static and dynamic analysis: basic concepts and...

Static and dynamic analysis: basic concepts andexamples

Ragnar Nymoen

Department of Economics, UiO

18 August 2009

ECON 3410/4410: Lecture 1

Lecture plan and web pages for this course

The lecture plan is at

http://folk.uio.no/rnymoen/ECON3410_h08_index.html,

which is the workpage of the course.The workpage is for practical posting of slides, exercises sets etc.But refer to the Department’s webpage

http://www.uio.no/studier/emner/sv/oekonomi/ECON4410/h09/

for all offi cial information: credits, overlap, exam dates and so on.


3 main topics

1 Concepts and methods of dynamic analysis.

Introductory Dynamic Macroeconomics (IDM), posted on theworkpage.

2 Medium term macro dynamics: The dynamic AD-AS model.

Introducing Advanced Macroeconomics (IAM) byBirch-Sørensen and Whitta Jacobsen

3 Critical assumptions of the standard model and alternativemodels of the supply-side.


The main focus: medium-run macro dynamics

Review of the "building blocks" of the dynamic AD-AS model

Closed economy AD-AS model:

The short run-and the long-run version of the model.

Full dynamic analysis

Application: Stabilization policy, rules versus discretion

A different perspective: The RBC model.

Open economy AD-AS model

Short-and long run (again)

Monetary policy regimes.


Dynamics is a typical feature of the real world (1)If it was not, what would the world look like?

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95.0

97.5

100.0

102.5

105.0

107.5

110.0

Blue line:a static variable determinedby many small shocks

Red line: static variable determinedby a single large shock.

Economic variableswould jumpwhenever incentiveschanged.

Time graphs wouldshow:

a step-wiseevolution, orvery erratic(volatile)behaviour, ora combination ifsome incentivesare huge, andsome are small.


Dynamics is a typical feature of the real world (2)

For some real world variables graphs look a little like the bluegraph in picture.

Daily data of stock prices, and exchange rates (under somemonetary poly regimes) are examples

But for most macroeconomic variables, persistence is adominant feature: It takes times before a change in incentives,or in legislation, or in policy, obtain full effect on macroeconomic variables.

Main sources of persistence (and therefore of dynamics) are:

Information and recognition lags,Adjustments cost,Uncertainty and expectations,Aggregation of individual decisions to the macro level.


Persistence in the response to a shock is typical ofdynamics

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98

99

100

101

102

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A static response pattern

A dynamic response pattern

The graph showsstatic (red) anddynamic (blue)responses to thesame sequence ofshocks

Note how dynamicsadd persistence tothe series, becauseshocks arepropagated throughtime


Some Norwegian economic variables: GDP per capita

1840 1860 1880 1900 1920 1940 1960 1980 2000

2

4

6

8

10

12

14

16GDP per capita relative to 1900

GDP per capita canfluctuate in themedium-run timeperspective

But in longerperspective thedominating trait isgrowth!


Unemployment (long time series)

1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020

0.01

0.02

0.03

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0.05

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Ledighetsrate

The unemploymentrate can fluctuatein the medium-runtime perspective

But in longer timeperspective, thedominating trait isno-growth!


Norway today: Inflation and unemployment

1980 1985 1990 1995 2000 2005 2010

2

4

6

8

10

12

Annual rate of inflation (AET) in Norway

Unemployment rate

The “end ofinflation” is typicalfor many countries

Note the financialcrisis at the end ofthe sample?


Norway and Sweden: Inflation and unemployment

19 80 19 85 19 90 19 95 20 00 20 05 20 10

0.0

2.5

5.0

7.5

10 .0

12 .5

Inflation (in red) and unemployment (in b lue). Norway: dashed lines.

Note again thetypical propagationof shocks.

In Swedishunemployment inparticular


Beliefs about dynamics and propagation mechanism givepremises for decision making

Norges Bank [The Norwegian Central Bank] is typical of manycentral banks’view:

“Monetary policy influences the economy with long andvariable lags. Norges Bank sets the interest rate with aview to stabilizing inflation at the target within areasonable time horizon, normally 1-3 years”

Policy decisions are based on Norges Banks beliefs about thedynamic nature of the monetary transmission mechanism.

