Question

Suppose that demand is given by the equation \(Q^{D}=500-50 P,\) where \(Q^{D}\) is quantity demanded, and \(P\) is the price of the good. Supply is described by the equation \(Q^{S}=50+25 P,\) where \(Q^{S}\) is quantity supplied. What is the equilibrium price and quantity? (See Appendix.)

Short answer

The equilibrium price (P_{e}) is 6 and the equilibrium quantity (Q_{e}) is 200.

Step-by-step solution

Set the Demand and Supply Equal

Set the demand equation equal to the supply equation. This gives: \(500 - 50P = 50 + 25P\).

Simplify the Equation

Simplify the equation to find the equilibrium price. This involves collecting like terms: \( 500 - 50P - 50 - 25P = 0\), which becomes \(450 - 75P = 0\).

Solve for Equilibrium Price

Solve for P by isolating it on one side of the equation. This results in: \( 75P = 450\), and then divide each side by 75 so that \(P = 450 / 75\). This simplifies to give \(P = 6\). This is the equilibrium price \(P_{e}\).

Find the Equilibrium Quantity

Substitute the equilibrium price into either the demand or supply equation to find the equilibrium quantity. Using the supply equation for instance, we get: \(Q^{S} = 50 + 25(6)\). Which simplifies to: \(Q^{S} = 200\). This is the equilibrium quantity \(Q_{e}\).

Key concepts

Demand and Supply Equations

Understanding how market dynamics work is pivotal for grasping economic concepts. Demand and supply equations are the mathematical representations of the behavior of buyers (demand) and sellers (supply) in the market. For example, in the given exercise, the demand equation is represented as (Q^{D}=500-50 P), where (Q^{D}) is the quantity demanded, and (P) is the price of the good. The -50 indicates that as the price increases, the quantity demanded decreases, following the law of demand.

Similarly, the supply equation (Q^{S}=50+25 P), where (Q^{S}) is quantity supplied, showcases that as the price increases, the quantity supplied increases, which complies with the law of supply. These equations are essential because they allow us to find the equilibrium price and quantity, where the amount that consumers are willing to buy equals the amount that producers are willing to sell.

Solving Equilibrium

Solving for equilibrium involves finding the price and quantity where the supply and demand in the market are balanced. This means no excess demand or excess supply exists. In our exercise, to find the equilibrium, we equate the demand equation to the supply equation and solve for the price ((P)). Doing this, we find that (500 - 50P = 50 + 25P) leads to an equilibrium price of (P = 6).

• Subtract (50) and add (50P) to both sides of the initial equilibrium equation.
• Simplify to get (450 - 75P = 0), and then isolate (P).
• You divide (450) by (75) to get the equilibrium price (P = 6).

Once we have the price, we substitute it back into either the demand or supply equation to obtain the equilibrium quantity, which we discovered to be (200) units. This process is crucial for both theoretical and practical applications, as it informs us about the market price and quantity that we can expect to see assuming there are no other external influences on the market.

Economic Equilibrium

Economic equilibrium is a state where economic forces such as supply and demand are balanced. In terms of the exercise provided, the equilibrium price is the price at which the quantity of goods supplied is equal to the quantity of goods demanded — this is at (P = 6). The economic equilibrium quantity is (200) units. These figures represent a stable state of the economy where there is no incentive for price to change because the market's needs are met.

It is important to understand that in the real world, multiple factors such as government policies, technological changes, or external shocks can cause shifts in the demand or supply curve. This, in turn, leads to a new equilibrium. This concept helps explain how markets self-regulate and find balance over time and is fundamental in understanding economic policies and market strategies.