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:

	GM	Ford	Chrys	Japan
GM	.75	.10	.10	.10
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 vector 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?

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 4 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 AX_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 : x₁+ x₂ + x₃ + x₄ = 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. Finally, 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?