Let’s break this down step by step. **Given:** - Average taxpayer income = $59,000 - Current average tax rate = 20% - Current average tax revenue per taxpayer = $59,000 × 20% = $11,800 - Congress wants an **8.5% increase** in tax revenue. **Let \( t \) = new average tax rate (as a decimal).** **Income Effect:** With the income effect, when the tax rate increases, people choose to work less, so taxable income **decreases** as the tax rate increases. But in this problem, it says "**If the income effect is in effect for all taxpayers**". This typically means that when tax rates increase, taxable income will decrease *proportionally* to the after-tax rate. **Standard Income Effect Formula (for small changes):** New income = Old income × (1 - new tax rate) / (1 - old tax rate) So: \[ \text{New Tax Revenue} = \text{New Tax Rate} \times \text{New Income} \] \[ \text{New Income} = \$59,000 \times \frac{1 - t}{1 - 0.20} \] \[ \text{New Tax Revenue} = t \times \$59,000 \times \frac{1 - t}{1 - 0.20} \] **We want this to be 8.5% higher:** \[ \text{New Tax Revenue} = 1.085 \times \$11,800 \] \[ t \times \$59,000 \times \frac{1 - t}{0.8} = 1.085 \times \$11,800 \] \[ t \times 59,000 \times (1 - t) = 0.8 \times 1.085 \times 11,800 \] \[ t \times 59,000 \times (1 - t) = 0.8 \times 12,803 = 10,242.4 \] \[ t \times (1 - t) = \frac{10,242.4}{59,000} \] \[ t \times (1 - t) = 0.1736 \] \[ t - t^2 = 0.1736 \] \[ t^2 - t + 0.1736 = 0 \] This is a quadratic equation in the form \( at^2 + bt + c = 0 \): \[ t^2 - t + 0.1736 = 0 \] **Solve using the quadratic formula:** \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = -1, c = 0.1736 \): \[ t = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times 0.1736}}{2 \times 1} \] \[ t = \frac{1 \pm \sqrt{1 - 0.6944}}{2} \] \[ t = \frac{1 \pm \sqrt{0.3056}}{2} \] \[ \sqrt{0.3056} \approx 0.5528 \] So: \[ t_1 = \frac{1 + 0.5528}{2} = \frac{1.5528}{2} = 0.7764 \] \[ t_2 = \frac{1 - 0.5528}{2} = \frac{0.4472}{2} = 0.2236 \] Only \( t_2 \) makes sense (since 77% is not a reasonable tax rate). ### **Final Answer (rounded to 2 decimal places):** \[ \boxed{22.36\%} \] **The average tax rate required to increase tax revenues by 8.5% (with the income effect) is 22.36%.**

Let's carefully work through the problem step by step. --- ### **Given Data:** - Current average income per taxpayer: **\$59,000** - Current average tax rate: **20% (0.20)** - Current tax revenue per taxpayer: \[ \$59,000 \times 0.20 = \$11,800 \] - Desired increase in tax revenue: **8.5%** --- ### **Step 1: Find the target tax revenue after increase** \[ \text{Target revenue} = \$11,800 \times (1 + 0.085) = \$11,800 \times 1.085 = \$12,803 \] --- ### **Step 2: Account for the income effect** When tax rates increase, taxable income decreases proportionally to the change in after-tax income. The income effect implies: \[ \text{New income} = \text{Original income} \times \frac{1 - t}{1 - 0.20} \] where: - \( t \) = new tax rate (unknown, as a decimal) --- ### **Step 3: Express new tax revenue** \[ \text{New tax revenue} = t \times \text{New income} \] \[ = t \times \$59,000 \times \frac{1 - t}{0.8} \] ### **Step 4: Set the new tax revenue equal to the target** \[ t \times \$59,000 \times \frac{1 - t}{0.8} = \$12,803 \] Multiply both sides by 0.8: \[ t \times \$59,000 \times (1 - t) = 0.8 \times 12,803 = 10,242.4 \] Divide both sides by 59,000: \[ t \times (1 - t) = \frac{10,242.4}{59,000} \approx 0.1736 \] --- ### **Step 5: Formulate the quadratic** \[ t(1 - t) = 0.1736 \] \[ t - t^2 = 0.1736 \] \[ t^2 - t + 0.1736 = 0 \] --- ### **Step 6: Solve the quadratic** Using the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where: - \( a = 1 \) - \( b = -1 \) - \( c = 0.1736 \) Calculate discriminant: \[ \Delta = (-1)^2 - 4 \times 1 \times 0.1736 = 1 - 0.6944 = 0.3056 \] Square root: \[ \sqrt{0.3056} \approx 0.5528 \] Find the roots: \[ t = \frac{1 \pm 0.5528}{2} \] - **First root:** \[ t = \frac{1 + 0.5528}{2} = \frac{1.5528}{2} = 0.7764 \] - **Second root:** \[ t = \frac{1 - 0.5528}{2} = \frac{0.4472}{2} = 0.2236 \] Since the current tax rate is 20%, the plausible new tax rate is **about 22.36%** (the second root), as the first root (77.64%) is unrealistic for this context. --- ### **### **Final Answer:** **The new average tax rate needed is approximately:** \[ \boxed{\mathbf{22.36\%}} \] --- **Summary:** To increase tax revenues by 8.5%, considering the income effect, the average tax rate should be approximately **22.36%**.

Let's carefully analyze the problem step by step. --- ### Given Data: - **Average income per taxpayer:** \$59,000 - **Current average tax rate:** 20% (0.20) - **Current tax revenue per taxpayer:** \[ \$59,000 \times 0.20 = \$11,800 \] - **Desired increase in tax revenue:** 8.5% --- ### Step 1: Calculate the target tax revenue \[ \text{Target revenue} = \$11,800 \times (1 + 0.085) = \$11,800 \times 1.085 = \$12,803 \] --- ### Step 2: Account for the income effect The income effect implies that when the tax rate increases from 20% to a new rate \( t \), taxable income **decreases proportionally** due to reduced work effort or income. This can be modeled as: \[ \text{New income} = \text{Original income} \times \frac{1 - t}{1 - 0.20} \] since the after-tax income at the original rate is \( 0.80 \times \$59,000 \), and at the new rate \( t \), it becomes \( (1 - t) \times \$59,000 \). --- ### Step 3: Express the new tax revenue \[ \text{New tax revenue} = t \times \text{New income} \] \[ = t \times \$59,000 \times \frac{1 - t}{0.8} \] Our goal: \[ t \times \$59,000 \times \frac{1 - t}{0.8} = \$12,803 \] Multiply both sides by 0.8: \[ t \times \$59,000 \times (1 - t) = 0.8 \times \$12,803 = \$10,242.4 \] Divide both sides by \$59,000: \[ t \times (1 - t) = \frac{\$10,242.4}{\$59,000} \approx 0.1736 \] --- ### Step 4: Formulate the quadratic equation \[ t(1 - t) = 0.1736 \] \[ t - t^2 = 0.1736 \] \[ t^2 - t + 0.1736 = 0 \] --- ### Step 5: Solve the quadratic Using the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a=1 \), \( b=-1 \), \( c=0.1736 \): \[ \Delta = (-1)^2 - 4 \times 1 \times 0.1736 = 1 - 0.6944 = 0.3056 \] \[ \sqrt{\Delta} \approx 0.5528 \] Calculate roots: \[ t = \frac{1 \pm 0.5528}{2} \] - **First root:** \[ t = \frac{1 + 0.5528}{2} = \frac{1.5528}{2} = 0.7764 \] - **Second root:** \[ t = \frac{1 - 0.5528}{2} = \frac{0.4472}{2} = 0.2236 \] ### Step 6: Interpret the solutions - The current rate is 20%. - The plausible new rate is **approximately 22.36%** (second root) because it is close to the current rate and makes sense economically. --- ### **Final Answer:** **The average tax rate that will result in an 8.5% increase in tax revenue, considering the income effect, is approximately:** \[ \boxed{\mathbf{22.36\%}} \] --- If you have any further questions or need clarification, feel free to ask!