In economics, beliefs means models, implicit or explicit.

Therefore the citation illustrates two theses: policy is modelbased, and policy models are dynamic.


Dynamic and static model: definition

Formal dynamic analysis in economics is a relatively newinvention.

Ragnar Frisch worked intensively with the foundations of thediscipline he dubbed macrodynamics in the early 1930s. Hisdefinition of dynamics was:

A dynamic theory or model is made up ofrelationships between variables that refer to different timeperiods. Conversely, when all the variables included in thetheory refer to the same time period (or, more generally,the model is conceptualized without time as an entity),the system of relationships is static.

In a dynamic model: time plays an essential role.


A static model demand schedule

A linear demand function is

Xt = aPt + b + εd ,t ,

with a < 0 and b > 0 as parameters.

The three variables: X and P,and εd (denoting a randomdemand shock) are all provided with time subscript t.

t might represent for example a year (the time period isannual); or a quarter (the period is quarterly); or month (theperiod is monthly).

This model of demand is static.

Note that the “appearance of time” (in the from of the timesubscript) is not enough to make the model dynamic, becausetime does not play an essential role!


A static equilibrium model

If we supplement the demand equation with a static supplyequation, we obtain the static market equilibrium model

Xt = a<0Pt + b + εd ,t ,

Xt = c>0Pt + d + εs ,t ,

determining the endogenous variables Xt and Pt for knownvalues of the exogenous variables εd ,t and εs ,t (and fixed andknown values of the 4 parameters a, b, c, d).

We assume that εd ,t and εs ,t are completely random variables.Their role is to represent shocks, or in Frischean terminology,impulses to the system. A variable that represents randomtechnology shocks is important in the Real Business Model(RBC) that we will discuss later in the course.


Solution of the static model (analytical)

The model written in structural form

1 · Xt − aPt = b + εd ,t

1 · Xt − cPt = d + εs ,t

Cramer’s rule gives:

Xt =1

a− c

∣∣∣∣ b + εd ,t −ad + εs ,t −c

∣∣∣∣ =ad − bc + aεs ,t − cεd ,t

a− c

Pt =1

a− c

∣∣∣∣ 1 b + εd ,t1 d + εs ,t

∣∣∣∣ =d − b + εs ,t − εd ,t

a− c


Solution of the static model (graphical)

Su p p ly cu rv e (av erag e p o sitio n )D eman d cu rv e(av erag e p o sitio n )

Pt

X t

A

B

X 0

P 0

C

P 1

X 1

D

The initial (beforethe shock)equilibrium is at A

B, C and D are newequilibria,corresponding todifferent types ofshocks.


Solution of the static model (numerical)

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Static marked equilibrium model

P t ε d , t − ε s , t

The graph shows 50simulatedequilibrium valuesfor Pt .

Pt is directreflection of “excessdemand”.


The effect of a single shock(A graph of a dynamic multiplier from a static model)

0 5 10 15 20

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1 .0 Price respon se to temp orary demand shock. Static marked equilib rium mo del

In the static model,the full effect of atemporary shockoccurs in the firstperiod.

In the periods afterthe shock, there areno responses in Pt .

The impactmultiplier isnon-zero, all otherdynamic multipliersare zero.


Summary of properties of the static model

The whole effect of a shock is contained in the equilibriumvalues of P and X in the period of the shock.

There are no spill-over effects of a shock in period t = 1 toperiod 2, 3, and later periods

We say that impulses in period 1 are not propagated to laterperiods

The time series of Pt (and Xt) are perfect mirror images of theshocks εd ,t and εst .The sequence of dynamic multipliers, for example ∂Pt/∂εs,1(t = 1, 2, 3...) are zero, expect for ∂P1/∂εs,1.


A dynamic equilibrium model

Xt = a<0Pt + b + εd ,t , demand, and

Xt = c>0Pt−1 + d + εs ,t , supply.

The only change is in the supply equation, where Pt−1replaces Pt .

Interpretation: In some markets supply is fixed in theshort-run. No matter how high or low the price is in thecurrent period, the supply of the good is ‘frozen’by decisionsof the past.

