Elastic vs. Inelastic Demand - Business Calculus

The price elasticity of demand for a product is given by the function $E(p) = \frac{p}{200-p}$ . The product is currently selling for $p = \$120$ . To increase total revenue, which of the following actions should the company take?
Answer: Decrease the price. When you encounter price elasticity of demand problems, remember that elasticity measures how responsive quantity demanded is to price changes, and it directly impacts revenue decisions. The key insight is that revenue increases when you move toward unit elasticity (where $|E| = 1$ ). First, let's calculate the current elasticity: $E(120) = \frac{120}{200-120} = \frac{120}{80} = 1.5$ . Since the elasticity is greater than 1 in absolute value, demand is elastic at this price point. When demand is elastic ( $|E| > 1$ ), a price decrease will increase total revenue because the percentage increase in quantity demanded exceeds the percentage decrease in price. Conversely, when demand is inelastic ( $|E| < 1$ ), price increases boost revenue. Revenue is maximized when $|E| = 1$ . Looking at each option: A) is correct because decreasing price when demand is elastic increases revenue. B) is wrong because increasing price when demand is elastic would decrease revenue - the large drop in quantity would outweigh the price gain. C) is incorrect because revenue isn't maximized; that would occur when $E = 1$ , which happens when $p = 100$ (solving $\frac{p}{200-p} = 1$ ). D) is wrong because production changes don't directly affect the price-revenue relationship that elasticity governs. Study tip: Remember the elasticity rule of thumb: when $|E| > 1$ (elastic), move price in the opposite direction of where you want revenue to go. When $|E| < 1$ (inelastic), move price in the same direction as your revenue goal.
A concert venue has estimated its demand function as $p = 150 - 0.25q$ where $p$ is the ticket price and $q$ is the number of tickets sold. The venue manager observes that when they reduced the price from $90 to $80, total revenue increased. However, when they further reduced the price from $80 to $70, total revenue decreased. What explains this pattern?
Answer: The venue moved from the elastic region to the inelastic region, passing through the revenue-maximizing point For p = 150 - 0.25q, revenue is maximized where MR = 0. Since MR = 150 - 0.5q and setting this to zero gives q = 300, so p* = $75. At $90 (above $75), demand is elastic, so reducing price increases revenue. At $80 (above $75), demand is still elastic. At $70 (below $75), demand is inelastic, so reducing price decreases revenue. The venue moved from elastic to inelastic demand, passing through the maximum at $75. Choice B incorrectly suggests the unit elastic point was between $90 and $80.
The weekly demand for a luxury good is given by the demand function $D(p) = 800 - 10\sqrt{p}$ , where $p$ is the price in dollars. At what price is the demand for this good unit elastic?
Answer: p = $2844.44 Demand is unit elastic when $E(p)=1$ . The elasticity function is $E(p) = -\frac{p \cdot D'(p)}{D(p)}$ . First, find the derivative: $D'(p) = -10 \cdot \frac{1}{2}p^{-1/2} = -5p^{-1/2}$ . Now, substitute into the elasticity formula: $E(p) = -\frac{p(-5p^{-1/2})}{800-10\sqrt{p}} = \frac{5\sqrt{p}}{800-10\sqrt{p}}$ . Set $E(p)=1$ : $1 = \frac{5\sqrt{p}}{800-10\sqrt{p}}$ . This gives $800-10\sqrt{p} = 5\sqrt{p}$ , so $15\sqrt{p} = 800$ . Then $\sqrt{p} = \frac{800}{15} = \frac{160}{3}$ . Finally, $p = (\frac{160}{3})^2 = \frac{25600}{9} \approx \$2844.44$ .
The demand function for a digital subscription service is given by $D(p) = 5000p^{-1.5}$ , where $p$ is the monthly price. Which statement accurately describes the relationship between price and revenue for this service?
Answer: Any price decrease will always lead to an increase in revenue. For a demand function of the form $D(p) = kp^{-n}$ , the elasticity of demand is constant and equal to $n$ . Here, $n=1.5$ . The elasticity of demand is $E(p) = -\frac{p \cdot D'(p)}{D(p)} = -\frac{p \cdot (5000(-1.5)p^{-2.5})}{5000p^{-1.5}} = 1.5$ . Since $E=1.5$ for all prices $p > 0$ , demand is always elastic. When demand is elastic, a decrease in price leads to an increase in revenue.
