Skip to content. | Skip to navigation

Personal tools
Log in
Sections
You are here: Home › Teaching › Intermediate Microeconomics › Problem Sets › HW2

HW2

next up previous
Next: About this document ...

Econ 300
Assignment 2

  1. Charlie likes both apples and bananas. He consumes nothing else. The consumption bundle where Charlie consumes $ x_{A}$ bushels of apples per year and $ x_{B}$ bushels of bananas per year is written as $ (x_{A},x_{B})$ . Last year, Charlie consumed 20 bushels of apples and 5 bushels of bananas. It happens that the set of consumption bundles $ (x_{A},x_{B})$ such that Charlie is indifferent between $ (x_{A},x_{B})$ and $ (20,5)$ is the set of all bundles such that $ x_{B}=\frac{100}{x_{A}}$ . The set of bundles $ % (x_{A},x_{B})$ such that Charlie is just indifferent between $ (x_{A},x_{B})$ and the bundle $ (10,15)$ is the set of bundles such that $ x_{B}=\frac{150}{% x_{A}}$ .

    1. Plot several point that lie on the indifference curve that passes through the point $ (20,5)$ and sketch this curve in blue. Do the same in red, for the indifference curve that passes through the point $ (10,15)$ . Plot apples on the x axis.

    2. Use red to shade in the set of commodity bundles that Charlie prefers to the bundle $ (10,15)$ . Use blue to shade in the set of commodity bundles such that Charlie prefers $ (20,5)$ to these bundles.

    3. Remember that Charlie's indifference curve through the point $ (20,5)$ has the equation $ x_{B}=\frac{100}{x_{A}}$ . Calculate the derivative of this, $ \frac{dx_{B}}{dx_{A}}$ . Find Charlie's marginal rate of substitution at the point $ (10,10)$ . (Hint: what do we know is true about the point $ (10,10)$ ?)

    4. Find Charlie's marginal rate of substitution at the point $ (5,20)$ .

    5. Find Charlie's marginal rate of substitution at the point $ (20,5)$ .

    6. Do the indifference curves you have drawn for Charlie exhibit diminishing marginal rate of substitution?

  2. Ambrose consumes only nuts and berries. Fortunately, he likes both goods. The set of consumption bundles $ (x_{1},x_{2})$ such that Ambrose is indifferent between $ (x_{1},x_{2})$ and $ (1,16)$ is the set of bundles such that $ x_{1}\geq 0,x_{2}\geq 0,$ and $ x_{2}=20-4x_{1}^{1/2}.$ The set of bundles $ (x_{1},x_{2})$ such that $ (x_{1},x_{2})$ $ \symbol{126}(36,0)$ is the set of bundles such that $ x_{1}\geq 0,x_{2}\geq 0,$ and $ % x_{2}=24-4x_{1}^{1/2}.$

    1. Draw a graph where nuts are on the x axis. Plot several points that lie on the indifference curve that passes through the point $ (1,16)$ and sketch this curve in blue.

    2. Do the same in red for the indifference curve passing through the point $ (36,0)$ .

    3. Use blue to shade in the set of commodity bundles that Ambrose weakly prefers to the bundle $ (1,16)$ . Use red to shade in the set of commodity bundles such that Ambrose weakly prefers $ (36,0)$ to these bundles. Is the set of bundles that Ambrose prefers to $ (1,16)$ a convex set?

    4. What is the slope of Ambrose's indifference curve at the point $ (9,8)$ ?

    5. What is the slope of Ambrose's indifference curve at the point $ % (4,12) $ ?

    6. Do the indifference curves you have drawn for Ambrose exhibit diminishing marginal rate of substitution?

    7. Does Ambrose have convex preferences?

  3. The notion of utility is an ''ordinal'' one for which it is assumed that people can rank combinations of goods as to their desirability but that they cannot assign a unique numerical (cardinal) scale for the goods that qunaitifies ''hom much'' one combination is preferred to another. For eadch of the following ranking systems, describe whether an ordinal or cardinal ranking is being used.

    1. military or academic ranks

    2. prices of vintage wines

    3. rankings of vintage wines by the French Wine Society

    4. press rankings of the ''Top Ten'' football teams

    5. results in the current US Open Golf Championships (in which players are ranked by stroke play)

    6. results of early US Open Golf Championships (which were conducted using match play)

  4. Two students are studying microeconomics trying to understand why the tangent conditions studied in this chapter means utility is at a maximum. Let's listen. Student A: If a person chooose a point on his or her budget constrain that is not tangent, it is clear that he or she can manage to get a higher utility by spending differently. Student B: I don't get it - how do you know he or she can do better instead of worse? Draw a graph and explain so that student B will understand.

