Question Statement

Find the consumer and producer surpluses for the following supply and demand functions:

(i) $S(x)=24,D(x)=100-2x$

(ii) $S(x)=x^2-4,D(x)=-x+8$

(iii) $S(x)=2x^2+3x,D(x)=36-x^2$

---

Background and Explanation

Consumer surplus represents the difference between what consumers are willing to pay and what they actually pay. Producer surplus represents the difference between what producers receive and their minimum acceptable price. Both are calculated using definite integrals at market equilibrium.

---

Solution

Part (i): $S(x)=24,D(x)=100-2x$

Finding Equilibrium Quantity

At equilibrium, supply equals demand:

$S(x)=D(x)$

$24=100-2x$

$2x=100-24=76$

$x=38$

Equilibrium quantity: $x_e=38$

Finding Equilibrium Price

Substituting $x=38$ into both functions:

$S(38)=24$

$D(38)=100-2(38)=100-76=24$

Equilibrium price: $p_e=24$

Consumer Surplus

$CS={\int }_0^{x_e}D(x)dx-x_e\cdot p_e$

$={\int }_0^{38}(100-2x)dx-(38)(24)$

$=100{\int }_0^{38}1dx-2{\int }_0^{38}xdx-912$

$=100{\left[x\right]}_0^{38}-2{\left[\frac{x^2}{2}\right]}_0^{38}-912$

$=100(38-0)-\left[(38{)}^2-0\right]-912$

$=3800-1444-912$

$\boxed{CS=1444}$

Producer Surplus

$PS=x_e\cdot p_e-{\int }_0^{x_e}S(x)dx$

$=(38)(24)-{\int }_0^{38}24dx$

$=912-24{\int }_0^{38}1dx$

$=912-24{\left[x\right]}_0^{38}$

$=912-24(38-0)$

$=912-912$

$\boxed{PS=0}$

---

Part (ii): $S(x)=x^2-4,D(x)=-x+8$

Finding Equilibrium Quantity

$S(x)=D(x)$

$x^2-4=-x+8$

$x^2+x-12=0$

$x^2-3x+4x-12=0$

$x(x-3)+4(x-3)=0$

$(x-3)(x+4)=0$

This gives $x=3$ or $x=-4$. Since quantity cannot be negative, we neglect $x=-4$.

Equilibrium quantity: $x_e=3$

Finding Equilibrium Price

$S(3)=(3{)}^2-4=9-4=5$

$D(3)=-3+8=5$

Equilibrium price: $p_e=5$

Consumer Surplus

$CS={\int }_0^{x_e}D(x)dx-x_e\cdot p_e$

$={\int }_0^3(-x+8)dx-(3)(5)$

$=-{\int }_0^{3}xdx+8{\int }_0^{3}1dx-15$

$=-{\left[\frac{x^2}{2}\right]}_0^3+8{\left[x\right]}_0^3-15$

$=-\frac{1}{2}\left[(3{)}^2-0\right]+8(3-0)-15$

$=-\frac{9}{2}+24-15$

$=-4.5+9$

$\boxed{CS=4.5}$

Producer Surplus

$PS=x_e\cdot p_e-{\int }_0^{x_e}S(x)dx$

$=(3)(5)-{\int }_0^3(x^2-4)dx$

$=15-{\int }_0^3{x}^{2}dx+4{\int }_0^{3}1dx$

$=15-{\left[\frac{x^3}{3}\right]}_0^3+4{\left[x\right]}_0^3$

$=15-\frac{1}{3}\left[(3{)}^3-0\right]+4(3-0)$

$=15-9+12$

$\boxed{PS=18}$

---

Part (iii): $S(x)=2x^2+3x,D(x)=36-x^2$

Finding Equilibrium Quantity

$S(x)=D(x)$

$2x^2+3x=36-x^2$

$3x^2+3x-36=0$

$3(x^2+x-12)=0$

$x^2+x-12=0$

$x^2-3x+4x-12=0$

$x(x-3)+4(x-3)=0$

$(x-3)(x+4)=0$

This gives $x=3$ or $x=-4$. We neglect the negative value.

Equilibrium quantity: $x_e=3$

Finding Equilibrium Price

$S(3)=2(3{)}^2+3(3)=18+9=27$

$D(3)=36-(3{)}^2=36-9=27$

Equilibrium price: $p_e=27$

Consumer Surplus

$CS={\int }_0^{x_e}D(x)dx-x_e\cdot p_e$

$={\int }_0^3(36-x^2)dx-(3)(27)$

$=36{\int }_0^{3}1dx-{\int }_0^3{x}^{2}dx-81$

$=36{\left[x\right]}_0^3-{\left[\frac{x^3}{3}\right]}_0^3-81$

$=36(3-0)-\frac{1}{3}\left[(3{)}^3-0\right]-81$

$=108-9-81$

$\boxed{CS=18}$

Producer Surplus

$PS=x_e\cdot p_e-{\int }_0^{x_e}S(x)dx$

$=(3)(27)-{\int }_0^3(2x^2+3x)dx$

$=81-2{\int }_0^3{x}^{2}dx-3{\int }_0^{3}xdx$

$=81-2{\left[\frac{x^3}{3}\right]}_0^3-3{\left[\frac{x^2}{2}\right]}_0^3$

$=81-\frac{2}{3}\left[(3{)}^3-0\right]-\frac{3}{2}\left[(3{)}^2-0\right]$

$=81-\frac{2}{3}(27)-\frac{3}{2}(9)$

$=81-18-13.5$

$\boxed{PS=49.5}$

---

Key Formulas or Methods Used

• Equilibrium condition: $S(x)=D(x)$
• Consumer Surplus: $CS={\int }_0^{x_e}D(x)dx-x_e\cdot p_e$
• Producer Surplus: $PS=x_e\cdot p_e-{\int }_0^{x_e}S(x)dx$
• Power rule for integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}$

---

Summary of Steps

1. Find equilibrium quantity by solving $S(x)=D(x)$
2. Discard negative solutions (quantity must be non-negative)
3. Find equilibrium price by substituting $x_e$ into either $S(x)$ or $D(x)$
4. Calculate Consumer Surplus: integrate demand from 0 to $x_e$, then subtract total expenditure $(x_e\cdot p_e)$
5. Calculate Producer Surplus: subtract integral of supply from 0 to $x_e$ from total revenue $(x_e\cdot p_e)$