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HW7

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Econ 300
Assignment 7

  1. Provide a brief description of the production function for each of the following firms. What is the firm's outputs? What inputs does it use? Can you think of any special features of the way production takes place in the firm?

    1. an Iowa wheat farm

    2. an Arizona vegetable farm

    3. U.S. Steel Corporation

    4. a local arc welding firm

    5. Sears

    6. Joe's Hot Dog Stand

    7. the Metropolitan Opera

    8. the Metropolitan Museum of Art

    9. the National Institutes of Health

    10. Dr. Smith's private practice

    11. Paul's lemonade stand

  2. In what ways are firms's isoquant maps and individual's indiffernce curve maps based on the same idea? What are the most important ways in which these concepts differ?

  3. A recent news headline read: ''Productivity rises by record amount as economy roars out of recession.'' Assuming that the ''productivity'' referred to this headline is the customary ''Average output per worker hou'' that is usually reported, how much would you evaluate whether this increase really is an increase in workers' marginal products.

  4. Marjorie Cplus wrote the following answer on her micro examination: ''Virtually every production function exhibits diminshing returns to scale because my professor said that all inputs have diminishing marginal productivities. So when all inputs are doubled, output must be less than double.'' How would you grade Marjorie's answer?

  5. Frisbees are porduced according to the production function

    $\displaystyle q=2K+1$    

    where
    $\displaystyle q$ $\displaystyle =$ output of Frisbees per hous  
    $\displaystyle K$ $\displaystyle =$ capital input per hour  
    $\displaystyle L$ $\displaystyle =$ labor input per hour  

    1. If $ K=10$ , how much $ L$ is needed to produce $ 100$ frisbees per hous?

    2. If $ K=25$ , how much $ L$ is needed to produce $ 100$ frisbees per hous?

    3. Graph the $ q=100$ isoquant. Indicate the points on the isoquant defined in $ a$ and $ b$ . What is the RTS along this isoquant? Explain why the RTS is the same at every point on the isoquant.

    4. Graph the $ q=50$ and $ q=200$ isoquants for this production function also. Describe the shape of the entire isoquant map.

    5. Suppose technical progress resulted in the production function for frisbees becomes

      $\displaystyle q=3K+1.5L.$    

      Answer $ a-d$ for this new production function and discuss how it compares to the previous case.

  6. Suppose that the hourly output of chili at a barbecue ($ q,$ measured in pounds) is characterized by

    $\displaystyle q=20\sqrt{KL}$    

    where $ K$ is the number of large pots used each hour and $ L$ is the number of worker hours employed.

    1. Graph the $ q=2000$ pounds per hour isoquant.

    2. The point $ K=100,L=100$ is one point on the $ q=2000$ isoquant. What value of $ K$ corresponds to $ L=101$ on that isoquant? What is the approximate value for the RTS at $ K=100,L=100?$

    3. The point $ K=25,L=400$ also lies on the $ q=2000$ isoquant. If $ L=401,$ what must $ K$ be for this input combination to lie on the $ q=2000$ isoquant? What is the approximate value of the RTS at $ K=25,L=400?$

    4. For this production function, it can be shown that a general formula for the RTS is

      $\displaystyle RTS=\frac{K}{L}.$    

      Compare the results from applying this formula to those you calculated in $ b$ and $ c.$ To convince yourself further, perform a similar calculation for the point $ K=200,L=50.$

  7. For the Cobb-Douglas production function

    $\displaystyle q=K^{\alpha }L^{\beta }$    

    it can be shown (using calculus) that
    $\displaystyle MP_{K}$ $\displaystyle =$ $\displaystyle \alpha K^{\alpha -1}L^{\beta }$  
    $\displaystyle MP_{L}$ $\displaystyle =$ $\displaystyle \beta K^{\alpha }L^{\beta -1}.$  

    If the Cobb-Douglas exhibits constant returns to scale $ \left( \alpha +\beta =1\right) $ , show that

