Write an equation that implies that the budget line is tangent to an indifference curve at (xA,xB). There are many solutions to this equation. Each of these solutions corresponds to a point on a different indifference curve. Use pencil to draw a line that passes through all of these points. The best bundle that Charlie can afford must lie somewhere on the line you just penciled in. It must also lie on his budget line. If the point is outside of his budget line, he can't afford it. If the point lies inside of his budget line, he can afford to do better by buying more of both goods. On your graph, label this best affordable bundle with an E. This happens where xA=____ and xB=____. 

Let's break down the process and create an equation that fits this scenario involving a budget line and indifference curves.

### 1. **Equation for the Budget Line**:
The budget line represents all the combinations of goods \( x_A \) and \( x_B \) that Charlie can afford given his income and the prices of the two goods. The equation for a budget line is:

\[
P_A x_A + P_B x_B = M
\]

Where:
- \( P_A \) is the price of good \( A \),
- \( P_B \) is the price of good \( B \),
- \( M \) is Charlie’s total income,
- \( x_A \) and \( x_B \) are the quantities of good \( A \) and good \( B \) respectively.

### 2. **Indifference Curve**:
An indifference curve represents all combinations of \( x_A \) and \( x_B \) that give Charlie the same level of satisfaction (utility). The slope of an indifference curve at any point is given by the **Marginal Rate of Substitution (MRS)**, which is:

\[
MRS = \frac{MU_A}{MU_B}
\]

Where:
- \( MU_A \) is the marginal utility of good \( A \),
- \( MU_B \) is the marginal utility of good \( B \).

At the tangency point between the budget line and the indifference curve, the slope of the budget line must equal the slope of the indifference curve, which implies that:

\[
\frac{P_A}{P_B} = \frac{MU_A}{MU_B}
\]

### 3. **Tangent Condition (Best Bundle)**:
At the tangency point, where the budget line is tangent to the indifference curve, the following condition holds:

\[
\frac{MU_A}{MU_B} = \frac{P_A}{P_B}
\]

This implies that at the best bundle \( E(x_A, x_B) \), the slope of the indifference curve equals the slope of the budget line, and Charlie maximizes his utility subject to his budget constraint.

### 4. **Graphing the Line Through Tangency Points**:
The tangency points between the budget line and various indifference curves form a locus, often called the **income-consumption curve**. This line can be drawn through all possible tangency points of different indifference curves with the budget line. Any point on this line represents a potential best bundle that Charlie can afford at different income levels or different prices.

### 5. **Best Affordable Bundle**:
The best affordable bundle \( E(x_A, x_B) \) occurs where the indifference curve just touches the budget line. This point must satisfy both the tangency condition and the budget constraint:

\[
P_A x_A + P_B x_B = M
\]
and
\[
\frac{MU_A}{MU_B} = \frac{P_A}{P_B}
\]

Thus, the best affordable bundle occurs when \( x_A = \) ___ and \( x_B = \) ___ are the optimal quantities that maximize utility subject to Charlie's budget.

Would you like more details on how to derive this or solve it with specific values? Here are some related questions to expand your understanding:

1. How do prices \( P_A \) and \( P_B \) affect the position of the budget line?
2. Can the MRS change along an indifference curve, or is it constant?
3. How does a change in income affect the budget line and the best affordable bundle?
4. What happens to the best bundle if the price of one of the goods increases?
5. How can you graphically show the difference between affordable and unaffordable bundles?

**Tip:** The tangency condition is crucial for utility maximization, as it ensures that the consumer is making the best possible trade-off between the two goods.

Write an equation that implies that the budget line is tangent to an indifference curve at (xA,xB). There are many solutions to this equation. Each of these solutions corresponds to a point on a different indifference curve. Use pencil to draw a line that passes through all of these points. The best bundle that Charlie can afford must lie somewhere on the line you just penciled in. It must also lie on his budget line. If the point is outside of his budget line, he can't afford it. If the point lies inside of his budget line, he can afford to do better by buying more of both goods. On your graph, label this best affordable bundle with an E. This happens where xA=____ and xB=____.

Understand the concept of the budget line being tangent to an indifference curve in microeconomics. This problem explores consumer choice theory, the marginal rate of substitution, and utility maximization. Learn how to graph the best affordable bundle where xA and xB maximize utility, subject to budget constraints.

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