A company sells a product with a linear demand function $p = 80 - 0.5q$ , where $p$ is the price in dollars and $q$ is the number of units. For what range of prices is the demand for this product elastic?
Answer: For prices between $40 and $80. For a linear demand curve, demand is unit elastic at the price that maximizes revenue. Revenue is $R(q) = p \cdot q = (80 - 0.5q)q = 80q - 0.5q^2$ . To maximize revenue, find where $R'(q)=0$ : $R'(q) = 80 - q = 0$ , which gives $q=80$ . The price corresponding to this quantity is $p = 80 - 0.5(80) = 80 - 40 = \$40$ . Demand is unit elastic at $p=\$40$ . The demand curve intersects the price axis at $p=\$80$ (when $q=0$ ). Demand is elastic on the upper half of the price range of the demand curve. Therefore, demand is elastic for prices between $40 and $80.
Two firms compete in the same market with identical linear demand curves $q = 400 - 2p$ . Firm A currently prices at $120 while Firm B prices at $80. Both firms are considering a 15% price reduction to gain market share. Based on elasticity analysis, which firm's strategy is more likely to succeed in increasing total revenue?
Answer: Firm B's strategy is better because reducing price in the elastic region yields larger revenue gains than in the inelastic region For q = 400 - 2p, revenue is maximized at p = $100 (where MR = 400 - 4p = 0). Firm A at $120 is in the inelastic region (above $100), while Firm B at $80 is in the elastic region (below $100). When reducing price in the elastic region, revenue increases, but when reducing price in the inelastic region, revenue decreases. Therefore, Firm B's strategy will increase revenue while Firm A's will decrease it. Choice A incorrectly states that inelastic price reductions increase revenue.
A coffee shop lowers the price of its signature latte by 3% and observes that its total revenue from the latte increases by approximately 1%. Which of the following statements about the demand for the latte at its original price is most accurate?
Answer: Demand is elastic, as a decrease in price led to an increase in total revenue. The relationship between revenue and elasticity dictates that if a price decrease leads to a revenue increase, the demand must be elastic ( $E>1$ ). The percentage change in quantity demanded must have been positive and greater in magnitude than the percentage change in price. Specifically, since $\%\Delta R \approx \%\Delta p + \%\Delta q$ , we have $1\% \approx -3\% + \%\Delta q$ , which implies $\%\Delta q \approx 4\%$ . The elasticity is $E \approx -\frac{\%\Delta q}{\%\Delta p} = -\frac{4\%}{-3\%} \approx 1.33$ , which is greater than 1.
A company's demand function is $p = 120 - 0.5q$ , where $p$ is the price per unit and $q$ is the quantity demanded. At what price level does the demand transition from elastic to inelastic, and what strategic pricing decision should the company make if their current price is $45?
Answer: The transition occurs at $60; since current demand is elastic, the company should increase price to maximize revenue First, find the elasticity transition point where |E| = 1. For linear demand p = a - bq, this occurs at p = a/2 = 120/2 = $60. At the current price of $45 (below $60), demand is elastic (|E| > 1), so revenue increases when price increases toward the unit elastic point. Choice B incorrectly states current demand is inelastic. Choices C and D incorrectly calculate the transition point as $75.
The demand for a specialized electronic component is given by the function $D(p) = 200(40-p)^{1.5}$ , where $p$ is the price in dollars. The component is currently priced at $p = \$24$ . If the company implements a small price increase, what will be the effect on total revenue?
Answer: Total revenue will decrease because demand is elastic. First, calculate the elasticity of demand, $E(p) = -\frac{p \cdot D'(p)}{D(p)}$ . The derivative of the demand function is $D'(p) = 200 \cdot 1.5(40-p)^{0.5}(-1) = -300\sqrt{40-p}$ . At $p=24$ , we have $D(24) = 200(16)^{1.5} = 200(64) = 12800$ and $D'(24) = -300\sqrt{16} = -300(4) = -1200$ . Plugging these into the elasticity formula: $E(24) = -\frac{24 \cdot (-1200)}{12800} = \frac{28800}{12800} = 2.25$ . Since $E = 2.25 > 1$ , demand is elastic. When demand is elastic, a price increase leads to a decrease in total revenue.