  5. Suppose that a person has preferences for apples (A) and oranges (O) given by

    $\displaystyle U=\left( AO\right) ^{\frac{1}{2}}$    

    1. If $ A=5$ and $ O=80$ , what will utility be?

    2. If $ A=10,$ what value for $ O$ will provide the same utility as in part $ a?$

    3. If $ A=20,$ what value for $ O$ will provide the same utility as in part $ a?$

    4. Graph the indifference curve implied by parts $ a$ through $ c.$

    5. Suppose a person has $8.00 to spend only on apples and oranges. Apples cost $0.40 each and oranges cost $0.10 each. Which of the points identified in parts $ a$ through $ c$ can be bought by this person?

    6. Show through some examples that every other way of allocating income provides less utility that does the point identified in $ b$ . Graph this utility maximizing situation.

  6. A common utility function used to illustrate economic examples is the Cobb-Douglas function where

    $\displaystyle U\left( X,Y\right) =X^{\alpha }Y^{\beta }$    

    where $ \alpha $ and $ \beta $ are fractional exponents that sum to 1.0 (that is for example, $ \alpha =0.3$ and $ \beta =0.7)$ .

    1. Explain why the utility problem used in problem $ 2$ , is a special case of this function.

    2. It can be shown that a person with this utility function will spend a fraction $ \alpha $ of his income on good $ X$ and a fraction $ \beta $ on good $ Y.$ Show that with this utility function, a person's total spending on good $ X$ will not change if the price of $ X$ changes so long as his income remains constant.

    3. Show that with this utility function, a change in the price of $ Y$ will not afffect the amount of $ X$ purchased.

    4. Show that with this utility function, a $ 50\%$ increase in income accompanied by no changes in the price of $ X$ or $ Y$ will cause purchases of both $ X$ and $ Y$ to rise by $ 50\%.$

  7. Charlie's utility function is $ U(x_{A},x_{B})=x_{A}x_{B}.$

    (a)
    Charlie has 40 apples $ (x_{A})$ and 5 bananas $ (x_{B})$ . Charlie's utility for the bundle $ (40,5)$ is $ U(40,5)=\_\_\_\_\_\_\_$ .

    (b)
    The indifference curve through $ (40,5)$ includes all commodity bundles $ (x_{A},x_{B})$ such that $ x_{A}x_{B}.=\_\_\_\_\_\_\_\_.$ So the indifference curve through $ (40,5)$ has the equation $ x_{B}=\_\_\_\_\_.$

    (c)
    Graph the indifference curve showing all the bundles that Charlie likes exactly as well as the bundle $ (40,5).$ Plot apples on the x axis.

    (d)
    Donna offers to give Charlie 15 bananas if he will give her 25 apples. Would Charlie have a bundle that he likes better than $ (40,5)$ if he makes this trade?

    (e)
    What is the largest number of apples that Donna could demand from Charlie in return for 15 bananas if she expects him to be wiling to trade or at least indifferent about trading? (Hint: if Donna gives Charlie 15 bananas, he will have a total of 20 bananas. If he has 20 bananas, how many apples does he need in order to be as well-off as he would without trade?)

  8. Recall Shirley Sixpack and Lorraine Quiche from HW1. Shirley thinks a 16-ounce can of beers is just as good as two 8-ounce cans. Lorraine only drinks 8 ounces at a time and hates stale beer, so she thinks a 16-ounce can is not better or worse than an 8-ounce can.

    (a)
    Write a utility function that represents Shirley's preferences between commodity bundles comprised of 8-ounce cans and 16-ounce cans of beer. Let $ x_{1}$ stand for the number of 8 ounce cans and $ x_{2}$ stand for the number of 16 ounce cans.

    (b)
    Write a utility function that represents Lorraine's preferences.

    (c)
    Would the utility function $ U(x_{1},x_{2})=100x_{1}+200x_{2}$ represent Shirley's preferences? Would the utility function $ % U(x_{1},x_{2})=(5x_{1}+10x_{2})^{2}$ represent her preferences? Would the utility function $ U(x_{1},x_{2})=(x_{1}+3x_{2})$ represent her preferences?

  9. Consider the utility function

    $\displaystyle U=\min \{4x_{1},x_{2}\}$    

    (a)
    Graph the indifference curves for $ U=4$ and $ U=8.$

    (b)
    Calculate the general form of $ x_{1}^{\ast }$ and $ x_{2}^{\ast }$

    (c)
    Suppose $ p_{1}=2$ , $ p_{2}=4,m=36.$ Calculate $ x_{1}^{\ast }$ and $ x_{2}^{\ast }.$

  10. Consider the utility function

    $\displaystyle U=2x_{1}+3x_{2}$    

    (a)
    Draw a few indifference curves. What is the slope of the indifference curve?

    (b)
    If $ x_{2}$ represents an 8 oz can of soda, what size of soda can is $ x_{1}?$

    (c)
    Write out the general solutions for this problem. You should show your answer graphically and explain the result intuitively.

    (d)
    Suppose $ p_{1}=1,p_{2}=1,$ and $ m=\$30.$ What is the solution to this problem?


next up previous
Next: About this document ...
Jenn Thacher 2008-08-25
Document Actions