Break it down how to solve the equation

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To break down how to solve for income elasticity, let's first understand the formula and what each part represents.

The Income Elasticity of Demand (ϵY\epsilon_Y) measures how responsive the quantity demanded of a good is to a change in consumers' income.

The formula is:

ϵY=Percentage Change in Quantity DemandedPercentage Change in Income\epsilon_Y = \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Income}}

This can be expanded into the point elasticity formula:

ϵY=ΔQ/QΔY/Y=ΔQΔY×YQ\epsilon_Y = \frac{\Delta Q / Q}{\Delta Y / Y} = \frac{\Delta Q}{\Delta Y} \times \frac{Y}{Q}

Where:

  • ΔQ\Delta Q (Delta Q) represents the change in quantity demanded.
  • QQ represents the initial quantity demanded.
  • ΔY\Delta Y (Delta Y) represents the change in income.
  • YY represents the initial income.

Let's use the example of the movement from Point A to Point B from your previous problem to illustrate the breakdown:

  • Point A (Initial): Income (Y1Y_1) = 100, Quantity (Q1Q_1) = 200
  • Point B (New): Income (Y2Y_2) = 200, Quantity (Q2Q_2) = 400

  1. Calculate the change in quantity demanded (ΔQ\Delta Q).

    This is the difference between the new quantity and the initial quantity.

    ΔQ=Q2−Q1=400−200=200\Delta Q = Q_2 - Q_1 = 400 - 200 = 200

  2. Calculate the change in income (ΔY\Delta Y).

    This is the difference between the new income and the initial income.

    ΔY=Y2−Y1=200−100=100\Delta Y = Y_2 - Y_1 = 200 - 100 = 100

  3. Identify the initial quantity (QQ) and initial income (YY).

    For the point elasticity formula, we use the values from the starting point (Point A in this case).

    Q=Q1=200Q = Q_1 = 200 Y=Y1=100Y = Y_1 = 100

  4. Substitute these values into the income elasticity formula.

    ϵY=ΔQΔY×YQ\epsilon_Y = \frac{\Delta Q}{\Delta Y} \times \frac{Y}{Q} ϵY=200100×100200\epsilon_Y = \frac{200}{100} \times \frac{100}{200}

  5. Perform the multiplication to find the income elasticity.

    First, simplify the fractions:

    ϵY=2×0.5\epsilon_Y = 2 \times 0.5

    Then, multiply:

    ϵY=1\epsilon_Y = 1

  6. Interpret the result.

    • If ϵY>0\epsilon_Y > 0 (positive), the good is a normal good. This means as income increases, demand for the good also increases.
      • If 0<ϵY<10 < \epsilon_Y < 1, it's a necessity.
      • If ϵY>1\epsilon_Y > 1, it's a luxury.
    • If ϵY<0\epsilon_Y < 0 (negative), the good is an inferior good. This means as income increases, demand for the good decreases.
    • If ϵY=0\epsilon_Y = 0, the good is an income-independent good. Demand does not change with income.

    In this case, ϵY=1\epsilon_Y = 1, which is positive. Therefore, the good is a normal good (specifically, unit elastic with respect to income).

    The income elasticity from A to B is 1.

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