Equilibrium Price & Quantity: Demand & Supply Functions

Let's dive into the fascinating world of economics, guys! In this article, we're going to break down a classic economic problem: finding the equilibrium price and quantity in a market. We'll be using supply and demand functions, which are the bread and butter of economic analysis. So, buckle up and let's get started!

Understanding Demand and Supply Functions

Before we jump into the calculations, it's crucial to understand what demand and supply functions actually represent. These functions are mathematical expressions that describe the relationship between the price of a good and the quantity demanded or supplied.

• Demand Function: The demand function shows how much of a good consumers are willing and able to buy at different prices. Generally, as the price of a good increases, the quantity demanded decreases, and vice versa. This inverse relationship is known as the law of demand. In our case, the demand function for good A is given as a quadratic equation: P = -4Q² + 40. This tells us that the price (P) is related to the quantity demanded (Q) in a non-linear way. As the quantity demanded increases, the price decreases, but at a decreasing rate due to the squared term. Understanding the demand function is critical because it serves as the backbone for figuring out consumer behavior and market dynamics. The shape of this curve, in our case a downward-sloping parabola, visually represents the inverse relationship between price and quantity demanded – a cornerstone principle in economics.

• Supply Function: The supply function, on the other hand, shows how much of a good producers are willing and able to supply at different prices. Generally, as the price of a good increases, the quantity supplied also increases. This is because producers are incentivized to supply more when they can sell at higher prices. This direct relationship is known as the law of supply. For good A, we're told that the supply function is linear and that when the price is 12 per unit, no suppliers offer the good. This crucial piece of information lets us formulate the supply function mathematically, setting the stage for determining market equilibrium. The supply function's upward slope mirrors the cost-benefit analysis suppliers perform: higher prices generally mean higher profits, encouraging increased production.

Setting Up the Problem

Alright, now let's get our hands dirty with the specifics of the problem. We're given the demand function for good A:

• P = -4Q² + 40

And we're told that the supply function is linear. We also have a key data point: when the price (P) is 12, the quantity supplied (Q) is 0. This is where our detective work begins! We need to find the supply function, which will be in the form of P = aQ + b, where 'a' and 'b' are constants we need to determine. To find the equilibrium price and quantity, we'll eventually need to equate the demand and supply functions. But first, let's focus on figuring out that supply function.

Finding the Supply Function

Since the supply function is linear, we can express it in the form P = aQ + b. We know one point on this line: (Q, P) = (0, 12). This means when the quantity supplied is 0, the price is 12. Let's plug this into our equation:

• 12 = a(0) + b

This simplifies to b = 12. Great! Now we have part of the supply function: P = aQ + 12. But we still need to find 'a', which represents the slope of the supply curve.

To determine the slope, we need another point on the supply curve. While we aren't given another point directly, we know that the supply curve generally slopes upwards. This implies that as price increases, the quantity supplied should also increase. We’ll need to use the equilibrium condition (demand equals supply) later to find another point. For now, let’s think about what 'a' represents. In the supply equation, 'a' is the slope, indicating how much the price needs to increase for each additional unit supplied. Economically, this reflects the marginal cost of production – how much it costs to produce one more unit. Without more information, we can’t definitively solve for 'a' just yet, so we'll hold onto this partial supply function and use it later when we equate supply and demand.

Determining Equilibrium: Where Demand Meets Supply

Now comes the exciting part: finding the equilibrium! The equilibrium price and quantity are the point where the demand and supply curves intersect. This is where the quantity demanded by consumers exactly equals the quantity supplied by producers. In other words, it's where the market clears – there's no surplus or shortage. To find this point, we need to set the demand function equal to the supply function.

We have:

• Demand: P = -4Q² + 40
• Supply: P = aQ + 12

Setting them equal gives us:

• -4Q² + 40 = aQ + 12

Now we have a quadratic equation in terms of Q. To solve for Q, we need to rearrange the equation into the standard quadratic form (Ax² + Bx + C = 0):

• 4Q² + aQ - 28 = 0

Solving the Quadratic Equation

To solve this quadratic equation, we can use the quadratic formula:

• Q = [-B ± √(B² - 4AC)] / (2A)

In our case, A = 4, B = a, and C = -28. Plugging these values into the formula gives us:

• Q = [-a ± √(a² - 4 * 4 * -28)] / (2 * 4)
• Q = [-a ± √(a² + 448)] / 8

We get two possible solutions for Q, but since quantity cannot be negative, we'll choose the positive solution. However, we still have 'a' in the equation, which is the slope of the supply curve we haven't fully determined yet. This highlights a crucial point: we need more information to definitively solve for the equilibrium quantity. We either need another point on the supply curve or some other information about the cost structure of the suppliers to determine 'a'.

A Critical Missing Piece

You might be thinking, “Hey, we've done a lot of work, but we're still stuck!”. And you're right. This is a common situation in economics problems. Sometimes, we don't have all the information we need upfront. In this case, the problem statement is missing a crucial piece of information that would allow us to determine the slope of the supply curve ('a').

Without knowing 'a', we can't fully solve the quadratic equation and find a numerical value for the equilibrium quantity (Q). Consequently, we can't plug that Q value back into either the demand or supply function to find the equilibrium price (P). We've reached a roadblock! This underscores the importance of complete information in economic modeling. We've correctly set up the problem and applied the economic principles, but a missing variable prevents us from reaching a final numerical solution.

Wrapping Up: The Importance of Complete Information

So, what have we learned, guys? We've taken a deep dive into finding the equilibrium price and quantity using demand and supply functions. We've successfully set up the equations, identified the key relationships, and even applied the quadratic formula. However, we've also encountered a common challenge in economic problem-solving: the need for complete information. In this case, the missing information about the supply curve's slope prevents us from finding a definitive numerical solution.

This doesn't mean our effort was wasted! We've gained valuable experience in understanding how demand and supply interact, how to set up and solve equations, and how to recognize when we're missing crucial information. Remember, real-world economic problems often involve incomplete data, so knowing how to identify these gaps is a vital skill. Keep practicing, keep learning, and you'll become economic problem-solving pros in no time!