Problem 9
Question
\(D(x)\) is the price, in dollars per unit, that consumers will pay for \(x\) units of an item, and \(S(x)\) is the price, in dollars per unit, that producers will accept for \(x\) units. Find (a) the equilibrium point, (b) the consumer surplus at the equilibrium point, and (c) the producer surplus at the equilibrium point. \(D(x)=7-x,\) for \(0 \leq x \leq 7 ; \quad S(x)=2 \sqrt{x+1}\)
Step-by-Step Solution
Verified Answer
The equilibrium point is (3, 4), consumer surplus is 4.5 dollars, and producer surplus is 2 dollars.
1Step 1: Finding the Equilibrium Point
To find the equilibrium point, equate the demand and supply functions. Solve the equation \(7-x = 2\sqrt{x+1}\). Square both sides to eliminate the square root: \((7-x)^2 = (2\sqrt{x+1})^2\). Simplifying gives \((7-x)^2 = 4(x+1)\). Expand and solve: \(49 - 14x + x^2 = 4x + 4\), resulting in \(x^2 - 18x + 45 = 0\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a=1, b=-18, c=45\), giving solutions \(x=15\) and \(x=3\). Only \(x=3\) is in the valid range \(0 \leq x \leq 7\). At \(x=3\), substitute in \(D(x)\) to find price: \(D(3)=4\). Thus, the equilibrium point is \((3,4)\).
2Step 2: Computing Consumer Surplus
Consumer surplus is the area between the demand curve and the price level up to the equilibrium quantity. It is given by the formula \(CS = \int_0^3 (D(x) - P) \, dx\), where \(P\) is the equilibrium price. Substituting the values: \(CS = \int_0^3 (7-x - 4) \, dx = \int_0^3 (3-x) \, dx\). Integrate to find \(CS = \left[3x - \frac{x^2}{2}\right]_0^3\). After evaluating, \(CS = (9 - 4.5) - (0 - 0) = 4.5\).
3Step 3: Computing Producer Surplus
Producer surplus is the area between the price level and the supply curve up to the equilibrium quantity. It is given by \(PS = \int_0^3 (P - S(x)) \, dx\). Substitute the equilibrium price: \(PS = \int_0^3 (4 - 2\sqrt{x+1}) \, dx\). Separate the integral: \(PS = \int_0^3 4 \, dx - \int_0^3 2\sqrt{x+1} \, dx\). Integrate: \(\left[4x\right]_0^3 = 12\); for the second integral, use substitution \(u = x+1\), then \(du = dx\), transforming it to \(\int_1^4 2\sqrt{u} \, du = \left[\frac{4}{3}u^{3/2}\right]_1^4\). Evaluate it to get \(\left[16\right]_1^4 = \frac{64}{3} - \frac{4}{3}\). Combine: \(PS = 12 - (\frac{60}{3}) = 12 - 20 = 2\).
4Step 4: Final Answer Summary
The equilibrium point is \((3, 4)\), the consumer surplus is \(4.5\) dollars, and the producer surplus is \(2\) dollars.
Key Concepts
Consumer SurplusProducer SurplusDemand and Supply Functions
Consumer Surplus
Consumer surplus represents the extra benefit or happiness that consumers receive when they pay a price lower than what they are willing to pay.
In simpler terms, it's the difference between the highest price consumers are ready to shell out and the market price they actually pay.
To quantify consumer surplus, we find the area under the demand curve down to the equilibrium price level, up to the equilibrium quantity.Imagine you're buying an item and you're willing to pay \(10, but you find it for \)7.
Your consumer surplus is $3 because you got a better deal than you expected.In the example exercise, the demand function is given by \(D(x) = 7 - x\).
The equilibrium point, where consumers and producers agree on quantity and price, is at \((3, 4)\).
The consumer surplus is calculated using the formula:- \(CS = \int_0^3 (D(x) - P) \, dx\)- Where \(P\) is the market price at equilibrium.The integration results in a consumer surplus of \(4.5\), highlighting the benefit consumers gain when prices are stable.
This shows how in markets, consumers can often end up with a little extra satisfaction from their purchases.
In simpler terms, it's the difference between the highest price consumers are ready to shell out and the market price they actually pay.
To quantify consumer surplus, we find the area under the demand curve down to the equilibrium price level, up to the equilibrium quantity.Imagine you're buying an item and you're willing to pay \(10, but you find it for \)7.