A software company's demand function is $q = \frac{1000}{p^{1.5}}$ where $q$ is the number of licenses sold and $p$ is the price per license. The company wants to determine if increasing price from $100 to $110 will increase or decrease total revenue. What is the correct analysis?
Answer: Revenue will decrease because for this demand function, elasticity is constant at -1.5, making demand always elastic For the demand function q = $1000p^{-1.5}$ , the elasticity is constant: $E_d$ = -(dq/dp)(p/q) = -(-1.5 × $1000p^{-2.5}$ )(p/( $1000p^{-1.5}$ )) = -1.5. Since | $E_d$ | = 1.5 > 1, demand is always elastic. When demand is elastic, price and revenue move in opposite directions, so increasing price decreases revenue. Choice A correctly identifies constant elasticity of -1.5 but incorrectly concludes revenue will increase. Choices C and D don't recognize the constant elasticity property.
A firm's weekly revenue function is $R(p) = p(100 - 2p)$ where $p$ is the price per unit. The marketing department reports that a 10% price increase from the current level resulted in a 12% decrease in total revenue. What does this indicate about the firm's current pricing strategy?
Answer: The firm is currently pricing in the elastic region and should increase price to move toward revenue maximization When analyzing pricing strategy using revenue functions, you need to understand the relationship between price changes and revenue changes, which connects to the concept of price elasticity of demand. Let's verify the reported data first. The revenue function $R(p) = p(100 - 2p) = 100p - 2p^2$ gives us $R'(p) = 100 - 4p$ . To find the revenue-maximizing price, set $R'(p) = 0$ : $100 - 4p = 0$ , so $p = 25$ . At this optimal price, $R(25) = 25(100 - 50) = 1250$ . Now, if a 10% price increase causes a 12% revenue decrease, this tells us the firm is currently pricing above the revenue-maximizing point. When you're in this region (called the elastic region), demand is highly responsive to price changes - a small price increase leads to a proportionally larger drop in quantity sold, reducing total revenue. To move toward revenue maximization, the firm should increase price until reaching $p = 25$ . Answer D is wrong because the reported data is actually consistent with the revenue function when the current price exceeds the optimal price. Answer A is incorrect because if the firm were at the revenue-maximizing price, a price increase wouldn't decrease revenue by 12%. Answer B misidentifies the pricing region - the firm is in the elastic region (above optimal price), not the inelastic region (below optimal price). Study tip: Remember that when price increases cause proportionally larger revenue decreases, you're in the elastic region above the revenue maximum. Counter-intuitively, you should continue increasing price to reach the peak.
A company finds that its revenue is maximized when it prices its product at $p = \$50$ . Based on the relationship between revenue and elasticity, what can be concluded about the elasticity of demand at prices of $p = \$49$ and $p = \$51$ ?
Answer: Demand is inelastic at $p = \$49$ and elastic at $p = \$51$ . Revenue is maximized at the price where demand is unit elastic ( $E=1$ ). For typical demand curves, at prices below the revenue-maximizing price, demand is inelastic ( $E<1$ ). At prices above the revenue-maximizing price, demand is elastic ( $E>1$ ). Therefore, at $p = \$49$ (which is less than $50), demand is inelastic, and at $p = \$51$ (which is greater than $50), demand is elastic.
The demand equation for a product is given by $q^2 + 2p = 600$ , where $p$ is the price per unit and $q$ is the number of units demanded. The company plans to set the price at the level that maximizes its total revenue. At this specific price, what is the price elasticity of demand?
Answer: $E = 1$ This is a conceptual question. Total revenue is maximized at the price (and corresponding quantity) where the price elasticity of demand is equal to 1. This condition is known as unit elasticity. No calculations are necessary to determine that $E=1$ at the point of maximum revenue.
