln M, define the natural level of output y n t , as the equilibrium level of output under flexible prices

a t − ln(1 − α)

[−mc − ln(1 − α)] α + ϕ + (1 − α)σ 1−α

1+ϕ (1 − α) [ln M − ln(1 − α)] =

(27) where ψ n

n t =ψ ya a t +ϑ y

> 0. Subtract (25) from (24)

ya = α+ϕ+(1−α)σ and ϑ y =−

(1−α)[ln M−ln(1−α)]

t −y t ) (28)

1 New Keynesian Model

that is, the log deviation of real marginal cost from its steady state is proportional to the log deviation of output from its natural level with flexible prices, named output gap (˜ y t ≡y t −y n t ).

Finally, equation (28) is used to remodel equation (23) (1 − θ)(1 − βθ) 1−α

or, simply

(29) where k = (1−θ)(1−βθ) 1−α

π t = βE t {π t+1 }+k˜ y t

1−α +σ This equation is the New Key- nesian Phillips Curve (NKPC), that relates inflation to its one period ahead forecast and the output gap, and represents one of the key equations of the New Keynesian model. The second key equation is derived by expressing equation (15) in terms of output gap, ˜ y t , given a path for exogenous natural rate, r n t , and the actual real rate, i t , as follows

The last two terms on the right hand-side of the latter expression can

be substituted by the natural rate of interest, r n t , as is shown below. Recall the definition of real interest rate as the expected real return on one period bond r t =i t −E t {π t+1 }, and substitute it in equation (15) to get

1 y t =E t {y t+1 }− (r t − ρ) σ

r t = σ (E t {y t+1 }−y t )+ρ = σ (E t {∆y t+1 }) + ρ

Similarly, the natural rate of interest becomes

t =σE t {∆y t+1 } +ρ

= σψ n ya E t {∆a t+1 }+ρ

1 New Keynesian Model

Therefore, the expression for output gap becomes

y ˜ t =E t {˜ y t+1 }− (i

t −E t {π t+1 }−r t ) (31)

The equation above is the Dynamic IS equation, DIS, that determines the output gap given a path for the (exogenous) natural rate and the actual real rate. The forward solution of DIS is

t+k+1 −r

t+k +E t {˜ y t+k+1 }

k=0

1 ∞ X =−

r t+k −r n t+k (32) σ k=0

where the last equation results from the assumption that the effects of nominal rigidities vanish asymptotically, lim k→∞ E t {˜ y t+k+1 } = 0. Equation (32) shows that output gap is proportional to the sum of current and anticipated deviations between the real and the natural real interest rates. Equations (29), (31) and (30) constitute the non-policy block of the basic New Keynesian model.

Having assumed in this model that prices are sticky, the Monetary policy is non-neutral, that is, the equilibrium path of real variables cannot be determined independently of monetary policy. Thus, in order to close the model, we need one or more equations determining how the nominal interest rate, i t evolves over time, i.e. how monetary policy is conducted.

1.5 Equilibrium under an Interest Rate Rule

Assume the simple interest rate rule of the form

i t =ρ+φ π π t +φ y y ˜ t +ϑ t

1 New Keynesian Model

where ϑ t is exogenous and possibly stochastic with E t {ϑ t } = 0. And assume that the monetary authority has the possibility to set the (non- negative) values for both the coefficients φ π , and φ y . In order to find the equilibrium conditions, we must solve the following system of difference equations, using (29), (31) and (33)

  π t = βE t {π t+1 }+k˜ y t   

    i t =ρ+φ π π t +φ y y ˜ t +ϑ t

Substitute π t and i t into ˜ y t

[(r t n −ρ)−ϑ t y ] ˜ = σ  t

σ+φ π k+φ y E t {˜ y t+1 }+ σ+φ π k+φ y E t {π t+1 }+ σ+φ π k+φ y       π = βE {π }+

k[(r n t −ρ)−ϑ t ] t

kσ

k(1−φ π β)

t t+1

σ+φ π k+φ y E t {˜ y t+1 }+ σ+φ π k+φ y E t {π t+1 }+ σ+φ π k+φ y

1 New Keynesian Model

[(r t n −ρ)−ϑ t y ] ˜ t = σ+φ π k+φ y E t {˜ y t+1 }+ σ+φ π k+φ y E t {π t+1 }+ σ+φ π k+φ y      

k[(r −ρ)−ϑ t  ] t n π t =

kσ

k+β(φ y +σ)

σ+φ π k+φ y E t {˜ y t+1 }+ σ+φ π k+φ y E t {π t+1 }+ σ+φ π k+φ y

We can write the above system in the following way 

 E t {˜ y t+1 } 

 n =A

+B T [(r t − ρ) − ϑ t ] (34) π t

Given that both the output gap and inflation are non-predetermined variables, the system (34) will have a unique local solution, if and only if, matrix A t has both eigen values (λ y ,λ π ) within the unit circle. The eigen values of a matrix are defined by A − λI = 0, where I is the identity matrix, or, furthermore, by (λ y − 1)(λ π − 1) > 0, that is λ y λ π − (λ y +λ π ) + 1 > 0, or, similarly, det(A) − trace(A) + 1 > 0, with λ y λ π = det(A) and λ y +λ π = trace(A), i.e. sum of the elements on the main diagonal.

