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 (`Q_e`) for each square foot (`Q_f`) of space so she always chooses:

    `Q_e/Q_f = 1/1` or `Q_e = Q_f` 

    Using that and her budget constraint to solve for her equilibrium choice:

    `M = P_e*Q_e + P_f*Q_f`

    `M = P_e*Q_e + P_f*Q_e` (substituting `Q_e` for `Q_f`)

    `M = (P_e+P_f)*Q_e`

    `Q_e = M/(P_e+P_f)`

    `Q_e = ($1000)/($0.1+$0.9) = 1000`

    `Q_f = Q_e = 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:

    `Q_e = M/(P_e+P_f)`

    `Q_e = ($1500)/($0.1+$0.9) = 1500`

    `Q_f = Q_e = 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 IC1 to IC2:

  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 `Q_x` and `Q_y` so that the following equation holds:

    `Q_x/Q_y = b/1` or `Q_x = b*Q_y`

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

    `M = P_x*Q_x + P_y*Q_y`

    `M = P_x*(b*Q_y) + P_y*Q_y`

    `M = (b*P_x + P_y)*Q_y`

    `Q_y = M/(b*P_x + P_y)`

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

    `Q_x = b*Q_y`

    `Q_x = (b*M)/(b*P_x + P_y)`

    To summarize, the finished demand equations are:

    `Q_x = (b*M)/(b*P_x + P_y)`

    `Q_y = M/(b*P_x + P_y)`
  2. There are two ways to answer this question. The quickest way is to calculate `Q_x`/`Q_y` for each household in each year. If a household has perfect-complements preferences and always chooses `Q_x = b*Q_y`, then `Q_x`/`Q_y` 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:

    `Q_y = M/(b*P_x + P_y)`

    `b*P_x + P_y = M/Q_y`

    `b = (M/Q_y - P_y)/P_x`

    Using this equation to calculate b from the 2007 data for each household produces the following:


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

    Household
    b
    `P_x` `P_y` M
    `Q_y` 
    Check
     A 2
    $6 $3 $330
    `($330)/(2*$6+$3) = 22`
    correct
     B 1.2
    $6 $3 $240
    `($240)/(1.2*$6+$3) = 23.5`
    wrong
     C 1.5
    $6 $3 $132
    `($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:

    `Q_x = (b*M)/(b*P_x + P_y) = (2*$360)/(2*$5 + $5) = 48`

    `Q_y = M/(b*P_x + P_y) = ($360)/(2*$5 + $5) = 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:

    `Q_x = (g*M)/P_x`

    `g = (P_x*Q_x)/M`

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

    Household
    M
    `P_x` `Q_x`
    `g`
    A
    $280
    $5 40
    `($5*40)/($280) = 0.71`
    B
    $200
    $5 24
    `($5*24)/($200) = 0.60`
    C
    $115
    $5 15
    `($5*15)/($115) = 0.65`

    Using these values to predict `Q_x` in 2008 gives:

    Household
    g
    `P_x` M
     `Q_x` Check
    A
    0.71
    $6 $330
    `(0.71*$330)/($6) = 39`
    wrong
    B
    0.60
    $6 $240
    `(0.60*$240)/($6) = 24`
    correct
    C
    0.65
    $6 $132
    `(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:

    `Q_x = (g*M)/P_x = (0.6*$300)/($5) = 36`

    `Q_y = ((1-g)*M)/P_y = ((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 10/24/2018