# 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:

URL: http://wilcoxen.maxwell.insightworks.com/pages/3017.html
Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 10/24/2018