For cobb-douglas utility function u(x1, x2) = xa
1 xb
2 , after calculation, we
got that optimal point x∗
1 = a
a+b
M
p1 . Calculate the income elasticity
Zihan Hu (SOE) Comparative Statics - Price and Income Effects Lec 5 7 / 22

The Cobb-Douglas utility function is given as \( u(x_1, x_2) = x_1^a x_2^b \), where \( a \) and \( b \) are positive constants representing the relative importance of goods \( x_1 \) and \( x_2 \) in the utility function.

You’ve already calculated the optimal point for \( x_1^* \), which is:

\[
x_1^* = \frac{a}{a+b} \cdot \frac{M}{p_1}
\]

### **Income Elasticity**

The income elasticity of demand measures how the quantity demanded of a good responds to a change in income. It is defined as:

\[
\epsilon_M = \frac{\partial x_1^*}{\partial M} \cdot \frac{M}{x_1^*}
\]

Let’s calculate the income elasticity for \( x_1 \).

#### 1. **Differentiate \( x_1^* \) with respect to \( M \):**
From the expression \( x_1^* = \frac{a}{a+b} \cdot \frac{M}{p_1} \), it’s clear that \( x_1^* \) is linear in \( M \). Thus,

\[
\frac{\partial x_1^*}{\partial M} = \frac{a}{a+b} \cdot \frac{1}{p_1}
\]

#### 2. **Apply the elasticity formula:**

Now, substitute this derivative and \( x_1^* \) into the income elasticity formula:

\[
\epsilon_M = \left( \frac{a}{a+b} \cdot \frac{1}{p_1} \right) \cdot \frac{M}{\frac{a}{a+b} \cdot \frac{M}{p_1}}
\]

Simplifying this:

\[
\epsilon_M = 1
\]

### **Conclusion:**

The income elasticity for the good \( x_1 \) is \( 1 \), meaning that \( x_1^* \) is unit elastic with respect to income. A 1% increase in income leads to a 1% increase in the quantity demanded of \( x_1 \).

---

Do you want any further details or clarification?

Here are 5 related questions you might find interesting:
1. What is the income elasticity for good \( x_2 \)?
2. How does the price elasticity of demand differ from income elasticity?
3. How would you interpret income elasticity values greater than or less than 1?
4. Can the income elasticity change if the utility function is not Cobb-Douglas?
5. How does the distribution of income between goods \( x_1 \) and \( x_2 \) depend on the parameters \( a \) and \( b \)?

**Tip:** In general, Cobb-Douglas utility functions exhibit unitary income elasticity, meaning that demand for each good changes proportionally with income.

For the Cobb-Douglas utility function u(x1, x2) = x1^a x2^b, after calculation, we got that optimal point x1* = (a / (a + b)) * (M / p1). Calculate the income elasticity.

Learn how to calculate the income elasticity of demand for the Cobb-Douglas utility function u(x1, x2) = x1^a x2^b. This solution explains key concepts such as utility functions, optimal points, and elasticity formulas, suitable for undergraduate economics students.

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