Problem 10 - Calculus and its applications

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. \(D(x)=5-x,\) for \(0 \leq x \leq 5 ; \quad S(x)=\sqrt{x+7}\)

Step-by-Step Solution

Answer

Equilibrium point is (2, 3); consumer surplus is 2; producer surplus is approximately 0.35.

Step 1: Find the Equilibrium Point

To find the equilibrium point, set the demand function \(D(x)\) equal to the supply function \(S(x)\). This means solving the equation \(5 - x = \sqrt{x+7}\). To eliminate the square root, square both sides of the equation, resulting in \((5 - x)^2 = x + 7\). Simplifying, we have:\[(5 - x)^2 = x + 7\]\[25 - 10x + x^2 = x + 7\]\[x^2 - 11x + 18 = 0\]Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -11\), and \(c = 18\). Solving gives:\[x = \frac{11 \pm \sqrt{121 - 72}}{2}\]\[x = \frac{11 \pm \sqrt{49}}{2}\]\[x = \frac{11 \pm 7}{2}\]This yields two solutions: \(x = 9\) and \(x = 2\). However, only \(x = 2\) falls within the allowable range \(0 \leq x \leq 5\). Thus, the equilibrium point is \((2, D(2)) = (2, 3)\).

Step 2: Calculate the Consumer Surplus

Consumer surplus is the area between the demand curve and the price level at equilibrium, up to the equilibrium quantity. Using the formula for consumer surplus:\[CS = \int_0^2 (5-x) \, dx - P_{eq} \cdot Q_{eq}\]First, calculate the integral:\[\int_0^2 (5-x) \, dx = \left[ 5x - \frac{x^2}{2} \right]_0^2\]Evaluating from 0 to 2:\[= \left(5(2) - \frac{2^2}{2}\right) - \left(5(0) - \frac{0^2}{2}\right)\]\[= 10 - 2\]\[= 8\]The equilibrium price is 3, so:\[CS = 8 - (3 \cdot 2)\]\[= 8 - 6\]\[= 2\]

Step 3: Calculate the Producer Surplus

Producer surplus is the area between the price level at equilibrium and the supply curve, up to the equilibrium quantity. Using the formula for producer surplus:\[PS = P_{eq} \cdot Q_{eq} - \int_0^2 \sqrt{x+7} \, dx\]The equilibrium price is 3 and the equilibrium quantity is 2.For the integral, make the substitution \(u = x+7\), \(du = dx\):\[\int_0^2 \sqrt{x+7} \, dx = \int_7^9 \sqrt{u} \, du = \left[\frac{2}{3}u^{3/2}\right]_7^9\]Evaluating:\[= \frac{2}{3}\left(9^{3/2} - 7^{3/2}\right)\]\[= \frac{2}{3}\left(27 - \sqrt{343}\right)\]Approximate \(\sqrt{343} \approx 18.52\), thus:\[= \frac{2}{3}(27 - 18.52)\]\[= \frac{2}{3}(8.48)\]\[\approx 5.65\]Therefore:\[PS = 6 - 5.65\]\[\approx 0.35\]

Key Concepts

Demand and Supply Functions

Understanding demand and supply functions is essential in economics. A demand function, such as \(D(x) = 5 - x\), indicates the price consumers are willing to pay for \(x\) units of a product. As \(x\) (the quantity) increases, the price \(D(x)\) consumers would pay tends to decrease, representing the law of demand. A supply function, like \(S(x) = \sqrt{x+7}\), shows the price at which producers are willing to supply \(x\) units. This function often increases with quantity, following the basic principle of supply that higher prices incentivize greater quantity produced.

Finding the equilibrium point involves setting \(D(x)\) equal to \(S(x)\), which depicts the price at which consumers' willingness to buy equals producers' willingness to sell. This is crucial because at this point (in our exercise, \((2, 3)\)), the market is considered balanced. No surplus product exists unsold, and no additional consumer demand is unmet. Both parties, buyers, and sellers, are satisfied, making it a vital concept in market dynamics.

Consumer Surplus

Consumer surplus measures the benefit consumers receive when they pay a price lower than what they were willing to pay. It is visually represented as the area beneath the demand curve and above the equilibrium price, up to the equilibrium quantity.

To calculate this surplus, you determine the area between the demand curve \(D(x) = 5 - x\) and the horizontal line at the equilibrium price. This involves integrating the demand function from 0 to the equilibrium quantity (in our case, 2), then subtracting the total revenue paid \((P_{eq} \cdot Q_{eq})\).

• In the given solution, this represents \(\int_0^2 (5-x) \, dx = 8\).
• After subtracting the total payment made by consumers, calculated as \(P_{eq} \cdot Q_{eq} = 6\) (since \(P_{eq} = 3\) and \(Q_{eq} = 2\)), we find the consumer surplus to be 2.

The result tells us that consumers collectively gain a net benefit of $2 by paying less than they were prepared to for the 2 units purchased.

Producer Surplus

Producer surplus is the benefit producers receive when they sell a product at a price higher than the minimum price they would be willing to accept. It's illustrated as the area above the supply curve and below the equilibrium price, up to the equilibrium quantity.

To find this area, we integrate the supply function \(S(x) = \sqrt{x+7}\) from 0 to 2, which corresponds to evaluating the actual revenue received minus this integral's value:

• First, compute \(\int_0^2 \sqrt{x+7} \, dx\) which, through substitution, approximates to about 5.65.
• Subtracting this from the total revenue \(P_{eq} \cdot Q_{eq} = 6\) results in a producer surplus of approximately 0.35.

This calculation shows that producers gain a small net benefit by selling the product at the market equilibrium price of $3, which is higher than what they might have originally accepted for the first two units produced.