To find the market equilibrium and calculate the price elasticity of demand and supply, let's follow these steps:
Part (a): Finding Market Equilibrium
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Set Demand equal to Supply: The demand equation is given as: \[ Q_d = 55 - P \] The supply equation is given as: \[ P = Q_s + 5 \] We can rearrange the supply equation to express \(Q_s\) as: \[ Q_s = P - 5 \]
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Equate the two: At equilibrium, \(Q_d = Q_s\): \[ 55 - P = P - 5 \]
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Solve for \(P\): Rearranging gives: \[ 55 + 5 = 2P \] \[ 60 = 2P \] \[ P = 30 \]
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Calculating the equilibrium quantity: Plugging the equilibrium price back into either the demand or supply equation to find \(Q\): Using the demand equation: \[ Q_d = 55 - 30 = 25 \]
Using the supply equation: \[ Q_s = 30 - 5 = 25 \]
Both equations confirm that the equilibrium quantity is \(Q = 25\).
Equilibrium Price: \(P = 30\)
Equilibrium Quantity: \(Q = 25\)
Part (b): Calculating Price Elasticity of Demand and Supply at the Equilibrium Point
To calculate the price elasticity of demand (\(E_d\)) and supply (\(E_s\)), we will use the following formulas:
\[ E_d = \frac{\Delta Q_d}{\Delta P} \cdot \frac{P}{Q_d} \] \[ E_s = \frac{\Delta Q_s}{\Delta P} \cdot \frac{P}{Q_s} \]
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Price elasticity of demand (\(E_d\)): From the demand equation, \(Q_d = 55 - P\), we find that: \[ \frac{\Delta Q_d}{\Delta P} = -1 \quad \text{(the slope of the demand curve)} \] At the equilibrium point: \[ P = 30, \quad Q_d = 25 \]
Plugging values: \[ E_d = -1 \cdot \frac{30}{25} = -1.2 \]
Interpretation: The price elasticity of demand is \(-1.2\), meaning that for a 1% increase in price, the quantity demanded decreases by approximately 1.2%.
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Price elasticity of supply (\(E_s\)): From the supply equation, we can rearrange it also to find the slope: \[ Q_s = P - 5 \Rightarrow \frac{\Delta Q_s}{\Delta P} = 1 \quad \text{(the slope of the supply curve)} \] At the equilibrium point: \[ P = 30, \quad Q_s = 25 \]
Plugging values: \[ E_s = 1 \cdot \frac{30}{25} = 1.2 \]
Interpretation: The price elasticity of supply is \(1.2\), indicating that for a 1% increase in price, the quantity supplied increases by approximately 1.2%.
Summary
- Equilibrium Price: \(30\)
- Equilibrium Quantity: \(25\)
- Price Elasticity of Demand: \(-1.2\) (elastic)
- Price Elasticity of Supply: \(1.2\) (elastic)