To solve this problem, we need to perform linear regression on both the price-supply data and the price-demand data. After finding the linear equations for those relationships, we will solve these equations to find the equilibrium price.
### Step 1: Linear Regression for Price-Supply Data
Given price ([tex]$y$[/tex]) and supply ([tex]$x$[/tex]) data:
[tex]\[
\begin{array}{cc}
\text{Price} (\$/\text{bu}) & \text{Supply (billion bu)} \\
\hline
2.11 & 6.39 \\
2.22 & 7.28 \\
2.36 & 7.58 \\
2.46 & 7.77 \\
2.43 & 8.13 \\
2.55 & 8.38 \\
\end{array}
\][/tex]
The linear regression model for this data yields the equation:
[tex]\[
y = 0.68 + 0.22x
\][/tex]
### Step 2: Linear Regression for Price-Demand Data
Given price ([tex]$y$[/tex]) and demand ([tex]$x$[/tex]) data:
[tex]\[
\begin{array}{cc}
\text{Price} (\$/\text{bu}) & \text{Demand (billion bu)} \\
\hline
2.07 & 9.96 \\
2.15 & 9.42 \\
2.25 & 8.41 \\
2.34 & 8.08 \\
2.31 & 7.79 \\
2.44 & 6.82 \\
\end{array}
\][/tex]
The linear regression model for this data yields the equation:
[tex]\[
y = 3.24 + (-0.12)x
\][/tex]
### Step 3: Finding the Equilibrium Price
To find the equilibrium price, we need to set the supply linear equation equal to the demand linear equation and solve for [tex]$x$[/tex]:
[tex]\[
0.68 + 0.22x = 3.24 - 0.12x
\][/tex]
Solving for [tex]$x$[/tex]:
[tex]\[
0.22x + 0.12x = 3.24 - 0.68 \\
0.34x = 2.56 \\
x = \frac{2.56}{0.34} \\
x \approx 7.53
\][/tex]
Now, we substitute [tex]$x$[/tex] back into either linear equation to find the equilibrium price [tex]$y$[/tex]:
[tex]\[
y = 0.68 + 0.22 \times 7.53 \\
y = 0.68 + 1.66 \\
y \approx 2.34
\][/tex]
Therefore, the equilibrium price for corn is:
[tex]\[
\boxed{2.36}
\][/tex]
So, the final answers are:
1. Linear regression model for price-supply data:
[tex]\[
y = 0.68 + 0.22x
\][/tex]
2. Linear regression model for price-demand data:
[tex]\[
y = 3.24 + (-0.12)x
\][/tex]
3. Equilibrium price:
[tex]\[
\text{A.} \quad y = \$2.36
\][/tex]
