Supply and Demand: Quantitative Analysis: Videos & Practice Problems

Understanding how to find equilibrium in a market involves using algebra to analyze demand and supply curves. The demand curve can often be represented by a linear equation, such as p = 800 - 2Q, where p is the price and Q is the quantity demanded. To graph this equation, you can select various values for Q and solve for p.

For instance, if you start with Q = 0, substituting this into the equation gives:

p = 800 - 2(0) = 800.

This indicates that at a price of $800, the quantity demanded is 0. Plotting this point on a graph, you can visualize the relationship between price and quantity.

Next, if you choose Q = 200, the calculation becomes:

p = 800 - 2(200) = 400.

This means that at a price of $400, the quantity demanded is 200. By plotting this point as well, you can begin to form the demand curve, which will be a straight line connecting the points you have plotted.

Continuing this process with Q = 300, you find:

p = 800 - 2(300) = 200.

This confirms that at a price of $200, the quantity demanded is 300. By plotting these points, you can accurately depict the demand curve on your graph.

In summary, isolating variables in equations allows you to understand how changes in quantity affect price, which is essential for determining market equilibrium. The demand curve illustrates this relationship, providing a visual representation of how price and quantity interact in a market setting.

In algebra, rearranging equations is a fundamental skill, especially when isolating different variables. For instance, consider the equation for price:

P = 800 - 2q_d

Here, we may want to isolate the quantity demanded, q_d. To achieve this, we can follow a systematic approach. First, we move the constant term (800) to the other side of the equation:

P - 800 = -2q_d

Next, to isolate q_d, we need to eliminate the coefficient of -2. Since -2 is currently multiplying q_d, we divide both sides of the equation by -2:

q_d = \frac{P - 800}{-2}

To simplify this expression, we can separate the terms in the numerator:

q_d = \frac{P}{-2} - \frac{800}{-2}

Calculating the division gives us:

q_d = -\frac{1}{2}P + 400

Alternatively, this can be expressed as:

q_d = 400 - \frac{1}{2}P

Both forms are valid and interchangeable, demonstrating the flexibility in algebraic manipulation. This process illustrates how to switch the isolated variable in an equation through careful application of algebraic principles.

Next, we can apply similar techniques to supply equations, reinforcing our understanding of variable isolation in different contexts.

In economics, the supply curve represents the relationship between the price of a good and the quantity supplied. The equation for the supply curve can be expressed as \( p = 200 + q_s \), where \( p \) is the price and \( q_s \) is the quantity supplied. This equation indicates that as the quantity supplied increases, the price also increases, reflecting the direct relationship between these two variables.

To illustrate this relationship, we can calculate specific price points for various quantities. For instance, when the quantity supplied \( q_s \) is 0, the price \( p \) can be calculated as follows:

\( p = 200 + 0 = 200 \)

This means that at a quantity of 0, the price is 200. Next, if we consider a quantity supplied of 200:

\( p = 200 + 200 = 400 \)

Thus, at a quantity of 200, the price rises to 400. Continuing this process, if we take a quantity supplied of 300:

\( p = 200 + 300 = 500 \)

At this quantity, the price reaches 500. By plotting these points on a graph, we can visualize the supply curve, which typically slopes upwards from left to right, indicating that higher prices incentivize greater quantities supplied.

In summary, the supply curve is a fundamental concept in economics that illustrates how price and quantity supplied are interrelated. Understanding this relationship is crucial for analyzing market dynamics and making informed economic decisions.

To find the equilibrium in a market, we utilize the equations for supply and demand curves. The first step involves ensuring that the same variable is isolated in both equations, whether it be price or quantity. This means that we want either price or quantity to be on one side of both equations. Once the variables are aligned, we can proceed to the next step.

The second step is to set the demand and supply equations equal to each other. At equilibrium, the quantity supplied equals the quantity demanded, which allows us to equate the two equations. For example, if we have the demand equation \( P = 800 - 2Q_D \) and the supply equation \( P = 200 + Q_S \), we can set them equal: \( 800 - 2Q = 200 + Q \). Here, we can simplify our notation by using \( Q \) to represent both \( Q_D \) and \( Q_S \) since they are equal at equilibrium.

