Leaks in the Spending Stream

Brightwater is an island that taxes income and buys goods from abroad. All amounts are millions of crowns per year:

• taxes: T = T_0 + t \, Y (a fixed part T0 plus a share t of income, written as a fraction, e.g. 25 % is 0.25)
• consumption: C = c_0 + c_1 \, (Y - T)
• imports: M = m \, Y (a share m of income is spent on foreign goods)
• planned investment I, government purchases G and exports X are fixed.

Output is in equilibrium when it equals planned spending on Brightwater's goods:

The multiplier is the change in equilibrium output per one-crown rise in G, all else fixed. Net exports are X - M.

Write leaky_economy(c0, c1, T0, t, m, I, G, X) that returns a tuple (Y, multiplier, taxes, net_exports) at the equilibrium.

Examples

Input:  c0 = 100, c1 = 0.8, T0 = 0, t = 0.25, m = 0.1, I = 200, G = 300, X = 150
Output: (1500.0, 2.0, 375.0, 0.0)
Explanation: at Y = 1500 taxes are 375, consumption is 100 + 0.8 * 1125 = 1000 and imports 150,
so planned spending is 1000 + 200 + 300 + 150 - 150 = 1500.

Input:  c0 = 50, c1 = 0.75, T0 = 20, t = 0.2, m = 0.2, I = 100, G = 150, X = 0
Output: (475.0, 1.6666666666666667, 115.0, -95.0)

Constraints

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

Goals

• Solve for equilibrium output when taxes and imports rise with income
• Derive the multiplier with income-dependent leakages
• Report the tax revenue and the trade balance at equilibrium