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Problem Set 6

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Problem Set 6

  1. For the $ CES$ production function

    $\displaystyle Q=A\left[ \delta K^{\gamma }+\left( 1-\delta \right) L^{\gamma }\right] ,$    $\displaystyle A>0,\gamma <1,0<\delta <1$ (1)

    1. Show that the isoquants are negatively sloped and convex
    2. Suppose $ \gamma <0.$ Sketch the isoquants.
    3. Now suppose that $ 0<\gamma <1.$ Sketch the isoquants.
    4. Use the Lagrangian multiplier method to determine the optimal values for each constrained optimization problem's choice variables and solve for the Lagrange multiplier, $ \lambda .$ Check the second order conditions and constraint qualifications. Find whether a slight relaxation of the constraint will increase or decrease the optimal value of the objective function.

  2. Optimize the following:
    1. $ U=x^{2}+2x+3z^{2}-6x+xz$ subject to $ 2x+2z=32$
    2. $ z=xy$ subject to $ x+2y=2$

  3. Find the maximum value of $ yz+xz$ subject to the two constraints
    $\displaystyle y+2z$ $\displaystyle =$ $\displaystyle 1$  
    $\displaystyle x+z$ $\displaystyle =$ $\displaystyle 3.$  

  4. A firm has a Cobb-Douglas production function

    $\displaystyle F\left( K,L\right) =AK^{\alpha }L^{\beta }$ (2)

    where $ A,\alpha ,\beta $ are positive constants. Let the prices of capital and labour be $ r$ and $ w$ respectively. The firm's output is $ q.$

    1. Express the firm's cost minimization problem as a constrained minimization problem.
    2. Write down the first-order conditions and explain why, in this case, they are sufficient for a constrained minimum. Hence, find the conditoinal input demand functions and the cost function.

  5. Assume that $ U=\left( x+2\right) \left( y+1\right) $ . No specific numerical values are assigned to the price and income parameters.
    1. Write the Lagrangian function
    2. Find the optimal $ x,y,\lambda $ .
    3. Check the SOC for a maximum.
    4. Solve for $ P_{x}=4,P_{y}=6,M=130.$
    5. Find all the comparative statics you can, evaluate their signs, and interpret their economic meaning.

  6. Suppose a production function $ y=f\left( L,K\right) $ is hod1.

    1. Show that
      $\displaystyle f_{LL}L+f_{LK}K$ $\displaystyle =$ 0  
      $\displaystyle f_{KL}L+f_{KK}K$ $\displaystyle =$ 0  

    2. Show that

      $\displaystyle f_{LL}L^{2}=f_{KK}K^{2}$    

  7. Show that if the marginal products are positive, the isoquants must be downward-sloping.

  8. Consider the CES utility function:

    $\displaystyle u\left( x_{1},x_{2}\right) =\left[ \alpha _{1}x_{1}^{\rho }+\alpha _{2}x_{2}^{\rho }\right] ^{\frac{1}{\rho }}$    

    1. Show that when $ \rho =1$ , indifference curves become linear.

    2. Show that as $ \rho \rightarrow 0$ , this utility function comes to represent the same preferences as the generalized Cobb-Douglas utility function $ u\left( x_{1},x_{2}\right) =x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}$

    3. Show that for the CES utility function, $ \sigma =\frac{1}{1-\rho }$

  9. In a model of monopolistic competition, Avinash Dixit and Joseph Stiglitz propose a utility function that reflects a preference for a range of differentiated products in consumption. In the three-good case, their utility function takes the form

    $\displaystyle U\left( X_{1},X_{2},X_{3}\right) =\left( X_{1}^{\alpha },+X_{2}^{\alpha }+X_{3}^{\alpha }\right) ^{\frac{1}{\alpha }}$    

    where $ 1>\alpha >0.$ Solve for the elasticity of substition between any two goods in this model.

  10. Consider the profit maximization problem:

    $\displaystyle \underset{x_{1},x_{2}}{Max}pf\left( x_{1},x_{2}\right) +w_{1}x_{1}+w_{2}x_{2}$    

    1. State the envelope theorem result of how a change in $ w_{2}$ affects maximized profit.

    2. Demonstrate that this result is true using typical maximization methods.

  11. Consider maximization models with the specification: maximize $ % y=f\left( x_{1},x_{2},\alpha \right) $ subject to $ g\left( x_{1},x_{2}\right) =k$ with

    $\displaystyle L=f\left( x_{1},x_{2},\alpha \right) +\lambda \left[ k-g\left( x_{1},x_{2}\right) \right]$    

    where $ x_{1},x_{2}$ are choice variables and $ \alpha ,k$ are parameters.

    1. Define $ \phi \left( \alpha ,k\right) =$ maximum value of $ y$ for given $ \alpha $ and $ k$ in this model. On a graph with $ \alpha $ on the horizontal axis and $ f$ and $ \phi $ on the vertical axis, explain geometrically the envelope results $ \phi _{\alpha }=f_{\alpha }$ and $ \phi _{\alpha a}>f_{\alpha a}.$

    2. On a similar graph, explain why it is not possible to carry out a similar procedure to the parameter $ k.$ How does this result relate to the appearance of refutable comparative statics theorems in economics?

    3. Using the results of $ \left( a\right) $ , prove that

      $\displaystyle f_{1\alpha }\frac{\partial x_{1}}{\partial \alpha } +f_{2\alpha }\frac{\partial x_{2}}{\partial \alpha }>0$    

  12. Using the primal-dual methodology, prove algebraically the envelope theorem results
    $\displaystyle \phi _{\alpha }$ $\displaystyle =$ $\displaystyle f_{\alpha }$  
    $\displaystyle \phi _{\alpha \alpha }$ $\displaystyle >$ $\displaystyle f_{\alpha \alpha }$  
    $\displaystyle \phi _{k}$ $\displaystyle =$ $\displaystyle \lambda ^{\ast }$  


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Jennifer Anne Thacher 2008-10-21
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