## Thursday, August 28, 2014

### The vehicle miles traveled variable in a gas expenditure simulation model.

Appendix One to the Two SUV Problem:

Introductory Comment:  A previous post used Excel simulation methods to provide estimates of the expected value and variance of expenditures on gasoline for two SUVS.

In the previous problem, models driven on the highway, miles driven in city and the price of gasoline were assumed to be random variables.   Both miles driven random variables were assumed normally distributed.    The gasoline price variable was assumed uniform.

The assumption of normally distributed highway miles may not be realistic because on some days people take very long trips on the highway and on other days people don’t drive on the highway at all.

This problem considers alternative assumptions on highway models driven and their impact on fuel expenditures.  First, I pose a methodological question.

Methodological Question:  The previous two-SUV problem assumed that daily highway models were normally distributed with mean 50 miles per day and standard deviation of 15 miles per day.  Why would it be inappropriate to change this assumption to N(50,40)?

Answer:  We can agree that negative miles traveled is not possible so the random variable picked to model vehicle miles traveled should preferably never have a negative outcome.   What is the likelihood that a vehicle miles day N(50,40) is negative?

Use Excel to get

Norm.Dist(0,50,40,True) = 0.10565.

In the original problem STD=15.   Note that

Norm.Dist(0,50,15,TRUE) = 0.0043.

This is pretty close to 0.   We could get rid of all negative vehicle miles traveled days by placing the NORM.INV function inside a MAX function VMT=MAX(0,NORM.INV). The problem can also be solved with a different distribution for VMT -- perhaps the log normal distribution.

Alternative approaches to the SUV simulation problem:  We modify the previous SUV problem by modifying the VMT highway miles driven assumption.

First, assume that VMT Highway miles driven is distributed by

VMTHWAY=Max(0,N(50,40).

Second assume that VMT is log normally distributed with mean log(50) and STD Log(5).

What are the mean and the standard deviation of the Subaru CrossTrek and the Honda CRV under the new highway VMT assumptions?

KEEP ALL OTHER ASSUMPTIONS FROM THE OLD PROBLEM INTACT.

Solutions with the Alternative Highway VMT assumptions:

Since I am keeping all assumptions from tbe old gas expenditure model intact I just open the old spreadsheet and modify the highway VMT model.  The spreadsheet automatically updates.

Note because the spreadsheet automatically updates you will not be able to exactly replicate my results.   Each simulation run is randomly generated If you recalculate your spreadsheet you will get new answers.

It is possible in some software to run the simulations many times and report the average and standard deviation of all simulations.  I am not set up to do this.

The new VMT formula for first alternative highway VMT assumption is:

VMT.HIGHWAY=MAX(0,NORM.INV(RAND(),50,40))

The new VMT formula for the lognormal assumption is

=LOGNORM.INV(RAND(),LN(50),LN(5))

 Simulation Results Under Alternative Assumptions for Highway Miles N(50,15) Subaru CrossTrek Honda CRV Average \$28.80 \$36.50 Standard Deviation \$4.10 \$5.10 MAX(0,N(50,40) Subaru CrossTrek Honda CRV Average \$29.29 \$37.00 Standard Deviation \$5.22 \$6.19 LOGNORM(ln(50),ln(5)) Subaru CrossTrek Honda CRV Average \$50.0 \$59.6 Standard Deviation \$99.6 \$108.9

A final note:  The Lognormal assumption needs to be tinkered with.  A smaller STD would be more realistic.