Equilibrium Quantity And Producer Surplus Calculation

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In economics, the concept of equilibrium is crucial for understanding how markets function. It represents a state where supply and demand balance, leading to a stable price and quantity. In this article, we'll dive into a specific scenario involving a demand function and a supply function, and we'll figure out the equilibrium quantity and the producer surplus at that equilibrium. So, let's put on our mathematical hats and get started, guys!

Understanding Demand and Supply Functions

Before we jump into the calculations, let's briefly touch on what demand and supply functions actually represent. The demand function, often denoted as d(x), illustrates the relationship between the price of a good or service and the quantity consumers are willing to purchase. Generally, as the price increases, the quantity demanded decreases, and vice versa. This inverse relationship is why demand curves are typically downward sloping. In our case, the demand function is given by:

d(x) = 157.5 - 0.5x2

Where d(x) represents the price at which x units can be sold. Notice the negative coefficient in front of the x2 term, indicating the downward-sloping nature of the demand curve. This means that as the quantity x increases, the price d(x) decreases. The constant term, 157.5, represents the maximum price at which the good or service could be sold if the quantity demanded were zero. The coefficient 0.5 scales the impact of the quantity on the price, showing how much the price changes for each unit increase in quantity.

On the other hand, the supply function, usually denoted as s(x), shows the relationship between the price of a good or service and the quantity producers are willing to supply. Typically, as the price increases, the quantity supplied also increases, leading to an upward-sloping supply curve. This positive relationship occurs because producers are more willing to supply goods or services at higher prices, as it increases their potential profit. In our scenario, the supply function is:

s(x) = 0.2x2

Here, s(x) represents the price at which producers are willing to supply x units. The positive coefficient 0.2 indicates the upward-sloping nature of the supply curve. As the quantity x increases, the price s(x) also increases, reflecting the higher prices needed to incentivize producers to supply more units. The absence of a constant term implies that the supply starts from zero; producers are not willing to supply any units if the price is zero. The coefficient 0.2 determines the steepness of the supply curve, showing how much the price changes for each unit increase in quantity supplied.

Finding the Equilibrium Quantity

The equilibrium quantity is the quantity at which the demand and supply curves intersect. In other words, it's the quantity where the quantity demanded equals the quantity supplied. At this point, the market is said to be in equilibrium, with no pressure for prices or quantities to change. To find the equilibrium quantity, we need to set the demand function equal to the supply function and solve for x:

d(x) = s(x)

157.5 - 0.5x2 = 0.2x2

Now, let's solve this equation step by step:

  1. Combine the x2 terms:

    157. 5 = 0.2x2 + 0.5x2

    157. 5 = 0.7x2

  2. Divide both sides by 0.7:

    x2 = 157.5 / 0.7

    x2 = 225

  3. Take the square root of both sides:

    x = ±√225

    x = ±15

Since quantity cannot be negative, we take the positive value. Therefore, the equilibrium quantity is:

x = 15

So, the equilibrium quantity is 15 units. This means that at the point where the market is in balance, 15 units of the good or service will be both demanded and supplied.

Calculating the Equilibrium Price

To confirm our understanding and provide a complete picture of the market equilibrium, it's essential to calculate the equilibrium price. The equilibrium price is the price at which the quantity demanded equals the quantity supplied. We've already found the equilibrium quantity (x = 15), so now we need to determine the price at which this quantity is traded in the market.

To find the equilibrium price, we can substitute the equilibrium quantity (x = 15) into either the demand function d(x) or the supply function s(x). At the equilibrium, both functions should give the same price since this is the point where demand and supply meet. Let's use both functions to double-check our calculations.

Using the Demand Function

Substitute x = 15 into the demand function d(x) = 157.5 - 0.5x2:

d(15) = 157.5 - 0.5(15)2

d(15) = 157.5 - 0.5(225)

d(15) = 157.5 - 112.5

d(15) = 45

So, according to the demand function, the equilibrium price is 45.

Using the Supply Function

Now, let's substitute x = 15 into the supply function s(x) = 0.2x2:

s(15) = 0.2(15)2

s(15) = 0.2(225)

s(15) = 45

As we can see, the supply function also gives us an equilibrium price of 45. This confirms that our calculations are correct, and we have found the true equilibrium price.

Interpretation

The equilibrium price of 45 is the price at which 15 units of the good or service will be bought and sold. At this price, there is neither excess supply (surplus) nor excess demand (shortage), indicating a stable market condition. This price balances the desires of consumers, who are willing to buy 15 units at this price, and producers, who are willing to supply 15 units. Understanding the equilibrium price is crucial for analyzing market efficiency and the impact of various economic policies.

Calculating Producer Surplus

The producer surplus is an economic measure that represents the difference between the price producers are willing to accept for a good or service and the price they actually receive. It's essentially the benefit or extra profit that producers gain from selling at the market price. Graphically, it's the area above the supply curve and below the equilibrium price.

To calculate the producer surplus, we need to find the area of the triangle formed by the supply curve, the equilibrium price, and the vertical axis. The formula for the area of a triangle is:

Area = (1/2) × base × height

In this context:

  • The base is the equilibrium quantity (x = 15).
  • The height is the difference between the equilibrium price and the price at which the first unit is supplied. Since the supply function is s(x) = 0.2x2, the price at which the first unit is supplied is s(0) = 0.2(0)2 = 0. Thus, the height is simply the equilibrium price (45) minus the supply price at zero quantity (0), so the height is 45.

However, since the supply curve is not a straight line, we need to use integration to find the exact area. The producer surplus can be calculated as the integral of the difference between the equilibrium price and the supply function, from 0 to the equilibrium quantity:

Producer Surplus = ∫015 [Equilibrium Price - s(x)] dx

Producer Surplus = ∫015 [45 - 0.2x2] dx

Now, let's evaluate the integral:

  1. Integrate the function:

    ∫ [45 - 0.2x2] dx = 45x - (0.2/3)x3 + C

  2. Evaluate the definite integral from 0 to 15:

    Producer Surplus = [45(15) - (0.2/3)(15)3] - [45(0) - (0.2/3)(0)3]

    Producer Surplus = [675 - (0.2/3)(3375)] - [0]

    Producer Surplus = 675 - 225

    Producer Surplus = 450

Therefore, the producer surplus at the equilibrium quantity is 450. This represents the total benefit that producers receive from selling their goods or services at the equilibrium price, above and beyond the minimum price they would be willing to accept.

Interpretation

The producer surplus of 450 indicates the economic welfare or benefit that producers gain by selling at the equilibrium price. It’s a measure of the producers' advantage in the market, showing how much more they receive compared to their minimum acceptable price. This surplus is a crucial concept in understanding market efficiency and the distribution of benefits in an economy. High producer surplus can incentivize producers to supply more, potentially leading to economic growth and development.

Conclusion

In this exploration, we've successfully navigated through the concepts of demand and supply functions, equilibrium quantity, and producer surplus. We determined that the equilibrium quantity for the given demand and supply functions is 15 units, and the producer surplus at this equilibrium is 450. Understanding these concepts is fundamental for analyzing market dynamics and the economic well-being of both consumers and producers. By understanding how these functions interact and how to calculate surpluses, we can better grasp the intricacies of market behavior and its implications for the economy. Keep up the great work, economics enthusiasts!