{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

midterm1_f09solutions2

# midterm1_f09solutions2 - Economics 105 Midterm 1 Solutions...

This preview shows pages 1–3. Sign up to view the full content.

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 α 1 + βx α 2 where 0 < α < 1 and 0 < β . (a) Calculate Lady Gaga’s marginal rate of substitution (2 pts) MRS = - ∂U/∂x 1 ∂U/∂x 2 = - αx α - 1 1 βαx α - 1 2 = - 1 β parenleftbigg x 1 x 2 parenrightbigg α - 1 (b) Calculate Lady Gaga’s elasticity of substitution (6 pts) MRS = - 1 β parenleftbigg x 1 x 2 parenrightbigg α - 1 = - 1 β parenleftbigg x 2 x 1 parenrightbigg 1 - α ln ( | MRS | ) = ln parenleftBigg 1 β parenleftbigg x 2 x 1 parenrightbigg 1 - α parenrightBigg = ln parenleftbigg 1 β parenrightbigg + (1 - α ) ln parenleftbigg x 2 x 1 parenrightbigg dln ( | MRS | ) = (1 - α ) dln parenleftbigg x 2 x 1 parenrightbigg dln parenleftBig x 2 x 1 parenrightBig dln ( | MRS | ) = σ = 1 1 - α (c) How does Lady Gaga’s elasticity of substitution depend upon α and β ? (2 pts) ∂σ/∂α = - 1 / (1 - α ) 2 < 0. It does not depend on β (d) Suppose Lady Gaga’s 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 σ > 1 then, then the substitution effect dominates the income effect and ∂x 2 /∂p 1 > 0 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. Stunned by Kanye’s appearance, Taylor forgets her preferences. All she can remember is that her expenditure function is ( p 1 + 4 p 2 ) u . (a) Find Taylor’s 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 ( τp 1 , τp 2 , τI ) = τI τp 1 + 4 τp 2 = I p 1 + 4 p 2 = v ( p 1 , p 2 , I ) (b) Find Taylor’s hicksian demands. Verify that h 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}