Classic example: agricultural products such as pork andwheat. Relevance today: “Salmon farming”, and China foodprice inflation; but also the market for oil and for rawmaterials.


Solution of the dynamic model (graphical)

Longrun supply curve

Demand curve(average position)

Pt

X t

A

B

X 0

P 0

C

X 2

P 1

P 2

D

The long-run supplyfunction isX = cP + d

After a temporarydemand shock, thesequence ofequilibria is A(t = 0), B (t = 1),C (t = 2), D(t = 3) and so onin a cobweb pattern

In the long-run, theequilibrium is backat A


Solution of the dynamic model (numerical)

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Static marked equilibrium model

market p r ice net d emand sh o ck

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1.00 Static marked equilibrium model

Dynamic mu ltip liers

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1.0 Dynamic marked equilibr ium model

Dynamic mu ltip liers

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0.050Dynamic marked equilibr ium model

market p r ice net d eman d sho ckGraph a) showssolution of Pt fromthe dynamic model,

b) shows thesequence ofdynamic responsesin Pt ( ∂Pt/∂εd ,1t = 1, 2, ...)

Graph c) and d)show thecorresponding forthe static model


Characteristic differences form the static model

In the dynamic model, the whole effect of a shock is notcontained in the equilibrium values of P and X in the periodof the shock.

The sequence of the dynamic multipliers, for example∂Pt/∂εd ,1 (t = 1, 2, 3...) are generally non-zero, but mayapproach zero for large values of t, if the dynamics is stable.

There are spill-over effects of a shock in period t = 1 toperiod 2, 3, ....

Impulses in period 1 are propagated to later periods

The solution doe Pt (and Xt) are not perfect mirror images ofthe shocks εd ,t and εst . in each period.

The cobweb pattern is however not general, as a secondexample will show.


A second dynamic model of market equilibrium

Xt = aPt + b1Xt−1 + b0 + εd ,t , demand

Xt = c0Pt + c1Pt−1 + d + εs ,t .supply

The demand function is now a dynamic equation. Theparameter b1 measures by how much an increase in Xt−1shifts the short-run demand curve. This can be rationalized byconsumer habits for example.

0 < b1 < 1.

In any given period, Xt−1 is determined from history andcannot be changed. Hence in this model there are twopre-determined variables: Pt−1 and Xt−1.

The supply equation is a generalization of the cobweb model:If we set c0 > 0, short run supply is no longer completelyinelastic as in the cobweb model.


Numerical solution of the model with habit formation

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1.0Dynamic market equilibrium model, cobweb.

Dynamic multipliers

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market price net demand shock

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Dynamic market equilibrium model, habit formation.

market price net demand shock

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1.00Dynamic market equilibrium model, habit formation.

dynamic multipliersCompare panel a)with c), and panelb) with d).

It is typical thatsmall changes inthe modelspecification cansignificantly affectthe solution of thedynamic model.


When is a static model relevant for the real world?

Frisch:“Hence it is clear that the static model world is best

suited to the type of phenomena whose mobility (speedof reaction) is in fact so great that the fact that thetransition from one situation to another takes a certainamount of time can be discarded. If mobility is for somereason diminished, making it necessary to take intoaccount the speed of reaction, one has crossed into therealm of dynamic theory.”

We would add: Static models are also relevant when we onlyclaim to analyze the very short-run effects (what we will callthe impact multiplier) of a shocks, i.e. we know that thedynamic effects of a shock “are there”, but we do not (knowhow to) analyze them.In this way we can interpret the Keynesian IS-LM model as ashort-run model.


The three steps in a dynamic analysis

The question we typically want to answer is: “What are thedynamic effects of a shock (of a certain type) on theendogenous variables of the model?”

It is often practical to break this question down to 3 “smaller”questions:

1 What are the short-run effects of the shock?2 What are the long-run effects of the shock, given that thedynamic adjustment process is stable?

3 What are the properties of the dynamic adjustment process(regarding stability in particular)?

To answer Q1 and Q2 we use two separate models!


