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 IV
The Phillips curve theory of inflation and related debates
The Phillips curve theory of inflation and related debates
Introduction
Introduction
The post-WWII American economics was dominated by ‘Keynesian economics.’ In the 1960s, a particular strand of Keynesian economics, the ‘Orthodox Keynesian economics’ (also known as the ‘Neoclassical-Keynesian synthesis’ approach to macroeconomic issues) emerged in the US academia. The leading figure in this school of thought was, among others, Paul Samuelson, James Tobin, and Robert Solow. They all embraced the ISLM interpretation of Keynes’s aggregate demand and the financial market analysis. They also adopted and used a particular inflation theory known as the Phillips curve theory of inflation for their macroeconomic policy framework.
The original Phillips curve is not truly an inflation theory. Rather it is merely an empirical observation of the correlation between the nominal wage rate and the unemployment rate for British data from 1861 to 1957 (Phillips 1958). Named after A.W. Phillips who presented a relatively stable negative correlation between the two variables, this particular observation became known as the Phillips curve.
In 1960, Paul Samuelson and Robert Solow replicated Phillips’s data-plotting exercise, this time using the US data from 1900 to 1960 (Samuleson and Solow 1960). They also discovered the similar downward slopping curve in between the change in the nominal wage rate and the unemployment rate. While presenting this correlation, Samuelson and Solow reinterpreted the change in the nominal wage rate as a proxy for the inflation rate, thereby cautiously arguing that there was a stable negative correlation between the inflation rate and the unemployment rate both in the UK and US. Since then, Phillips-Samuelson-Solow’s empirical findings became known as the Phillips curve theory of inflation that directly tied the change in the unemployment rate to the change in the price level.
For many Keynesian economists, the ascendance of the Phillips curve theory of inflation was not a pleasant development: There was no particular reason to exclusively focus on the correlation between the nominal wage rate and the unemployment rate from Keynes’s point of view. As we saw earlier, Keynes’s cost of production approach to inflation is a lot more general and inclusive: It accounts for the change in the cost of buying, leasing, and installing the capital equipment for production; it also includes the possible change in the price of intermediate goods and raw materials; the change in labor cost, wages and salaries, along the business cycle is also an important component that constitutes the cost of production that affects the changing price level. Therefore, treating the change in the nominal wage rate as if it is the most important production cost is not necessarily justifiable from the perspective of Keynes’s general theory of price. Similarly, the change in the unemployment rate is interpreted in the Phillips curve theory as if it is a good approximation of the changing aggregate output-income. But this is not readily justifiable from Keynes’s point of view, either, because the labor market outcome measured by employment and unemployment rate is merely one ex post indicator of changing aggregate output-income, which is also affected by technological efficiencies of the production activity.
Nonetheless, the Phillips curve theory of inflation quickly assumed the supreme position in Keynesian economics circle (broadly defined) and became one of the most controversial issues in modern macroeconomic debates. Many advocates of the Phillips curve believed at that time that, if there was a stable negative correlation between these two variables, this curve could provide a coherent framework for explaining the change in the general price level, too. They also believed that this stable negative correlation between the two variables had a very important policy implication: If there is a negative correlation between the two variables, economic policy makers can choose a pair of a particular level of unemployment rate and inflation rate, whenever they make both monetary and fiscal policy decision. If the economy is in a prolonged recession, for example, the government can adopt an expansionary fiscal and monetary policy to lower the unemployment rate. But this expansionary policy choice would lead to a higher inflation rate, and vice versa. This notion of “tradeoff” in macroeconomic policy making was quite influential and pervasive throughout 1960s and 1970s (Snowdon and Vane 2006).
Soon after our orthodox Keynesian economists proposed this ‘tradeoff’ idea, however, the US and many advanced capitalist market economies encountered historically unprecedented economic situations, the situation where high unemployment rate went hand in hand with higher rate of inflation (in the 1970s and early 1980s). Obviously, during this “stagflation era,” the Phillips curve relation was not observable, and the alleged tradeoff broke down. IN this theoretical juncture, both Milton Friedman and Robert Lucas stepped in this debate and introduced one powerful concept of (rational) expectations. They argued that rational economic agents could form (adaptive or rational) expectations about the future inflation rate, accounting for the possible government’s policy action. As soon as the rational economic agent’s inflation expectation is accounted for, the actual inflation rate is also affected by the agent’s inflation expectation. Simply put, the Phillips-Samuelson-Solow curve of inflation rate-unemployment rate becomes no longer valid in the face of economic agents’ inflation expectations (Friedman 1968).
