To find the consumers' surplus and the producers' surplus, we need to follow these steps:
1. Determine the Equilibrium Quantity and Price:
- The demand function is given by [tex]\( p_{\text{demand}} = -0.2x^2 + 160 \)[/tex].
- The supply function is given by [tex]\( p_{\text{supply}} = 0.1x^2 + 5x + 80 \)[/tex].
To find the equilibrium quantity [tex]\( x \)[/tex], set [tex]\( p_{\text{demand}} = p_{\text{supply}} \)[/tex]:
[tex]\[
-0.2x^2 + 160 = 0.1x^2 + 5x + 80
\][/tex]
Solving for [tex]\( x \)[/tex], the equilibrium quantity is [tex]\( x \approx -26.67 \)[/tex].
2. Calculate the Equilibrium Price:
- Substitute the equilibrium quantity [tex]\( x = -26.67 \)[/tex] back into either the demand or supply function. Using the demand function:
[tex]\[
p = -0.2(-26.67)^2 + 160 \approx 17.78
\][/tex]
3. Calculate Consumer Surplus:
- Consumer surplus is the area between the demand curve and the equilibrium price, from [tex]\( x = 0 \)[/tex] to the equilibrium quantity.
- To compute consumer surplus, integrate the demand function from [tex]\( x = 0 \)[/tex] to the equilibrium quantity and subtract the total revenue at the equilibrium price:
[tex]\[
\text{Consumer Surplus} = \int_{0}^{x_{\text{equilibrium}}} (-0.2x^2 + 160) \, dx - x_{\text{equilibrium}} \times p_{\text{equilibrium}}
\][/tex]
Results in [tex]\( \text{Consumer Surplus} \approx -2528 \)[/tex].
4. Calculate Producer Surplus:
- Producer surplus is the area between the supply curve and the equilibrium price, from [tex]\( x = 0 \)[/tex] to the equilibrium quantity.
- To compute producer surplus, Integrate the supply function from [tex]\( x = 0 \)[/tex] to the equilibrium quantity and subtract it from the total revenue at the equilibrium price:
[tex]\[
\text{Producer Surplus} = x_{\text{equilibrium}} \times p_{\text{equilibrium}} - \int_{0}^{x_{\text{equilibrium}}} (0.1x^2 + 5x + 80) \, dx
\][/tex]
Results in [tex]\( \text{Producer Surplus} \approx 514 \)[/tex].
Finally rounding to the nearest dollar, we find:
- Consumer's surplus is approximately [tex]\(-$2528\)[/tex].
- Producer's surplus is approximately [tex]\($514\)[/tex].
