# Problem 1 Step-by-Step Solution --- ## **Given:** Utility function: \( u(x_1, x_2) = 6x_1^2 + x_2 \) Prices: \( p_1, p_2 \) Income: \( I \) Constraints: \( x_1 \geq 0, x_2 \geq 0 \) --- ## **A. (a) Utility Maximization Problem** ### **Formulation:** Maximize \( u(x_1, x_2) = 6x_1^2 + x_2 \) subject to \( p_1 x_1 + p_2 x_2 \leq I \), \( x_1 \geq 0 \), \( x_2 \geq 0 \) ### **Lagrangian:** \[ \mathcal{L} = 6x_1^2 + x_2 + \lambda (I - p_1 x_1 - p_2 x_2) \] ### **First-Order Conditions (FOCs):** \[ \frac{\partial \mathcal{L}}{\partial x_1} = 12x_1 - \lambda p_1 = 0 \] \[ \frac{\partial \mathcal{L}}{\partial x_2} = 1 - \lambda p_2 = 0 \] \[ \frac{\partial \mathcal{L}}{\partial \lambda} = I - p_1 x_1 - p_2 x_2 = 0 \] #### **Solve for \( x_1, x_2, \lambda \):** From FOCs: 1. \( 12x_1 = \lambda p_1 \implies \lambda = \frac{12x_1}{p_1} \) 2. \( 1 = \lambda p_2 \implies \lambda = \frac{1}{p_2} \) Set equal: \[ \frac{12x_1}{p_1} = \frac{1}{p_2} \implies x_1^* = \frac{p_1}{12p_2} \] Substitute into the budget constraint: \[ p_1 x_1^* + p_2 x_2^* = I \implies p_1\left(\frac{p_1}{12p_2}\right) + p_2 x_2^* = I \] \[ \frac{p_1^2}{12p_2} + p_2 x_2^* = I \] \[ p_2 x_2^* = I - \frac{p_1^2}{12p_2} \] \[ x_2^* = \frac{I}{p_2} - \frac{p_1^2}{12p_2^2} \] ### **Corner Solutions:** - If \( x_2^* < 0 \), set \( x_2^* = 0 \), solve for \( x_1 \) using \( p_1 x_1 = I \implies x_1 = \frac{I}{p_1} \). - If \( x_1^* < 0 \), set \( x_1^* = 0 \), solve for \( x_2 \) using \( p_2 x_2 = I \implies x_2 = \frac{I}{p_2} \). --- ### **Summary of Demand Functions:** \[ x_1^*(p_1, p_2, I) = \max\left\{0, \frac{p_1}{12p_2}\right\} \] \[ x_2^*(p_1, p_2, I) = \max\left\{0, \frac{I}{p_2} - \frac{p_1^2}{12p_2^2}\right\} \] --- ## **A. (b) Lagrange Multiplier Dependence** - From above: \( \lambda = \frac{1}{p_2} \) - **Depends only on \( p_2 \), not on \( p_1 \) or \( I \).** - **Reason:** Marginal utility of income (shadow price of wealth) is equal to the marginal utility of \( x_2 \) divided by its price. Since utility is linear in \( x_2 \), the marginal utility is 1, so \( \lambda = \frac{1}{p_2} \). --- ## **A. (c) Does \( x_1^* \) depend on income \( I \)?** - From earlier: \( x_1^* = \frac{p_1}{12p_2} \), **which does not depend on \( I \)**. - **Interpretation:** The demand for good 1 is independent of income—it's a "quasi-linear" preference structure. --- ## **A. (d) Indifference Curve Map and Features** - Indifference curves: \( 6x_1^2 + x_2 = \text{constant} \) - Rearranged: \( x_2 = \text{constant} - 6x_1^2 \) - **Shape:** Downward-sloping, convex to the origin. - **Special feature:** All indifference curves are vertically shifted parabolas opening downward. - **Implication:** Marginal rate of substitution depends only on \( x_1 \), not \( x_2 \). --- ## **B. (a) Expenditure Minimization Problem** ### **Formulation:** Minimize \( p_1 x_1 + p_2 x_2 \) subject to \( 6x_1^2 + x_2 \geq \bar{U} \), \( x_1 \geq 0 \), \( x_2 \geq 0 \) ### **Lagrangian:** \[ \mathcal{L} = p_1 x_1 + p_2 x_2 + \mu (\bar{U} - 6x_1^2 - x_2) \] ### **FOCs:** \[ \frac{\partial \mathcal{L}}{\partial x_1} = p_1 - \mu \cdot 12x_1 = 0 \implies \mu = \frac{p_1}{12x_1} \] \[ \frac{\partial \mathcal{L}}{\partial x_2} = p_2 - \mu = 0 \implies \mu = p_2 \] Set equal: \[ \frac{p_1}{12x_1} = p_2 \implies x_1^e = \frac{p_1}{12p_2} \] \[ 6(x_1^e)^2 + x_2^e = \bar{U} \implies x_2^e = \bar{U} - 6(x_1^e)^2 = \bar{U} - 6\left(\frac{p_1^2}{144p_2^2}\right) = \bar{U} - \frac{p_1^2}{24p_2^2} \] - If \( x_2^e < 0 \), set \( x_2^e = 0 \), solve for \( x_1 \) using \( 6x_1^2 = \bar{U} \implies x_1 = \sqrt{\frac{\bar{U}}{6}} \). --- ## **B. (b) Slutsky Equation** ### **Interior Solution: Both goods consumed** - **Slutsky Equation for good 1:** \[ \frac{\partial x_1^*}{\partial p_1} = \frac{\partial x_1^e}{\partial p_1} - x_1^* \frac{\partial x_1^*}{\partial I} \] But \( x_1^* \) does **not** depend on \( I \), so \( \frac{\partial x_1^*}{\partial I} = 0 \). \[ x_1^* = x_1^e = \frac{p_1}{12p_2} \implies \frac{\partial x_1^*}{\partial p_1} = \frac{1}{12p_2} \] So, the Slutsky equation holds: \[ \frac{\partial x_1^*}{\partial p_1} = \frac{\partial x_1^e}{\partial p_1} \] **because \( x_1^* \) is independent of income.** --- ## **Summary Table** | Part | Key Results | |------|-------------| | A(a) | \( x_1^* = \frac{p_1}{12p_2} \), \( x_2^* = \frac{I}{p_2} - \frac{p_1^2}{12p_2^2} \) | | A(b) | \( \lambda = \frac{1}{p_2} \), depends only on \( p_2 \) | | A(c) | \( x_1^* \) independent of income \( I \) | | A(d) | Indifference curves: downward-opening parabolas, \( x_2 = c - 6x_1^2 \) | | B(a) | \( x_1^e = \frac{p_1}{12p_2} \), \( x_2^e = \bar{U} - \frac{p_1^2}{24p_2^2} \) | | B(b) | Slutsky equation holds for \( x_1 \) since \( x_1^* \) is independent of \( I \) | --- **If you need graphical illustrations or further discussion for any part, let me know!**

# Graphical Illustration and Further Discussion --- ## **Indifference Curves** ### **Indifference Curve Representation** The indifference curves for Sorah's utility function \( u(x_1, x_2) = 6x_1^2 + x_2 \) take the form: \[ x_2 = U - 6x_1^2 \] where \( U \) is a constant level of utility. ### **Graphical Representation** - **Axes:** - \( x_1 \) on the horizontal axis - \( x_2 \) on the vertical axis - **Shape:** - The curves are downward-opening parabolas. - Each curve represents different utility levels, with higher curves corresponding to higher utility levels. ### **Key Features of Indifference Curves** 1. **Downward Sloping:** As \( x_1 \) increases, \( x_2 \) must decrease to maintain the same utility level. 2. **Convex to the Origin:** Indicates diminishing marginal rate of substitution (MRS) between \( x_1 \) and \( x_2 \). 3. **No Intersection:** Each utility level is distinct; curves cannot cross. ### **Graph Example**  *Example graph of indifference curves showing downward-opening parabolas for different utility levels.* --- ## **Discussion of Indifference Curves** - **Special Nature:** The unique quadratic form of the utility function leads to parabolic indifference curves, reflecting how the two goods interact in terms of utility. - **Marginal Rate of Substitution:** The MRS, given by the slope of the curve, changes depending on the quantity of \( x_1 \) consumed: \[ MRS = -\frac{\partial u/\partial x_1}{\partial u/\partial x_2} = -\frac{12x_1}{1} = -12x_1 \] This suggests that as more of \( x_1 \) is consumed, the consumer is willing to give up less of \( x_2 \) for an additional unit of \( x_1 \). --- ## **Conclusion** The analysis of Sorah's utility maximization problem and expenditure minimization problem highlights the nature of consumer preferences and how they shape demand. The unique form of the utility function leads to distinct demand characteristics and indifference curves, demonstrating the importance of understanding consumer behavior in economic theory. If further visualizations or clarifications on any aspect are needed, please let me know!

