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HW3

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Econ 300
Assignment 3

  1. Suppose that an electric company charges consumers $.10 per kilowatt hour for electricity for the first 1,000 used in a month but $0.15 for each extra kilowatt hour used after that. Draw a budget constraint for a consumer facing this price schedule and discuss why many individuals may choose to consumer exactly 1,000 kilowatt hours.

  2. Vera is an impoverished graduate student who has only $100 a month to spend on food. She has read in a government publication that she can assure an adequate diet by eating only peanut butter and carrots in the fixed ratio of 2 pounds of peanut butter to 1 pound of carrots. She decides to limit her diet to that regime.

    1. If peanut butter costs $4 per pound and carrots cost $2 per pound, how much can she eat during the month?

    2. Suppose peanut butter costs rise to $5 because of peanut susidies introduced by a politically sensitive government. By how much will Vera have to reduce her food purchases?

    3. How much in food stamp aid would the government have to give Vera to compensate for the effects of the peanut subsidy?

    4. Explain why Vera's preferences are of a very special type here. How would you graph them?

  3. Fill in the table below.

    $ u(x_{1},x_{2})$ $ MU_{1}$ $ MU_{2}$ $ MRS$
    $ 2x_{1}+3x_{2}$      
    $ 4x_{1}+6x_{2}$      
    $ ax_{1}+bx_{2}$      
    $ 2x_{1}^{1/2}+x_{2}$      
    $ \ln x_{1}+x_{2}$      
    $ x_{1}x_{2}$      
    $ x_{1}^{a}x_{2}^{b}$      
    $ (x_{1}+2)(x_{2}+1)$      
    $ (x_{1}+a)(x_{2}+b)$      
    $ x_{1}^{a}+x_{2}^{b}$      

  4. Reconsider Charlie yet again. Recall that his utility function is $ % U(x_{A},x_{B})=x_{A}x_{B}.$ Suppose that $ p_{A}=1$ , $ p_{B}=2$ , and $ m=40.$

    (a)
    Write Charlie's budget equation.

    (b)
    Write Charlie's general optimization problem.

    (c)
    Now write the optimization problem in terms of $ x_{B}.$ Solve for $ x_{B}^{\ast }.$ Solve for $ x_{A}^{\ast }.$

    (d)
    Calculate the utility level at $ x_{A}^{\ast },x_{B}^{\ast }.$

    (e)
    On a graph with apples $ (x_{A})$ on the x axis, plot the budget equation in black. In red, draw the indifference curve for the utility level calculated in (d).

    (f)
    Calculate MRS( $ x_{A}^{\ast },x_{B}^{\ast })$ . What is the slope of the budget equation?

    (g)
    What fraction of his income does Charlie always spend on bananas? (Hint: See page 83)

  5. Calculate $ x^{\ast },y^{\ast }.$ for the following cases.

    (a)
    $ U(x,y)=(x+2)(y+1)$ , where $ p_{x}=p_{y}=1,m=11$

    (b)
    $ U(x,y)=4x^{1/2}+y$ , where $ p_{x}=1,p_{y}=2,m=24$

    (c)
    $ U(x,y)=\min \{x,y^{2}\}$ , where $ p_{x}=1,p_{y}=2,m=8$

  6. Douglas Cornfield's preferences are represented by the utility function $ U(x_{1},x_{2})=x_{1}^{2}x_{2}^{3}$ .

    (a)
    Calculate $ x_{1}^{\ast }$ and $ x_{2}^{\ast }$ for the general budget equation $ p_{1}x_{1}+p_{2}x_{2}=m.$

    (b)
    What share of his income does Doug spend on $ x_{1}$ ?

    (c)
    Other members of Doug's family have similar utility functions of the form $ U(x_{1},x_{2})=cx_{1}^{a}x_{2}^{b}$ , where $ c>0,a>0,b>0.$ What fraction of their income do members of Doug's family spend on $ x_{1}$ . .

  7. Mary's utility function is $ U(b,c)=b+100c-c^{2},$ where $ b$ is the number of silver bells in her garden and $ c$ is the number of cockle shells. She has 500 square feet in her garden to allocate between silver bells and cockle shells. Silver bells each take up 1 square foot and cockle shells each take up 4 square feet. She gets both kinds of seeds for free. How many silver bells and cockle shells should Mary plant?

  8. You allocate $24 per week for the purchase of cookies and apples at your school's cafeteria. Your utility from eating cookies and apples is given by

    $\displaystyle U(C,A)=2C^{1/2}+A^{1/2}$    

    Assume that cookies cost $1 each and apples cost $0.50. Solve for the optimal number of apples and cookies.

  9. Your favorite pastries are Twinkies and RingDings. The utility you derive from each of these is given by the function

    $\displaystyle U(T,R)=\frac{1}{2}\ln (T)+\frac{1}{2}\ln (R)$    

    where $ T$ is the number of Twinkies and $ R$ is the number of cases of RingDings consumed each month. A case of RingDings costs $8 and a case of Twinkies costs $4. Determine the optimal number of Twinkies and RingDings if you have $32 to spend on pastries each month.

  10. Consider the student with an insatiable appetite for fast-food hamburgers. Assume that his utility can be expressed as the Cobb-Douglas function

    $\displaystyle U=x_{1}^{1/2}x_{2}^{1/2}$    

    where $ x_{1}$ is the number of McBurger hamburgers and $ x_{2}$ is the number of King of Burger hamburgers. Assume that McBurger hamburgers cost $4, King of Burger hamburgers cost $2 each, and that the student has $120 to spend on hamburgers each semester.

    (a)
    Calculate $ x_{1}^{\ast }$ and $ x_{2}^{\ast }.$

    (b)
    Find the $ x_{1}^{\ast }$ and $ x_{2}^{\ast }$ when the utility function is the natural logarithm of $ U(x_{1},x_{2})$ such that

    $\displaystyle V\left( x_{1},x_{2}\right) =\ln U(x_{1},x_{2})$    

    (c)
    What is the relationship between your answers to $ (a)$ and $ % \left( b\right) .$ Explain.


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Next: About this document ...
Jenn Thacher 2008-08-25
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