In the late 1900 s, the relationship between the domestic...

In the late 1900 s, the relationship between the domestic market shares of three major U.S. manufacturers of cars and light trucks could be modeled by $$ x_{3}=0.66-2.2 x_{1}-0.02 x_{2} $$ where $x_{1}, x_{2}$, and $x_{3}$ are, respectively, the fractions of the market held by Chrysler, Ford, and General Motors. $^{3}$ Thinking of General Motors' market share as a function of the shares of the other two manufacturers, describe the graph of the resulting function. How are the different slices by $x_{1}=$ constant related to one another? What does this say about market share?

In the late 1900 s, the relationship between the domestic market shares of three major U.S. manufacturers of cars and light trucks could be modeled by
$$
x_{3}=0.66-2.2 x_{1}-0.02 x_{2}
$$
where $x_{1}, x_{2}$, and $x_{3}$ are, respectively, the fractions of the market held by Chrysler, Ford, and General Motors. $^{3}$ Thinking of General Motors' market share as a function of the shares of the other two manufacturers, describe the graph of the resulting function. How are the different slices by $x_{1}=$ constant related to one another? What does this say about market share?

Key Concepts

Affine Function

An affine function is a linear function plus a constant term. In this context, General Motors’ market share is modeled as an affine function of Chrysler’s and Ford’s market shares. This means that the relationship is represented by a plane where changes in the independent variables (the competitors’ market shares) lead to proportional changes in the dependent variable (General Motors’ share), while the constant term sets the level of the baseline value.

Graph of a Plane

When a function of two variables is represented in three-dimensional space, its graph is a plane. Here, considering the market shares of Chrysler and Ford as the independent variables and General Motors’ share as the dependent variable, the model describes a plane. This geometric interpretation helps in visualizing how shifts in one or both of the competitors’ shares will impact General Motors’ share in a predictable, linear manner.

Slices and Cross Sections

Taking slices of the plane by fixing one variable (for instance, setting Chrysler’s market share at a constant value) results in a line in the plane that shows the relationship between Ford’s share and General Motors’ share for that fixed level of Chrysler’s share. These slices are parallel to each other because the coefficient associated with Ford’s share remains unchanged regardless of the value fixed for Chrysler’s share. This indicates a consistent trade-off in market share: the effect of a change in one manufacturer’s share on General Motors’ share is independent of the other manufacturer’s market share.

Transcript

00:01 In this problem of functions of several variables we have given the problem based on market share in 90s.

00:08 So, market share in 1900s.

00:16 For chrysler, ford general motors.

00:19 So here, chrysler, ford and general motor we can write it as gm motors.

00:31 In the late 90s, the relationship between the domestic market share of the three major us manufacture of cars, and light trucks could be modeled by x3 is equals to 0 .66 minus 2 .22, this is 2 .2 actually multiplied with x1 minus 0 .02 multiplied with x2.

00:54 Where x1 x2 x3 are respectively except here x1 x2 x3 are respectively the fractions of the market held by chrysler ford and say here this is x1, this is represented by x2 and general motor is represented by x3.

01:16 Market share as a function of share of other two manufacturers.

01:21 Describe the graph of the resulting funser and how are these slices by x1 is equal to constant so here x1 is equal to constant related to one another and what does this say about the market share? so, since we have x3 the power is 1, x1 power is 1 and x2 is a power 1.

01:48 So we can say the highest degree in this polynomial is equal to 1.

01:52 So graph would be a plane.

01:54 So we can say the graph is a plane.

01:57 So we can say graph is a plane.

02:03 And now in plain what we can do we can find x1 intercept and x2 intercept and x3 intercept.

02:09 So, we can find x1 intercept, so by finding x1 intercept.

02:17 That means we have to put other two quantities equals to 0, so putting x3 and x2 equals to 0.

02:23 So we get in this equation, so x1 is equals to 0 .3.

02:29 Similarly we can find x2 intercept, so x2 intercept.

02:34 So this is found by putting x1 and x3 is equal to 0.

02:38 So now putting x1 and x3 both equals 0, so we get x2 is equals to 33.

02:46 And also we can find the x3 intercept.