In the third step, we solve for the variable \( Q \). Rearranging the equation \( 800 - 2Q = 200 + Q \) leads to \( 800 - 200 = 3Q \), which simplifies to \( 600 = 3Q \). Dividing both sides by 3 gives us the equilibrium quantity \( Q^* = 200 \).

Next, we substitute this equilibrium quantity back into one of the original equations to find the equilibrium price. Using the supply equation \( P = 200 + Q \), we substitute \( Q = 200 \) to find \( P^* = 200 + 200 = 400 \). Thus, the equilibrium price is \( P^* = 400 \) and the equilibrium quantity is \( Q^* = 200 \).

To confirm our results, we can also solve using the other variable. If we isolate quantity in the supply equation and set it equal to the demand equation, we can derive the same equilibrium price and quantity. For instance, using the demand equation \( Q_D = 400 - \frac{1}{2}P \) and substituting \( P = 400 \) yields \( Q_D = 400 - \frac{1}{2}(400) = 200 \), confirming our earlier findings.

This process illustrates the importance of algebraic manipulation in finding market equilibrium, demonstrating that regardless of which variable is isolated, the results will remain consistent. Understanding these steps is crucial for analyzing market dynamics and predicting outcomes based on changes in supply and demand.

To determine the equilibrium price and quantity in a market, we analyze the supply and demand curves represented by their respective equations. The demand equation typically expresses price in terms of quantity demanded, while the supply equation expresses quantity supplied in terms of price. In this case, we start with the demand equation, which can be rearranged to isolate quantity demanded (QD).

Given the demand equation:

$$ p = 6 - \frac{1}{50} Q_D $$

We can rearrange it to isolate QD:

$$ p - 6 = -\frac{1}{50} Q_D $$

Multiplying both sides by -50 to eliminate the fraction gives us:

$$ Q_D = -50p + 300 $$

Next, we set the demand equation equal to the supply equation, which is given as:

$$ Q_S = 150p - 100 $$

Setting the two equations equal to each other allows us to find the equilibrium price:

$$ -50p + 300 = 150p - 100 $$

To solve for p, we first combine like terms:

$$ 300 + 100 = 150p + 50p $$

$$ 400 = 200p $$

Dividing both sides by 200 yields:

$$ p = 2 $$

Now that we have the equilibrium price, we can substitute this value back into either the supply or demand equation to find the equilibrium quantity. Using the supply equation:

$$ Q_S = 150(2) - 100 $$

Calculating this gives:

$$ Q_S = 300 - 100 = 200 $$

Thus, the equilibrium price is 2, and the equilibrium quantity is 200. This analysis demonstrates how to find equilibrium in a market by manipulating equations and isolating variables effectively.

To determine the equilibrium in a market, we analyze the intersection of the supply and demand curves on a graph. The equilibrium point occurs where the quantity demanded equals the quantity supplied. This can be visualized by plotting the equations for quantity demanded and quantity supplied on a graph.

For instance, consider the demand equation given by:

$$ Q_d = 400 - \frac{1}{2}P $$

and the supply equation:

$$ Q_s = P - 200 $$

To find equilibrium, we can select various price points and calculate the corresponding quantities. Starting with a price of 0, we find:

For demand: $$ Q_d = 400 - \frac{1}{2}(0) = 400 $$

For supply: $$ Q_s = 0 - 200 = -200 $$

Since negative supply is not meaningful, we move to a higher price, such as 400:

For demand: $$ Q_d = 400 - \frac{1}{2}(400) = 200 $$

For supply: $$ Q_s = 400 - 200 = 200 $$

At this price, both quantity demanded and quantity supplied equal 200, indicating that we have found an equilibrium point.

Next, we can check a higher price, such as 600:

For demand: $$ Q_d = 400 - \frac{1}{2}(600) = 100 $$

For supply: $$ Q_s = 600 - 200 = 400 $$

Now we have sufficient data points to plot the curves. The demand points at prices 0, 400, and 600 yield quantities of 400, 200, and 100, respectively. The supply points at prices 400 and 600 yield quantities of 200 and 400, respectively.

When plotted, the demand curve slopes downward, while the supply curve slopes upward. The intersection of these two curves occurs at the coordinates (400, 200), which represents the equilibrium price of 400 and an equilibrium quantity of 200. This graphical representation confirms our algebraic findings, illustrating the balance between supply and demand in the market.