# Graph of Indifference Curves Below is an example graph illustrating the indifference curves for Sorah's utility function \( u(x_1, x_2) = 6x_1^2 + x_2 \). ### **Indifference Curves Graph**  *Example graph of indifference curves showing downward-opening parabolas for different utility levels.* ### **Description of the Graph** - **Axes:** - The horizontal axis represents good \( x_1 \). - The vertical axis represents good \( x_2 \). - **Curves:** - Each curve corresponds to a different level of utility \( U \). - The curves are downward-opening parabolas, illustrating how as one good increases, the other must decrease to maintain the same level of utility. ### **Key Observations** 1. **Utility Levels:** Higher curves represent higher levels of utility. 2. **Shape:** The parabolic shape indicates the unique relationship between \( x_1 \) and \( x_2 \) in terms of utility. 3. **Diminishing Marginal Rate of Substitution:** The slope of the curve decreases as \( x_1 \) increases, reflecting the diminishing willingness to substitute \( x_2 \) for additional units of \( x_1 \). If you require more details or specific adjustments to the graph, please let me know!

# Graph of Indifference Curves Below is a graph illustrating the indifference curves for Sorah's utility function \( u(x_1, x_2) = 6x_1^2 + x_2 \). ### **Indifference Curves Graph**  *Example graph of indifference curves showing downward-opening parabolas for different utility levels.* ### **Description of the Graph** - **Axes:** - The horizontal axis represents good \( x_1 \). - The vertical axis represents good \( x_2 \). - **Curves:** - Each curve corresponds to a different level of utility \( U \). - The curves are downward-opening parabolas, illustrating how as one good increases, the other must decrease to maintain the same level of utility. ### **Key Observations** 1. **Utility Levels:** Higher curves represent higher levels of utility. 2. **Shape:** The parabolic shape indicates the unique relationship between \( x_1 \) and \( x_2 \) in terms of utility. 3. **Diminishing Marginal Rate of Substitution:** The slope of the curve decreases as \( x_1 \) increases, reflecting the diminishing willingness to substitute \( x_2 \) for additional units of \( x_1 \). If you require more details or specific adjustments to the graph, please let me know!

# Problem 1 Step-by-Step Solution --- ## **Given:** Utility function: \( u(x_1, x_2) = 6x_1^2 + x_2 \) Prices: \( p_1, p_2 \) Income: \( I \) Constraints: \( x_1 \geq 0, x_2 \geq 0 \) --- ## **A. (a) Utility Maximization Problem** ### **Formulation:** Maximize \( u(x_1, x_2) = 6x_1^2 + x_2 \) subject to \( p_1 x_1 + p_2 x_2 \leq I \), \( x_1 \geq 0 \), \( x_2 \geq 0 \) ### **Lagrangian:** \[ \mathcal{L} = 6x_1^2 + x_2 + \lambda (I - p_1 x_1 - p_2 x_2) \] ### **First-Order Conditions (FOCs):** 1. \(\frac{\partial \mathcal{L}}{\partial x_1} = 12x_1 - \lambda p_1 = 0\) 2. \(\frac{\partial \mathcal{L}}{\partial x_2} = 1 - \lambda p_2 = 0\) 3. \(\frac{\partial \mathcal{L}}{\partial \lambda} = I - p_1 x_1 - p_2 x_2 = 0\) #### **Solve for \( x_1, x_2, \lambda \):** From FOCs: 1. \( 12x_1 = \lambda p_1 \implies \lambda = \frac{12x_1}{p_1} \) 2. \( 1 = \lambda p_2 \implies \lambda = \frac{1}{p_2} \) Set equal: \[ \frac{12x_1}{p_1} = \frac{1}{p_2} \implies x_1^* = \frac{p_1}{12p_2} \] Substitute into the budget constraint: \[ p_1 x_1^* + p_2 x_2^* = I \implies p_1\left(\frac{p_1}{12p_2}\right) + p_2 x_2^* = I \] \[ \frac{p_1^2}{12p_2} + p_2 x_2^* = I \] \[ p_2 x_2^* = I - \frac{p_1^2}{12p_2} \] \[ x_2^* = \frac{I}{p_2} - \frac{p_1^2}{12p_2^2} \] ### **Corner Solutions:** - If \( x_2^* < 0 \), set \( x_2^* = 0 \) and solve for \( x_1 \): \( p_1 x_1 = I \implies x_1 = \frac{I}{p_1} \). - If \( x_1^* < 0 \), set \( x_1^* = 0 \) and solve for \( x_2 \): \( p_2 x_2 = I \implies x_2 = \frac{I}{p_2} \). --- ### **Summary of Demand Functions:** \[ x_1^*(p_1, p_2, I) = \max\left\{0, \frac{p_1}{12p_2}\right\} \] \[ x_2^*(p_1, p_2, I) = \max\left\{0, \frac{I}{p_2} - \frac{p_1^2}{12p_2^2}\right\} \] --- ## **A. (b) Lagrange Multiplier Dependence** - From above: \( \lambda = \frac{1}{p_2} \) - **Depends only on \( p_2 \), not on \( p_1 \) or \( I \).** - **Reason:** Marginal utility of income (shadow price of wealth) is equal to the marginal utility of \( x_2 \) divided by its price. Since utility is linear in \( x_2 \), the marginal utility is 1, so \( \lambda = \frac{1}{p_2} \). --- ## **A. (c) Does \( x_1^* \) depend on income \( I \)?** - From earlier: \( x_1^* = \frac{p_1}{12p_2} \), **which does not depend on \( I \)**. - **Interpretation:** The demand for good 1 is independent of income—it's a "quasi-linear" preference structure. --- ## **A. (d) Indifference Curve Map and Features** - Indifference curves: \( 6x_1^2 + x_2 = \text{constant} \) - Rearranged: \( x_2 = \text{constant} - 6x_1^2 \) - **Shape:** Downward-sloping, convex to the origin. - **Special feature:** All indifference curves are vertically shifted parabolas opening downward. - **Implication:** Marginal rate of substitution depends only on \( x_1 \), not \( x_2 \). --- ## **B. (a) Expenditure Minimization Problem** ### **Formulation:** Minimize \( p_1 x_1 + p_2 x_2 \) subject to \( 6x_1^2 + x_2 \geq \bar{U} \), \( x_1 \geq 0 \), \( x_2 \geq 0 \) ### **Lagrangian:** \[ \mathcal{L} = p_1 x_1 + p_2 x_2 + \mu (\bar{U} - 6x_1^2 - x_2) \] ### **FOCs:** 1. \(\frac{\partial \mathcal{L}}{\partial x_1} = p_1 - \mu \cdot 12x_1 = 0 \implies \mu = \frac{p_1}{12x_1}\) 2. \(\frac{\partial \mathcal{L}}{\partial x_2} = p_2 - \mu = 0 \implies \mu = p_2\) Set equal: \[ \frac{p_1}{12x_1} = p_2 \implies x_1^e = \frac{p_1}{12p_2} \] Substituting into the utility constraint: \[ 6(x_1^e)^2 + x_2^e = \bar{U} \implies x_2^e = \bar{U} - 6\left(\frac{p_1}{12p_2}\right)^2 = \bar{U} - \frac{p_1^2}{24p_2^2} \] - If \( x_2^e < 0 \), set \( x_2^e = 0 \), solve for \( x_1 \): \( 6x_1^2 = \bar{U} \implies x_1 = \sqrt{\frac{\bar{U}}{6}} \). --- ## **B. (b) Slutsky Equation** ### **Interior Solution: Both goods consumed** - **Slutsky Equation for good 1:** \[ \frac{\partial x_1^*}{\partial p_1} = \frac{\partial x_1^e}{\partial p_1} - x_1^* \frac{\partial x_1^*}{\partial I} \] But \( x_1^* \) does **not** depend on \( I \), so \( \frac{\partial x_1^*}{\partial I} = 0 \). Thus, we conclude: \[ \frac{\partial x_1^*}{\partial p_1} = \frac{\partial x_1^e}{\partial p_1} \] **because \( x_1^* \) is independent of income.** --- ## **Summary Table** | Part | Key Results | |------|-------------| | A(a) | \( x_1^* = \frac{p_1}{12p_2} \), \( x_2^* = \frac{I}{p_2} - \frac{p_1^2}{12p_2^2} \) | | A(b) | \( \lambda = \frac{1}{p_2} \), depends only on \( p_2 \) | | A(c) | \( x_1^* \) independent of income \( I \) | | A(d) | Indifference curves: downward-opening parabolas \( x_2 = c - 6x_1^2 \) | | B(a) | \( x_1^e = \frac{p_1}{12p_2} \), \( x_2^e = \bar{U} - \frac{p_1^2}{24p_2^2} \) | | B(b) | Slutsky equation holds for \( x_1 \) since \( x_1^* \) is independent of \( I \) | --- If you need further clarifications or additional details, feel free to ask!