1 New Keynesian Model

In our case, we have σ [k + β(σ + φ y )] − σk(1 − βφ π )

det(A) =

βσ(σ + φ π k+φ y ) = (σ + φ π k+φ y ) 2

βσ = σ+φ π k+φ y

σ + k + βσ + βφ y

trace(A) =

σ+φ π k+φ y

Therefore, det(A) − trace(A) + 1 > 0 becomes

σ + k + βσ + βφ y

+1>0 σ+φ π k+φ y

σ+φ π k+φ y

βσ − σ − k − βσ − βφ y +σ+φ π k+φ y >0 k(φ π − 1) + φ y (1 − β) > 0

(35) Expression (35) is referred to as Taylor Principle and represents a nec-

essary and sufficient condition for unique solution.

1.5.1 The effects of a monetary policy shock We will now analyze the response of economy’s equilibrium to an exogenous

monetary policy shock, under the interest rate rule given in (33) . Assume the exogenous component,ϑ t , in the interest rate rule follows an AR(1) process

(36) where ρ ϑ ∈ [0, 1), and such that, a positive realization of ε ϑ t means a

+ε t ϑ ϑ t−1 t

contractionary monetary policy shock with a consequent rise in the nominal interest rate; similarly, a negative realization of ε ϑ t means an expansionary monetary policy shock with a consequent decline in the nominal interest rate, given inflation and the output gap. We assume that the natural interest rate, r n

t , is not affected by monetary policy, so that we can set r t − ρ = 0 for all t. And we guess that the solution takes the form ˜ y t =ψ yϑ ϑ t , and π t =ψ πϑ ϑ t ,

where ψ yϑ and ψ πϑ are the coefficients to be determined using the method of undetermined coefficients.

Hence, start by substituting (31) and (33) into (29), in order to find an

1 New Keynesian Model

expression for π t in terms of E t {π t+1 } and E t {˜ y t+1 }

1 t = k (π t − βE t {π t+1 }), and substitute it into the latter expression

Explicit (29) for ˜ y t , that becomes ˜ y

σ + kφ π +φ y (37)

Now, we find a relation for y t in terms of E t {π t+1 } and E t {˜ y t+1 }. Sub- stitute (33) into (31), as follows

y ˜ t =E t {˜ y t+1 }− [φ π π t +φ y y ˜ t +ϑ t −E t {π t+1 } − (r n t − ρ)]

1+ y ˜ t =E t {˜ y t+1 }− (φ π π t +ϑ t −E t {π t+1 }) σ

In the latter expression substitute (29) for π t , so that φ y

σ+φ y + kφ π (38)

1 New Keynesian Model

Now, recall the form of the solutions we guessed, that is

˜ y t =ψ yϑ ϑ t π t =ψ πϑ ϑ t

Forward them one period, and take the expectations, as follows E t {˜ y t+1 }= ψ yϑ ϑ t+1 , and E t {π t+1 }=ψ πϑ ϑ t+1 . In the same manner, forward one period equation (36), that becomes ϑ

t+1 =ρ ϑ ϑ t , being E t {ε t } = 0. Therefore, we end up with

E t {˜ y t+1 }=ψ yϑ ρ ϑ ϑ t

E t {π t+1 }=ψ πϑ ρ ϑ ϑ t

Use these four expressions into equations (37) and (38). The first one becomes

σ + kφ π +φ y −ρ ϑ (βσ + k + βφ y ) σ + kφ π +φ y −ρ ϑ (βσ + k + βφ y ) (39)

and the latter becomes σ

(40) σ+φ y + kφ π − σρ ϑ

σ+φ y + kφ π − σρ ϑ

Combine equations (39) and(40)

kσρ ϑ

ψ πϑ + σ + kφ π +φ y −ρ ϑ (βσ + k + βφ y ) σ+φ y + kφ π − σρ ϑ

kσρ ϑ

+ σ + kφ π +φ y −ρ ϑ (βσ + k + βφ y ) σ+φ y + kφ π − σρ ϑ

− σ + kφ π +φ y −ρ ϑ (βσ + k + βφ y )

1 New Keynesian Model

[σ + kφ π +φ y −ρ ϑ (βσ + k + βφ y )] (σ + φ y + kφ π − σρ ϑ ) − kσρ 2 ϑ (1 − βφ π )

ψ πϑ = [σ + kφ π +φ y −ρ ϑ (βσ + k + βφ y )] (σ + φ y + kφ π − σρ ϑ )

−kσρ ϑ − k (σ + φ y + kφ π − σρ ϑ ) = [σ + kφ π +φ y −ρ ϑ (βσ + k + βφ y )] (σ + φ y + kφ π − σρ ϑ )

