PAI 723 Economics for Public Decisions > Exercise 6

Solution

Here are notes on the solution. It's terse in places so stop by if you have questions.

Part A

  1. She likes 1 kWh of electricity (Qe) for each square foot (Qf) of space so she always chooses Qe/Qf = 1/1 or Qe = Qf.  Using that and her budget constraint to solve for her equilibrium choice:

    M = Pe*Qe + Pf*Qf

    M = Pe*Qe + Pf*(Qe)

    M = (Pe+Pf)*Qe

    Qe = M/(Pe+Pf)

    Qe = $1000/($0.1+$0.9) = 1000

    Qf = Qe = 1000

    Thus, she consumes 1000 sq ft of floor space and 1000 kWh of electricity per month. Her equilibrium looks like this:

  2. After she gets the raise, she can increase her consumption of both goods.  The amounts can be computed using the equations above:

    Qe = M/(Pe+Pf)

    Qe = $1500/($0.1+$0.9) = 1500

    Qf = Qe = 1500

    Thus, her floor space rises to 1500 sq ft and her consumption of electricity rises to 1500 kWh. The new graph is shown below and she's moved from IC2 to IC4:

  3. When the price of electricity rises to $0.20, the process above can be repeated to show that her consumption of floor space and kWh will drop to 909.

 

Part B

  1. If the household wants b units of X for each unit of Y, it will always choose Qx and Qy so that the following equation holds:

    Qx/Qy = b/1 or Qx = b*Qy

    That equation can be used with the household's budget constraint to derive its demand curve for Y as follows:

    M = Px*Qx + Py*Qy

    M = Px*(b*Qy) + Py*Qy

    M = (b*Px + Py)*Qy

    Qy = M/(b*Px + Py)

    Using this equation, the household's demand for X can be obtained as follows:

    Qx = b*Qy

    Qx = b*M/(b*Px + Py)
  2. There are two ways to answer this question.  The quickest way is to calculate Qx/Qy for each household in each year.  If a household has perfect-complements preferences and always chooses Qx = b*Qy, then Qx/Qy will equal b and will therefore be the same from one year to the next.  For household A, the ratio is 2 in both years, so it has perfect complements preferences with b=2.  For household B the ratio is 1.2 in 2007 and 0.75 in 2008, and for household C the ratios are 1.5 and 0.6 so neither has perfect complements preferences.

    An alternative approach is to use the demand equations to determine b from the 2007 data and then use that value to predict 2008 behavior.   It takes a few more steps but unlike the trick above, it works for other kinds of preferences, too. Either demand equation can be used; the results below use the equation for Y:

    Qy = M/(b*Px + Py)

    (b*Px + Py) = M/Qy

    b = ((M/Qy) - Py)/Px

    Using this equation to calculate b from the 2007 data for each household produces the following (note that Px and Py are not shown separately in the table because they are the same for all three households):


    Household
    M
    Qy
    Equation
    b
     A $280
    20
     b = (($280/20) - $4)/$5 2
     B $200
    20
     b = (($200/20) - $4)/$5 1.2
     C $115
    10
     b = (($115/10) - $4)/$5 1.5
    Using these values to predict Qy in 2008 gives:

    Household
    b
    M
    Equation 
    Qy
     A 2
    $330
    Qy = $330/(2*$6+$3)
    22 (correct)
     B 1.2
    $240
    Qy = $240/(1.2*$6+$3)
    23.5 (wrong)
     C 1.5
    $132
    Qy = $132/(1.5*$6+$3)
    11 (wrong)
    Since the 2007 data leads to a correct 2008 prediction for household A but not for households B and C, it is possible to conclude that only household A has perfect-complements preferences.  In addition, household A's value of parameter b is 2.
  3. Household A's 2009 consumption can be calculated using its demand equations and the 2009 data:

    Qx = b*M/(b*Px + Py)

    Qy = M/(b*Px + Py)

    Qx = 2*$360/(2*$5 + $5)

    Qy = $360/(2*$5 + $5)

    Qx = 48

    Qy = 24

    Checking via the budget constraint:

    $5*48 + $5*24 = $360

    The household's 2009 equilibrium is shown below:

Part C

  1. As in Part B, either demand equation can be used to determine the unknown parameter.  In this case, the demand for X will be used.  Rearranging the demand equation to solve for g:

    Qx = g*M/Px

    g = Px*Qx/M

  2. Using this equation to calculate g from the 2007 data for each household produces the following:

    Household
    M
    Qx
    Equation
    g
    A
    $280
    40
    g = $5*40/$280
    0.71
    B
    $200
    24
    g = $5*24/$200
    0.60
    C
    $115
    15
    g = $5*15/$115
    0.65



    Using these values to predict Qx in 2008 gives:

    Household
    g
    M
     Equation Qx
    A
    0.71
    $330
    Qx = 0.71*$330/$6
    39 (wrong)
    B
    0.60
    $240
    Qx = 0.60*$240/$6
    24 (correct)
    C
    0.65
    $132
    Qx = 0.65*$132/$6
    14.3 (wrong)
    Since the 2007 data leads to a correct 2008 prediction for household B but not for households A and C, it is possible to conclude: (1) that only household B has Cobb-Douglas preferences, and (2) that household B's value of parameter g is 0.60. 
  3. Using the value of g obtained above, household B's 2009 consumption can be calculated as follows:

    Qx = 0.6*$300/$5 = 36

    Qy = (1-0.6)*$300/$5 =24

    Checking via the budget constraint:

    $5*36 + $5*24 = $180 + $120 = $300

    The household's 2009 equilibrium is shown below:


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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 10/26/2016