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HW5

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Econ 300
Problem Set 5

  1. Irene's demand for pizza is given by $ Q=\frac{0.31}{P}$ where Q is the weekly quantity of pizza bought (in prices), I is weekly income, and P is the price of pizza. Using this demand function, answer the following:

    1. Graph this function for the case $ I=200.$

    2. One problem in using this function to study consumer surplus is that Q never reaches zero no matter how high P is. Hence, suppose that the function holds only for $ P\leq 10$ and that $ Q=0$ for $ P>10.$ How should your graph in $ b$ be adjusted to fit this assumption?

    3. With this demand function $ ($ and $ I=200)$ , it can be shown that the area of consumer surplus is approximately $ CS=198-6P-60\ln P$ , where $ ''\ln \left( P\right) ''$ refers to the natural logarithm of $ P.$ Show that if $ % P=10,$ $ CS=0$ .

    4. Suppose $ P=3.$ How much pizza is demanded and how much consumer surplus does Irene receive? Given an economic interpretation to this magnitude.

    5. If $ P$ were to increase to $ 4,$ how much would Irene demand and what would her consumer surplus be? Give an economic interpretation to why the value of CS has fallen.

  2. An economist hired by a home building firm has been asked to estimate a demand curve for homes. He gathers data on the price of new houses and on the number sold form the top 100 metropolitan areas in the United States. He plots these data, draws a line that seems to pass near the points, and labels that line ''demand''. How many problems can you identify in this approach to estimating the demand for houses?

  3. Suppose that the demand curve for garbanzo beans is given by

    $\displaystyle Q=20-P$    

    where $ Q$ is thousands of pounds of beans bought per week and $ P$ is the price in dollars per pound.

    1. How many beans will be bought at P=0?

    2. At what price does the quantity demanded of beans become zero?

    3. Calculate total expenditures $ \left( P\ast Q\right) $ for beans of each whole dollar price between the prices identified in $ a$ and $ b.$

    4. What price for beans yields the highest total expenditures?

    5. Suppose the demand for beans shifted to

      $\displaystyle Q=40-2P.$    

      How would your answers to $ a-d$ change? Explain the differences intuitively and with a graph.

  4. Suppose the quantity of good $ X$ demanded by individual $ 1$ is given by
    $\displaystyle X_{1}$ $\displaystyle =$ $\displaystyle 10-2P_{X}+0.01I_{1}+0.4P_{Y}$  
    $\displaystyle X_{2}$ $\displaystyle =$ $\displaystyle 5-P_{X}+0.02I_{2}+0.2P_{Y}$  

    1. What is the market demand function for total $ X$ $ \left( =X_{1}+X_{2}\right) $ as a function of $ P_{x},I_{1},I_{2}$ , and $ P_{y}.$

    2. Graph the two individual demand curves (with $ X$ on the horizontal axis, $ P_{x}$ on the vertical axis) for the case $ I_{1}=1000,$ $ I_{2}=1000,$ and $ P_{y}=10.$

    3. Using these individual demand curves, construct the market demand curve for total $ X$ . What is the algebraic equation for this curve?

    4. Now suppose $ I_{1}$ increases to $ 1,100$ and $ I_{2}$ decreases to $ % 900.$ How would the market demand curve shift? How would the individual dmenad curves shift? Graph these new curves?

  5. In Gas Pump, South Dakota, there are two kinds of consumers, Buick owners and Dodge owners. Every Buick owner has a demand function for gasoline $ D_{B}(p)=20-5p$ for $ p\leq 4$ and $ D_{B}(p)=0$ if $ p>4$ . Every Dodge owner has a demand function for gasoline $ D_{B}(p)=15-3p$ for $ p\leq 5$ and $ D_{B}(p)=0$ if $ p>5$ . (Quantities are measured in gallons every week and price is measured in dollars.) Suppose that Gas Pump has 150 consumers, 100 Buick owners, and 50 Dodge owners.

    (a)
    If the price is $3, what is the total amount demanded by each individual Buick owner? And by each individual Dodge owner?

    (b)
    At that price, what is the total amount demanded by all Buick owners? And by all Dodge owners?

    (c)
    At $3, what is the total amount demanded by all consumers in Gas Pump?

    (d)
    On a graph, use blue to draw the demand curve representing total demand by Buick owners. Use black to draw the demand curve representing total demand by Dodge owners. Use red to draw the market demand curve for the entire town.

    (e)
    At what prices does the market demand curve have kinks?

    (f)
    When the price of gasoline is $1 per gallon, how much does the weekly quantity demanded fall when price rises by 10 cents?

    (g)
    When the price of gasoline is $4.50 per gallon, how much does the weekly quantity demanded fall when price rises by 10 cents?

  6. For the case of a normal good, draw a graph that illustrates the SE, IE, and TE for an increase in price. Use 3 separate colors.

  7. For the case of an inferior but non-Giffen good, draw a graph that illustrates the SE, IE, and TE for an increase in price. Use 3 separate colors.

  8. For the case of a Giffen good, draw a graph that illustrates the SE, IE, and TE for an increase in price. Use 3 separate colors.

  9. Sir Plus consumes mead and his demand function for tankards of mead is given by $ D(p)=100-p$ , where $ p$ is the price of mead in shillings. The original price of mead is 50 shillings per tankard. Suppose the price of mead increases to 60 shillings per tankard. Calculate the change in consumer surplus. (Try doing it using both integration and non-integration methods).

  10. An economist is using econometrics to estimate the demand for wood stoves in the United States. She uses the following model:

    $\displaystyle q_{ws}=f\left( p_{ws},p_{w},p_{kh,}p_{eh},I\right)$    

    where $ q_{ws}$ is the quantity demanded of wood stoves, $ p_{ws}$ is the price of wood stoves, $ p_{w}$ is the price of a cord of firewood, $ p_{kh}$ is the price of kerosene heaters, $ p_{eh}$ is the price of electric room heaters, and $ I$ is per capita consumer income. Does the model include all of the major factors you would expect to influence the demand for wood stoves? Would you expect this model to yield a good econometric estimate of the demand for wood stoves? Why or why not?


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Jenn Thacher 2008-08-25
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