Cancel out denominators and develope the multiplications (σ 2 +φ σ + kφ σ−ρ σ 2 2 y 2 π ϑ + kφ π σ + kφ π φ

y +k φ π − kφ π ρ ϑ σ+φ y σ+φ y + kφ π φ y −φ y ρ ϑ σ+ − βρ ϑ σ 2 − βφ y ρ ϑ σ − βσρ ϑ φ π k + βσ 2 ρ 2 ϑ − kρ ϑ σ − kρ ϑ φ y −k 2 ρ φ + kρ 2 σ − βφ ρ σ − βφ ϑ 2 π ϑ y ϑ y ρ ϑ + − βφ ρ φ k + βφ ρ 2 σ − σkρ 2 + σkρ y 2 ϑ π y ϑ ϑ ϑ βφ π )ψ πϑ = −kσρ ϑ − kσ − kφ y −k 2 φ π + kσρ ϑ

Cancel out ±kρ 2 ϑ σ on the left hand-side of the equality, and ±kσρ ϑ on the right hand one.

2 y ρ ϑ φ π k + βφ y ρ 2 ϑ σ + σkρ ϑ βφ π )ψ πϑ = −k (σ + φ y + kφ π ) Now, collect the terms on the left hand-side of the equality

[σ 2 +φ 2 y +k 2 φ 2 π + 2σφ y + 2σφ π k + 2φ y φ π k + σ(−σρ ϑ )+φ π k(−σρ ϑ )+φ y (−σρ ϑ )+

+ σ(βσρ 2 ϑ )+φ π k(βσρ 2 ϑ )+φ y (βσρ 2 ϑ ) + σ(−kρ ϑ )+φ π k(−kρ ϑ )+φ y (−kρ ϑ ) + σ(−βφ y ρ ϑ )+ φ π k(−βφ y ρ ϑ )+φ y (−βφ y ρ ϑ ) + σ(−βσρ ϑ )+φ π k(−βσρ ϑ )+φ y (−βσρ ϑ )]ψ πϑ = −k (σ + φ y + kφ π )

[(σ + φ y + kφ π ) 2 − σρ ϑ (σ + φ π k+φ y ) + βσρ 2 ϑ (σ + φ π k+φ y ) − kρ ϑ (σ + φ π k+φ y )+ − βφ y ρ ϑ (σ + φ π k+φ y ) − βσρ ϑ (σ + φ π k+φ y )]ψ πϑ = −k (σ + φ y + kφ π )

(σ + φ y + kφ π )[σ + φ y + kφ π − σρ ϑ + βσρ 2 ϑ − kρ ϑ − βφ y ρ ϑ − βσρ ϑ ]ψ πϑ = −k (σ + φ y + kφ π ) [σ(1 − βρ ϑ ) − σρ ϑ (1 − βρ ϑ )+φ y (1 − βρ ϑ ) + k(φ π −ρ ϑ )] ψ πϑ = −k {(1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )} ψ πϑ = −k

−k

ψ πϑ = (1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

(1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

1 New Keynesian Model

Next, determine the coefficient for output gap. Substitute ψ πϑ into (40) ρ ϑ (1 − βφ π )

1 ψ yϑ =

−k

− σ+φ y + kφ π − σρ ϑ (1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

σ+φ y + kφ π − σρ ϑ −kρ

ϑ + kβφ π ρ ϑ − [σ + φ y + kφ π − σρ ϑ + βσρ − kρ ϑ − βφ y ρ ϑ − βσρ ϑ ] =

(σ + φ y + kφ π − σρ ϑ ){(1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )} σ(−1) + φ y (−1) + kφ π (−1) − σρ ϑ (−1) + σ(βρ ϑ )+φ y (βρ ϑ ) + kφ π (βρ ϑ ) − σρ ϑ (βρ ϑ )

= (σ + φ y + kφ π − σρ ϑ ){(1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )}

(σ + φ y + kφ π − σρ ϑ )(βρ ϑ − 1)

= (σ + φ y + kφ π − σρ ϑ ){(1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )}

ψ yϑ =− (1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

The response of output gap becomes

y ˜ t =−

(42) (1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

As long as Taylor principle is satisfied, (35), denominators of both the coefficients, ψ πϑ ,ψ yϑ , are strictly greater than zero. Hence, an exogenous increase in the interest rate (ε ϑ t > 0), leads to a persistant decline in the output gap and inflation. Because the natural level of output, see equation (27), is not affected by the monetary policy shock, the response of output maches that of the output gap, ˜ y ≡y −y n t t t ≡y t .

Now we want to derive the deviation of real interest rate from its steady state value, that is ˆ r t =r t −r n t . Replace r t =i t −E t {π t+1 }, and the guessed solutions for ˜ y t and E t {˜ y t+1 } in equation (31)

(43) (1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

Equation (43) shows that the deviation of real inrìterest rate from its steady state value increases in response to an exogenous increase in the nominal interest rate (ε v t > 0 implys ϑ t > 0, which induce i t to increase).

