$\mathrm{D}(\mathrm{x})$ is the price, in dollars per unit, that consumers are willing to pay for $\mathrm{x}$ units of an item, and $\mathrm{S}(\mathrm{x})$ is the price, in dollars per unit, that producers are willing to accept for $\mathrm{x}$ units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. $\newline$ $D(x)=-\frac{5}{6} x+13, S(x)=\frac{1}{2} x+5$

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Algebra 2

Find inverse functions and relations

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\mathrm{D}(\mathrm{x})

is the price, in dollars per unit, that consumers are willing to pay for

\mathrm{x}

units of an item, and

\mathrm{S}(\mathrm{x})

is the price, in dollars per unit, that producers are willing to accept for

\mathrm{x}

units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point.

\newline

D(x)=-\frac{5}{6} x+13, S(x)=\frac{1}{2} x+5

Identify Equilibrium Point: Identify the equilibrium point by setting $D(x)$ equal to $S(x)$ . $\newline$ Calculation: $\newline$ $D(x) = S(x)$ $\newline$ $-\frac{5}{6}x + 13 = \frac{1}{2}x + 5$ $\newline$ Combine like terms: $\newline$ $-\frac{5}{6}x - \frac{1}{2}x = 5 - 13$ $\newline$ Convert $\frac{1}{2}$ to $\frac{3}{6}$ to simplify: $\newline$ $-\frac{5}{6}x - \frac{3}{6}x = -8$ $\newline$ $-\frac{8}{6}x = -8$ $\newline$ x = $6$
Calculate Equilibrium Price: Calculate the equilibrium price by substituting $x = 6$ into either $D(x)$ or $S(x)$ . $\newline$ Calculation: $\newline$ Using $D(x)$ : $\newline$ $D(6) = -\left(\frac{5}{6}\right)(6) + 13 = -5 + 13 = 8$
Calculate Consumer Surplus: Calculate the consumer surplus at the equilibrium point. $\newline$ Consumer surplus is the area between the demand curve and the price level at the equilibrium quantity. $\newline$ Calculation: $\newline$ Consumer Surplus = $0.5 \times \text{base} \times \text{height}$ $\newline$ Base = $6$ (equilibrium quantity) $\newline$ Height = $13$ (maximum price willing to pay) - $8$ (equilibrium price) $\newline$ Consumer Surplus = $0.5 \times 6 \times (13 - 8) = 0.5 \times 6 \times 5 = 15$
Calculate Producer Surplus: Calculate the producer surplus at the equilibrium point. $\newline$ Producer surplus is the area between the supply curve and the price level at the equilibrium quantity. $\newline$ Calculation: $\newline$ Producer Surplus = $0.5 \times \text{base} \times \text{height}$ $\newline$ Base = $6$ (equilibrium quantity) $\newline$ Height = $8$ (equilibrium price) - $5$ (minimum price willing to accept) $\newline$ Producer Surplus = $0.5 \times 6 \times (8 - 5) = 0.5 \times 6 \times 3 = 9$