This notebook series is designed to introduce selected dynamic economic models to both advanced undergraduate and graduate students. Economists have relied on a series of economic models to better understand the otherwise highly complex causal relationship among many economic variables. Dynamic economic models set up by systems of either difference or differential equations are particularly useful in tracing the change in the quantity of unknown economic variables (e.g., the capital stock, demand for labor, aggregate output, etc.) and their interactions over time.
This notebook series aims to help students better understand the depth, usefulness, and limitations of major existing micro and macroeconomic models by demonstrating how we can use Wolfram Mathematica to replicate and reconstruct the core ideas and find dynamic solution of the model.
The model introduced in this notebook series include, among others, (1) the cobweb model of the quantity and price adjustment, (2) Keynes’s aggregate demand model, (3) Hicks’s ISLM model, (4) Samuelson’s investment accelerator model, (5) Kalecki’s investment delay model, (6) the predator-prey interaction model, (7) the epidemic model, (8) Ricardo’s three dimensional business cycle model, (9) Marx-Goodwin business cycle and growth model, (10) Smith’s and Ricardo’s growth model, (11) the Harrod-Domar growth model, (12) the Solow growth model, (13) the RBC model, (14) Romer’s new growth model, and (15) the circuit of capital model.
Dynamic inflation models II
The expectation-based monetarist inflation dynamics
The expectation-based monetarist inflation dynamics
Introduction
Introduction
Cagan’s 1956 model is a prototype monetarist inflation model in which the change in money supply and the agent’s expectation about the future inflation positively feedback with one another. Cagan model of inflation is one of many possible ways of reconstructing classical quantity of money theory of inflation. The novel feature of his model is that it allows us to formulate and experiment with various inflation expectations formation process. Other than that, Cagan model of inflation shares the common assumptions and aspects of classical monetarist inflation models: The velocity and aggregate output do not change. The inflation is fundamentally driven by the change in the quantity of money in disregard to any non-monetary variables.
In this section, we present how to derive crucial variables and parameters used in this model and show the dynamic interaction between inflation expectations and the actual inflation rates. This part of note is greatly aided by Raghavendra and Piiroinen (2023)’s textbook style presentation.
The model setup and derivation
The model setup and derivation
One crucial variable in Cagan’s model is the money demand. Cagan proposes the idea that the public’s demand for money is a decreasing function of expected inflation. That is, the demand for money, Md[t] is negatively correlated with the expected rate of inflation in the future period, π e[t+1].
Md[t]=
-απe[t+1]
(
1
)where the expected inflation rate is the expected growth rate of price level between time t+1 and t. That is, π e[t+1] = (Pe[t+1]- P[t])/P[t]
The money supply Ms[t] is assumed to be exogenously given in the model, and the real money supply is defined as:
Ms[t]/P[t]
(
2
)If the real money demand equals the real money supply, we get the following equation:
Md[t]=Ms[t]/P[t]=
-απe[t+1]
(
3
)By taking logarithm,
ln(M[t])-ln(P[t])=-απe[t+1]
(
4
)The inflation expectation can be also approximated by
πe[t+1]=(Pe[t+1]-P[t])/P[t]≈ln(Pe[t+1]-P[t])
(
5
)Therefore, the equation (4) can be written by combining the equation (5):
ln(M[t])-ln(P[t])=-α(ln(Pe[t+1]-P[t]))
(
6
)If we rearrange, we get:
ln(P[t])=ln(Pe[t+1])+ln(M[t])
α
1+α
1
1+α
(
7
)which is the same as
p[t]=pe[t+1]+m[t]
α
1+α
1
1+α
(
8
)where lower case p[t] = ln (P[t]) and m[t] = ln(M[t]).
This is a linear first-order difference equation, whose solution can be determined by the given parameters, α, and the exogenously given quantity of money, m[t]. But a more crucial variable is the price expectation, Pe[t+1].
Equations for the price expectation
Equations for the price expectation
Following conventional ways of formulating the expectation variable, we can choose one of the following three price expectation process:
Naive expectation
Naive expectation
The first case is to set the price expectation at time t + 1 equal to the price level at time t and some adjustment in the past two price levels at time t and t-1. Economic agents expect and adjust their expectation about the price level as a function of the past two price levels. Mathematically, we can write pe[t+1] = p[t] + γ (p[t]-p[t-1]) where γ is an adjustment factor. This equation can be rewritten as:
pe[t+1]-p[t]=γ(p[t]-p[t-1])
Perfect foresight
Perfect foresight
The perfect foresight model means that the expected price at time t+1 is equal to the actual price level at the same time. That is:
pe[t+1]=p[t+1]
Rational expectation
Rational expectation
Finally, the rational expectation requires that economic agents expect that the price level at time t+1 is equal to the change in money supply augmented by some random behaviors.
pe[t+1]=E[t][p[t+1]]=m[t]+ϵ[t]
We can solve the original price/inflation equation by replacing the price expectation variable with the three possible equations.
Numerical solutions and visualizations with fixed parameters
Numerical solutions and visualizations with fixed parameters
Naive expectation solution
Naive expectation solution
In[]:=
Clear[p,pe,m,t,α,γ];Solvep[t]==pe[t+1]+m[t]/.pe[t+1]->p[t]+γ(p[t]-p[t-1]),p[t]//Expand
α
1+α
1
1+α
Out[]=
p[t]-
m[t]
(1+α)1--
α
1+α
αγ
1+α
αγp[-1+t]
(1+α)1--
α
1+α
αγ
1+α
If m[t] = m is given, and the initial value of p[0]=p0, then the solution curve is:
In[]:=
RSolvep[t]==-,p[0]==p0,p[t],t
m
(1+α)1--
α
1+α
αγ
1+α
αγp[-1+t]
(1+α)1--
α
1+α
αγ
1+α
Out[]=
p[t]m-m+p0
t
αγ
-1+αγ
t
αγ
-1+αγ
Suppose m =10, α=0.5, γ=0.5, and p0 =11.
