You now have two equations, the budget equation and the tangency equation, that must be satisfied by the bundle demanded. Solve these two equations for xA and xB. Charlie's demand function for apples is xA(pA,pB,m)=____and his demand function for bananas is xB(pA,pB,m)=_____.

To solve for Charlie's demand functions for apples \( x_A \) and bananas \( x_B \), we typically work with the **budget equation** and the **tangency equation** that arise from a utility maximization problem in consumer theory. Here's a step-by-step approach to solving for \( x_A \) and \( x_B \).

### Step 1: Budget Equation
The budget equation represents Charlie's income (or budget constraint) and is expressed as:
\[
p_A x_A + p_B x_B = m
\]
Where:
- \( p_A \) is the price of apples,
- \( p_B \) is the price of bananas,
- \( x_A \) is the quantity of apples,
- \( x_B \) is the quantity of bananas,
- \( m \) is Charlie's income.

### Step 2: Tangency Condition (Utility Maximization)
The tangency condition comes from the fact that Charlie maximizes his utility subject to his budget constraint. The tangency equation is derived from the condition where the **Marginal Rate of Substitution (MRS)** equals the price ratio:
\[
\frac{MU_A}{MU_B} = \frac{p_A}{p_B}
\]
Where:
- \( MU_A \) and \( MU_B \) are the marginal utilities of apples and bananas, respectively.
- \( \frac{p_A}{p_B} \) is the price ratio of apples to bananas.

If Charlie’s utility function takes the form of a **Cobb-Douglas** utility function, for example:
\[
U(x_A, x_B) = x_A^\alpha x_B^{1-\alpha}
\]
The marginal utilities would be:
\[
MU_A = \alpha x_A^{\alpha-1} x_B^{1-\alpha}, \quad MU_B = (1-\alpha) x_A^\alpha x_B^{-\alpha}
\]
Using the tangency condition:
\[
\frac{\alpha}{1-\alpha} \cdot \frac{x_B}{x_A} = \frac{p_A}{p_B}
\]
Solving for \( x_B \) in terms of \( x_A \):
\[
x_B = \frac{1-\alpha}{\alpha} \cdot \frac{p_A}{p_B} \cdot x_A
\]

### Step 3: Substitute into the Budget Equation
Now substitute this expression for \( x_B

Explore how to derive Charlie's demand functions for apples and bananas using key concepts in consumer theory. This detailed solution covers the budget equation, tangency condition, and Cobb-Douglas utility function to help solve for optimal quantities of goods.

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