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**Unformatted text preview: **Economics 105 Midterm 1 Solutions 1. Lady Gaga values outlandish clothes ( x 1 ) and special effects ( x 2 ). Here utility function is given by: u ( x 1 ,x 2 ) = x 1 + x 2 where 0 < < 1 and 0 < . (a) Calculate Lady Gagas marginal rate of substitution (2 pts) MRS =- U/x 1 U/x 2 =- x - 1 1 x - 1 2 =- 1 parenleftbigg x 1 x 2 parenrightbigg - 1 (b) Calculate Lady Gagas elasticity of substitution (6 pts) MRS =- 1 parenleftbigg x 1 x 2 parenrightbigg - 1 =- 1 parenleftbigg x 2 x 1 parenrightbigg 1- ln ( | MRS | ) = ln parenleftBigg 1 parenleftbigg x 2 x 1 parenrightbigg 1- parenrightBigg = ln parenleftbigg 1 parenrightbigg + (1- ) ln parenleftbigg x 2 x 1 parenrightbigg dln ( | MRS | ) = (1- ) dln parenleftbigg x 2 x 1 parenrightbigg dln parenleftBig x 2 x 1 parenrightBig dln ( | MRS | ) = = 1 1- (c) How does Lady Gagas elasticity of substitution depend upon and ? (2 pts) / =- 1 / (1- ) 2 < 0. It does not depend on (d) Suppose Lady Gagas elasticity of substitution was greater than 1. What would this tell you about how an increase in p 1 affected demand for x 2 ? Explain your answer. (2 pts) If > 1 then, then the substitution effect dominates the income effect and x 2 /p 1 > 1 2. Stunned by Kanyes appearance, Taylor forgets her preferences. All she can remember is that her expenditure function is ( p 1 + 4 p 2 ) u . (a) Find Taylors indirect utility function. Verify that it is homogeneous of degree in (3 pts) The indirect utility function is the inverse of the expenditure function. So: v ( p 1 ,p 2 ,I ) = I p 1 + 4 p 2 Indirect utility functions are homogeneous of degree zero in income and prices v ( p 1 ,p 2 ,I ) = I p 1 + 4 p 2 = I p 1 + 4 p 2 = v ( p 1 ,p 2 ,I...

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