A working husband and wife are considering setting up a home gym.  They are considering five pieces of equipment, with the calorie burn (based on intensity) and cost listed in the data file.

The wife will work out from 6 am to 9 am while the husband gets the children ready and sent to school. She prefers to take an hour of relaxation time and then work out at a higher intensity.  The family will cook and share a common meal until 6 pm.  The husband will work out at a lower intensity from 6 pm to 9 pm and then will join his wife in getting the children to bed. Both the husband and wife have the luxury of starting work at 10 am and leaving work at 4 pm, with no more than a half hour commute for each.

The wife has decided to do  workout of moderate intensity for two hours and use an hour to relax prior to work.  The husband has decided to do  workout of low intensity for all three hours after dinner.

Requirements

1. The husband and wife have targets for calories burned during their daily workouts and preference functions over the usage of various equipment during their workout that are also given in the data file.  Of course, you can only purchase whole pieces of equipment and, given that no equipment will be used simultaneously, you will only purchase one piece of equipment if it is in the home gym.  Formulate this set-up and calculate the set of equipment (that will be in the home gym) that will minimize cost.  Be sure to discuss and interpret this result.
2. Each partner does have a gym that is on the way to work so that each partner could work out in the gym rather than workout at home (dinner would be shifted to a more civilized hour of 7:30 pm).  The CO2 of producing each of machine is given in the data file.  If the whole set of machines were in the gym, they would support 15 customers per day in the gym.
   
   Currently, a carbon tax is applied to producers and these are passed on to the consumers.  Many have pointed out that this is problematic since consumers are unaware of their carbon footprint on purchases (versus usage).  For example, individuals may know the carbon footprint of their car usage, but do not know the carbon footprint of producing the car (CO2 in excess of 6.5 metric tons for a midsize SUV and in excess of 9.3 metric tons for an all-electric midsize SUV).  Thus, some advocates for sustainable use claim that a direct CO2 tax on consumers would be more successful at changing behavior.
   
   Assume that a carbon tax is proposed for purchasers that is $500 per metric ton of CO2.  How much higher would the CO2 tax on the home gym in Part One be than the CO2 tax passed onto the couple via gym membership if they were to opt to work out in the gym?  Do you think this tax would change the couple's behavior?  Be sure to explain your calculation, interpret the results, and provide a reasoned argument on whether the behavior would change.

Data

Exercise Equipment Cost

All equipment is commercial grade.

Exercise Item	Cost
Elliptical	$1,800
Inclining Treadmill	$2,100
Rowing Machine	$2,200
Stationary Bicycle	$1,100
Weight (Circuit) Machine	$1,800

Calorie Burn Rates

All burn rates are per hour.

Equipment	Effort	Wife	Husband
Elliptical	Low Intensity	266	389
	Moderate Intensity	310	452
Inclining Treadmill	Low Intensity	240	355
	Moderate Intensity	333	490
Rowing Machine	Low Intensity	413	604
	Moderate Intensity	502	733
Stationary Bicycle	Low Intensity	325	480
	Moderate Intensity	413	604
Weight Machine	Low Intensity	177	260
	Moderate Intensity	354	518

Preferences

1. The husband wants to burn no less than 1,100 calories during his low intensity three-hour workout.
2. The wife wants to burn no less than 830 calories during her moderate intensity two-hour workout.
3. The husband has the following preference cardinal measure:  0.00*Elliptical minutes+0.10*Inclining Treadmill minutes+0.05*Rowing Machine Minutes+0.12*Stationary Bicycle Minutes+19*Weight Machine Minutes+0.005*Max(Calories Burned-1100,0).  The husband demands a preference score that is no less than 2,200.
4. The wife has the following preference cardinal measure:  0.05*Elliptical minutes+0.15*Inclining Treadmill minutes+0.05*Rowing Machine Minutes+7*Stationary Bicycle Minutes+0.00*Weight Machine Minutes+0.005*Max(Calories Burned-830,0).  The wife demands a preference score that is no less than 730.

Carbon Footprint

The following are the CO2 emissions incurred in producing each piece of exercise equipment.  All units are in metric tons ("tonnes").

Exercise Item	CO2 (Tonnes)
Elliptical	0.68
Inclining Treadmill	0.85
Rowing Machine	0.70
Stationary Bicycle	0.75
Weight (Circuit) Machine	0.90