The response of nominal interest rate combines both direct effect of ϑ t

1 New Keynesian Model

and the variation induced by lower output gap and inflation, as shown below ˆi t =ˆ r t +E t {π t+1 }=ˆ r t +ψ πϑ ρ ϑ ϑ t

σ(1 − ρ ϑ )(1 − βρ ϑ ) − kρ ˆi ϑ t =

ϑ t (44) (1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

If the persistance of monetary policy shock, ρ ϑ , is sufficiently high, the nominal rate will decrease in response to a rise in ϑ t . This is because the indirect effect of a positive realization of ε ϑ t , that is the decline in inflation and output gap, more than offset the direct effect of higher ϑ t , resulting in

a downward adjustment in the nominal rate. In that case, and despite the lower nominal rate, the policy (positive) shock still has contractionary effect on output, because the latter is inversely related to the real rate, which goes up unambiguously.

Assume a log-linear money demand equation of the form

(45) where η is the interest semi-elasticity of money demand. The response

m t −p t =y t − ηi t

of m t to change in interest rate, using (41), (42) and (44), is dm t

−k − (1 − βρ ϑ ) − η[σ(1 − ρ ϑ )(1 − βρ ϑ ) − kρ ϑ ] = (1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

(1 − βρ ϑ )[1 + ησ(1 − ρ ϑ )] + k(1 − ηρ ϑ ) =−

(46) (1 − βρ ϑ ) [σ(1 − ρ ϑ )+φ y ] + k(φ π −ρ ϑ )

Hence, The sign of the change in the money supply that supports the exogenous policy intervention is ambiguous. Note that di t /dε ϑ t > 0 is a sufficient condition for a contraction in the money supply, as well as for the presence of a liquidity effect(i.e. a negative short-run comovement of the nominal rate and the money supply in response to an exogenous monetary policy shock).

1 New Keynesian Model

Appendix 1: Derivation of log-linear money demand equation Assume the following utility function

(47) P t

and a sequence of flow budget constraints

P t C t +Q t B t +M t ≤B t−1 +M t−1 +W t N t +T t

Call A t =B t−1 +M t−1 total financial wealth. Budget constraint becomes P t C t +Q t B t +M t +Q t M t −Q t M t ≤B t−1 +M t−1 +W t N t +T t

P t C t +Q t A t + (1 − Q t )M t ≤A t−1 +W t N t +T t (48) We want to maximize (47) with respect to consumption C t and nominal

money demand M t , subject to (48). Hence, the Lagrangian is

The marginal rate of substitution between nominal money demand and consumtion is

So, the demand for real balances becomes M t

(1 − exp{−i }) t −1/ϑ t t t

= C −σ (1 − Q ) ϑ =C

(49) P t

where Q −1

= exp{i t }, or 1−Q t = 1−exp{−i t }∼ =i t . The log-linearizatino

1 New Keynesian Model

of (49) is as follows σ

ln M t − ln P t = ln C t − ln(1 − exp{−i t }) ϑ

1 exp{−i t } ∗ (M t −M )− ∗ (P t −P )=

∗ (C t −C )−

ϑ C ϑ 1 − exp{−i t }

Note that exp{−i t }

1 1 1 exp{i t } =

1 − exp{−i t } exp{i t }

exp{i t }

exp{i t } exp{i t }−1

exp{i t }−1

So, the log-linearization becomes

1 ϑ(exp{i t }−1) = η is the implied interest semi-elasticity of money demand. In the particular case of ϑ = σ, that implies a unit elasticity

where 1 ϑi ∗ =

with respect to consumtion, a conventional linear demand for real balances becomes

m t −p t =c t − ηi t

(51) where the latter equality uses the market clearing condition C t =Y t , that

=y t − ηi t

is all output is consumed. The money demand equation (51) is usefull to determine the quantity of money that the central bank will need to supply in order to support, in equilibrium, the nominal interest rate implied be the policy rule.

1 New Keynesian Model

1.5.2 The effects of a technology shock Assume an AR(1) process for the technology parameter a t

(52) where ρ a ∈ [0, 1] and ε a t is a zero mean white noise process. Note that,

Given (30), the implied natural rate expressed in terms of deviations from steady state (ˆ r n t =r n t − ρ), is given by

r n t = σψ n ya E t {∆a t+1 }+ρ r n t − ρ = σψ n ya (ρ a − 1)a t

(53) In this case we set ϑ t = 0, for all t, that is we turn off the monetary

t = −σψ ya (1 − ρ a )a t

shocks, and we guess that output gap and inflation are proportional to ˆ r n t . The guessed solutions will take the following form

Use the method of undetermined coefficients to find the solutions. Recall the two equations from the system (34)

