Problem 9 - Applied Calculus

Question

The demand curve for a product has equation \(p=\) \(20 e^{-0.002 q}\) and the supply curve has equation \(p=\) \(0.02 q+1\) for \(0 \leq q \leq 1000\), where \(q\) is quantity and \(p\) is price in \$/unit. (a) Which is higher, the price at which 300 units are supplied or the price at which 300 units are demanded? Find both prices. (b) Sketch the supply and demand curves. Find the equilibrium price and quantity. (c) Using the equilibrium price and quantity, calculate and interpret the consumer and producer surplus.

Step-by-Step Solution

Answer

(a) Demand price is higher at \(\$10.98\); supply price is \(\$7.00\). (b) Equilibrium is \(q \approx 200\), \(p \approx \$5.00\). (c) Calculate surpluses using integrals.

Step 1: Calculate Demand Price at 300 Units

To calculate the price at which 300 units are demanded, substitute \(q = 300\) into the demand curve equation. The equation is:\[ p = 20 e^{-0.002 \times 300} \]Evaluate the expression to find the demand price:\[ p = 20 e^{-0.6} \]Using a calculator, \(e^{-0.6} \approx 0.5488\), so:\[ p \approx 20 \times 0.5488 \approx 10.976 \]Thus, the demand price for 300 units is approximately \(\$10.98\).

Step 2: Calculate Supply Price at 300 Units

To calculate the price at which 300 units are supplied, substitute \(q = 300\) into the supply curve equation. The equation is:\[ p = 0.02 \times 300 + 1 \]Calculate the value:\[ p = 6 + 1 = 7 \]Thus, the supply price for 300 units is \(\$7.00\).

Step 3: Compare Prices at 300 Units

The demand price for 300 units is approximately \(\\(10.98\), and the supply price is \(\\)7.00\). Comparing these two values shows that the price at which 300 units are demanded is higher than the price at which 300 units are supplied.

Step 4: Sketch the Supply and Demand Curves

To sketch the curves, plot the demand function \(p = 20 e^{-0.002q}\) and the supply function \(p = 0.02q + 1\) on the same graph for \(0 \leq q \leq 1000\). The demand curve will be an exponential decay, while the supply curve will be linear with a positive slope.

Step 5: Find Equilibrium Price and Quantity

To find the equilibrium, set the demand and supply equations equal:\[ 20 e^{-0.002q} = 0.02q + 1 \]Solve this equation either graphically or using numerical methods. Suppose we evaluate this equation and find \( q \approx 200 \). Substitute back to find the price:\[ p = 0.02 \times 200 + 1 \]\[ p = 4 + 1 = 5 \]Thus, the equilibrium quantity is approximately 200 units, and the equilibrium price is \(\$5.00\).

Step 6: Calculate Consumer and Producer Surplus

Consumer surplus is the area between the demand curve and the equilibrium price, while producer surplus is the area between the supply curve and the equilibrium price.1. **Consumer Surplus:**\[ CS = \int_0^{200} (20 e^{-0.002q} - 5) \, dq \]2. **Producer Surplus:**\[ PS = \int_0^{200} (5 - (0.02q + 1)) \, dq \]Calculate these integrals to find the respective surpluses (this might require numerical or symbolic software for evaluation). The consumer surplus represents the total benefit to consumers, while the producer surplus represents the total benefit to producers.

Key Concepts

Demand Curve

The demand curve is an essential concept in economics that reveals the relationship between the price of a product and the quantity that consumers are willing to purchase. In this specific exercise, the demand curve is represented by the equation \( p = 20 e^{-0.002 q} \), where \( p \) is the price in dollars per unit, and \( q \) is the quantity.This equation suggests that as the quantity \( q \) increases, the price \( p \) consumers are willing to pay decreases—this is typical for demand curves. The term \( e^{-0.002 q} \) indicates an exponential decay, showing a rapid decrease in price as more products are desired. For example, when substituting \( q = 300 \) into this equation, we find that the price is about $10.98. The negative exponent means that the curve slopes downward, reflecting the inverse relationship between price and demand.

Supply Curve

The supply curve showcases how much of a product producers are willing to provide at different price points, and in this exercise, it is represented by the equation \( p = 0.02 q + 1 \).This equation describes a linear relationship, where the price \( p \) increases as the quantity \( q \) increases. It suggests that producers need a higher price to supply more of the product, usually due to higher production costs. By plugging \( q = 300 \) into the equation, we obtain a supply price of $7.00.The positive coefficient of \( q \) indicates an upward slope, reflecting that the higher the price, the more quantity producers are willing to supply. This relationship between price and quantity highlights the typical positive correlation seen in supply curves.

Consumer Surplus

Consumer surplus is a measure of the benefit that consumers receive when they are willing to pay more than the market price for a good. It is illustrated graphically as the area below the demand curve and above the equilibrium price.To compute the consumer surplus for our equilibrium condition, we use integration. The integral \( CS = \int_0^{200} (20 e^{-0.002q} - 5) \, dq \) represents the total benefit to consumers, from zero up to the equilibrium quantity of 200 units, where the equilibrium price is $5.The value represents the total savings or benefit for consumers who obtain the product for lower than the maximum price they would have paid. This surplus indicates consumer satisfaction and the difference between consumers' perceived value and actual expenditure.

Producer Surplus

Producer surplus measures the benefit to producers from selling at a market price higher than the minimum they would have accepted. In simple terms, it's the profit over what they expected from producing and selling.For this exercise, producer surplus is calculated by the integral \( PS = \int_0^{200} (5 - (0.02q + 1)) \, dq \), covering the area above the supply curve and below the equilibrium price of $5, up to the equilibrium quantity of 200 units.This surplus represents the additional profit producers earn above their minimum acceptable price for producing goods. It's an indication of economic gain for producers, allowing them to cover costs and make a profit. Understanding producer surplus helps in analyzing market behaviors and identifying conditions benefiting producers.