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The demand function for a certain USB key is $p = 490 - 6x$ , where $p$ is the price (in $\$$ ) per unit and $x$ is the number of units (in thousands), and the supply function is $p = 1130 + 18x$ . Determine the consumer's and the producer's surplus when the market unit price is set at the equilibrium price

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Write linear functions: word problems

Full solution

Q. The demand function for a certain USB key is

p = 490 - 6x

, where

p

is the price (in

\$

) per unit and

x

is the number of units (in thousands), and the supply function is

p = 1130 + 18x

. Determine the consumer's and the producer's surplus when the market unit price is set at the equilibrium price

Find Equilibrium Price: First, find the equilibrium price by setting the demand function equal to the supply function. $\newline$ $490 - 6x = 1130 + 18x$
Combine Like Terms: Combine like terms. $24x = 640$
Solve for x: Divide both sides by $24$ to solve for x. $\newline$ $x = \frac{640}{24}$ $\newline$ $x = 26.67$ (in thousands)
Calculate Equilibrium Price: Plug the value of $x$ into either the demand or supply function to find the equilibrium price. $p = 490 - 6(26.67)$
Calculate Consumer's Surplus: Calculate the equilibrium price. $\newline$ $p = 490 - 160.02$ $\newline$ $p = 329.98$
Calculate Height: Now, calculate the consumer's surplus, which is the area above the price and below the demand curve up to the equilibrium quantity. $\newline$ Consumer's Surplus = $\frac{1}{2} \times \text{base} \times \text{height}$ $\newline$ Base = equilibrium quantity = $26.67$ $\newline$ Height = difference between the maximum price consumers are willing to pay and the equilibrium price. $\newline$ Maximum price = $490$ when $x = 0$ $\newline$ Height = $490 - 329.98$
Calculate Surplus: Calculate the height for the consumer's surplus. $\newline$ Height = $490 - 329.98$ $\newline$ Height = $160.02$
Perform Multiplication: Calculate the consumer's surplus. $\newline$ Consumer's Surplus = $\frac{1}{2} \times 26.67 \times 160.02$
Perform Multiplication: Calculate the consumer's surplus. $\newline$ Consumer's Surplus = $\frac{1}{2} \times 26.67 \times 160.02$ Perform the multiplication for the consumer's surplus. $\newline$ Consumer's Surplus = $\frac{1}{2} \times 26.67 \times 160.02$ $\newline$ Consumer's Surplus = $2133.67$

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