Use the duality theorem to derive (he Hicksian demand function from 
the expenditure function
Discuss the uniqueness in relation to the 
duality theorem. 

### Deriving the Hicksian Demand Function from the Expenditure Function Using the Duality Theorem

The **Hicksian demand function** (or compensated demand function) represents the quantity of goods a consumer will demand when they seek to achieve a specific level of utility at the lowest possible cost, given a set of prices. The **expenditure function** gives the minimum expenditure required to achieve a given utility level for a given set of prices. The duality theorem connects these two concepts.

#### Step 1: Define the Expenditure Function
The expenditure function \( E(p, u) \) is defined as:
\[
E(p, u) = \min_x \{ p \cdot x \, : \, u(x) \geq u \}
\]
where:
- \( p \) is the vector of prices.
- \( u(x) \) is the utility function.
- \( u \) is the utility level.

This function represents the minimum expenditure needed to reach the utility level \( u \) given the prices \( p \).

#### Step 2: Apply Shephard's Lemma
Shephard's Lemma states that the Hicksian demand function can be derived by differentiating the expenditure function with respect to prices. Specifically:
\[
h_i(p, u) = \frac{\partial E(p, u)}{\partial p_i}
\]
where \( h_i(p, u) \) is the Hicksian demand for good \( i \).

#### Step 3: Derive the Hicksian Demand Function
To derive the Hicksian demand function \( h(p, u) \), differentiate the expenditure function with respect to each price \( p_i \):
\[
h_i(p, u) = \frac{\partial E(p, u)}{\partial p_i}
\]
This gives the amount of good \( i \) that the consumer will demand to achieve the utility level \( u \) at the minimum cost, given the prices.

### Discussing Uniqueness in Relation to the Duality Theorem

The **duality theorem** in consumer theory establishes a one-to-one correspondence between the utility maximization problem (leading to Marshallian demand) and the expenditure minimization problem (leading to Hicksian demand).

- **Uniqueness of the Expenditure Function**: The expenditure function is unique for a given utility function \( u(x) \) and price vector \( p \). This is because for any given utility level \( u \) and prices \( p \), there is a unique minimal expenditure required to achieve that utility.

- **Uniqueness of the Hicksian Demand Function**: Since the Hicksian demand function is derived from the expenditure function using Shephard's Lemma, and the expenditure function is unique, the Hicksian demand function is also unique for a given utility function and price vector. There can only be one set of demand quantities that minimize expenditure for a given utility level and price vector.

- **Duality Theorem**: The duality theorem ensures that the Hicksian demand function derived from the expenditure function corresponds uniquely to the Marshallian demand function derived from the utility maximization problem. This guarantees consistency between the two approaches in consumer theory.

In summary, the duality theorem ensures that the expenditure function and the Hicksian demand function derived from it are unique and consistent with the utility maximization problem, providing a solid foundation for consumer choice theory.

### Questions to Explore Further:

1. How does the Marshallian demand function differ from the Hicksian demand function?
2. What is the significance of Shephard's Lemma in consumer theory?
3. Can you explain the relationship between the expenditure function and indirect utility function?
4. How does the Slutsky equation relate to the Hicksian demand function?
5. What are the key differences between compensated and uncompensated demand curves?

**Tip**: Understanding the relationship between the expenditure function and the Hicksian demand function is crucial for mastering consumer theory, especially in advanced microeconomics.

Explore how to derive the Hicksian demand function from the expenditure function using the duality theorem in consumer theory. Learn about the uniqueness of the Hicksian demand function and its relationship with the duality theorem.

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