To find the producer's surplus given the supply function [tex]\( S(q) = q^{7/2} + 2q^{5/2} + 50 \)[/tex] at the equilibrium quantity [tex]\( q = 25 \)[/tex], follow these steps:
1. Calculate the equilibrium supply price:
Evaluate the supply function [tex]\( S(q) \)[/tex] at [tex]\( q = 25 \)[/tex].
[tex]\[
S(25) = 25^{7/2} + 2 \cdot 25^{5/2} + 50
\][/tex]
First, simplify each term:
[tex]\[
25^{7/2} = (25^{1/2})^7 = 5^7 = 78125
\][/tex]
[tex]\[
25^{5/2} = (25^{1/2})^5 = 5^5 = 3125
\][/tex]
Now, substitute back:
[tex]\[
S(25) = 78125 + 2 \cdot 3125 + 50
\][/tex]
[tex]\[
S(25) = 78125 + 6250 + 50
\][/tex]
[tex]\[
S(25) = 84425
\][/tex]
2. Calculate the total revenue at equilibrium:
Multiply the equilibrium quantity ([tex]\( q = 25 \)[/tex]) by the equilibrium supply price:
[tex]\[
\text{Total Revenue} = S(25) \times q = 84425 \times 25
\][/tex]
[tex]\[
\text{Total Revenue} = 2110625
\][/tex]
3. Calculate the area under the supply curve from [tex]\( q = 0 \)[/tex] to [tex]\( q = 25 \)[/tex]:
This requires integrating the supply function [tex]\( S(q) \)[/tex] from 0 to 25.
[tex]\[
\int_{0}^{25} S(q) \, dq = \int_{0}^{25} \left( q^{7/2} + 2 q^{5/2} + 50 \right) \, dq
\][/tex]
Evaluate each term separately:
[tex]\[
\int_{0}^{25} q^{7/2} \, dq = \left[ \frac{2}{9} q^{9/2} \right]_{0}^{25} = \frac{2}{9} \left( 25^{9/2} \right) = \frac{2}{9} (5^9) = \frac{2}{9} (1953125) = 434027.78
\][/tex]
[tex]\[
\int_{0}^{25} 2 q^{5/2} \, dq = 2 \left[ \frac{2}{7} q^{7/2} \right]_{0}^{25} = \frac{4}{7} \left( 25^{7/2} \right) = \frac{4}{7} (5^7) = \frac{4}{7} (78125) = 44642.86
\][/tex]
[tex]\[
\int_{0}^{25} 50 \, dq = \left[ 50q \right]_{0}^{25} = 50 \times 25 = 1250
\][/tex]
Adding these components together:
[tex]\[
\int_{0}^{25} S(q) \, dq = 434027.78 + 44642.86 + 1250 = 480920.64
\][/tex]
4. Calculate the producer's surplus:
The producer surplus is the area between the supply curve and the equilibrium price line up to the equilibrium quantity, which is the difference between the total revenue and the area under the supply curve.
[tex]\[
\text{Producer Surplus} = \text{Total Revenue} - \int_{0}^{25} S(q) \, dq
\][/tex]
[tex]\[
\text{Producer Surplus} = 2110625 - 480920.64 = 1629704.36
\][/tex]
Therefore, the producer's surplus is approximately [tex]\( \$ 1629704.36 \)[/tex].