Market equilibrium: the short-run model

Xt = aPt + b1Xt−1 + b0 + εd ,t , (1)

Xt = c0Pt + c1Pt−1 + d + εs ,t . (2)

Since Pt−1 and Xt−1 are pre-determined from history in eachperiod t, they are exogenous in this short-run model. Theanalytical solution:

Xt =ad − b0c0 − c0b1Xt−1 + ac1Pt−1 + aεs ,t − c0εd ,t

a− c0Pt =

d − b0 − b1Xt−1 + c1Pt−1 + εs ,t − εd ,ta− c0

gives the short-run effects of the shocks εd ,t and εs ,t asderivatives,see Table 1.1 in IDM


Market equilibrium: The long-run model

The long-run model applies to a hypothetical (or counterfactual)stationary situation where there are no new shocks, and all pastshocks have worked their way through the system.The long-run model is therefore defined for the situation:Xt = Xt−1 = X̄ ,Pt = Pt−1 = P̄ and εd ,t = ε̄d , εs ,t = ε̄s . Themodel is given by

X̄ = aP̄ + b1X̄ + b0 + ε̄d , long-run demand

X̄ = c0P̄ + c1P̄ + d + ε̄s , long-run supply

or

X̄ =a

1− b1P̄ +

11− b1

(b0 + ε̄d ),

X̄ = (c0 + c1)P̄ + (d + ε̄s ).

Solve to obtain analytical expressions for long-run effects, seeTable 1.1 in IDM.


Lo n g ru n su p p ly cu rv e

Lo n g ru n d eman d cu rv e

P

X

Sh o rt ru n d eman d cu rv e

Sh o rt ru n su p p ly cu rv e

A

BC

Graphically, we canrepresent theshort-run andlong-run models inone diagram,

Lines with differentslopes define theshort-run and thelong-run.

We can thenanalyze theshort-run effect of ashock, as well asthe long-run effect.


The third question

The short-run model and the long-run model of themacroeconomy will be important tools in the following, inparticular for the medium-term AS-AS model covered by theIAM book.

But we will also address systematically the third question:

What are the properties of the dynamic adjustment process(regarding stability in particular)?

To do this we need to develop several concepts more preciselythan we have done in this introduction.

We do that within a class of dynamic equations which wideenough to cover many economic interpretations as specialcases (Ch 2 of IDM).


Discrete or continuous time?

The distinction between static models and dynamic modes isfundamental.

Whether dynamic models are expressed in terms of discretetime or continuous time is however not fundamental.

Often theories are expressed in continuous time, but sinceactual data series are recorded in discrete time, choosingdiscrete time keeps the theory closer to applications.

Refer to Box 1.1 in IDM for example. The point is that fordynamics to occur, time must play an essential role in model(discrete/continuous time is a secondary issue).


Stock and flow variables

Dynamic models often include both flow and stock variables.

Flow: in units of (for example) million kroner per year

Stock: in units of (for example) million of kroner at aparticular period in time (for example start or end of the year).

Population size , and capital stock are examples of stockvariables. But so are also price indices: Pt may represent thevalue of the Norwegian CPI in period t (a month, a quarter ora year), and indicators of the wage level.

In practice: the values of P will be index numbers. Thenumber 100 (often 1 is used instead) refers to the base periodof the index. If Pt > 100 it means that relative to the baseperiod, prices are higher in period t.


A flow variable is often a change in a stock variable

Starting from a stock variable like Pt , a flow variable results fromobtaining the change of that variable, hence

xt = Pt − Pt−1, the (absolute) change

yt =Pt − Pt−1Pt−1

, the relative change, and

zt = lnPt − lnPt−1 the approximate relative change

are examples of flow variables. Note that:

yt × 100 is inflation in percentage points. In this course weoften use to the rate formulation (hence, we omit the scalingby 100)

zt ≈ yt by the properties of the (natural) logarithmic function,see for example the appendix of IDM, if in doubt.


A stock variable is the cumulated sum of a flow

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75 The Norwegian current account

Bill

ion

kron

er

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5 00

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Norwegian net foreign debt

Bill

ion

kron

er

debt = −current account+ lagged debt.

If there is a primaryaccount surplus for sometime, this will lead to agradual reduction ofdebt– or an increase inthe nation’s net wealth.Conversely, a consistentcurrent account deficitraises a nation’s debt.


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