Dynamic AD and Dynamic AS · Dynamic IS curve (DIS) has been developed, and is currently used in modern New Keynesian models. I To be consistent with modern macro, the demand curve

Dynamic AD and Dynamic AS

Pedro Serodio

July 21, 2016

Inadequacy of the IS curve

I The IS curve remains ’Keynesian’ in nature. It is ’static’ andnot explicitly microfounded. An alternative, microfounded,Dynamic IS curve (DIS) has been developed, and is currentlyused in modern New Keynesian models.

I To be consistent with modern macro, the demand curve1. must incorporate agents forward-looking behaviour2. so, must be based on a dynamic (not just static) model3. old IS-LM model has a Keynesian IS curve that embodies only

static response of the level of investment today to the level ofthe real interest rate.

I The New Keynesian dynamic IS (DIS) curve is the basis of allmodern macro models.

I It formalises the intertemporal-optimising insights we haveverbally and diagrammatically incorporated into our basicIS-MP model:↑ r → current consumption via reallocation of spending overtime.



















Foundations of the dynamic IS schedule

I Because the DIS is such an important component of modernmacro, it is worth understanding its formal derivation.

I Deriving the DIS is fairly straightforward and helps clarify whymodern New Keynesian macro emphasises consumption ratherthan the Keynesian investment channel in the monetarytransmission mechanism.

I We model consumers’ intertemporal optimisation using thenormal budget constraint and indifference curves.

I From these, we derive an IS curve in which current incomedepends on future income and the real interest rate.

I Iterating the resulting recursive expression forward shows thatin this dynamic model it is not just current income thatmatters, it is all future expected income (which embodies allfuture expected real interest rates).

























Aggregate Demand and Aggregate Supply

C2

C1

I 2-period model, t = 1, 2. Initialwealth is W .

I Consumption per period isCt , t = 1, 2.

I Income per period is Yt , t = 1, 2.

I Real interest rate is r on savingsfrom t1 to t2. According to thebudget constraint,

C2 = (W + Y1 − C1)(1 + r) + Y2

I Slope of the budget constraint:

MRT :dC2

dC1= −(1 + r)


C2

C1





C2 = (W + Y1 − C1)(1 + r) + Y2


MRT :dC2

dC1= −(1 + r)


C2

C1





C2 = (W + Y1 − C1)(1 + r) + Y2


MRT :dC2

dC1= −(1 + r)


C2

C1





C2 = (W + Y1 − C1)(1 + r) + Y2


MRT :dC2

dC1= −(1 + r)


C2

C1





C2 = (W + Y1 − C1)(1 + r) + Y2


MRT :dC2

dC1= −(1 + r)

Indifference curves

C2

C1

U

I Individual’s discount factor is β.

I Utility, looking forward from period1, is U = U(C1) + βU(C2).

I We can define the implicit functionU = 0 and use the implicit functiontheorem to find:

∂C2

∂C1= − 1

β

U ′C1

U ′C2= MRS

which is the slope of theindifference curve.

Indifference curves

C2

C1

U




∂C2

∂C1= − 1

β

U ′C1

U ′C2= MRS


Indifference curves

C2

C1

U




∂C2

∂C1= − 1

β

U ′C1

U ′C2= MRS


Optimisation

C2

C1

U

C∗2

C∗1

I In order to maximise utility, agentswill equate the marginal rate ofsubstitution (MRS) with themarginal rate of transformation(MRT), which yields:

MRS =1

β

U ′C1

U ′C2= (1 + r) = MRT

I This gives rise to the Eulerequation for consumption, anexpression that characterises theoptimal path of consumption overtime:

U ′C1 = (1 + r)βU ′C2

Optimisation

C2

C1

U

C∗2

C∗1

I In order to maximise utility, agentswill equate the marginal rate ofsubstitution (MRS) with themarginal rate of transformation(MRT), which yields:

MRS =1

β

U ′C1

U ′C2= (1 + r) = MRT

I This gives rise to the Eulerequation for consumption, anexpression that characterises theoptimal path of consumption overtime:

U ′C1 = (1 + r)βU ′C2

Deriving the dynamic IS

I Assume a particular form for the utility function that implies:U = C−σ, where σ is known as the (constant) coefficient ofrelative risk aversion (CRRA) (implies an isoelastic, orconstant elasticity of substitution (CES), utility function).

I Using this functional form and linearising along the long runvalues, we get:

−σct = −r − σcet+1 + rt .

I We can use the fact that the income-expenditure equationimplies: yt = ct + it + gt (in log terms) and replace that intothe expression for consumption. In the New Keynesian model,investment is a function of the capital stock and,consequently, responds to changes to the real interest rate(just like the standard IS). Here, we make the assumptionthat investment simply replaces depleted stock, so that it isalways constant.












I That means we can write the dynamic IS schedule as:

−σ(yt − gt − it) = −r − σ(y et+1 − g et+1 − iet+1) + rt .

yt = gt + y et+1 − g et+1 −

1

σ(rt − r)

I This equation again implies a negative relationship betweenoutput and the real interest rate, just like the standard ISschedule.


I That means we can write the dynamic IS schedule as:

−σ(yt − gt − it) = −r − σ(y et+1 − g et+1 − iet+1) + rt .

yt = gt + y et+1 − g et+1 −

1

σ(rt − r)

I This equation again implies a negative relationship betweenoutput and the real interest rate, just like the standard ISschedule.


To summarise, so far:

I A New Keynesian dynamic IS (DIS) curve can be derived froma simple microeconomic model of an intertemporallyoptimising representative consumer with rational expectations.

