Exercise 1 (15+10+15=40 points ) Suppose a consumer with homothetic preferences over bundles of goods in R+n ie this consumer has a utility function u that is homogeneous of degree 1 Assume that the usual assumptions about utility functions hold (Assumption 12) and that u is C2 (a) Show that the indirect utility function v is linear in income ie v( p y)= y v( p 1) for p R++n y q 0 (b) Show that also the Marshallian demands are linear in income (c) Show that for the demand functions of this consumer we have that xi( p y) pj= xj( p y) pi for i j 12 n Hint First show that xj( p y) xi( p y) y=xi( p y) xj( p y) y then apply the Slutsky equation

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Exercise 1 (15+10+15=40 points ) Suppose a consumer with homothetic preferences over bundles of goods in R+n ie this consumer has a utility function u that is homogeneous of degree 1 Assume that the usual assumptions about utility functions hold (Assumption 12) and that u is C2 (a) Show that the indirect utility function v is linear in income ie v( p y)= y v( p 1) for p R++n y q 0 (b) Show that also the Marshallian demands are linear in income (c) Show that for the demand functions of this consumer we have that  xi( p y) pj= xj( p y) pi for i j 12 n Hint First show that xj( p y) xi( p y) y=xi( p y) xj( p y) y then apply the Slutsky equation

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Solution By Steps

Part (a): Indirect Utility Function is Linear in Income

Step 1: Define Homothetic Preferences  
Homothetic preferences imply that if a utility function $u(\mathbf{x})$ is homogeneous of degree 1, then for any $\alpha > 0$, $u(\alpha \mathbf{x}) = \alpha u(\mathbf{x})$.

Step 2: Define Indirect Utility Function  
The indirect utility function $v(\mathbf{p}, y)$ represents the maximum utility achievable given prices $\mathbf{p}$ and income $y$.

Step 3: Apply Homogeneity to Indirect Utility  
Since $u$ is homogeneous of degree 1, scaling the bundle $\mathbf{x}$ by income $y$ scales utility by $y$. Thus, $v(\mathbf{p}, y) = y v(\mathbf{p}, 1)$, showing linearity in income.

Part (b): Marshallian Demands are Linear in Income

Step 1: Define Marshallian Demand  
Marshallian demand functions $x_i(\mathbf{p}, y)$ give the quantity of good $i$ demanded given prices $\mathbf{p}$ and income $y$.

Step 2: Apply Homogeneity to Demand  
Given homothetic preferences, scaling income $y$ scales the demand for goods linearly. Thus, $x_i(\mathbf{p}, y) = y x_i(\mathbf{p}, 1)$, showing demands are linear in income.

Part (c): Symmetry of Cross-Price Derivatives

Step 1: Show Proportionality of Demand and Income Derivatives  
First, establish $x_{j}(\mathbf{p}, y) \frac{\partial x_{i}(\mathbf{p}, y)}{\partial y} = x_{i}(\mathbf{p}, y) \frac{\partial x_{j}(\mathbf{p}, y)}{\partial y}$, indicating that the change in demand for one good with respect to income is proportional to the demand for another good.

Step 2: Apply Slutsky Equation  
The Slutsky equation relates the change in demand for a good with respect to its price to the substitution effect and the income effect. Given the proportionality from Step 1 and the Slutsky equation, it follows that $\frac{\partial x_{i}(\mathbf{p}, y)}{\partial p_{j}} = \frac{\partial x_{j}(\mathbf{p}, y)}{\partial p_{i}}$, showing symmetry in cross-price derivatives.

Final Answer
(a) $v(\mathbf{p}, y) = y v(\mathbf{p}, 1)$  
(b) $x_i(\mathbf{p}, y) = y x_i(\mathbf{p}, 1)$  
(c) $\frac{\partial x_{i}(\mathbf{p}, y)}{\partial p_{j}} = \frac{\partial x_{j}(\mathbf{p}, y)}{\partial p_{i}}$

Key Concept
Homothetic Preferences

Key Concept Explanation
Homothetic preferences imply that utility functions are homogeneous of degree 1, meaning if all inputs are scaled by a factor, utility scales by the same factor. This leads to linear relationships in indirect utility and demand functions with respect to income, and symmetry in cross-price derivatives of demand functions.

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Step 1: Indirect Utility Function 
The indirect utility function $v(\mathbf{p}, y)$ is defined as $v(\mathbf{p}, y) = \max_{\mathbf{x} \in \mathbb{R}_{+}^{n}} \left\{ \mathbf{p} \cdot \mathbf{x} : u(\mathbf{x}) \geq u(\mathbf{0}), \mathbf{p} \cdot \mathbf{x} \leq y \}$.

Step 2: Homogeneity of Degree 1 
Since the utility function $u$ is homogeneous of degree 1, we have $u(\lambda \mathbf{x}) = \lambda u(\mathbf{x})$ for all $\lambda > 0$.

Step 3: Linear in Income 
Using the homogeneity property, we can show that $v(\mathbf{p}, y) = y v(\mathbf{p}, 1)$ for $\mathbf{p} \in \mathbb{R}_{++}^{n}, y \geq 0$.

Step 4: Marshallian Demands 
The Marshallian demand functions $x_i(\mathbf{p}, y)$ represent the optimal bundle chosen by the consumer at prices $\mathbf{p}$ and income $y$.

Step 5: Linear in Income 
Similar to the indirect utility function, the Marshallian demands are also linear in income, i.e., $x_i(\mathbf{p}, y) = y x_i(\mathbf{p}, 1)$.

Step 6: Symmetry of Demand Functions 
To show $\frac{\partial x_{i}(\mathbf{p}, y)}{\partial p_{j}} = \frac{\partial x_{j}(\mathbf{p}, y)}{\partial p_{i}}$, first establish $x_{j}(\mathbf{p}, y) \frac{\partial x_{i}(\mathbf{p}, y)}{\partial y} = x_{i}(\mathbf{p}, y) \frac{\partial x_{j}(\mathbf{p}, y)}{\partial y}$.

Step 7: Apply Slutsky Equation 
Utilize the Slutsky equation to derive the symmetry property of the demand functions.

Final Answer
(a) $v(\mathbf{p}, y) = y v(\mathbf{p}, 1)$  
(b) $x_i(\mathbf{p}, y) = y x_i(\mathbf{p}, 1)$  
(c) $\frac{\partial x_{i}(\mathbf{p}, y)}{\partial p_{j}} = \frac{\partial x_{j}(\mathbf{p}, y)}{\partial p_{i}}$

Key Concept
Homothetic Preferences

Key Concept Explanation
Homothetic preferences imply that the consumer's preferences are consistent with a utility function that is homogeneous of degree 1. This property leads to linear relationships in the indirect utility function, Marshallian demands, and symmetry in demand functions, simplifying the analysis of consumer behavior and market interactions.

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