# Problem 1 Summary --- ## **Given:** - **Utility function:** \( u(x_1, x_2) = 6x_1^2 + x_2 \) - **Prices:** \( p_1, p_2 \) - **Income:** \( I \) - **Constraints:** \( x_1 \geq 0, x_2 \geq 0 \) --- ## **A. Utility Maximization** ### **(a) Problem Formulation** - **Objective:** Maximize \( u(x_1, x_2) \) - **Subject to:** \( p_1 x_1 + p_2 x_2 \leq I \) ### **Lagrangian:** \[ \mathcal{L} = 6x_1^2 + x_2 + \lambda (I - p_1 x_1 - p_2 x_2) \] ### **First-Order Conditions:** 1. \( 12x_1 - \lambda p_1 = 0 \) 2. \( 1 - \lambda p_2 = 0 \) 3. \( I - p_1 x_1 - p_2 x_2 = 0 \) ### **Demand Functions:** \[ x_1^* = \max\left\{0, \frac{p_1}{12p_2}\right\} \] \[ x_2^* = \max\left\{0, \frac{I}{p_2} - \frac{p_1^2}{12p_2^2}\right\} \] --- ### **(b) Lagrange Multiplier Dependence** - \( \lambda = \frac{1}{p_2} \) - **Dependence:** Only on \( p_2 \); reflects marginal utility of wealth. --- ### **(c) Income Dependence of \( x_1^* \)** - **Result:** \( x_1^* = \frac{p_1}{12p_2} \) does not depend on \( I \). --- ### **(d) Indifference Curves** - **Shape:** Downward-opening parabolas. - **Features:** Convex to the origin; MRS depends solely on \( x_1 \). --- ## **B. Expenditure Minimization** ### **(a) Problem Formulation** - **Objective:** Minimize \( p_1 x_1 + p_2 x_2 \) - **Subject to:** \( 6x_1^2 + x_2 \geq \bar{U} \) ### **Demand Functions:** \[ x_1^e = \frac{p_1}{12p_2}, \quad x_2^e = \bar{U} - \frac{p_1^2}{24p_2^2} \] --- ### **(b) Slutsky Equation** - Holds for good 1 since \( x_1^* \) is independent of income: \[ \frac{\partial x_1^*}{\partial p_1} = \frac{\partial x_1^e}{\partial p_1} \] --- ## **Summary Table** | Part | Key Results | |------|-------------| | A(a) | \( x_1^* = \frac{p_1}{12p_2} \), \( x_2^* = \frac{I}{p_2} - \frac{p_1^2}{12p_2^2} \) | | A(b) | \( \lambda = \frac{1}{p_2} \), depends only on \( p_2 \) | | A(c) | \( x_1^* \) independent of income \( I \) | | A(d) | Indifference curves: downward-opening parabolas \( x_2 = U - 6x_1^2 \) | | B(a) | \( x_1^e = \frac{p_1}{12p_2} \), \( x_2^e = \bar{U} - \frac{p_1^2}{24p_2^2} \) | | B(b) | Slutsky equation holds for \( x_1 \) since \( x_1^* \) is independent of \( I \) | --- For further inquiries or additional details, feel free to ask!