Which of the following is true when optimizing a Cobb-Douglas utility function given by u=xayb
, where x
 and y
 are consumption goods and a
 and b
 are constants, assuming a,b>0
?


Choose one:
A.  	Each good's demand depends on income and on its own price.
B.  	Neither good's demand depends on income.
C.  	Each good's demand depends on income and on the prices of both goods.
D.  	One good is an inferior good and the other is a normal good.
E.  	One good's demand never depends on income.

To analyze this, let’s go through the characteristics of the Cobb-Douglas utility function \( u = x^a y^b \) for goods \( x \) and \( y \), where \( a \) and \( b \) are constants greater than zero. 

### Step-by-Step Reasoning:

1. **Demand Dependence on Income**: For a Cobb-Douglas utility function, the demand for each good is proportional to the consumer's income. This is because, when maximizing utility subject to a budget constraint, we find that the optimal demand for each good is derived from the consumer's budget and the relative prices of the goods.

2. **Demand Dependence on Prices of Both Goods**: In a Cobb-Douglas utility function, the demand for each good depends on the prices of both goods. This is because, when determining the optimal quantities of \( x \) and \( y \), the consumer considers both prices to allocate their budget efficiently.

3. **Normal Goods**: In the Cobb-Douglas utility function, both goods are normal goods because an increase in income leads to an increase in the demand for both goods. 

4. **No Inferior Goods**: Cobb-Douglas preferences generally do not lead to one good being inferior because both goods' demand is positively correlated with income.

### Answer:

Based on these points, the correct answer is:

**C. Each good's demand depends on income and on the prices of both goods.**

This is because, in Cobb-Douglas utility functions, the demand for each good is influenced by the income level and the prices of both goods.

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Would you like further details on how the demand functions for each good are derived? 

Here are five follow-up questions that build on this topic:

1. How do we derive the demand functions for goods in a Cobb-Douglas utility function?
2. What role does the budget constraint play in determining optimal consumption for a Cobb-Douglas utility?
3. Can the Cobb-Douglas utility function represent complements or substitutes?
4. How does the elasticity of substitution affect choices in a Cobb-Douglas function?
5. What happens if the parameters \(a\) and \(b\) are not positive in a Cobb-Douglas utility?

**Tip:** In Cobb-Douglas functions, the parameters \( a \) and \( b \) determine the proportion of income spent on each good.

Which of the following is true when optimizing a Cobb-Douglas utility function given by u = x^a y^b, where x and y are consumption goods and a and b are constants, assuming a, b > 0?

Explore the optimization of a Cobb-Douglas utility function, u = x^a y^b, for consumption goods x and y, where constants a and b are positive. This solution explains how demand depends on income and the prices of both goods, covering essential economic and mathematical principles. Ideal for college-level students in economics and mathematics.

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