12 Interactive: Keynesian cross

Move the sliders. Watch the equilibrium Y^* and the multiplier respond.

viewof c = Inputs.range([0.1, 0.95], {value: 0.75, step: 0.01, label: "MPC (c)"})
viewof C0 = Inputs.range([0, 200], {value: 100, step: 5, label: "Subsistence consumption (C₀)"})
viewof Id = Inputs.range([0, 200], {value: 25, step: 5, label: "Investment Id"})

Ystar = (C0 + Id) / (1 - c)
multiplier = 1 / (1 - c)

md`### Live results

  - **Equilibrium output:** \`Y* = (${C0} + ${Id}) / (1 − ${c.toFixed(2)}) = ${Ystar.toFixed(1)}\`
  - **Spending multiplier:** \`K = 1 / (1 − ${c.toFixed(2)}) = ${multiplier.toFixed(2)}\`
  - **Tax multiplier:** \`K_T = −${c.toFixed(2)} / (1 − ${c.toFixed(2)}) = ${(-c/(1-c)).toFixed(2)}\`
`

Plot.plot({
  width: 600,
  height: 400,
  marginLeft: 60,
  marginBottom: 50,
  x: {label: "Aggregate output / income, Y", domain: [0, Math.max(800, Ystar*1.4)]},
  y: {label: "Aggregate desired expenditure, AE^d", domain: [0, Math.max(800, Ystar*1.4)]},
  marks: [
    Plot.line(d3.range(0, Math.max(800, Ystar*1.4), 5).map(y => ({y: y, ae: y})), {x: "y", y: "ae", stroke: "#888", strokeWidth: 1.2}),
    Plot.line(d3.range(0, Math.max(800, Ystar*1.4), 5).map(y => ({y: y, ae: C0 + Id + c*y})), {x: "y", y: "ae", stroke: "#0e7c7b", strokeWidth: 2.2}),
    Plot.dot([{y: Ystar, ae: Ystar}], {x: "y", y: "ae", fill: "#b7410e", r: 6}),
    Plot.text([{y: Ystar, ae: Ystar, l: `Y* = ${Ystar.toFixed(0)}`}], {x: "y", y: "ae", text: "l", dx: 12, dy: -8, fill: "#b7410e", fontSize: 12, fontWeight: "bold"})
  ]
})

12.1 Things to play with

Set c = 0.5. Note K = 2. Now move c to 0.9. K jumps to 10. The multiplier is wildly nonlinear in MPC. A small change near 1 produces a huge change in K.
Hold c fixed at 0.75. Move \overline{I} from 25 to 30 (a \Delta\overline{I} = 5). Watch Y^* jump by 20. That’s K_I \cdot \Delta\overline{I} = 4 \cdot 5 = 20.
Move \overline{C_0} from 100 to 50 (people more frugal). Y^* falls. The paradox of thrift in action.

12.2 What this tells you about exam problems

Most equilibrium-Y^* exam questions can be checked here in 30 seconds. If your hand calculation gives a different answer than this widget, the widget is right and your math has a typo somewhere.