Sophia consumes avocados and bananas. Her utility function is given by
𝑈(𝑥1, 𝑥2) = 𝑥1
𝑐 𝑥2
𝑑,
where 𝑐, 𝑑 > 0, 𝑥1 is the quantity of avocados she consumes in pounds, and 𝑥2 is the quantity of bananas
she consumes in pounds. Suppose when Sophia has 16 pounds of avocados and 4 pounds of bananas,
her marginal rate of substitution is 1
3 (of a pound of bananas for an additional pound of avocados).
Determine the mathematical relationship between 𝑐 and 𝑑.

To determine the mathematical relationship between \( c \) and \( d \), we need to compute the marginal rate of substitution (MRS) from Sophia's utility function and then apply the given condition that her MRS is \( \frac{1}{3} \) when she consumes 16 pounds of avocados (\( x_1 = 16 \)) and 4 pounds of bananas (\( x_2 = 4 \)).

### Step 1: Utility Function
Sophia's utility function is given as:
\[
U(x_1, x_2) = x_1^c x_2^d
\]
where \( x_1 \) is the quantity of avocados, and \( x_2 \) is the quantity of bananas. The constants \( c \) and \( d \) are positive parameters.

### Step 2: Marginal Rate of Substitution (MRS)
The MRS is the rate at which Sophia is willing to trade bananas for avocados, keeping her utility constant. It is the ratio of the marginal utility of avocados to the marginal utility of bananas.

The marginal utility of avocados (\( MU_{x_1} \)) and the marginal utility of bananas (\( MU_{x_2} \)) can be derived by taking partial derivatives of the utility function with respect to \( x_1 \) and \( x_2 \), respectively.

- Marginal utility of avocados:
\[
MU_{x_1} = \frac{\partial U}{\partial x_1} = c x_1^{c-1} x_2^d
\]

- Marginal utility of bananas:
\[
MU_{x_2} = \frac{\partial U}{\partial x_2} = d x_1^c x_2^{d-1}
\]

Thus, the MRS is:
\[
MRS = \frac{MU_{x_1}}{MU_{x_2}} = \frac{c x_1^{c-1} x_2^d}{d x_1^c x_2^{d-1}}
\]

Simplifying the expression:
\[
MRS = \frac{c}{d} \cdot \frac{x_2}{x_1}
\]

### Step 3: Apply Given Condition
The problem states that when \( x_1 = 16 \) and \( x_2 = 4 \), the MRS is \( \frac{1}{3} \). So we set:
\[
\frac{c}{d} \cdot \frac{4}{16} = \frac{1}{3}
\]

Simplifying this equation:
\[
\frac{c}{d} \cdot \frac{1}{4} = \frac{1}{3}
\]

Multiplying both sides by 4:
\[
\frac{c}{d} = \frac{4}{3}
\]

### Step 4: Conclusion
The mathematical relationship between \( c \) and \( d \) is:
\[
\frac{c}{d} = \frac{4}{3}
\]
or equivalently:
\[
c = \frac{4}{3} d
\]

This is the required relationship between the parameters \( c \) and \( d \).

---

Do you want more details on how marginal rate of substitution works, or have any further questions?

### Related Questions:
1. How is the utility maximization problem solved for a general Cobb-Douglas utility function?
2. What are the economic interpretations of \( c \) and \( d \) in the utility function?
3. How does a change in the price of avocados affect Sophia's consumption choices?
4. Can you derive the demand functions for avocados and bananas using the Lagrangian method?
5. How would the MRS change if the quantities of avocados and bananas are different?

**Tip:** The marginal rate of substitution (MRS) reveals preferences for goods and how willing someone is to substitute between them. Understanding it is key in consumer theory.

Sophia consumes avocados and bananas. Her utility function is given by U(x1, x2) = x1^c x2^d, where c, d > 0, x1 is the quantity of avocados she consumes in pounds, and x2 is the quantity of bananas she consumes in pounds. Suppose when Sophia has 16 pounds of avocados and 4 pounds of bananas, her marginal rate of substitution is 1/3 (of a pound of bananas for an additional pound of avocados). Determine the mathematical relationship between c and d.

Learn how to calculate the marginal rate of substitution for avocados and bananas with a utility function U(x1, x2) = x1^c x2^d. Given Sophia consumes 16 pounds of avocados and 4 pounds of bananas, this problem helps determine the relationship between c and d. Ideal for undergraduate economics students.

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