Home Gym vs Public Gym Analysis

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 a workout of moderate intensity for two hours and use an hour to relax prior to work. The husband has decided to do a 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.