Production Costs Suarez-Villa and Karlsson ^31 studiedthe...

Production Costs Suarez-Villa and Karlsson $^{31}$ studiedthe relationship between the sales in Sweden's electronic industry and production costs (per unit value of product sales). Their data is presented in the following table. $$ \begin{array}{|c|ccccc|} \hline x & 7 & 13 & 14 & 15 & 17 \\ \hline y & 0.91 & 0.72 & 0.91 & 0.81 & 0.72 \\ \hline x & 20 & 25 & 27 & 35 & 45 \\ \hline y & 0.90 & 0.77 & 0.65 & 0.73 & 0.70 \\ \hline x & 45 & 63 & 65 & 82 & \\ \hline y & 0.78 & 0.82 & 0.92 & 0.96 & \\ \hline \end{array} $$ Here $x$ is product sales in millions of krona, and $y$ is production costs per unit value of product sales. a. Use cubic regression to find the best-fitting cubic to the data. Graph. b. Find where the function is increasing and where it is decreasing. What does this say about economies of scale?

Production Costs Suarez-Villa and Karlsson $^{31}$ studiedthe relationship between the sales in Sweden's electronic industry and production costs (per unit value of product sales). Their data is presented in the following table.
$$
\begin{array}{|c|ccccc|}
\hline x & 7 & 13 & 14 & 15 & 17 \\
\hline y & 0.91 & 0.72 & 0.91 & 0.81 & 0.72 \\
\hline x & 20 & 25 & 27 & 35 & 45 \\
\hline y & 0.90 & 0.77 & 0.65 & 0.73 & 0.70 \\
\hline x & 45 & 63 & 65 & 82 & \\
\hline y & 0.78 & 0.82 & 0.92 & 0.96 & \\
\hline
\end{array}
$$
Here $x$ is product sales in millions of krona, and $y$ is production costs per unit value of product sales.
a. Use cubic regression to find the best-fitting cubic to the data. Graph.
b. Find where the function is increasing and where it is decreasing. What does this say about economies of scale?

Key Concepts

Economies of Scale

Economies of scale are cost advantages that occur when production increases, leading to a lower cost per unit of output. Analyzing where a function representing production costs is increasing or decreasing can provide insights into the presence and extent of economies of scale. If costs per unit decrease as production increases, it is indicative of economies of scale; conversely, if costs increase or do not decrease sufficiently, economies of scale may not be achieved.

Monotonicity and Derivative Analysis

Monotonicity refers to whether a function is consistently increasing or decreasing over a specific interval. By taking the derivative of the fitted polynomial, one can analyze where the derivative is positive (indicating the function is increasing) or negative (indicating it is decreasing). This analysis helps in understanding the behavior of the function across different intervals of the independent variable.

Cubic Regression

Cubic regression is a form of polynomial regression where a third-degree polynomial is fitted to a set of data points. This method involves estimating coefficients for a polynomial of the form ax³ + bx² + cx + d such that the curve best approximates the observed data. It is particularly useful when the relationship between the independent and dependent variables is expected to have curvature that a linear or quadratic model cannot capture.

Least Squares Method

The least squares method is a standard approach in regression analysis to determine the coefficients of the model. It works by minimizing the sum of the squared differences between the observed data points and the values predicted by the model. This technique ensures that the fitted curve has the smallest possible overall deviation from the actual data, leading to the best possible fit under the assumption of normally distributed errors.

Graphical Analysis

Graphical analysis involves plotting both the observed data points and the best-fitting curve obtained from the regression model. This visualization helps in assessing the accuracy and appropriateness of the model by showing how closely the model follows the trend in the data. It also makes it easier to identify patterns such as turning points or areas where the model may not capture the underlying trend well.

Transcript

00:02 Hello, so we have this problem where a store manager wants to know the demand for a product as a function of the price.

00:08 And we're given this table where we have different prices that are represented by x, and we have the demand, which is y, or the daily sales.

00:23 So for part a, we had to find the lee squared regression line by solving the system given to us.

00:31 So let's start doing that.

00:34 For a, we have 3b plus 3 .7a, which equals to 105.

00:49 Now, mind you, this 3 .00 is the same as 3 .7.

00:55 At 3 .70 is 3 .7.

00:58 I just made it simpler to write.

01:02 So we're going to use this first equation and solve for a.

01:06 Now, to do that, first we're going to subtract 3b on both sides.

01:14 We'll get 105 minus 3b, which is equal to 3 .7a.

01:20 Then we're going to divide both sides by 3 .7, which is a equals 105 minus 3b over 3 .7.

01:35 Now, we know that a decimal can't be on the bottom, but we're just going to leave it there for now and handle it a little later.

01:44 So we're going to use a and plug it into the second equation.

01:49 So we have 3 .7b plus 4 .69 times 105 minus 3b and this is over 3 .7, which equals to 123 .9.

02:09 Now we're going to distribute the 4 .69 into the parentheses.

02:20 To do so, we're going to transform the decimals into fractions.

02:27 So we have 469 over 100.

02:33 We're going to times that by 105 minus 3b.

02:39 This is going to be over 37 over 10.

02:48 Sorry.

02:51 Now, notice how there's a fraction on the bottom.

02:56 So we're just going to do keep change flip, meaning we're going to keep the numerator, change the division to multiplication, and then flip this fraction.

03:07 So it's going to be 105 minus 3b times 10.

03:13 10 and this is going to be divided by 37.

03:20 Now bring down 469 over 100.

03:28 Well, we know that 10 divided by 100.

03:33 We could just do this and we're left with 10 on the bottom here.

03:40 Now 10 times 37 is 370 and then on top we're going to have 460.

03:48 Times 105 minus 3b.

03:54 And just so we remember, let's bring down our other terms, 123 .9, and then our 3 .7b.

04:10 If we distribute 469 in the parentheses, we'll get 49 ,245 minus 1 ,107, or 1 ,407, 1 ,407b and this is over 370 and we got 3 .7b and then the 123 .9.

04:41 Now that's settled.

04:45 We're going to transform this 3 .7b by multiplying it by 370 over 370 so we could combine it with this numerator...

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