[(r − ρ) − ϑ t ] y ˜ t =

E t {˜ y t+1 }+

E t {π t+1 }+

σ+φ y + kφ π

σ+φ y + kφ π βσ + k + βφ

σ+φ y + kφ π

k [(r t − ρ) − ϑ t ] π t =

kσ

E t {˜ y t+1 }+ σ + kφ π +φ y

E t {π t+1 }+

σ + kφ π +φ y

1 New Keynesian Model

Substitute the guessed solutions, and put ϑ t = 0, as follows

Collect the terms σ

Now, solve for ψ yr and ψ πr

σ + kφ π +φ y − βσρ a − kρ a − βφ y ρ a σ + kφ π +φ y − βσρ a − kρ a − βφ y ρ a Start with ψ πr , that is

kσρ a (1 − βφ π )ρ a 1 ψ πr =

ψ πr +

+ σ + kφ π +φ y − βσρ a − kρ a − βφ y ρ a σ+φ y + kφ π − σρ a σ+φ y + kφ π − σρ a

σ + kφ π +φ y − βσρ a − kρ a − βφ y ρ a

kσρ 2 a (1 − βφ π )

ψ πr =

(σ + kφ π +φ y − βσρ a − kρ a − βφ y ρ a )(σ + φ y + kφ π − σρ a ) kσρ a

(σ + kφ π +φ y − βσρ a − kρ a − βφ y ρ a )(σ + φ y + kφ π − σρ a ) k

σ + kφ π +φ y − βσρ a − kρ a − βφ y ρ a

(σ + kφ 2 π +φ y − βσρ a − kρ a − βφ y ρ a )(σ + φ y + kφ π − σρ a ) − kσρ a (1 − βφ π ) ψ πr =

= kσρ a + k(σ + φ y + kφ π − σρ a )

1 New Keynesian Model

σ 2 +k 2 φ 2 π +φ 2 y + 2σφ y + 2σkφ π + 2kφ π φ y + σ(−σρ a ) + kφ π (−σρ a )+φ y (−σρ a ) + σ(−βσρ a )+ +φ (−βσρ ) + kφ (−βσρ ) + σ(−kρ )+φ (−kρ ) + kφ (−kρ ) + σ(βσρ 2 y a π a a y a π a a )+φ y (βσρ 2 a )+ + [kφ π (βσρ 2 a ) + σ(−βφ y ρ a )+φ y (−βφ y ρ a ) + kφ π (−βφ y ρ a )]ψ πr = k(σ + φ y + kφ π )

[(σ + kφ π +φ y ) 2 − σρ a (σ + kφ π +φ y ) − βσρ a (σ + kφ π +φ y ) − kρ a (σ + kφ π +φ y )+

+ βσρ 2 a (σ + kφ π +φ y ) − βφ y ρ a (σ + kφ π +φ y )]ψ πr = k(σ + φ y + kφ π )

(σ + φ y + kφ π )[(σ + φ y + kφ π ) − σρ a − βσρ a − kρ a + βσρ 2 a − βφ y ρ a ]ψ πr = k(σ + φ y + kφ π )

[βρ a (−σ + σρ a −φ y ) + k(φ π −ρ a ) + σ(1 − ρ a )+φ y ]ψ πr =k

{−βρ a [σ(1 − ρ a )+φ y ] + k(φ π −ρ a ) + σ(1 − ρ a )+φ y }ψ πr =k

ψ πr =

(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

1 New Keynesian Model

Continuing with ψ yr , it becomes

1 ψ yr =

(1 − βφ π )ρ a k

σ+φ y + kφ π − σρ a (1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

σ+φ y + kφ π − σρ a

(1 − βφ π )ρ a k + (1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

= (σ + φ y + kφ π − σρ a ){(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )}

ρ a k − βφ π ρ a k+σ+φ y + kφ π − σρ a − βσρ a − kρ a + βσρ 2 a − βφ y ρ a = (σ + φ y + kφ π − σρ a ){(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )}

φ π k(1 − βρ a )+φ y (1 − βρ a ) − σρ a (1 − βρ a ) + σ(1 − βρ a )

= (σ + φ y + kφ π − σρ a ){(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )}

(1 − βρ a )(σ + φ y + kφ π − σρ a )

= (σ + φ y + kφ π − σρ a ){(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )}

1 − βρ a

ψ yr =

(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

Note that, these coefficients are the same as in case of a monetary policy shock (ˆ r n t = 0), but with opposite sign. This, in fact, can be seen from the system (34), where ϑ t and ˆ r n t enter both the equations for π t and ˜ y t with the opposite sign in case of monetary policy shock and technology shock, respectively. Denominator of both ψ yr and ψ πr is strictly greater than zero.

The guessed solutions for inflation and output gap are, respectively

(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

1 − βρ a n

y ˜ t =ψ yr ˆ r t =

(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a ) t

(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

a < 1, a positive technology shock (ε t > 0) leads to a persistant decline both in inflation and the output gap. In this case, the

As long as ρ

natural level of output will be affected by technology shock, see equation (27). The implied equilibrium response of output is given by

a t (1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

(1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

1 New Keynesian Model

And the equilibrium response of employment, starting from equation (18), is

a t −a t (1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

σ(1 − ρ a )(1 − βρ a ) = (ψ ya − 1) − ψ ya

a t (1 − βρ a )[σ(1 − ρ a )+φ y ] + k(φ π −ρ a )

(57) The sign of the response of output and employment to a positive tech-

nology shock is in general ambiguous, depending on the configuration of parameter values.

