13  Interactive: multiplier dashboard

Drag the MPC slider. Every multiplier updates.

viewof mpc = Inputs.range([0.05, 0.99], {value: 0.75, step: 0.01, label: "MPC (c)"})

mults = ({
  K_G: 1 / (1 - mpc),
  K_I: 1 / (1 - mpc),
  K_C0: 1 / (1 - mpc),
  K_EX: 1 / (1 - mpc),
  K_T: -mpc / (1 - mpc),
  K_IM: -1 / (1 - mpc),
  K_BB: 1
})

Inputs.table([
  {name: "K_G  (gov spending)", value: mults.K_G, sign: "+"},
  {name: "K_I  (investment)", value: mults.K_I, sign: "+"},
  {name: "K_C0  (subsistence consumption)", value: mults.K_C0, sign: "+"},
  {name: "K_EX  (exports)", value: mults.K_EX, sign: "+"},
  {name: "K_T  (taxes)", value: mults.K_T, sign: "−"},
  {name: "K_IM  (autonomous imports)", value: mults.K_IM, sign: "−"},
  {name: "K_BB  (balanced budget)", value: mults.K_BB, sign: "+"},
], {
  format: {value: x => x.toFixed(2)},
  width: {name: 280, value: 100, sign: 60}
})

md`### To close a recessionary gap of \$100 with each instrument

| instrument | required change |
|---|---:|
| Increase G alone | \$${(100/mults.K_G).toFixed(2)} |
| Cut T alone | \$${(100/mults.K_T).toFixed(2)} |
| Balanced budget (raise both G and T equally) | \$${(100/mults.K_BB).toFixed(2)} each |
`

13.1 What to notice

• As MPC approaches 1, every multiplier blows up. K_G at c=0.95 is 20. At c=0.99 it’s 100. Real economies don’t behave this way because of leakages, so the textbook formula is the upper bound.
• Tax multiplier is always smaller in magnitude than the spending multiplier, by exactly 1. The K_{BB} = K_G + K_T = 1 identity falls out of this.
• At MPC = 0.75, the values you’ll see most: K_G = 4, K_T = -3, K_{BB} = 1. Memorize these.

13.2 Why |K_T| < K_G

A dollar of G enters AE^d in full immediately. A dollar of tax cut adds only c to round-one spending — households save (1-c) of the tax cut. The lost first round is exactly the gap.

K_G - |K_T| = \frac{1}{1-c} - \frac{c}{1-c} = \frac{1-c}{1-c} = 1

That gap is also the balanced-budget multiplier. Same identity, two ways to read it.