Equilibrium Price And Quantity Calculation For Good 1 With Given Supply And Demand Functions

October 8, 2025 by Scholario Team 93 views

Hey guys! Let's dive into a classic economics problem: calculating the equilibrium price and quantity for a good. This is super important for understanding how markets work. We're going to use the supply and demand functions for good 1 to figure this out. So, stick around, and let's get started!

Understanding Supply and Demand Functions

Before we jump into the math, let's quickly recap what supply and demand functions are all about. These functions basically describe how the quantity of a good that consumers want to buy (demand) and the quantity that producers are willing to sell (supply) change with price and other factors.

Demand Function (D1): This tells us how much of good 1 consumers will demand at different prices (p1), given the price of another good (p2) and consumer income (R). In our case, the demand function is:
```
D1 = 25 - 0.5p1 - p2 + 0.35R
```
Think of it like this: as the price of good 1 (p1) goes up, the demand (D1) usually goes down. But if consumer income (R) goes up, the demand for good 1 might increase. The price of the other good (p2) also plays a role, which we'll discuss later.
Supply Function (S1): This function shows how much of good 1 producers are willing to supply at different prices (p1). Our supply function is:
```
S1 = 0.7p1
```
Simple enough, right? As the price of good 1 (p1) goes up, producers are usually willing to supply more of it.

Equilibrium is the point where the quantity demanded equals the quantity supplied. It's like the sweet spot in the market where everyone's happy – buyers can buy what they want, and sellers can sell what they want. To find this point, we need to set the demand function equal to the supply function and solve for the price (p1).

Setting Up the Equations

Alright, let's get our hands dirty with the math! We've got the following:

Demand Function: D1 = 25 - 0.5p1 - p2 + 0.35R
Supply Function: S1 = 0.7p1
Consumer Income: R = 1200
Price of Good 2: p2 = 20

Our goal is to find the equilibrium price (p1) and the equilibrium quantity (D1 or S1, since they're equal at equilibrium). First, we need to plug in the values for R and p2 into the demand function:

D1 = 25 - 0.5p1 - 20 + 0.35 * 1200

Now, let's simplify this a bit:

D1 = 25 - 0.5p1 - 20 + 420
D1 = 425 - 0.5p1

Great! Now we have a simplified demand function. The next step is to set the demand function equal to the supply function:

425 - 0.5p1 = 0.7p1

Solving for Equilibrium Price (p1)

Time to solve for p1! We need to get all the p1 terms on one side of the equation. Let's add 0.5p1 to both sides:

425 = 0.7p1 + 0.5p1
425 = 1.2p1

Now, to isolate p1, we'll divide both sides by 1.2:

p1 = 425 / 1.2
p1 ≈ 354.17

So, the equilibrium price (p1) is approximately 354.17. That's the price at which the quantity demanded and the quantity supplied are equal.

Calculating Equilibrium Quantity

Now that we've got the equilibrium price, let's find the equilibrium quantity. We can plug the equilibrium price (p1 ≈ 354.17) into either the demand function or the supply function. Let's use the supply function since it's simpler:

S1 = 0.7p1
S1 = 0.7 * 354.17
S1 ≈ 247.92

So, the equilibrium quantity is approximately 247.92. This means that at a price of around 354.17, both consumers want to buy about 247.92 units of good 1, and producers are willing to supply about the same amount.

Putting It All Together

We've successfully calculated the equilibrium price and quantity for good 1! Here's what we found:

Equilibrium Price (p1): ≈ 354.17
Equilibrium Quantity: ≈ 247.92

This is a crucial concept in economics. The equilibrium price and quantity represent the point where the market is in balance. If the price were higher than the equilibrium price, there would be a surplus of the good, and producers would have to lower the price to sell it. If the price were lower than the equilibrium price, there would be a shortage, and consumers would be willing to pay more for the good.

Key Takeaway: The equilibrium price and quantity are determined by the interaction of supply and demand in the market.

The Impact of Income and Prices of Related Goods

Let's quickly chat about how changes in consumer income (R) and the price of related goods (p2) can influence the equilibrium. Remember, in our example:

R (Consumer Income) = 1200
p2 (Price of Good 2) = 20

What if these values changed? How would that affect the demand for good 1 and, consequently, the equilibrium price and quantity?

Change in Consumer Income (R)

Consumer income plays a crucial role in shaping the demand for various goods. A rise in income often leads to an increase in the demand for normal goods, while it might decrease the demand for inferior goods.

In our demand function:

D1 = 25 - 0.5p1 - p2 + 0.35R

The term 0.35R indicates that an increase in income (R) will increase the demand for good 1 (D1), assuming good 1 is a normal good. If income rises, consumers have more money to spend, and they are likely to purchase more of good 1 at each price level. This shift in the demand curve to the right will result in a higher equilibrium price and quantity for good 1.

Conversely, if income decreases, the demand for good 1 will fall (assuming it's a normal good). This will shift the demand curve to the left, leading to a lower equilibrium price and quantity.

Change in the Price of Good 2 (p2)

The price of related goods can significantly influence the demand for a specific good. Related goods can be either substitutes or complements.

Substitutes: These are goods that can be used in place of each other (e.g., coffee and tea). If the price of a substitute good increases, consumers may switch to the original good, increasing its demand.
Complements: These are goods that are often consumed together (e.g., cars and gasoline). If the price of a complementary good increases, the demand for the original good may decrease.

In our demand function:

D1 = 25 - 0.5p1 - p2 + 0.35R

The term - p2 indicates that good 2 is a substitute for good 1. If the price of good 2 (p2) increases, the demand for good 1 (D1) will increase, as consumers may switch from good 2 to good 1. This shift in the demand curve will result in a higher equilibrium price and quantity for good 1.

Conversely, if the price of good 2 decreases, the demand for good 1 will fall, leading to a lower equilibrium price and quantity.

Real-World Applications

Understanding equilibrium price and quantity isn't just some abstract economic concept. It has tons of real-world applications! For example:

Businesses: Companies use supply and demand analysis to make decisions about pricing, production levels, and inventory management. If they know the demand for their product is high, they might increase production and even raise prices.
Governments: Policymakers use these concepts to understand the effects of taxes, subsidies, and regulations on different markets. For example, a tax on a product can shift the supply curve and lead to a higher equilibrium price.
Consumers: Even as consumers, we're implicitly using these ideas when we decide whether to buy something or not. If the price is too high, we might wait for a sale or look for a substitute.

Conclusion

So, there you have it! We've walked through how to calculate the equilibrium price and quantity for a good using supply and demand functions. We also explored how changes in income and the prices of related goods can affect the market equilibrium. These concepts are fundamental to understanding how markets work and making informed economic decisions.

Remember, economics might seem intimidating at first, but breaking it down step by step can make it much easier to grasp. Keep practicing, and you'll be an economics whiz in no time! Keep your curiosity up, and keep learning! You've got this!