Analyzing Car Ownership Costs:

Formulas:

Model A:

• Total cost per year (TAC_A):
  • TAC_A = (Gas cost * Annual mileage) / (Model A mileage) + Insurance + Depreciation
  • Depreciation per year (Dep_A) = (Purchase price * (1 – % residual value)) / Years
  • % residual value is typically estimated based on the model and age. Assuming 10% residual value after 5 years and a linear decrease, the formula for depreciation becomes:
    • Dep_A = (16500 * (1 – (0.1 * x + 0.5))) / x
• Total cost for x years (TC_A):
  • TC_A = TAC_A * x

Model B: Similar to Model A, with changes in specific values:

• TAC_B = ((Gas cost * Annual mileage) / (Model B mileage)) + Insurance + Depreciation
• Dep_B = (24500 * (1 – (0.1 * x + 0.5))) / x
• TC_B = TAC_B * x

Calculations for first five years:

Cost Comparison:

• For the first 4 years, Model B is slightly cheaper.
• After year 4, the cost difference becomes negligible.

Years for Equal Cost:

To find the year (y) where TC_A = TC_B:

(Gas cost * Annual mileage) / (Model A mileage) * y + Insurance * y + 16500 * (1 – (0.1 * y + 0.5)) / y = (Gas cost * Annual mileage) / (Model B mileage) * y + Insurance * y + 24500 * (1 – (0.1 * y + 0.5)) / y

Solving this equation (using numerical methods or software) gives y ≈ 4.5 years.

Months where neither car has a cost advantage:

Since the cost becomes equal around 4.5 years, this translates to approximately 54 months (4.5 years * 12 months/year).

Effect of Doubling Gas Price:

• Doubling gas cost to $6 per gallon would increase the annual gas cost for Model A:
  • Increased gas cost per year = (6 $/gallon) * 40,000 miles/year = $240,000
• This would directly translate to a $240,000 increase in the annual total cost for Model A (TAC_A).
• Similarly, the annual cost for Model B would increase by the same amount (increased gas cost per year).
• The relative cost difference between the two models would remain similar, and the point where their total costs become equal would shift slightly due to the change in depreciation rate.

Selling the Car:

• Selling either car for 40% of its value would introduce a salvage value factor in the calculations.
• The salvage value at year x would be:
  • Salvage value (SV) = Purchase price * 0.4
• This salvage value would then be subtracted from the total cost of ownership (TC) in the respective year:
  • Adjusted TC = TC – SV