This note shows a simple inflation model that captures the core ideas of both the original Phillips-Samuelson-Solow curve and its one variant, the so-called expectation-augmented Phillips curve, partly aided by Snowdon and Vane’s textbook presentation of this issue (Snowdon and Vane 2006).
A simple model of the original Phillips curve
A simple model of the original Phillips curve
The original Phillips curve
The original Phillips curve
Both the original and expectation-augmented Phillips curve model assumes that the nominal wage rate is sorely determined by the change in the unemployment rate. Therefore, relevant variables in these models are (1) an equation that describes the change in unemployment rate in the labor market, f(u), and (2) the change in nominal wage, W(t). Since the nominal wage rate is treated as a proxy for the change in the general price level, P(t), we can also treat both W(t) and P(t) as being identical.
Following notational conventions, the change in any of these variables are denoted by the differential form. That is, the change in the nominal wage is defined by = and the change in the price level by =.
W
t
W
P
t
P
As a starter, suppose that there is no meaningful role played by an inflation expectations in the wage and price level determination. Suppose that the current nominal wage or the price level is sorely determined by the going labor market condition captured by a general unemployment rate function f(u) = 1 - α u[t]. Then we can write =f[u]=1-αu[t] where u[t] is unemployment rate at time t, while α represents how strongly the change in the unemployment rate would affect the change in the nominal wage rate.
W
Clear[W,α,u,t,eqn1];eqn1={W'[t]==(1-αu[t])};DSolve[eqn1,W[t],t]
Out[]=
W[t]+(1-αu[K[1]])K[1]
1
t
∫
1
Since it is assumed that the change in the nominal wage is a proxy for the change in the general price level, we can replace W’[t] with P’[t] as follows:
Clear[P,α,u,t,eqn2];eqn2={P'[t]==(1-αu[t])};DSolve[eqn2,P[t],t]
Out[]=
P[t]+(1-αu[K[1]])K[1]
1
t
∫
1
If the initial value of P[t] at t = 0 is given by P0, we can add P[0] = P0 into the equation. If we further assume that we can normalize this initial value to be 1, P0 = 1, the equation and its solution become:
Clear[P,α,u,t,P0,eqn3];P0=1;eqn3={P'[t]==(1-αu[t]),P[0]==P0};DSolve[eqn3,P[t],t]
Out[]=
P[t]1+(1-αu[K[1]])K[1]
t
∫
0
The price level at time t, which starts from 1, is determined by the integrated value of the unemployment rate over the period. The solution to this linear first-order differential equation takes the integral sign because u[t] is not yet fully defined. The exact behavior of the solution curve depends on the time-dependent trajectory of the unemployment rate, u[t], and the parameter, α, that shows how strongly the unemployment rate would affect the current price level.
The expectation-augmented Phillips curve
The expectation-augmented Phillips curve
A differential equation model
A differential equation model
Suppose now that the price level is also determined by workers’ collective desire and action to increase their nominal wage level at the same time, which is, in turn, affected by their inflation expectations. Then we can write an equation, =f[u]+ where (replaced by PE in the Mathematica syntax below) is the inflation expectations variable and the parameter β captures how strongly the change in the nominal wage rate is influenced by workers’ inflation expectations. If we further assume that the general unemployment rate function is a simple linear one such as f[u] = 1 - α u[t] as before, then we can write the following syntax:
W
e
βP
e
P
Clear[P,PE,W,u,t,α,β,eqn4];eqn4={P'[t]==(1-αu[t])+βPE'[t]};DSolve[eqn4,P[t],t]
Out[]=
P[t]+(1-αu[K[1]]+β[K[1]])K[1]
1
t
∫
1
′
PE
If the initial value of P[t] at t = 0 is given by P0, we can add P[0] = P0 = 1 into the equation, whose solution becomes:
Clear[P,PE,P0,u,t,α,β,eqn5];P0=1;eqn5={P'[t]==(1-αu[t])+βPE'[t],P[0]==P0};DSolve[eqn5,P[t],t]
Out[]=
P[t]1+(1-αu[K[1]]+β[K[1]])K[1]
t
∫
0
′
PE
The price level at time t, which starts from P0=1, is determined by both the integrated value of the price expectations and the integrated value of unemployment rate. The solution to this linear first-order differential equation takes the integral sign because both PE[t] and u[t] are not yet fully defined.
