Quantity Theory & Inflation

Money, prices, and inflation

The quantity equation is an identity: \(M \cdot V = P \cdot Y\).

• \(M\) = money supply, \(V\) = velocity (times the average dollar is spent), \(P\) = price level, \(Y\) = real output.
• In growth rates (writing \(\Delta\log X\) for the growth rate of \(X\)): \(\Delta\log M + \Delta\log V = \Delta\log P + \Delta\log Y\). With inflation \(\pi = \Delta\log P\),

\[\pi = \Delta\log M + \Delta\log V - \Delta\log Y\]

• Quantity theory: if \(V\) is roughly constant (\(\Delta\log V \approx 0\)), then \(\pi = \Delta\log M - \Delta\log Y\) — sustained inflation comes from money growth outrunning real growth.
• Classical dichotomy: in the long run money moves only nominal things (P), not real output.

Try it. Money growth above real growth → inflation. Faster output growth absorbs money and lowers inflation. Rising velocity adds to inflation.

#| standalone: true
#| viewerHeight: 520

library(shiny)
ui <- fluidPage(
  tags$head(tags$style(HTML("body{font-family:'Inter',system-ui,sans-serif;}
    .sb{background:#f0f4f8;border-radius:6px;padding:12px 14px;margin-top:10px;font-size:15px;line-height:1.85;} .sb b{color:#1f3b73;}"))),
  sidebarLayout(
    sidebarPanel(width=4,
      sliderInput("gM","Money growth  g_M (%):",min=0,max=20,value=5,step=1),
      sliderInput("gY","Output growth  g_Y (%):",min=0,max=6,value=2,step=1),
      sliderInput("gV","Velocity growth  g_V (%):",min=-3,max=3,value=0,step=1),
      uiOutput("sb")),
    mainPanel(width=8, plotOutput("plot",height="440px"))))
server <- function(input,output,session){
  pit <- reactive(input$gM + input$gV - input$gY)
  output$plot <- renderPlot({
    yrs <- 0:10; P <- (1+pit()/100)^yrs
    par(mar=c(4.2,4.6,1.6,1))
    plot(yrs,P,type="o",col="#b5462a",lwd=3,pch=19,xlab="Years",ylab="Price level (start = 1)",
         las=1,bty="l",cex.lab=1.1,main=sprintf("Inflation pi = %d + %d - %d = %d%% per year",input$gM,input$gV,input$gY,pit()))
    abline(h=1,col="#9aa4b2",lty=3)
  })
  output$sb <- renderUI(HTML(sprintf("<div class='sb'>Inflation<br><b>pi = g_M + g_V - g_Y = %d%%</b><br>%s</div>",
    pit(), if(pit()>0) "Prices rising." else if(pit()<0) "Deflation." else "Stable prices.")))
}
shinyApp(ui,server)

What to notice

• Set \(g_V = 0\): inflation is exactly \(g_M - g_Y\) — the pure quantity theory.
• Faster output growth lowers inflation (more goods absorb the money).
• A negative velocity growth (people hold money longer) pulls inflation down even if money grows.