**Project #1 Market Share in the Auto Industry **

**Introduction**

The auto industry long exemplified the capabilities of American manufacturing, beginning at the turn of the century. From the end of World War II throughout the 1960s, with minor exceptions, the market was essentially closed, with little external competition. The Big Three; Ford, General Motors, and Chrysler, had things to themselves. As a result, they grew confident and complacent, while quality and productivity quietly slipped, volume and profit being the driving forces. In the 1970s, new factors appeared. In 1972, the first Honda Civic came to the US, followed by compact cars from Datsun (now Nissan), Toyota and others. These cars were of modest quality, small, and low priced. Their impact was minimal as Americans were enthralled with bigger, powerful cars (the V-8 being the standard engine). However, things changed in late 1973 with the first Arab Oil Embargo. Within a month, gas lines appeared, gas prices tripled, and driving, long taken for granted, became an adventure. The fuel efficient, front wheel drive foreign cars drew much attention as a possible solution (Honda Civics getting 35 mpg hiway, while large V-8s struggled in the upper teens.). Ford's Pinto was perhaps the best American economy car, but it was plagued by poor engine quality (camshafts and bearings), rust, and the sedan version was prone to fire if impacted from the rear. Thus a combination of factors from three continents changed permanently the American auto industry.

**Mathematical Background** The reader should be familiar with Markov
Chains, covered in section 8.3 of Kolman.

**Market Share **During the same period of time, the methods of measuring
business success changed. Instead of measuring the success of a company
in terms of only net profit, a different metric, market share, became popular
(the interested reader is referred to Lester Thurow's book, *Head to
Head *). Simply put, as the name implies, the market share a company
has is simply the fraction of all sales for a given product that the company
has made. In 1972, GM had approximately 40% of the customers for new cars,
Ford had 30%, Chrysler had one quarter, and the total for all Japanese
manufacturers was only 5%.

At that time, studies of car buyers found the following trends** **for
a one year period**:**

__from__

GM Ford Chyrs Japan

GM.75 .10 .10 .10

** to Ford**
.10 .70
.05 .10

** Chrys**
.05 .10
.65 .05

** Japan **
.10 .10
.20
.75

**Work to be done **

**Preliminary **Set up the __transition matrix __as well as the
distribution matrix for** **1972. What interpretation can you make of
the fact that all of the columns in the transition matrix sum to one?

Which company is best at retaining its own customers? Which is best at attracting customers from other companies? Which is losing the most customers?

For a single company, there are 7 entries in the table related to it.
What activities **by** that company effect those entries? Which different
components of the company are involved in those activities? To the best
extent you can, relate your answers to your own particular major

**1**. Predict what the market shares will look like in 1978, 1985,1990
and 1995 based upon the above information.

**2**. Perform some *sensitivity analysis* by picking 2 different
entries of the transition matrix and, one by one, changing them by 2%.
What change, in each case, does this cause in the projections? Do you feel
the model is "sensitive"? (Three decimal place accuracy is acceptable
for all work).

Since any mathematical model will have constants in it some degree of
uncertainty or error, it is important to see if the model performs in essentially
the same manner despite small changes in the parameters. If this is the
case, then the model may be called *robust*.

**3**. Look at the distribution for several relatively "large"
numbers of years (transitions). What conclusion can you draw? What role
does the initial distribution play in the "long term" distribution
of the system? (ascertain this by running the simulation for 3 or more
different initial distributions).

**4**. Now explore the behavior of **powers** of the transition
matrix by looking at them for several "large" values (the same
ones as the part **3** should suffice). What conclusion can you draw?

**5**. Find the fixed point, or "**steady state**" distribution;
the solution of A**x**_{s} = **x**_{s}. This can
be done in two ways:

1. by directly solving the homogeneous system, remembering that the solution must be a distribution.

2. by adding an additional equation : x_{1}+ x_{2}
+ x_{3} + x_{4} = 1 and then directly solving.

Having done this and examined the steady state distribution relative
to results obtained in parts **3** and **4** , what conclusions can
you draw?

**6**. Suppose we decide to refine the model, having for each group
two sub categories, __regular__ and __luxury__ (such as in the case
of Honda and Acura). What would the model look like now? (realizing that
you cannot provide all numerical values). What information would you request
in order to set up an actual transition matrix?

**7**. What technology in the near future do you feel might have
a significant impact on automotive market place? What will need to happen
for the examples you have chosen to make it from the design phase to the
marketplace?

**References**

Halberstam, David. *The Reckoning.* 1986.New York: Avon Books.

Thurow, Lester *Head to Head *.1992 New York: William Morrow
and Company.