Let's break down the problem step by step: **Given:** - **Income (M):** 32 - **Price of peanut butter (\(P_{pb}\)):** 1 - **Price of jelly (\(P_{j}\)):** 4 (initial), then 6 (after increase) - **Utility function:** Standard Cobb-Douglas: \(U = x^{a} y^{b}\) Here, \(a = 0.6\) (peanut butter), \(b = 0.4\) (jelly) We are asked: **When the price of jelly rises from 4 to 6, how much income would you need to lose (keeping all else constant) to feel as bad as you do after the price increase?** This is called the **Equivalent Variation** (EV) in consumer theory. --- ## Step 1: Find Initial Optimal Bundle The utility-maximizing bundle for Cobb-Douglas with budget constraint is: - Spend fraction \(a\) of income on peanut butter - Spend fraction \(b\) of income on jelly Let \(x\) = peanut butter, \(y\) = jelly \[ x^* = \frac{a \cdot M}{P_{pb}} \] \[ y^* = \frac{b \cdot M}{P_{j}} \] **At initial prices:** \[ x_0 = \frac{0.6 \times 32}{1} = 19.2 \] \[ y_0 = \frac{0.4 \times 32}{4} = 3.2 \] --- ## Step 2: Calculate Initial Utility \[ U_0 = (x_0)^{0.6} (y_0)^{0.4} \] \[ U_0 = (19.2)^{0.6} \cdot (3.2)^{0.4} \] Let's calculate this step by step: - \(\ln(19.2) = 2.9549\), so \(0.6 \times 2.9549 = 1.7729\), \(e^{1.7729} = 5.891\) - \(\ln(3.2) = 1.1632\), so \(0.4 \times 1.1632 = 0.4653\), \(e^{0.4653} = 1.592\) So, \[ U_0 = 5.891 \times 1.592 = 9.374 \] --- ## Step 3: New Prices, Find New Optimal Bundle Now, price of jelly rises to 6. \[ x_1 = \frac{0.6 \times 32}{1} = 19.2 \] \[ y_1 = \frac{0.4 \times 32}{6} = \frac{12.8}{6} = 2.1333 \] --- ## Step 4: Calculate New Utility \[ U_1 = (19.2)^{0.6} \times (2.1333)^{0.4} \] - \(\ln(2.1333) = 0.7572\), \(0.4 \times 0.7572 = 0.3029\), \(e^{0.3029} = 1.353\) So, \[ U_1 = 5.891 \times 1.353 = 7.972 \] --- ## Step 5: Equivalent Variation Calculation Equivalent Variation (EV) = Amount of income you'd have to take away (at the original prices) to reduce utility to the new, lower level \(U_1\). So, **how much income \(M_{EV}\) at original prices gets you utility \(U_1\)?** Let \(M_{EV}\) be the income such that utility at original prices is \(U_1\): \[ U_1 = \left( \frac{0.6 M_{EV}}{1} \right)^{0.6} \left( \frac{0.4 M_{EV}}{4} \right)^{0.4} \] Let’s simplify: \[ x_{EV} = 0.6 M_{EV} \] \[ y_{EV} = \frac{0.4 M_{EV}}{4} = 0.1 M_{EV} \] \[ U_1 = (0.6 M_{EV})^{0.6} \times (0.1 M_{EV})^{0.4} \] Combine: \[ = (0.6^{0.6} \times 0.1^{0.4}) \times M_{EV}^{0.6 + 0.4} \] \[ = (0.6^{0.6} \times 0.1^{0.4}) \times M_{EV} \] Similarly, initial utility: \[ U_0 = (0.6^{0.6} \times 0.1^{0.4}) \times M \] So the utility function is **linear in income** for Cobb-Douglas when you spend fractions \(a\) and \(b\) at fixed prices. Therefore: \[ \frac{U_1}{U_0} = \frac{M_{EV}}{M} \] \[ M_{EV} = M \times \frac{U_1}{U_0} \] Let’s compute: \[ M_{EV} = 32 \times \frac{7.972}{9.374} \] \[ = 32 \times 0.8507 = 27.223 \] --- ## Step 6: Income Loss Loss = Original income - EV income \[ \text{Income Loss} = 32 - 27.223 = 4.777 \] --- ## **Final Answer (rounded to the nearest cent):** \[ \boxed{4.78} \] --- ### **Summary Table** | Step | Value | |------------------------------|---------| | Initial Utility (\(U_0\)) | 9.374 | | New Utility (\(U_1\)) | 7.972 | | Equivalent Income (\(M_{EV}\))| 27.22 | | Income Loss (rounded) | **4.78**| **Therefore, you feel like you lost \$4.78 in income.**