Now suppose for simplicity that the change in inflation happens to be equal to the expected inflation for some reasons. That is, P’[t] = PE’[t]. Then the equation and its general solution become:
Clear[P,PE,P0,u,t,α,β,eqn6];eqn6={(1-β)P'[t]==(1-αu[t]),P[0]==P0};P0=1;DSolve[eqn6,P[t],t]
Out[]=
P[t]1+K[1]
t
∫
0
-1+αu[K[1]]
-1+β
In this case, the exact behavior of the solution curve depends on time-dependent trajectory of the unemployment rate and the two parameters that show how strongly the unemployment rate and inflation expectations would have on the current price level. If the unemployment rate at time t, u(t), is defined, then u(t) can be specified as a function of P(t) and the related parameters. The unemployment rate that is defined in this way is called the natural rate of unemployment, the rate of unemployment that is consistent with P’[t] = PE’[t]. This natural rate of unemployment is also called NAIRU (non-accelerating inflation rate of unemployment).
A difference equation model
A difference equation model
In some contexts, a difference equation model is more insightful than the differential equation one because the former enables us to explicitly describe the expectation formation process. For example, suppose that economic agents form their expectations about the future inflation rate by looking at the difference between the inflation rates observed in the past two time periods. In this case, we can set PE[t] = β (P[t-1]-P[t-2]) where the parameter β represents how agents react to the change in the inflation rate between the two time periods.
Using this idea, we can rewrite the previous differential equation-based model in the following way:
Clear[P,PE,P0,u,t,α,β,eqn7];P0=1;eqn7={P[t]==(1-αu[t-1])+β(P[t-1]-P[t-2]),P[0]==P0};RSolve[eqn7,P[t],t]
Out[]=
P[t]---+-(-1+αu[1+K[1]])-(-1+αu[1+K[2]])
-t
2
t
-4+β
β
+βt
-
-4+β
β
+β
1
t
-4+β
β
+β
1
t
-
-4+β
β
+β-1+t
∑
K[1]=-1
1
-4+β
β-1+K[1]
2
-4+β
+β
-K[1]
-
-4+β
β
+βt
-4+β
β
+β-1+t
∑
K[2]=-1
1
-4+β
β-1+K[2]
2
-4+β
-β
-K[2]
-4+β
β
+βThe solution curve is qualitatively the same as before. The only difference is that now the inflation expectations variable, PE[t], is replaced by some lagged reaction to the discrete difference in the two actual inflation rates. Once we introduce the equation that explains the change in u(t), we can find a more concrete analytical solution for this difference equation, as well.
Conclusion
Conclusion
One fundamental assumption in the Phillips curve theory of inflation is that the price level is sorely determined by the change in the unemployment rate. If we introduce some additional non-wage cost variables, such as the utilization of the fixed capital equipment and intermediate goods and their varying cost, the original Phillips curve model should be modified accordingly. In this note, we have confined our focus of analysis to merely reconstruct the original Phillips curve and the expectation-augmented Phillips curve model using both difference and differential equation.
One crucial variable that is not properly specified in this note is the unemployment rate. We can easily write down a reasonable behavioral equation for this variable, once we consider relevant macroeconomic variables such as the change in aggregate output, investment spending, the capital stock, the capital-labor ratio, and the growth of labor force.
Reference
Reference
A.W. Phillips, “The relation between unemployment and the rate of change of money wage rate in the United Kingdom, 1861-1957,” Economica, New Series, Vol. 25, No. 100 (Nov. 1958), pp.283-299.
Brian Snowdon and Howard Vane, Modern Economics - Its Origins. Development and Current State. Cheltenham, UK and Northampton, MA, USA: Edward Elgar, 2006, pp.135-144 and pp.174-187.
John M. Keynes, The General Theory of Employment, Interest and Money. London: Macmillan, 1997 [1936].
Milton Friedman, “The role of monetary policy,” The American Economic Review, Vol. 58, No. 1 (March 1968), pp.1-17.
Paul A. Samuelson, and Robert M. Solow, “Analytical aspects of anti-inflation policy,” The American Economic Review, Vol. 50, No. 2 (May 1960), pp.177-194.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Introduction to dynamic economic models: dynamic inflation models IV
by Hee-Young Shin
Wolfram Community, STAFF PICKS, December 18, 2025
https://community.wolfram.com/groups/-/m/t/3593669
by Hee-Young Shin
Wolfram Community, STAFF PICKS, December 18, 2025
https://community.wolfram.com/groups/-/m/t/3593669