Let's go through **Section A** step-by-step for the utility function \( U(X, Y) = X^{\alpha} Y^{\beta} \): --- ## **A. Utility maximization and demand functions** ### **1. Derive the Marshallian demand curve for X (\(X^M\)):** #### **Budget Constraint:** \[ P_X X + P_Y Y = I \] #### **Utility Function:** \[ U(X, Y) = X^{\alpha} Y^{\beta} \] #### **Lagrangian:** \[ \mathcal{L} = X^{\alpha} Y^{\beta} + \lambda (I - P_X X - P_Y Y) \] #### **First-Order Conditions (FOCs):** - With respect to \(X\): \[ \frac{\partial \mathcal{L}}{\partial X} = \alpha X^{\alpha-1} Y^{\beta} - \lambda P_X = \] - With respect to \(Y\): \[ \frac{\partial \mathcal{L}}{\partial Y} = \beta X^{\alpha} Y^{\beta-1} - \lambda P_Y = \] - With respect to \(\lambda\): \[ \frac{\partial \mathcal{L}}{\partial \lambda} = I - P_X X - P_Y Y = \] #### **Find \(X\) and \(Y\):** Divide FOC for \(X\) by FOC for \(Y\): \[ \frac{\alpha X^{\alpha-1} Y^{\beta}}{\beta X^{\alpha} Y^{\beta-1}} = \frac{P_X}{P_Y} \implies \frac{\alpha}{\beta} \cdot \frac{Y}{X} = \frac{P_X}{P_Y} \implies \frac{Y}{X} = \frac{\beta}{\alpha} \cdot \frac{P_X}{P_Y} \implies Y = X \cdot \frac{\beta}{\alpha} \cdot \frac{P_X}{P_Y} \] Substitute \(Y\) into the budget constraint: \[ P_X X + P_Y Y = I \\ P_X X + P_Y \left( X \cdot \frac{\beta}{\alpha} \cdot \frac{P_X}{P_Y} \right) = I \\ P_X X + X \cdot \frac{\beta}{\alpha} P_X = I \\ P_X X \left(1 + \frac{\beta}{\alpha}\right) = I \\ P_X X \left(\frac{\alpha + \beta}{\alpha}\right) = I \\ X = \frac{I \alpha}{P_X (\alpha + \beta)} \] **Marshallian demand for X:** \[ \boxed{ X^M = \frac{I \alpha}{P_X (\alpha + \beta)} } \] Similarly, \[ Y^M = \frac{I \beta}{P_Y (\alpha + \beta)} \] --- ### **2. Derive the Hicksian demand curve for X (\(X^H\)):** Given utility level \(\overline{U}\): Minimize \(P_X X + P_Y Y\) s.t. \(X^{\alpha} Y^{\beta} = \overline{U}\). Set up the Lagrangian: \[ \mathcal{L} = P_X X + P_Y Y + \lambda (\overline{U} - X^{\alpha} Y^{\beta}) \] FOCs: - \(\frac{\partial \mathcal{L}}{\partial X} = P_X - \lambda \alpha X^{\alpha-1} Y^{\beta} = \) - \(\frac{\partial \mathcal{L}}{\partial Y} = P_Y - \lambda \beta X^{\alpha} Y^{\beta-1} = \) - \(\frac{\partial \mathcal{L}}{\partial \lambda} = \overline{U} - X^{\alpha} Y^{\beta} = \) Use the ratio method as before: \[ \frac{P_X}{P_Y} = \frac{\alpha}{\beta} \cdot \frac{Y}{X} \implies Y = X \cdot \frac{\beta}{\alpha} \cdot \frac{P_X}{P_Y} \] Substitute into the utility constraint: \[ X^{\alpha} \left[ X \cdot \frac{\beta}{\alpha} \cdot \frac{P_X}{P_Y} \right]^{\beta} = \overline{U} \implies X^{\alpha + \beta} \left( \frac{\beta}{\alpha} \cdot \frac{P_X}{P_Y} \right)^{\beta} = \overline{U} \implies X^{\alpha + \beta} = \overline{U} \left( \frac{\beta}{\alpha} \cdot \frac{P_X}{P_Y} \right)^{-\beta} \] \[ X^H = \left[ \overline{U} \left( \frac{\beta}{\alpha} \cdot \frac{P_X}{P_Y} \right)^{-\beta} \right]^{\frac{1}{\alpha + \beta}} \] --- ### **3. What is the own-price elasticity for X^M?** From Marshallian demand: \[ X^M = \frac{I \alpha}{P_X (\alpha + \beta)} \] Own-price elasticity: \[ \epsilon_{X, P_X} = \frac{\partial X^M}{\partial P_X} \cdot \frac{P_X}{X^M} \] \[ \frac{\partial X^M}{\partial P_X} = -\frac{I \alpha}{P_X^2 (\alpha + \beta)} = -\frac{X^M}{P_X} \] \[ \epsilon_{X,P_X} = -1 \] **Own-price elasticity is \(-1\)** --- ### **4. What is the own-price elasticity for X^H?** For Cobb-Douglas, the Hicksian and Marshallian demands have the same price elasticity (since there is no income effect beyond the compensated effect): \[ \epsilon_{X,P_X}^{H} = -1 \] --- ### **5. What is the income elasticity of X?** \[ X^M = \frac{I \alpha}{P_X (\alpha + \beta)} \] \[ \frac{\partial X^M}{\partial I} = \frac{\alpha}{P_X (\alpha + \beta)} = \frac{X^M}{I} \] \[ \text{Income elasticity} = \frac{\partial X^M}{\partial I} \frac{I}{X^M} = 1 \] **Income elasticity = 1** --- ### **6. What is the price elasticity of X with respect to \(P_Y\) (\(\epsilon_{X, P_Y}\))?