Formula for Change in Output from an Autonomous Spending Shock

An economy's equilibrium output is initially described by the equation Y₁ = k(c₀ + I). After an increase in investment (ΔI), the new equilibrium is Y₂ = k(c₀ + I + ΔI). To find the total change in output (ΔY), we calculate ΔY = Y₂ - Y₁. Which algebraic step is the most direct and crucial for simplifying the resulting expression, ΔY = k(c₀ + I + ΔI) - k(c₀ + I), to its final, most concise form?

Arrange the following algebraic expressions in the correct logical sequence to derive the relationship between a change in investment (ΔI) and the resulting change in equilibrium output (ΔY).

Analysis of the Output Change Derivation

In the process of determining the total change in equilibrium output (ΔY) following a change in investment (ΔI), the algebraic derivation starts with ΔY = k(c₀ + I + ΔI) - k(c₀ + I) and simplifies to ΔY = k * ΔI. What is the key economic insight revealed by the fact that the k(c₀ + I) terms cancel out during this simplification?