Question

Find the equilibrium solution for each of the following models: (a) \(Q_{d}=Q_{\text {s }}\) \(Q_{d}=3-p^{2}\) \(Q_{\mathrm{s}}=6 P-4\) (b) \(Q_{d}=Q_{s}\) \(Q_{d}=8-p^{2}\) \(Q_{3}=p^{2}-2\)

Short answer

(a) Equilibrium: \(p = 1, Q = 2\). (b) Equilibrium: \(p = \sqrt{5}, Q = 3\).

Step-by-step solution

Set the Quantity Equations Equal (a)

For the first model, set the demand equation equal to the supply equation to find the equilibrium price: \[ 3 - p^2 = 6p - 4 \]

Simplify and Solve for p (a)

Rearrange the equation from Step 1 into a standard quadratic form: \[ p^2 + 6p - 7 = 0 \]Now solve the quadratic equation using the quadratic formula, where \(a = 1\), \(b = 6\), and \(c = -7\):\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]\[ p = \frac{-6 \pm \sqrt{36 + 28}}{2} \]\[ p = \frac{-6 \pm \sqrt{64}}{2} \]\[ p = \frac{-6 \pm 8}{2} \]This gives two potential solutions for \(p\): \(p = 1\) and \(p = -7\). Only \(p = 1\) makes economic sense as a price.

Find Corresponding Quantity (a)

Once \(p = 1\) is found, substitute back into either the demand or supply equation:\[ Q_d = 3 - (1)^2 = 2 \]Thus, the equilibrium quantity \(Q = 2\).

Set the Quantity Equations Equal (b)

For the second model, again set the demand equation equal to the supply equation: \[ 8 - p^2 = p^2 - 2 \]

Simplify and Solve for p (b)

Rearrange to isolate \(p^2\):\[ 8 + 2 = p^2 + p^2 \]\[ 10 = 2p^2 \]Divide both sides by 2:\[ p^2 = 5 \]Taking the square root gives \(p = \sqrt{5}\) or \(p = -\sqrt{5}\). We consider only \(p = \sqrt{5}\) since the price must be positive.

Find Corresponding Quantity (b)

Substitute \(p = \sqrt{5}\) into either equation to find the equilibrium quantity:\[ Q_d = 8 - (\sqrt{5})^2 = 8 - 5 = 3 \]Thus, the equilibrium quantity \(Q = 3\).

Key concepts

Supply and Demand

The concepts of supply and demand are foundational to economics. They describe the quantity of goods consumers want to buy and the amount producers are willing to sell, respectively. Supply and demand are visually represented through graphs which typically show demand sloping downwards and supply sloping upwards from left to right.

When these two curves intersect, it signifies a balance or agreement in the market. This intersection is essential because it determines prices and quantities in different markets. Here's a quick look at what influences them:

• Demand: Influenced by needs, preferences, income levels, and the price of related goods.
• Supply: Affected by production costs, technology, and the number of sellers.

Understanding these principles allows economists and businesses to predict how changes can affect market prices and quantities. By studying supply and demand curves, they can make calculated decisions to enhance profitability and efficiency.

Equilibrium Price

Equilibrium price is a crucial concept where supply equals demand. It means the market operates efficiently, and there's no surplus or shortage of goods. The consumers' willingness to pay matches the sellers' willingness to sell. Equilibrium is balanced when demand's downward pressure and supply's upward push meet perfectly.

In real markets, equilibrium might shift due to several factors, such as changing consumer preferences or production costs. Noticeable effects of shifts include:

• Increase in Demand: Prices are likely to rise.
• Increase in Supply: Prices generally fall.

When adjusting to new equilibriums, markets naturally correct themselves over time. This flexibility absorbs shocks and maintains a stable economic environment. Identifying equilibrium prices helps stakeholders make informed choices about production volumes and pricing strategies.

Quadratic Equations

Quadratic equations are vital for modeling situations where variables interact in a nonlinear way, such as in finding equilibrium prices. These equations typically take the form:\[a x^2 + b x + c = 0\]They require solving for 'x' using methods like factoring, completing the square, or applying the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]These solutions can involve complex numbers or imaginary parts if the discriminant (\(b^2 - 4ac\)) is negative.

In economic models, particularly concerning supply and demand, quadratic equations solve the equilibrium condition where quantity demanded equals quantity supplied. Accurately solving these equations can:

• Predict price levels.
• Identify stable market conditions.

Mastery of quadratic equations expands one's toolkit in addressing various economic and mathematical challenges.

Economic Modeling

Economic modeling is a process that simplifies real-world economic situations into mathematical forms to predict future behaviors or outcomes. It helps economists and policymakers understand complex scenarios. Models vary from simple linear equations to intricate systems incorporating multiple variables and constraints.

Key characteristics of economic models include:

• Assumptions: Setting simplifying assumptions helps manage complex real-world systems.
• Variables: Core measurable factors, like price or quantity, reflect economic activities.

Models also help simulate different scenarios, observe outcomes, and make strategic decisions. For example, assessing equilibrium using supply and demand models determines resource allocation and pricing strategies in a competitive market. The skills developed through economic modeling are invaluable for addressing economic issues with analytical precision.