The demand for a product is modeled by the equation $p = 100e^{-0.02q}$ , where $p$ is the price per unit and $q$ is the quantity demanded. At a production level of $q=40$ units, is the demand elastic or inelastic, and what is the revenue implication of increasing production?
Answer: Elastic; increasing production will increase total revenue. Since price is given as a function of quantity, we can use the elasticity formula $E(q) = -\frac{p/q}{dp/dq}$ . First, find $dp/dq = 100e^{-0.02q}(-0.02) = -2e^{-0.02q}$ . Then $E(q) = -\frac{100e^{-0.02q}/q}{-2e^{-0.02q}} = \frac{50}{q}$ . At $q=40$ , the elasticity is $E(40) = \frac{50}{40} = 1.25$ . Since $E > 1$ , demand is elastic. Increasing production ( $q$ ) requires a decrease in price ( $p$ ). When demand is elastic, a price decrease leads to an increase in total revenue. Therefore, increasing production will increase revenue.
The price elasticity of demand for a product is $E_d = -\frac{2p}{100-p}$ , where $p$ is the price. If the company currently charges $25 and is considering whether to raise or lower prices to increase revenue, which analysis is correct?
Answer: At $25, | $E_d$ | = 0.67 < 1, so demand is inelastic and the company should raise prices to increase revenue Substituting p = 25: $E_d$ = -2(25)/(100-25) = -50/75 = -2/3. So | $E_d$ | = 2/3 ≈ 0.67 < 1, meaning demand is inelastic. When demand is inelastic, price and revenue move in the same direction, so raising prices increases revenue. Choice B correctly calculates elasticity but incorrectly identifies it as elastic. Choices C and D incorrectly calculate | $E_d$ | = 1.5.
A manufacturer's marketing department estimates that the price elasticity of demand for its flagship product is currently $E = 1.8$ . The company is considering a 5% price increase to improve its profit margins. What is the most likely immediate effect of this price increase on the company's total revenue?
Answer: Total revenue will decrease because the percentage decrease in quantity demanded will be greater than 5%. The elasticity of demand is $E=1.8$ , which is greater than 1. This means demand is elastic. For elastic demand, price and total revenue have an inverse relationship. An increase in price will lead to a decrease in total revenue. The definition of elasticity $E = -\frac{\%\Delta q}{\%\Delta p}$ implies that $\%\Delta q = -E \cdot \%\Delta p = -1.8 \cdot (5\%) = -9\%$ . Since the percentage decrease in quantity (9%) is larger in magnitude than the percentage increase in price (5%), total revenue will fall.
A subscription service has a demand function $q = 500e^{-0.02p}$ where $q$ is the number of subscribers and $p$ is the monthly price. If the company's current price yields an elasticity of demand of -0.8, what pricing adjustment should the company make to maximize revenue?
Answer: Increase the price because demand is inelastic and higher prices will increase revenue until elasticity reaches -1 When you encounter price elasticity and revenue optimization problems, remember that revenue is maximized when the price elasticity of demand equals exactly -1. This is because elasticity measures how responsive quantity demanded is to price changes, and at elasticity = -1, the percentage change in quantity exactly offsets the percentage change in price. Since the current elasticity is -0.8, demand is inelastic (absolute value less than 1). This means subscribers are relatively unresponsive to price changes. When demand is inelastic, increasing price will increase total revenue because the price increase more than compensates for the small decrease in quantity demanded. The company should increase price until the elasticity reaches -1, at which point revenue will be maximized. Answer C correctly identifies this strategy. Answer A is wrong because -0.8 indicates the company hasn't reached the revenue-maximizing point yet. Revenue is only maximized when elasticity equals -1, not -0.8. Answer B incorrectly classifies -0.8 as elastic demand. Demand is only elastic when the absolute value exceeds 1. Additionally, it suggests decreasing price, which would move further away from the revenue-maximizing elasticity of -1. Answer D is incorrect because an elasticity of -0.8 is perfectly reasonable for an exponential demand function. Elasticity values between 0 and -1 indicate inelastic demand, which is common for many products and services. Study tip: Remember the revenue rule: if |elasticity| < 1 (inelastic), raise prices; if |elasticity| > 1 (elastic), lower prices. Stop when you reach |elasticity| = 1.

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