1.6 Matlab Codes What follows are some possible codes configuration in Matlab of the model

described above. The same calibration of parameters as in the book of Jordi Galì is used.

Therefore, the coding of linearized model in case of an exogenous shock on monetary policy can be written as follows Matlab Code for Policy Monetary Shock in the NK model with

a Taylor Rule ///////////////////////////////////////////////////

//DECLARATION OF ENDOGENOUS VARIABLES// ////////////////////////////////////////////////// var pi y_tilde i v mr y rn y_n a r;

///////////////////////////////////////////////// //DECLARATION OF EXOGENOUS VARIABLES// //////////////////////////////////////////////// varexo varepsilon_v varepsilon_a;

////////////////////////////////////// //DECLARATION OF PARAMETERS// ///////////////////////////////////// parameters betta sigma varphi alpha varepsilon eta theta phi_pi phi_y rho_v rho_a k lambda THETA psi_yan rho lambda_v;

///////////////////////////////////// //CALIBRATION OF PARAMETERS// /////////////////////////////////////

1 New Keynesian Model

betta=0.99; //implies a steady state real return on financial as- sests of about 4%. sigma=1; //implies a log utility. varphi=1; //implies a unitary Frisch elasticity of labour supply. alpha=1/3; varepsilon=6; eta=4; theta=2/3; //implies an average price duration of three quarters. phi_pi=1.5; phi_y=0.5/4; rho_v=0.5; //implies a moderately persistent shock. rho_a=0.9; THETA = (1 − alpha)/(1 − alpha + alpha · varepsilon); lambda = (((1 − theta) · (1 − betta · theta))/theta) · THETA; k = lambda · (sigma + ((varphi + alpha)/(1 − alpha))); psi_yan = (1 + varphi)/(sigma · (1 − alpha) + varphi + alpha); rho=-log(betta); lambda_a = 1/((1−betta·rho_a)·(sigma·(1−rho_a)+phi_y)+ k · (phi_pi − rho_a)); lambda_v = 1/((1−betta·rho_v)·(sigma·(1−rho_v)+phi_y)+ k · (phi_pi − rho_v));

///////////////////////// //DSGE-Model-equations// //////////////////////// model(linear);

//New Keynesian Phillips Curve eq. 29. pi = betta · pi(+1) + k · y_tilde;

//Dynamic IS eq. 31. y_tilde = y_tilde(+1) − (1/sigma) · (i − pi(+1) − rn);

//Output Gap. y_n=y-y_tilde;

//Natural Output eq. 27. y_n = psi_yan · a;

//Natural Rate of Interest eq. 30. rn = rho + sigma · (y_n(+1) − y_n);

//Taylor Interest Rate Rule eq. 33.

1 New Keynesian Model

//Real Interest Rate r=i-pi(+1); //or, similarly // r = sigma · (y(+1) − y) + rho;

//Real Money Demand eq. 45. mr = y − eta · i; //where mr=m-p,

//AR(1) for monetary policy shock eq. 36. v = rho_v · v(−1) + varepsilon_v;

//AR(1) for technology shock eq. 52.

a = rho_a · a(−1) + varepsilon_a; end ; ////////////////////////////

//SHOCK SPECIFICATION// //////////////////////////// shocks; var varepsilon_v; stderr 0.25; var varepsilon_a; stderr 0; end;

steady; check; stoch_simul (order=1, irf=12)y_tilde pi i r mr v;

figure(’Name’,’Effects of a Monetary Policy Shock’,’NumberTitle’,’off’); subplot(3,2,1);

plot(y_tilde_varepsilon_v, ’-o’); title(’Output Gap’); axis([0,12,-0.4,0]);

subplot(3,2,2); plot(4 · pi_varepsilon_v, ′ −o ′ ); title(’Inflation’); axis([0,12,-0.4,0]);

subplot(3,2,3);

1 New Keynesian Model

title(’Nominal Interest Rate’); axis([0,12,0,0.8]);

subplot(3,2,4); plot(4 · r_varepsilon_v, ′ −o ′ ); title(’Real Interest Rate’); axis([0,12,0,0.8]);

subplot(3,2,5); plot(mr_varepsilon_v, ’-o’); title(’Nominal Money Growth’); axis([0,12,-4,2]);

subplot(3,2,6); plot(v_varepsilon_v, ’-o’); title(’Monetary Shock’); axis([0,12,0,0.4]);

The impulse Response Functions of the variables of interest are (compare these with Figure 3.1 from the book of Galì).

