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 xx, we use the relationship given by the Hicksian demand function for xx:

hx=(pypx)45uˉh_x = \left(\frac{p_y}{p_x}\right)^{\frac{4}{5}} \bar{u}

where:

  • pyp_y and pxp_x are the prices of goods yy and xx, respectively.
  • uˉ\bar{u} represents the utility level.

Additionally, we know:

  • Mathias spends 15\frac{1}{5} of his income on good xx, which implies that the expenditure share for xx is αx=0.2\alpha_x = 0.2.
  • The income elasticity for good xx, denoted by ex,Ie_{x,I}, is given as 11.

Step 1: Expressing Own Price Elasticity of Demand

The own price elasticity of demand for good xx, denoted by ex,xe_{x,x}, can generally be expressed as:

ex,x=Income elasticity of demand×(Expenditure share1)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 xx, we use the relationship given by the Hicksian demand function for xx:

hx=(pypx)45uˉh_x = \left(\frac{p_y}{p_x}\right)^{\frac{4}{5}} \bar{u}

where:

  • pyp_y and pxp_x are the prices of goods yy and xx, respectively.
  • uˉ\bar{u} represents the utility level.

Additionally, we know:

  • Mathias spends 15\frac{1}{5} of his income on good xx, which implies that the expenditure share for xx is αx=0.2\alpha_x = 0.2.
  • The income elasticity for good xx, denoted by ex,Ie_{x,I}, is given as 11.

Step 1: Expressing Own Price Elasticity of Demand

The own price elasticity of demand for good xx, denoted by ex,xe_{x,x}, can generally be expressed as:

ex,x=Income elasticity of demand×(Expenditure share1)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 xx (ex,Ie_{x,I}) is 11.
  • Expenditure share on xx (αx\alpha_x) is 0.20.2.

Substitute these values:

ex,x=1×(0.21)e_{x,x} = 1 \times (0.2 - 1)

Calculating inside the parentheses:

ex,x=1×0.8=0.8e_{x,x} = 1 \times -0.8 = -0.8

Conclusion

The own price elasticity of demand for good xx is 0.8-0.8.

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 11?
  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.

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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

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Suitable Grade Level

Undergraduate Economics