** \[ X^M = \frac{I \alpha}{P_X (\alpha + \beta)} \] No \(P_Y\) in the expression for \(X^M\): \[ \epsilon_{X,P_Y} = \] --- ## **Summary Table** | Elasticity | Value | |----------------------------------|-------| | Own-price (Marshallian, \(X^M\))| -1 | | Own-price (Hicksian, \(X^H\)) | -1 | | Income elasticity (\(X^M\)) | 1 | | Cross-price (\(P_Y\)) | | --- ## **B. Consumer Demand Curve: Direction of Shift** a. **Own price increases?** → Demand curve moves **left** (decreases quantity demanded). b. **Price of substitute goods increases?** → Demand curve moves **right** (increases demand for X). c. **Price of complement goods increases?** → Demand curve moves **left** (decreases demand for X). d. **Income increases if the product is inferior?** → Demand curve moves **left**. e. **Income increases if the product is luxurious?** → Demand curve moves **right**. --- ## **C. Degree of Homogeneity of Production Functions** Recall: If \(f(\lambda K, \lambda L) = \lambda^r f(K, L)\), then degree of homogeneity is \(r\). a. \(K^{\alpha}L^{\beta}\): **\(\alpha + \beta\)** b. \(aK + bL\): **1** c. \(aK^{\alpha} + bL^{\beta}\): **No definite degree** (unless \(\alpha = \beta\)) d. \(K/\alpha + L/\beta\): **1** e. \([K^{1/\alpha}, L^{1/\beta}]\): **Take min. Degree is \(1/\alpha\) if both terms scale the same.** f. \(\min(aK, bL)\): **1** --- ## **D. Budget Line and Indifference Map** Let \(P_X\) be the price of X, \(P_Y\) the price of Y, and \(I\) the income. ### **a. Draw the budget line and indifference map** Budget line: \(P_X X + P_Y Y = I\) - X-intercept: \(X = I/P_X\) - Y-intercept: \(Y = I/P_Y\) Indifference curves: Level curves of \(U(X, Y)\). ### **b. Mark the utility-maximizing X and Y** Occurs where the budget line is tangent to the highest possible indifference curve. ### **c. If \(P_X\) increases, show X and Y's income and substitution effects when X is inferior** - When \(P_X\) increases: - **Substitution effect**: X decreases, Y increases (movement along indifference curve). - **Income effect** (since X is inferior): X increases (opposite direction to substitution effect), Y decreases. - **Total effect**: Depends on which is stronger. --- If you’d like any of these steps worked out in more detail, just ask!

## Utility Function Definition A utility function represents a consumer's preferences for a set of goods. It assigns a numerical value to different combinations of goods, reflecting the satisfaction or happiness derived from them. In the case of Cobb-Douglas utility functions, the form is \( U(X, Y) = X^{\alpha} Y^{\beta} \), where \(\alpha\) and \(\beta\) are parameters that indicate the importance of each good. ### Own-Price Elasticity for Hicksian Demand (\(X^H\)) The own-price elasticity of Hicksian demand (\( \epsilon_{X, P_X}^H \)) is \(-1\) for Cobb-Douglas utility functions. This indicates that a 1% increase in the price of good \(X\) results in a 1% decrease in the quantity demanded, demonstrating constant elasticity regardless of income effects.

## Summary of Budget Line and Indifference Map Analysis The budget line is defined by the equation \(P_X X + P_Y Y = I\), with intercepts at \(X = \frac{I}{P_X}\) (X-intercept) and \(Y = \frac{I}{P_Y}\) (Y-intercept). Convex indifference curves illustrate various utility levels, with the utility-maximizing bundle marked at the tangent point between the highest indifference curve and the budget line \((X^*, Y^*)\). When the price of good \(X\) increases, the budget line rotates inward, reducing the X-intercept and altering the optimal consumption bundle. The substitution effect indicates a movement along the initial indifference curve, showing reduced consumption of \(X\) and increased consumption of \(Y\) as \(X\) becomes relatively more expensive. Simultaneously, the income effect occurs since \(X\) is an inferior good, resulting in increased consumption of \(X\). Arrows on the graph distinguish these effects, illustrating the total impact on consumption after considering both substitution and income effects.