Math Problem Statement
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?
Solution
To determine the own price elasticity of demand for good , we use the relationship given by the Hicksian demand function for :
where:
- and are the prices of goods and , respectively.
- represents the utility level.
Additionally, we know:
- Mathias spends of his income on good , which implies that the expenditure share for is .
- The income elasticity for good , denoted by , is given as .
Step 1: Expressing Own Price Elasticity of Demand
The own price elasticity of demand for good , denoted by , can generally be expressed as:
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 , we use the relationship given by the Hicksian demand function for :
where:
- and are the prices of goods and , respectively.
- represents the utility level.
Additionally, we know:
- Mathias spends of his income on good , which implies that the expenditure share for is .
- The income elasticity for good , denoted by , is given as .
Step 1: Expressing Own Price Elasticity of Demand
The own price elasticity of demand for good , denoted by , can generally be expressed as:
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 () is .
- Expenditure share on () is .
Substitute these values:
Calculating inside the parentheses:
Conclusion
The own price elasticity of demand for good is .
Would you like further details or clarifications on any part of this calculation?
Expanding Questions:
- How would the elasticity change if the expenditure share were higher?
- What would happen to the elasticity if the income elasticity was less than ?
- How does the Hicksian demand function differ from the Marshallian demand function in terms of elasticity?
- Can this elasticity be applied directly to a compensated demand curve interpretation?
- 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.
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Math Problem Analysis
Mathematical Concepts
Elasticity of Demand
Income Elasticity
Expenditure Share
Hicksian Demand Function
Formulas
Own Price Elasticity of Demand: e_x,x = Income Elasticity * (Expenditure Share - 1)
Hicksian Demand Function: h_x = (p_y / p_x)^(4/5) * ū
Theorems
-
Suitable Grade Level
Undergraduate Economics
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