Solving for equilibrium price and quantity mathematically

Summary

This video tutorial guides viewers through the mathematical process of determining equilibrium price and quantity in microeconomics. It outlines four key steps: expressing demand and supply functions in terms of quantity, setting supply equal to demand to find the equilibrium point, solving for the equilibrium price, and verifying by calculating the equilibrium quantity. An example is used to illustrate these steps, showing how to manipulate equations to solve for 'q' and 'p', and ensuring accuracy by checking both the supply and demand functions.

Takeaways

• 📚 The video provides a step-by-step guide to finding equilibrium price and quantity in microeconomics.
• 🔢 The first step is to express the demand and supply functions in terms of quantity (Q) rather than price (P).
• ✏️ Example: Transforming the demand function from q_d = 10,000 - 80p to p = (10,000 - q_d) / 80.
• 🔄 The second step involves setting the quantity supplied equal to the quantity demanded to find the equilibrium point.
• 💡 Solving for equilibrium price (p) is the third step, which is done by equating the supply and demand equations and solving the resulting equation.
• 📉 The video explains that the equilibrium price is where the supply and demand curves intersect on a typical graph.
• 🔍 The fourth and final step is to plug the equilibrium price back into the supply and demand functions to find the equilibrium quantity.
• 🔗 It's important to verify the equilibrium quantity by checking it against both the supply and demand functions to ensure accuracy.
• 📈 The example given demonstrates how the law of demand (quantity demanded decreases as price increases) and the law of supply (quantity supplied increases as price increases) are reflected in the equations.
• 📝 The video script serves as a tutorial for students learning introductory microeconomics, emphasizing the importance of algebraic manipulation and graph interpretation.

Q & A

What is the main topic of the video?

The main topic of the video is how to find equilibrium price and quantity mathematically for an introductory microeconomics course.

How many steps are involved in finding equilibrium price and quantity according to the video?

There are four steps involved in finding equilibrium price and quantity as described in the video.

What is the first step in finding equilibrium price and quantity?

The first step is to solve the demand and supply functions in terms of quantity, meaning to express the functions with quantity (q) as the subject.

What does setting quantity of supply equal to quantity demanded represent?

Setting quantity of supply equal to quantity demanded represents the condition at the equilibrium point where the supply and demand curves intersect on a typical supply and demand graph.

How do you solve for the equilibrium price (p)?

To solve for the equilibrium price (p), you set the quantity supplied equal to the quantity demanded, then solve the resulting equation for p.

What is the purpose of plugging the equilibrium price back into the supply and demand functions?

Plugging the equilibrium price back into the supply and demand functions is done to solve for the equilibrium quantity and to check that the calculations are correct.

What is an example of a demand function given in the video?

An example of a demand function given in the video is qd = 10,000 - 80p, which is already solved for in terms of quantity (qd).

How does the video demonstrate converting a demand function into terms of quantity?

The video demonstrates converting a demand function into terms of quantity by manipulating the equation to isolate qd, and then solving for p to get the function in terms of q.

What is the example supply function given in the video?

The example supply function given in the video is 20p, which confirms the law of supply where quantity supplied increases as price increases.

How does the video ensure the accuracy of the calculated equilibrium price and quantity?

The video ensures the accuracy by plugging the calculated equilibrium price into both the supply and demand functions and verifying that the same equilibrium quantity is obtained.

What is the final equilibrium price and quantity found in the video example?

In the video example, the final equilibrium price is found to be $100, and the equilibrium quantity is 2,000 units.

Outlines

00:00

📚 Introduction to Finding Equilibrium Price and Quantity

This paragraph introduces a four-step method to find the equilibrium price and quantity in microeconomics. The first step is to express the demand and supply functions in terms of quantity (q) rather than price (p). An example is given where the demand function is already in terms of q, and the process to solve for p in such a function is explained. The second step is to set the quantity supplied equal to the quantity demanded, which leads to the third step of solving for the equilibrium price (p). The final step is to verify the solution by plugging the equilibrium price back into the supply and demand functions to find the equilibrium quantity. The paragraph emphasizes the importance of this method for checking the accuracy of the calculations.

