Problem 246727 · medium · Economics · Level 02 Choices and Multipliers

Three Levers for the Treasury

output gap · spending multiplier · tax multiplier · balanced budget · inverse problem

Equilibrium output in Marrowby (millions of crowns per year) satisfies

Y = c_0 + c_1 \, (Y - T) + I + G

with a fixed tax T, planned investment I, government purchases G and a marginal propensity to consume c1 with 0 <= c1 < 1. The treasury wants equilibrium output to equal target, and it may pull one lever:

  • "G": change government purchases by some amount x;
  • "T": change taxes by some amount x;
  • "both": change purchases and taxes by the same amount x, so the budget balance T - G stays as it is.

Write lever(c0, c1, I, G, T, target, tool) that returns the change x that moves equilibrium output exactly to target (negative for a cut). Return None if no change of the chosen lever can do it: if the lever has no effect on output (even when output is already at the target), or if the change needed would make purchases or taxes negative (ending exactly at 0 is allowed).

Examples

Input:  c0 = 80, c1 = 0.75, I = 120, G = 150, T = 120, target = 1200, tool = "G"
Output: 40.0
Explanation: output is now (80 - 90 + 120 + 150) / 0.25 = 1040, so it must rise by 160;
each crown of purchases raises it by 4.

Input:  c0 = 80, c1 = 0.75, I = 120, G = 150, T = 120, target = 1200, tool = "T"
Output: -53.333333333333336

Input:  c0 = 80, c1 = 0.75, I = 120, G = 150, T = 120, target = 1500, tool = "T"
Output: None
Explanation: the tax cut needed, 153.33, is more than the 120 collected.

Constraints

  • answers are compared with a tolerance of 1e-6; do not round

Goals

  • Turn a desired change in output into the policy change that produces it
  • Use the spending, tax and balanced-budget multipliers side by side
  • Detect policy changes that are impossible in the model
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