Fig. 1: Effects of a Monetary Policy Shock with Interest Rate Rule

1 New Keynesian Model

First of all, note that the responses for inflation and the two interest rates are expressed in annual terms. For this purpose their IRFs have been multiplied by 4. Figure 1, hence, illustrates the dynamic effects of an expansionary monetary policy shock, which corresponds to an increase in ε v t of 0.25. In the absence of a further change induced by the response of inflation or the output gap, this would imply an increase of 100 basis points in the annualized nominal interest rate. The policy shock generates a decrease both in output gap, which effectively corresponds to output, because the natural level of output is not affected by monetary policy shock, and in inflation, and an increase in the real rate. The nominal interest rate increases, too, but by less than its exogenous component because of a downward adjustment induced by the decline in output gap and inflation. Moreover, the response of the real interest rate is larger than that of the nominal rate due to a decrease in expected inflation. Finally, the model displays a liquidity effect resulting form the actions taken by the Central Bank, which must implement a reduction in money supply given the observed interest rate response.

Matlab Code for Technology Shock in the NK model with a Taylor Rule

/////////////////////////////////////////////////// //DECLARATION OF ENDOGENOUS VARIABLES// ////////////////////////////////////////////////// var pi y_tilde i v mr y rn y_n a r n;

///////////////////////////////////////////////// //DECLARATION OF EXOGENOUS VARIABLES// //////////////////////////////////////////////// varexo varepsilon_v varepsilon_a;

///////////////////////////////////// //CALIBRATION OF PARAMETERS// ///////////////////////////////////// betta=0.99; //implies a steady state real return on financial as- sests of about 4%.

1 New Keynesian Model

varphi=1; //implies a unitary Frisch elasticity of labour supply. alpha=1/3; varepsilon=6; eta=4; theta=2/3; //implies an average price duration of three quarters. phi_pi=1.5; phi_y=0.5/4; rho_v=0.5; //implies a moderately persistent shock. rho_a=0.9; THETA = (1 − alpha)/(1 − alpha + alpha · varepsilon); lambda = (((1 − theta) · (1 − betta · theta))/theta) · THETA; k = lambda · (sigma + ((varphi + alpha)/(1 − alpha))); psi_yan = (1 + varphi)/(sigma · (1 − alpha) + varphi + alpha); rho=-log(betta); lambda_a = 1/((1−betta·rho_a)·(sigma·(1−rho_a)+phi_y)+ k · (phi_pi − rho_a)); lambda_v = 1/((1−betta·rho_v)·(sigma·(1−rho_v)+phi_y)+ k · (phi_pi − rho_v));

///////////////////////// //DSGE-Model-equations// //////////////////////// model(linear);

//New Keynesian Phillips Curve eq. 29. pi = betta · pi(+1) + k · y_tilde;

//Dynamic IS eq. 31. y_tilde = y_tilde(+1) − (1/sigma) · (i − pi(+1) − rn);

//Output Gap. y_n=y-y_tilde;

//Natural Output eq. 27. y_n = psi_yan · a;

//Natural Rate of Interest eq. 30. rn = rho + sigma · psi_yan · (a(+1) − a);

//Taylor Interest Rate Rule eq. 33.

i = rho + phi_pi · pi + phi_y · y_tilde + v; //Real Interest Rate

1 New Keynesian Model

//or, similarly // r = sigma · (y(+1) − y) + rho;

//Real Money Demand eq. 45. mr = y − eta · i; //where mr=m-p,

//Economic Equilibrium Employment eq. 18. n = (1/(1 − alpha)) · (y − a);

//AR(1) for monetary policy shock eq. 36. v = rho_v · v(−1) + varepsilon_v;

//AR(1) for technology shock eq. 52.

a = rho_a · a(−1) + varepsilon_a; end ; ////////////////////////////

//SHOCK SPECIFICATION// //////////////////////////// shocks; var varepsilon_v; stderr 0; var varepsilon_a; stderr 1; end;

steady; check; stoch_simul (order=1, irf=12)y_tilde pi y n i r mr a;

figure(’Name’,’Effects of a Technology Shock’,’NumberTitle’,’off’); subplot(4,2,1);

plot(y_tilde_varepsilon_a, ’-o’); title(’Output Gap’); axis([0,12,-0.2,0]);

subplot(4,2,2); plot(4 · pi_varepsilon_a, ′ −o ′ ); title(’Inflation’); axis([0,12,-1,0]);

subplot(4,2,3);

1 New Keynesian Model

title(’Output’); axis([0,12,0,1]);

subplot(4,2,4); plot(n_varepsilon_a, ′ −o ′ ); title(’Employment’); axis([0,12,-0.2,0]);

subplot(4,2,5); plot(4 · i_varepsilon_a, ′ −o ′ ); title(’Nominal Interest Rate’); axis([0,12,-1,0]);

subplot(4,2,6); plot(4 · r_varepsilon_a, ′ −o ′ ); title(’Real Interest Rate’); axis([0,12,-0.4,0]);

subplot(4,2,7); plot(mr_varepsilon_a, ’-o’); title(’Nominal Money Growth’); axis([0,12,-10,10]);

subplot(4,2,8); plot(a_varepsilon_a, ’-o’); title(’Technology Shock’); axis([0,12,0,1]);

The impulse Response Functions of the variables of interest are illus- trated in Figure 2 (compare these with Figure 3.2 from the book of Galì).