05:03

🔍 Calculating Equilibrium in a Supply and Demand Example

The paragraph presents a practical example to calculate equilibrium in a supply and demand scenario. It begins by setting the demand and supply functions equal to each other to find the equilibrium quantity. The example uses a demand function of 10,000 - 80p and a supply function of 20p. By solving the equations, the equilibrium price is determined to be $100. The paragraph then demonstrates how to plug this price back into both the supply and demand functions to verify the equilibrium quantity, which is found to be 2,000 units. The process confirms that the quantity demanded equals the quantity supplied at the equilibrium price, indicating that the calculations were correct. The paragraph concludes with a reminder to follow these steps for solving equilibrium problems and to check the work by plugging the equilibrium price into both functions.

Keywords

💡Equilibrium Price

Equilibrium price refers to the price at which the quantity of a good or service that buyers are willing to purchase is equal to the quantity that sellers are willing to sell. In the context of the video, finding the equilibrium price is a key step in solving supply and demand problems. The script uses the example where the equilibrium price is calculated to be $100, demonstrating how the price where supply equals demand is determined.

💡Equilibrium Quantity

Equilibrium quantity is the amount of a good or service that is bought and sold at the equilibrium price. The video emphasizes the importance of calculating the equilibrium quantity to ensure that the market is in balance. In the example provided, the equilibrium quantity is found to be 2,000 units, illustrating the point where the quantity supplied equals the quantity demanded.

💡Supply Function

A supply function is a mathematical representation of the relationship between the price of a good and the quantity of that good that producers are willing to supply. In the video, the supply function is given as '20P', indicating that the quantity supplied is directly proportional to the price. This function is used to find the equilibrium quantity by setting it equal to the demand function.

💡Demand Function

A demand function represents the quantity of a product that consumers are willing to purchase at various price levels. The video script provides an example of a demand function as '10,000 - 80P', showing how the quantity demanded decreases as the price increases. This function is crucial for determining the market equilibrium.

💡Quantity Supplied

Quantity supplied is the amount of a good or service that producers are willing to offer for sale at a given price. The video explains how to solve for the quantity supplied by setting it equal to the quantity demanded, which is essential for finding the equilibrium point in the market.

💡Quantity Demanded

Quantity demanded is the amount of a good or service that consumers are willing and able to purchase at a given price. The video script uses the concept of quantity demanded to demonstrate how to set it equal to the quantity supplied to find the equilibrium price and quantity in a market.

💡Law of Demand

The law of demand states that, all else being equal, the quantity demanded of a good tends to decrease as the price of the good increases. The video script confirms this law by showing a downward-sloping demand curve and a demand function where the quantity demanded decreases as price increases.

💡Law of Supply

The law of supply suggests that, all else being equal, the quantity supplied of a good tends to increase as the price of the good increases. The video script illustrates this law with an upward-sloping supply curve and a supply function where the quantity supplied increases with the price.

💡Solving for Q

Solving for Q in the context of the video means expressing the demand and supply functions in terms of quantity rather than price. This is an essential step in the process of finding the equilibrium price and quantity. The video provides an example of how to algebraically manipulate an equation to express it solely in terms of quantity.

💡Setting Quantities Equal

In the video, setting quantities equal is a method used to find the equilibrium point where the quantity supplied by producers is equal to the quantity demanded by consumers. This is done by equating the supply and demand functions and solving for the price, which is a fundamental step in determining market equilibrium.

💡Checking Answers

Checking answers in the video script refers to the process of verifying the calculated equilibrium price and quantity by plugging them back into the supply and demand functions. This step ensures the accuracy of the calculations and confirms that the market is in equilibrium, as demonstrated by the script where the equilibrium price and quantity are tested in both the supply and demand functions.

Highlights

Introduction to finding equilibrium price and quantity in microeconomics.

Four-step method to solve for equilibrium in economics.

Step 1: Solve demand and supply functions in terms of quantity (q).

Example of solving a demand function for q.

Manipulation of equations to express in terms of q.

Step 2: Set quantity of supply equal to quantity demanded.

Explanation of why setting quantities equal is essential.

Step 3: Solve for equilibrium price (p).

Solving for p by setting supply and demand equations equal.

Step 4: Plug equilibrium price into supply and demand functions to find equilibrium quantity.

Verification of equilibrium by plugging values back into equations.

Example supply function given as 20p.

