Suppose Mathias only consumes two goods x and y. The Hicksian demand function for
good x is: hx =
 py
px
1
4
4/5
 ̄u. Also we know that Mathias spends 1
5 of his income on good
x and her income elasticity for good x is ex,I = 1. What is the own price elasticity of
demand for good x?

To determine the own price elasticity of demand for good \( x \), we use the relationship given by the Hicksian demand function for \( x \):

\[
h_x = \left(\frac{p_y}{p_x}\right)^{\frac{4}{5}} \bar{u}
\]

where:
- \( p_y \) and \( p_x \) are the prices of goods \( y \) and \( x \), respectively.
- \( \bar{u} \) represents the utility level.

Additionally, we know:
- Mathias spends \(\frac{1}{5}\) of his income on good \( x \), which implies that the expenditure share for \( x \) is \( \alpha_x = 0.2 \).
- The income elasticity for good \( x \), denoted by \( e_{x,I} \), is given as \( 1 \).

### Step 1: Expressing Own Price Elasticity of Demand
The own price elasticity of demand for good \( x \), denoted by \( e_{x,x} \), can generally be expressed as:

\[
e_{x,x} = \text{Income elasticity of demand} \times \left(\text{Expenditure share} - 1\right)
\]

This elasticity formula arises from the decomposition of the elasticity for a good in relation to its budget share and responsiveness to income.

### Step 2: Substitute Known Values
From the given dataTo determine the own price elasticity of demand for good \( x \), we use the relationship given by the Hicksian demand function for \( x \):

\[
h_x = \left(\frac{p_y}{p_x}\right)^{\frac{4}{5}} \bar{u}
\]

where:
- \( p_y \) and \( p_x \) are the prices of goods \( y \) and \( x \), respectively.
- \( \bar{u} \) represents the utility level.

Additionally, we know:
- Mathias spends \(\frac{1}{5}\) of his income on good \( x \), which implies that the expenditure share for \( x \) is \( \alpha_x = 0.2 \).
- The income elasticity for good \( x \), denoted by \( e_{x,I} \), is given as \( 1 \).

### Step 1: Expressing Own Price Elasticity of Demand
The own price elasticity of demand for good \( x \), denoted by \( e_{x,x} \), can generally be expressed as:

\[
e_{x,x} = \text{Income elasticity of demand} \times \left(\text{Expenditure share} - 1\right)
\]

This elasticity formula arises from the decomposition of the elasticity for a good in relation to its budget share and responsiveness to income.

### Step 2: Substitute Known Values
From the given 
- Income elasticity of demand for \( x \) (\( e_{x,I} \)) is \( 1 \).
- Expenditure share on \( x \) (\( \alpha_x \)) is \( 0.2 \).

Substitute these values:

\[
e_{x,x} = 1 \times (0.2 - 1)
\]

Calculating inside the parentheses:

\[
e_{x,x} = 1 \times -0.8 = -0.8
\]

### Conclusion
The own price elasticity of demand for good \( x \) is \( -0.8 \).

Would you like further details or clarifications on any part of this calculation?

---

**Expanding Questions:**
1. How would the elasticity change if the expenditure share were higher?
2. What would happen to the elasticity if the income elasticity was less than \(1\)?
3. How does the Hicksian demand function differ from the Marshallian demand function in terms of elasticity?
4. Can this elasticity be applied directly to a compensated demand curve interpretation?
5. What role does the expenditure share play in determining the price elasticity of a good?

**Tip:** Remember, a good with an own-price elasticity less than -1 is considered elastic, while if it's between -1 and 0, it is inelastic.

Suppose Mathias only consumes two goods x and y. The Hicksian demand function for good x is: hx = (py/px)^(4/5) * ū. Also we know that Mathias spends 1/5 of his income on good x and her income elasticity for good x is ex,I = 1. What is the own price elasticity of demand for good x?

This solution explains how to determine the own price elasticity of demand for good x, given Mathias's Hicksian demand function and expenditure share on x. Learn about elasticity formulas, income elasticity, and expenditure share calculations, essential for understanding consumer demand theory.

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