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Problem Set 3

Problem Set 3
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Problem Set 3

 

 

  1. Find the partial derivative, second partial derivatives, and cross partial derivative of each function.

     

    1. $ f\left( x_{1},x_{2}\right) =12x_{1}^{4}-6x_{1}^{2}x_{2}+4x_{2}^{3}$

       

    2. $ f\left( x_{1},x_{2}\right) =\left( 3x_{1}^{2}+5x_{1}+1\right) \left( x_{2}+4\right) $

     

  2. Find $ \frac{dy}{dz}$ .

     

    1. $ y=f\left( x,w\right) =3x^{2}-2xw+w^{2}$ where $ x=8z-18$ and $ w=4z$

       

    2. $ y=f\left( x,z\right) =4x^{3}+\frac{1}{4}xz^{2}-2z$ where $ x=z^{-2}$

     

  3. Find $ \frac{\partial Z}{\partial u}$ and $ \frac{\partial Z}{\partial v}$

     

    1. $ Z=f\left( x,y\right) =4x^{2}+2xy+y^{2}$ where $ x=3u^{2}$ and $ y=u-2v$

       

    2. $ Z=f\left( x,y\right) =ax^{3}-bx^{2}y+cy$ where $ x=\gamma u+\theta v$ and $ y=\theta u-\gamma v^{2}$

     

  4. Consider the Cobb-Douglas production function, $ y=L^{\alpha }K^{1-\alpha }$ where $ y=$ output, $ L=$ labor, $ K=$ capital, and $ 0<\alpha <1.$

     

    1. Calculate $ MP_{L}$ and $ MP_{K}$

       

    2. Find the rate of change of each due to changes in labor and capital.

       

    3. Show that $ f_{L}L+f_{K}K=y$

     

  5. Consider $ U=x_{1}^{1/3}x_{2}^{2/3}.$ The demand curves associated with this utility function are $ x_{1}=\frac{M}{3p_{1}}$ and $ x_{2}=\frac{2M}{% 3p_{2}}.$ Find the rate of change of $ U$ with respect to changes in each price and money income. Interpret.

     

  6. Consider the production function $ Q=K^{3/4}L^{3/4}$ . Show that marginal productivity of each factor is diminshing. Show, however, that for any strictly positive input combination, if the input combination is doubled, then output more than doubles.

     

  7. Consider the following three utility functions: $ U=x_{1}x_{2}$ , $ % V=x_{1}^{2}x_{2}^{2}$ , $ W=\log x_{1}+\log x_{2}$

     

    1. Find the MU of $ x_{1}$ and $ x_{2}$ for each utility function

       

    2. Find the rates of change of MU of one good with respect to a change in consumption of the other good for each utility function. Verify that, for these functions, the change in the MU of one good due to a change in the other good is the same, no matter which good is chosen first.

       

    3. Find the MRS of $ x_{1}$ for $ x_{2}$ for each utility function

       

    4. What do (b) and (c) imply for economic analysis?

     

  8. Let $ U=f\left( x_{1},x_{2}\right) $ be a utility function and let $ % V\left( x_{1},x_{2}\right) =F\left( U\right) ,$ where $ F^{\prime }\left( U\right) >0$

     

    1. Show that $ \frac{V_{1}}{V_{2}}=\frac{U_{1}}{U_{2}}$

       

    2. Find $ V_{ij}$ in terms of $ U_{ij}$ $ i,j=1,2$ . Show that in general $ % U_{ij}$ and $ V_{ij}$ need not have the same sign.

     

 

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Jennifer Anne Thacher 2008-09-15
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