Douglas’s preferences are represented by the utility function u(x1,x2)=(x1^2)*(x2^3). The prices of x1 and x2 are p1 and p2, respectively. What is the slope of his indifference curve at the point (x1,x2)? 

Douglas's utility function is \( u(x_1, x_2) = x_1^2 x_2^3 \), where \( x_1 \) and \( x_2 \) represent the quantities of goods 1 and 2, respectively. The slope of the indifference curve is given by the **Marginal Rate of Substitution (MRS)**, which is the negative ratio of the marginal utilities of \( x_1 \) and \( x_2 \).

### Step 1: Find the Marginal Utility of \( x_1 \) (MU\(_{x_1}\))
The marginal utility of \( x_1 \) is obtained by taking the partial derivative of the utility function with respect to \( x_1 \):

\[
MU_{x_1} = \frac{\partial u(x_1, x_2)}{\partial x_1} = 2x_1 x_2^3
\]

### Step 2: Find the Marginal Utility of \( x_2 \) (MU\(_{x_2}\))
Similarly, the marginal utility of \( x_2 \) is obtained by taking the partial derivative of the utility function with respect to \( x_2 \):

\[
MU_{x_2} = \frac{\partial u(x_1, x_2)}{\partial x_2} = 3x_1^2 x_2^2
\]

### Step 3: Compute the Marginal Rate of Substitution (MRS)
The MRS is the ratio of the marginal utilities:

\[
MRS = -\frac{MU_{x_1}}{MU_{x_2}} = -\frac{2x_1 x_2^3}{3x_1^2 x_2^2}
\]

### Step 4: Simplify the MRS
Simplifying the expression for the MRS:

\[
MRS = -\frac{2x_2}{3x_1}
\]

Thus, the slope of the indifference curve at the point \( (x_1, x_2) \) is \( \frac{-2x_2}{3x_1} \).

Would you like more details or further clarifications on this?

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Here are five related questions to deepen your understanding:
1. How does the MRS change as \( x_1 \) and \( x_2 \) increase or decrease?
2. What does the indifference curve look like for different values of \( x_1 \) and \( x_2 \)?
3. What happens to the MRS if the utility function changes to a different form, like \( u(x_1, x_2) = \ln(x_1) + \ln(x_2) \)?
4. How do changes in prices \( p_1 \) and \( p_2 \) affect the consumer's optimal choice of \( x_1 \) and \( x_2 \)?
5. Can the MRS ever become positive, and what would that imply in an economic context?

**Tip:** The Marginal Rate of Substitution reveals how much of one good a consumer is willing to give up to gain more of another good while maintaining the same level of utility.

Douglas’s preferences are represented by the utility function u(x1,x2) = (x1^2)(x2^3). The prices of x1 and x2 are p1 and p2, respectively. What is the slope of his indifference curve at the point (x1,x2)?

Discover how to find the slope of the indifference curve for the utility function u(x1, x2) = (x1^2)(x2^3). This step-by-step solution uses partial derivatives and the Marginal Rate of Substitution (MRS) concept. Suitable for undergraduate economics students.

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