    1. Both marginal productivites are diminishing

    2. The RTS for this function is given by

      $\displaystyle RTS=\frac{\beta K}{\alpha L}.$    

    3. The function exhibits a diminishing RTS.

  8. Fill in the table below. Some of the calculations are used in the problems below.

    $ f(x_{1},x_{2})$ $ MP_{1}$ $ MP_{2}$ $ TRS$
    $ x_{1}+2x_{2}$      
    $ ax_{1}+bx_{2}$      
    $ 50x_{1}x_{2}$      
    $ x_{1}^{0.25}x_{2}^{0.75}$      
    $ Cx_{1}^{a}x_{2}^{b}$      
    $ (x_{1}+2)(x_{2}+1)$      
    $ (x_{1}+a)(x_{2}+b)$      
    $ ax_{1}+bx_{2}^{0.5}$      
    $ x_{1}^{a}+x_{2}^{a}$      
    $ (x_{1}^{a}+x_{2}^{a})^{b}$      

  9. Suppose that the production function has the form $ % f(x_{1},x_{2},x_{3})=Ax_{1}^{a}x_{2}^{b}x_{3}^{c}$ where $ a+b+c>1$ . Prove that there are increasing returns to scale.

  10. The estimation results in this table are based on the specification $ % Q=AL^{\alpha _{1}}K^{\alpha _{2}}M^{\alpha _{3}},$ where Q=output, A=constant, L=labor, K=capital, and M=raw materials.

    Industry Country $ \alpha _{1}$ $ \alpha _{2}$ $ \alpha _{3}$ Returns to Scale
    Gas France .83 .10    
    Railroads US .89 .12 .28  
    Coal UK .79 .29    
    Food US .72 .35    
    Metals/Machinery US .71 .26    
    Cotton India .92 .12    
    Sugar India .59 .33    
    Coal India .71 .44    
    Paper India .64 .45    
    Chemicals India .80 .37    
    Electricity India .20 .67    
    Paper US .62 .37    
    Telephone Canada .70 .41    
    Chemicals US .54 .38 .11  
    Aircraft US .79 .18 .04  

    (a)
    Fill in the above table with the appropriate conclusion about returns to scale: increasing, decreasing, constant.

    (b)
    Consider the case of gas production in France. If labor was increased by 1% and capital was held constant, how much would output increase?

    (c)
    Present a possible explanation for the different returns to scale for paper in the US and India.

  11. Prunella raises peaches. Where $ L$ is the number of units of labor she uses and $ T$ is the number of units of land she uses, her output is $ % f(L,T)=L^{0.5}T^{0.5}$ bushels of peaches.

    (a)
    Plot input combinations that give here an output of 4 bushels. (Plot L on the x axis) Sketch a production isoquant that runs through these points. The points on the isoquant that give here an output of 4 bushels all satisfy the equation $ T=\_\_\_\_\_\_\_\_\_\_\_\_\_$

    (b)
    This production function exhibits what type of returns to scale?

    (c)
    In the short run, Prunella can't vary the amount of land she uses. On the graph below, in blue, draw a curve showing Prunella's output as a function of labor input if she has 1 unit of land. What happens to the slope of this function as labor increases?

    (d)
    Calculate the MPL (marginal product of labor).

    (e)
    In the long run, Prunella can change her input of land as well as of labor. Suppose that she increases the size of her orchard to 4 units of land. Use red ink to draw a new curve on the graph, showing output as a function of labor input. Also use red ink to draw a curve showing marginal product of labor as a function of labor input when the amount of land is fixed at 4.

  12. Suppose the production function is Cobb-Douglas and $ % f(x_{1},x_{2})=x_{1}^{0.5}x_{2}^{1.5}$ .

    (a)
    Calculate $ MP_{x_{1}}$ . Does $ MP_{x_{1}}$ increase, decrease, or stay constant for small increases in $ x_{1}$ , holding $ x_{2}$ constant?

    (b)
    Calculate $ MP_{x_{2}}$ . Does $ MP_{x_{2}}$ increase, decrease, or stay constant for small increases in $ x_{2}$ , holding $ x_{1}$ constant?

    (c)
    Does $ MP_{x_{1}}$ increase, decrease, or stay constant for small increases in $ x_{2}$ ?

    (d)
    Calculate the technical rate of substitution between $ x_{1}$ and $ x_{2}.$

    (e)
    Does the technology have diminishing technical rate of substitution?

    (f)
    This technology demonstrates __________ returns to scale.


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Jenn Thacher 2008-08-25
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