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: