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Problem Set 8

Problem Set 8

Problem Set 8

Compute the Marshallian demand funcitons, indirect utility function, and expenditure function for the constant elasticity of substitution (CES) utility function $U(x_{1},x_{2})=(x_{1}^{r} + x_{2}^{r})^{1/r}$
Consider the utility function

$\displaystyle U=2x_{1}^{\frac{1}{2}}+4x_{2}^{\frac{1}{2}}$ (1)
1. Find the Marshallian demand functions for goods and as they depend on prices and income.
2. Find the Hicksian demand functions.
3. Find the expdenditure function and verify that
  
  $\displaystyle x_{1}^{H}\left( p_{1},p_{2},U^{0}\right)$ $\displaystyle =$ $\displaystyle \frac{\partial e\left( p_{1},p_{2},U^{0}\right) }{\partial p_{1}}$
  
  $\displaystyle x_{2}^{H}\left( p_{1},p_{2},U^{0}\right)$ $\displaystyle =$ $\displaystyle \frac{\partial e\left( p_{1},p_{2},U^{0}\right) }{\partial p_{2}}$
4. Find the indirect utility function and verify Roy's identity.
Consider the utility maximization problem

$\displaystyle Max$ $\displaystyle U\left( x_{1},x_{2}\right)$ s.t. $\displaystyle p_{1}x_{1}+p_{2}x_{2}=1$ (2)

where prices have been normalized by setting Let $U^{\ast }\left( p_{1},p_{2}\right)$ be the indirect utility function and $\lambda$ be the lagrange multiplier.
1. Show that $\lambda ^{M}=\frac{\partial U^{\ast }}{\partial x_{1}}% x_{1}^{\ast }+\frac{\partial U^{\ast }}{\partial x_{2}}x_{2}^{\ast }.$
2. Show that
  
  $\displaystyle \frac{\partial U^{\ast }}{\partial p_{1}}$ $\displaystyle =$ $\displaystyle -\lambda ^{M}x_{1}^{\ast }$
  
  $\displaystyle \frac{\partial U^{\ast }}{\partial p_{2}}$ $\displaystyle =$ $\displaystyle -\lambda ^{M}x_{2}^{\ast }$
3. Show that $\lambda ^{M}=-\left[ \frac{\partial U^{\ast }}{\partial p_{1}}p_{1}+\frac{\partial U^{\ast }}{\partial p_{2}}p_{2}\right]$
4. Prove that if $U\left( x_{1},x_{2}\right)$ is hodr in $\left(x_{1},x_{2}\right)$ then $U^{\ast }\left( p_{1},p_{2}\right)$ is hod(-r) in $\left( p_{1},p_{2}\right) .$
Consider the class of utility functions that are ''additively separable'', i.e.,

$\displaystyle U\left( x_{1},x_{2}\right) =U^{1}\left( x_{1}\right) +U^{2}\left( x_{2}\right)$ (3)
1. Find the first- and second-order conditions for utility maximization for these utility functions. Show that diminishing marginal utility in at least one good is implied.
2. Show that if there is diminishing marginal utility in each good then both goods are ''normal'', ie., not inferior.
3. Show that this specification does not imply
  
  $\displaystyle \frac{\partial x_{i}^{M}}{\partial p_{j}}=0,$ $\displaystyle i\neq j$ (4)
4. Show, however, that if $\frac{\partial x_{i}^{M}}{\partial p_{j}}=% \frac{\partial x_{j}^{M}}{\partial p_{i}}=0$ then $U\left( x_{1},x_{2}\right) =\alpha _{1}\log x_{1}+$ $\alpha _{2}\log x_{1}.$
$x^{M}\left( p,M\right)$ satisifies the following properties:
- It is hod0 in and
- $\underset{i}{\sum }p_{i}x_{i}=M$
- $x^{M}\left( p,M\right)$ is a convex set.
1. For $U\left( x_{1},x_{2}\right) =kx_{1}^{\alpha }x_{2}^{1-\alpha },$ where , $0<\alpha <1,$ verify the above properties.
For the utility function $U\left( x_{1},x_{2}\right) =x_{1}^{\alpha}x_{2}^{1-\alpha }$ , calculate the expenditure function.
Consider the utility function $U\left( x_{1},x_{2},x_{3}\right) =\left( x_{1}-b_{1}\right) ^{\alpha }\left( x_{2}-b_{2}\right) ^{\beta }\left( x_{3}-b_{3}\right) ^{\gamma }.$
1. Derive the Hicksian demand and expenditure functions.
2. Show that the derivatives of the expenditure function are the Hicksian demand function
3. Verify that the Slutsky equation holds.

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Jennifer Anne Thacher 2008-11-13

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