It shows that a positive technology shock leads to a persistent decline in both inflation and output gap. Instead, the responses of output and employment to a positive technology shock are, in general, ambiguous, depending on the calibration of the parameters values. Given the calibration adopted here, a technological improvement leads to

a persistent employment decline, while actual output increses, though less than its natural counterpart. The improvement in technology is partly accomodated by the Central Bank, which lowers nominal and real interest rates, while increasing the quantity of money in circulation. This policy, however, is not sufficient to close the negative output gap, that is responsible for the decline in inflation.

1 New Keynesian Model

Fig. 2: Effects of a Technology Shock with Interest Rate Rule

1.7 Equilibrium under an Exogenous Money Supply The equilibrium dynamics of the basic New Keynesian model is, now, ana-

lyzed under an exogenous path for the growth rate of money supply, ∆m t . Rewrite the money market equilibrium condition (45) in terms of output gap (˜ y t =y t −y n t ), as follwos

m t −p t =y t − ηi t m t −p t =˜ y t +y n t − ηi t ˜ y t − ηi t = (m t −p t )−y n t

(59) Substitute equation (59) in the dynamic IS equation (31)

(1 + ησ) ˜ n y t = ησE t {˜ y t+1 } + ηE t {π t+1 } + (m t −p t ) + ηr t −y t (60)

1 New Keynesian Model

that is a difference equation for output gap. In order to complete the system and to find the equilibrium solutions, we need two more equations. The one is the NKPC described in equation (29), the other is an expression for real balances (m t −p t ) in relation with inflation and money growth, as shown below

The log-linearization of this identity becomes ln M t − ln M t−1 = (ln M t − ln P t ) − (ln M t−1 − ln P t−1 ) + (ln P t − ln P t−1 )

m t −m t−1 = (m t −p t ) − (m t−1 −p t−1 ) + (p t −p t−1 ) ∆m t = (m t −p t ) − (m t−1 −p t−1 )+π t

(61) Therefore, the three equations that form the system are

   (1 + ησ) ˜ y t = ησE t {˜ y t+1 } + ηE t {π t+1 } + (m t −p t ) + ηr n

n t −y t π t = βE t {π t+1 }+k˜ y t

The system to be solved is 

∆m t In this system we have one predetermined variable (m t−1 −p t−1 ), and

0 01 (m t −p t )

two non-predetermined variables (˜ y t ,π t ). Accordingly, a stationary solution

1 New Keynesian Model

will exist and be unique, if and only if, matrix 

0 01 has one eigenvalue outside (or on) the unit circle, and two eigenvalues

inside the unit circle. It can be shown, that this condition is always satisfied, so the equilibrium is always determined under an exogenous path for the money supply (in contrast with the interest rate rule, where the Taylor Principle must hold).

1.7.1 The effects of a monetary policy shock We want here to analyze the responses of the economy to an exogenous

monetary policy shock to the money supply. Assume, for this purpose, an AR(1) process for the growth of money

(63) where ρ ∈ [0, 1), and ε m m t is a zero mean white noise process. A positive

∆m m

t =ρ m ∆m t−1 +ε t

realization of ε m t implies an expansionary monetary policy shock. Combine equation (63) with the dynamic system (62)

and guess that the solutions will take the following form

˜ y t =ψ ym ∆m t π t =ψ πm ∆m t

(m t−1 −p t−1 )=ψ lm ∆m t

Forward one period, the AR(1) process for money growth becomes

E t {∆m t+1 }=ρ m ∆m t

Therefore, forward one period, the guessed solutions are

E t {˜ y t+1 }=ψ ym E t {∆m t+1 }=ψ ym ρ m ∆m t

E t {π t+1 }=ψ πm E t {∆m t+1 }=ψ πm ρ m ∆m t m t −p t =ψ lm E t {∆m t+1 }=ψ lm ρ m ∆m t

1 New Keynesian Model

Using the method of undetermined coefficients one can find the response functions for output gap, inflation, nominal and real rates, and real balances. Note that in this framework it must be set ˆ r n t =y n t = 0, for all t.

1.7.2 The effects of a technology shock Assume once again the technology parameter a t follows the stationary pro-

cess given in (52). Combining this with equations (27) and (30) a path is determined for ˆ r n t and y n t as function of a t , which is then used to solve system (62). Note that, in this case it must be set ∆m t = 0 for all t.

1 New Keynesian Model

References [1] Guillermo A. Calvo: "Staggered prices in a utility-maximizing frame-

work". Volume 12, Issue 3, Journal of Monetary Economics, 1983. [2] Avinash K. Dixit, Joseph E. Stiglitz: "Monopolistic Competition and

Optimum Product Diversity". Vol. 67, No. 3, pp. 297-308, The American Economic Review, 1977.

Basic New Keynesian Model docx