Confirming the law of demand and law of supply with the example functions.

Setting up the equation to find equilibrium price.

Calculation of equilibrium price as 100.

Testing the equilibrium price in both supply and demand functions.

Conclusion of equilibrium quantity being 2,000 and price being 100.

Advice on checking work by solving for equilibrium quantity in both functions.

Transcripts

[all] [right] this video is going to go over how to find equilibrium price and quantity

Mathematically for your introductory microeconomics course so there are four steps

Here that you'll need to go through [and] you should be able [to] solve any of these problems, [so] [the] first step

is to solve

the demand and

supply functions

in terms of quantity

So by this, I [mean] you have q

equals blah blah blah blah blah

Instead of p equals blah blah blah blah blah, and we'll go through an example in a second, the second step

is to set

Quantity of supply equal to quantity demanded or you set the quantities

equal to each other

The third Step here is to solve for [p]

Or equilibrium price

And the last step in this method is to plug in your equilibrium price

To your quantity supplied in your quantity demanded functions

to solve for equilibrium quantity

and

Then by plugging it into both of these and checking you'll make sure that you did your math right and that you didn't mess anything up

so it's a good way to check your answers, too

So let's go through [an] example

Let's imagine that our quantity demanded

is equal to

10,000

minus 80 p

So this is already solved for in terms of Q

It's possible

That this isn't solved for in terms of Q so for that method you would have to do some

Manipulation to get this solved in terms of Q so as an example. Let's take this function and solve for p

To solve this for p. We would have to add

80 p to both sides so we would get 80 p

equal

80 p plus our Qd equals

10,000 and

then subtract Qd from both sides and then

[divide] both sides by this 80 and that would give us p equals

10,000 over 80 minus Qd over 80

So if you were given something that looks like this you would want to solve for this qD

so to do that you would want to get qd by itself so you could add qd over 80 to both sides so

[you'd] get qd over 80 plus p

equals 10,000 over 80

You would then subtract p from both sides to get rid of it here and subtract it over there

And then you would multiply everything by 80 to get Qd equals

10,000 minus

80 P

And so that's how you would get your demand function in terms of Q instead of in terms of p

So this is our demand function

Our supply function in this example is going to be

20 P and

So note [that] these both confirm the law of demand and the law of supply, so this price goes up quantity

Demanded goes down as price goes up quantity supplied goes up

So for a typical graph we would get downward sloping demand

and

upward sloping supply

so we've solved for these two guys in terms of q we now have to set these quantities equal to each other and

The reason we do this is because in our typical supply and demand graph

we only have one equilibrium point and

that occurs at equilibrium price and

equilibrium quantity

So we know that we're going to be at the same equilibrium quantity on both the supply curve

represented by [our] supply function and the demand curve

represented by our demand function

So if we set those two guys equal to each other we'll have 10,000

minus 80 p

equals 20 p

so now we can add 80 p

to both sides and we [will] get

10,000

equals 100 p

so we get 10,000 equals 100 p if we divide both sides by a hundred we get a

Hundred equals p. So we now know that our equilibrium price is a hundred

And we know this represents the price where these two guys cross, so if we plug in this p-value now into our supply

function and our demand function

We should get that same quantity back out, so let's test it and see what happens

We know the equilibrium price

equals 100 our supply function

equal 20 times p

So if we plug in our equilibrium price we get 20

times

120 times 100 is 2,000

So our equilibrium quantity

should be 2,000

But let's plug it into our demand function just to make sure so [our] demand function was 10,000

minus 80 p

so if we plug in that hundred

for the p we get 10,000

minus 8,000 we got the 8,000 by multiplying 80 times p and

10,000 minus 8,000 is

Again 2,000 so our quantity demanded equals our quantity supplied. Which is good it means

We did it right and so our equilibrium

Quantity is 2,000 and our equilibrium price is 100

So remember these steps when solving for the equilibrium?

First solve your supply and demand function in terms of quantity so q equals something

Then set your quantity supplied equal to your quantity demanded

Once you've done this solve for p that will be your equilibrium price

Then plug those peas back into your quantity supplied and quantity demand functions to solve for equilibrium

quantity

You can do it for one if you're confident in your math, but I recommend doing it for both

Just to double-Check your methods and make sure you did your Algebra correctly