A consumer has the utility function u(x,y) = x 0.5
y 0.5 for baskets (x,y). The consumer’s income is M = $200
and the prices are Px = $4 and Py = $6.
a. Find the optimal basket?

We are tasked with finding the optimal basket $(x^*, y^*)$ for a consumer who maximizes their utility, given the utility function:

\[
u(x, y) = x^{0.5}y^{0.5}
\]

### Step 1: Formulate the budget constraint
The consumer has an income $M = 200$, and the prices of the goods are $P_x = 4$ and $P_y = 6$. The budget constraint is:

\[
4x + 6y = 200
\]

### Step 2: Use the Cobb-Douglas utility function to find the demand
For Cobb-Douglas utility functions like $u(x, y) = x^{0.5}y^{0.5}$, the optimal consumption of each good is proportional to the consumer's income. The general formula for the demand for each good is:

\[
x^* = \frac{\alpha M}{P_x}, \quad y^* = \frac{(1-\alpha)M}{P_y}
\]

In our case, both goods $x$ and $y$ have equal exponents (0.5), which means that $\alpha = 0.5$. Thus, the consumer spends half of their income on each good.

### Step 3: Calculate the optimal values of $x^*$ and $y^*$

1. **Optimal quantity of $x$**:
   \[
   x^* = \frac{0.5 \times 200}{4} = \frac{100}{4} = 25
   \]

2. **Optimal quantity of $y$**:
   \[
   y^* = \frac{0.5 \times 200}{6} = \frac{100}{6} \approx 16.67
   \]

### Conclusion:
The optimal basket is $(x^*, y^*) = (25, 16.67)$.

Do you want more details on the derivation, or have any questions?

Here are 5 questions that extend this information:
1. How does the consumer's optimal basket change if their income increases to $300?
2. What happens to the optimal basket if the price of good $x$ rises to $5?
3. How would the consumer allocate their budget if the utility function changes to $u(x, y) = x^{0.7}y^{0.3}$?
4. How does the budget constraint change if the prices of both goods double?
5. What is the consumer’s utility at the optimal basket?

**Tip**: In Cobb-Douglas utility functions, the exponents on goods represent the proportion of income spent on each good.

A consumer has the utility function u(x,y) = x^0.5 * y^0.5 for baskets (x,y). The consumer’s income is M = $200 and the prices are Px = $4 and Py = $6. Find the optimal basket.

Discover how to find the optimal basket for a consumer with a Cobb-Douglas utility function u(x, y) = x^0.5 * y^0.5, given an income of $200, and prices Px = $4, Py = $6. This detailed step-by-step solution explains the budget constraint and utility maximization techniques used to calculate the optimal quantities of goods x and y.

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