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.