In[]:=
m=10;α=0.5;γ=0.5;p0=11;RSolvep[t]==-,p[0]==p0,p[t],t[[1]]
m
(1+α)1--
α
1+α
αγ
1+α
αγp[-1+t]
(1+α)1--
α
1+α
αγ
1+α
Out[]=
{p[t]30.(0.333333+0.366667+)}
t
(-0.333333)
1+t
(-0.333333)
In[]:=
DiscretePlot30.000000000000007`(0.33333333333333326`+0.3666666666666666`+),{t,0,10},PlotRange->Full,Joined->False
t
(-0.33333333333333326`)
1+t
(-0.33333333333333326`)
Out[]=
Suppose this time that m =10, α=0.8, γ=0.8, and p0 =11.
In[]:=
m=10;α=0.8;γ=0.8;p0=11;RSolvep[t]==-,p[0]==p0,p[t],t[[1]]
m
(1+α)1--
α
1+α
αγ
1+α
αγp[-1+t]
(1+α)1--
α
1+α
αγ
1+α
Out[]=
{p[t]11.(-0.909091+0.909091+1.)}
-3.t
2.
t
(-14.2222)
3.t
2.
t
(-1.77778)
3.t
2.
In[]:=
p[t]/.{p[t]11.(-0.909091+0.909091+1.)}
-3.t
2.
t
(-14.2222)
3.t
2.
t
(-1.77778)
3.t
2.
Out[]=
11.(-0.909091+0.909091+1.)
-3.t
2.
t
(-14.2222)
3.t
2.
t
(-1.77778)
3.t
2.
In[]:=
DiscretePlot11.`-0.909090909090909`+0.909090909090909`+1.`,{t,0,10},PlotRange->Full,Joined->False
-3.`t
2.`
t
(-14.222222222222227`)
3.`t
2.`
t
(-1.7777777777777783`)
3.`t
2.`
Out[]=
Finally, if α=4 and γ=0.8,
In[]:=
m=10;α=4;γ=0.8;p0=11;RSolvep[t]==-,p[0]==p0,p[t],t[[1]]
m
(1+α)1--
α
1+α
αγ
1+α
αγp[-1+t]
(1+α)1--
α
1+α
αγ
1+α
Out[]=
{p[t](10.+11.-10.)}
-1.t
5.16037×
13
10
1.t
0.6875
t
7.506×
13
10
1.t
7.506×
13
10
1.t
0.6875
t
1.09178×
14
10
In[]:=
p[t]/.{p[t](10.+11.-10.)}
-1.t
5.16037×
13
10
1.t
0.6875
t
7.506×
13
10
1.t
7.506×
13
10
1.t
0.6875
t
1.09178×
14
10
Out[]=
-1.t
5.16037×
13
10
1.t
0.6875
t
7.506×
13
10
1.t
7.506×
13
10
1.t
0.6875
t
1.09178×
14
10
In[]:=
DiscretePlot(10.000000000000005`+11.`-10.000000000000005`),{t,0,10},PlotRange->Full,Joined->False
-1.`t
5.160374573028745`*^13
1.`t
0.6875000000000002`
t
7.5059993789509`*^13
1.`t
7.5059993789509`*^13
1.`t
0.6875000000000002`
t
1.0917817278474033`*^14
Out[]=
Perfect foresight solution
Perfect foresight solution
If the agent has perfect foresight, that is, pe[t+1] =p[t+1], the inflation dynamics and its solution becomes:
Suppose m = 10, α = 4, and p0=11, then the solution curve behaves:
Rational expectation solution
Rational expectation solution
The rational expectation hypothesis states that the expected price level at t+1 is equal to its notional expected value of the price level at the same period, pe[t+1] = E[t][p[t+1]]. If the agent believes that the money supply process follows a stochastic process such as m[t] = m+ϵ[t] where ϵ[t] is a random variable with zero mean, the original price dynamics becomes a linear equation with the stochastic term as follows:
If m = 10, α = 4, p0=11 and ϵ ∼N(0,1),
Conclusion
Conclusion
This section reconstructs one famous monetarist inflation dynamic model with the three possible formulations of price/inflation expectations. One fundamental assumption in this model, just as in many other monetarist inflation models, is that the money supply play a crucial role in affecting the agent’s price/inflation expectations without reference to the change in aggregate output and demand. Therefore, this model does not allow us to see another important inflation mechanism, in which the interaction between the change in aggregate output and demand drives up the price level and inflation rate. The change in the cost of production in the economy, not the quantity of money, can substantially affect the overall price level and its change. But none of these factors appears in this expectation-based monetarist theory of inflation.
Reference
Reference
Srinivas Raghavendra and Petri T. Piiroinen, An Introduction to Economic Dynamics - Modelling, Analysis and Simulation, Routledge: 2023, pp.91-108.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Introduction to dynamic economic models: dynamic inflation models II
by Hee-Young Shin
Wolfram Community, STAFF PICKS, December 17, 2025
https://community.wolfram.com/groups/-/m/t/3592670
by Hee-Young Shin
Wolfram Community, STAFF PICKS, December 17, 2025
https://community.wolfram.com/groups/-/m/t/3592670