I Basically, the DIS says that the output gap depends onexpectations of future output gaps and today’s real interestrate.

I The big difference is the forward-looking nature of DIS.Expectations matter, and policy can operate through theirmanipulation.











Deriving the dynamic AD

I We can use the monetary policy rule we used in the IS-MPmodel to characterise the conduct of monetary policy in theNew Keynesian model.

I Doing so allows us to derive a relationship between inflationand output similar to the one derived in previous models.

I The two equilibrium conditions are:

yt − y = gt + (y et+1 − y)− g et+1 −

1

σ(rt − r)

rt = r + φπ(πt − πT ) + φy (yt − y)

I While the expression for output is:

y∗t =1

σ + φy

[σ(gt + y et+1 − g e

t+1)− φπ(π − πT ) + φy y]

Note that there is a negative relationship between inflationand output, as in the IS-MP model discussed before.





yt − y = gt + (y et+1 − y)− g et+1 −

1

σ(rt − r)



y∗t =1

σ + φy


t+1)− φπ(π − πT ) + φy y]






yt − y = gt + (y et+1 − y)− g et+1 −

1

σ(rt − r)



y∗t =1

σ + φy


t+1)− φπ(π − πT ) + φy y]






yt − y = gt + (y et+1 − y)− g et+1 −

1

σ(rt − r)



y∗t =1

σ + φy


t+1)− φπ(π − πT ) + φy y]


Dynamic AS

I We saw in the preceding section how the Phillips curve allowedus to derive an upward sloping aggregate supply schedule.

I In standard New Keynesian models prices, rather than wages,are assumed to be sticky due to a variety of possible reasons.

I Menu costs: the increase in profits from changing prices issmall for any given firm but, due to aggregate demandexternalities, this cost can be much larger for the economy asa whole. Firms don’t change prices frequently. (problem: if allfirms are the same, they will all either adjust or not adjust so,every period, prices are either infinitely flexible or completelyfixed)

Dynamic AS




Dynamic AS




Dynamic AS

I Information asymmetries I: firms receive incompleteinformation regarding whether prices reflect demand for theirown product or an increase in the general price level. Theywill respond by changing prices somewhat, but not enough toprevent all of the additional demand, and will thereforeincrease production too.

I Information asymmetries II: firms do not receive a signal toadjust their prices and, therefore, keep them constant for anumber of periods.

Dynamic AS

I Information asymmetries I: firms receive incompleteinformation regarding whether prices reflect demand for theirown product or an increase in the general price level. Theywill respond by changing prices somewhat, but not enough toprevent all of the additional demand, and will thereforeincrease production too.

I Information asymmetries II: firms do not receive a signal toadjust their prices and, therefore, keep them constant for anumber of periods.

Dynamic AS

I The latter case is by far the most common in the literature.

I Because only some firms update their price, the general pricewill reflect that fact: some firms will change prices to reflecteconomic conditions while other firms simply keep lastperiod’s prices constant. This implies:

pt = θp∗t + (1− θ)pt−1

where θ is the fraction of firms changing their price and p∗ isthe optimal price.

I The problem of finding the optimal price, p∗, is very complex,but it can be shown that, in combination with the expressionabove, that leads to the so-called New Keynesian PhillipsCurve:

πt = βπet+1 + κ(yt − y)

Dynamic AS

I The latter case is by far the most common in the literature.I Because only some firms update their price, the general price

will reflect that fact: some firms will change prices to reflecteconomic conditions while other firms simply keep lastperiod’s prices constant. This implies:

pt = θp∗t + (1− θ)pt−1




Dynamic AS

I The latter case is by far the most common in the literature.I Because only some firms update their price, the general price

will reflect that fact: some firms will change prices to reflecteconomic conditions while other firms simply keep lastperiod’s prices constant. This implies:

pt = θp∗t + (1− θ)pt−1




I We can combine the new Phillips curve with the dynamic ISschedule to complete the DAD-DAS model.


y∗t =1

σ + φy


t+1)− φπ(π − πT ) + φy y]



yt =1

σ + φy


t+1)− φπ(π − πT ) + φy y]

I This model has a representation in (y , π) space that is exactlyidentical to the IS-MP model described above.

I Why is it a more useful model than any of those that wediscussed previously? In short, because this version of themodel allows us to think about the impact of expectations onmacroeconomic variables, and how policy makers can attemptto manage these in order to achieve their policy goals (e.g.,forward guidance).



yt =1

σ + φy


t+1)− φπ(π − πT ) + φy y]





yt =1

σ + φy


t+1)− φπ(π − πT ) + φy y]



Equilibrium

r

y

DIS0

DIS1

MP0

MP1

r

r1

y

π

y

y

LRAS

DAD0

DAD1

DAS0

DAS1

πT

πT

1

I Let’s examine the impact of apolicy where the central bankannounces that it intends to raiseits inflation target.

I If the announcement is believed(and even if the central bank doesnot adjust their policy rule), thenDIS shifts out to DIS1.

I This leads to a shift in DAD toDAD1 and DAS to DAS1. Inflationis higher even though the centralbank has not actually changedpolicy.

Equilibrium

r

y

DIS0

DIS1

MP0

MP1

r

r1

y

π

y

y

LRAS

DAD0

DAD1

DAS0

DAS1

πT

πT

1




Equilibrium

r

y

DIS0

DIS1

MP0

MP1

r

r1

y

π

y

y

LRAS

DAD0

DAD1

DAS0

DAS1

πT

πT

1




Dynamic AD and Dynamic AS · Dynamic IS curve (DIS) has been developed, and is currently used in modern New Keynesian models. I To be consistent with modern macro, the demand curve

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