Your consumer surplus is $3 because you got a better deal than you expected.In the example exercise, the demand function is given by \(D(x) = 7 - x\).
The equilibrium point, where consumers and producers agree on quantity and price, is at \((3, 4)\).
The consumer surplus is calculated using the formula:- \(CS = \int_0^3 (D(x) - P) \, dx\)- Where \(P\) is the market price at equilibrium.The integration results in a consumer surplus of \(4.5\), highlighting the benefit consumers gain when prices are stable.
This shows how in markets, consumers can often end up with a little extra satisfaction from their purchases.
Producer Surplus
Producer surplus is the flip side of consumer surplus, describing the additional earnings producers make when they sell at a market price higher than the minimum they would accept.
It's essentially the profit producers make above their minimum desired price.To visualize, think of an apple farmer willing to sell apples at \(2 each but selling them at \)3.
His producer surplus per apple is $1.From the solution, the supply function \(S(x) = 2 \sqrt{x+1}\) leads to the producer surplus when evaluated from the supply curve up to the equilibrium price and quantity.
This surplus is obtained by calculating:- \(PS = \int_0^3 (P - S(x)) \, dx\), where \(P = 4\) at equilibrium.The result shows the producer surplus as \(2\).
This surplus reflects the profit margin producers get, indicating the positive gaps between what they sell for and the baseline price they accept.
It's essentially the profit producers make above their minimum desired price.To visualize, think of an apple farmer willing to sell apples at \(2 each but selling them at \)3.
His producer surplus per apple is $1.From the solution, the supply function \(S(x) = 2 \sqrt{x+1}\) leads to the producer surplus when evaluated from the supply curve up to the equilibrium price and quantity.
This surplus is obtained by calculating:- \(PS = \int_0^3 (P - S(x)) \, dx\), where \(P = 4\) at equilibrium.The result shows the producer surplus as \(2\).
This surplus reflects the profit margin producers get, indicating the positive gaps between what they sell for and the baseline price they accept.
Demand and Supply Functions
Understanding demand and supply functions is crucial in finding the equilibrium and calculating surpluses.
They are mathematical expressions that estimate how much of a good will be bought or sold at different price levels.The **demand function** is the relationship between the quantity consumers are willing to purchase and the price.
In this case, it's \(D(x) = 7 - x\), suggesting that as prices fall, demand increases.**Supply functions** indicate the quantity of a good that producers will provide at different prices.
Here, it is represented as \(S(x) = 2 \sqrt{x+1}\), which shows that supply generally increases as the price rises.Understanding these functions helps determine the **equilibrium point** where the quantity supplied equals the quantity demanded.
In the solved problem, setting \(D(x) = S(x)\) finds this point, which is critical for determining both the consumer and producer surplus.
Ultimately, these functions not only guide pricing but also help evaluate market efficiency by balancing supply and demand.
They are mathematical expressions that estimate how much of a good will be bought or sold at different price levels.The **demand function** is the relationship between the quantity consumers are willing to purchase and the price.
In this case, it's \(D(x) = 7 - x\), suggesting that as prices fall, demand increases.**Supply functions** indicate the quantity of a good that producers will provide at different prices.
Here, it is represented as \(S(x) = 2 \sqrt{x+1}\), which shows that supply generally increases as the price rises.Understanding these functions helps determine the **equilibrium point** where the quantity supplied equals the quantity demanded.
In the solved problem, setting \(D(x) = S(x)\) finds this point, which is critical for determining both the consumer and producer surplus.
Ultimately, these functions not only guide pricing but also help evaluate market efficiency by balancing supply and demand.
Other exercises in this chapter
Problem 9
Verify Property 2 of the definition of a probability density function over the given interval. $$ f(x)=\frac{1}{3} x^{2}, \quad[-2,1] $$
View solution Problem 9
Determine whether each improper integral is convergent or divergent, and find its value if it is convergent. $$ \int_{0}^{\infty} \frac{d x}{2+x} $$
View solution Problem 10
For each probability density function, over the given interval, find \(\mathrm{E}(x), \mathrm{E}\left(x^{2}\right),\) the mean, the variance, and the standard d
View solution Problem 10
Show that \(y=-2 e^{x}+x e^{x}\) is a solution of \(y^{\prime \prime}-2 y^{\prime}+y=0\)
View solution