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To address the various parts of the question using the given demand and supply functions [tex]\( Q_d = 12 - P \)[/tex] and [tex]\( Q_s = 8.4 + 0.5P \)[/tex], let's go through each part step-by-step.
### i. Calculate the Market Clearing Price and Quantity
The market clearing price (equilibrium price) is determined where the quantity demanded equals the quantity supplied ([tex]\( Q_d = Q_s \)[/tex]).
Set the demand function equal to the supply function:
[tex]\[ 12 - P = 8.4 + 0.5P \][/tex]
Solving for [tex]\( P \)[/tex]:
[tex]\[ 12 - 8.4 = 0.5P + P \][/tex]
[tex]\[ 3.6 = 1.5P \][/tex]
[tex]\[ P = \frac{3.6}{1.5} \][/tex]
[tex]\[ P = 2.4 \][/tex]
This is the equilibrium price. Plug it back into either the demand or supply function to find the equilibrium quantity.
Using the demand function [tex]\( Q_d = 12 - P \)[/tex]:
[tex]\[ Q_d = 12 - 2.4 \][/tex]
[tex]\[ Q_d = 9.6 \][/tex]
Hence, the market clearing price is [tex]\( P = 2.4 \)[/tex] and the market clearing quantity is [tex]\( Q = 9.6 \)[/tex].
### ii. Determine Surplus or Shortage at Price Levels of 2 Birr and 3 Birr
At [tex]\( P = 2 \)[/tex]:
- Quantity demanded: [tex]\( Q_d = 12 - 2 = 10 \)[/tex]
- Quantity supplied: [tex]\( Q_s = 8.4 + 0.5(2) = 9.4 \)[/tex]
Surplus/Shortage:
[tex]\[ Q_s - Q_d = 9.4 - 10 = -0.6 \][/tex]
Since [tex]\( Q_d > Q_s \)[/tex], there is a shortage of 0.6 units at the price of 2 birr.
At [tex]\( P = 3 \)[/tex]:
- Quantity demanded: [tex]\( Q_d = 12 - 3 = 9 \)[/tex]
- Quantity supplied: [tex]\( Q_s = 8.4 + 0.5(3) = 9.9 \)[/tex]
Surplus/Shortage:
[tex]\[ Q_s - Q_d = 9.9 - 9 = 0.9 \][/tex]
Since [tex]\( Q_s > Q_d \)[/tex], there is a surplus of 0.9 units at the price of 3 birr.
### iii. Compute Price Elasticity of Demand and Supply at the Equilibrium Level
The price elasticity of demand (PED) and price elasticity of supply (PES) at the equilibrium level can be calculated using the equilibrium price ([tex]\( P = 2.4 \)[/tex]) and quantity ([tex]\( Q = 9.6 \)[/tex]).
For demand elasticity:
[tex]\[ \text{PED} = \left( \frac{dQ_d}{dP} \right) \left( \frac{P}{Q_d} \right) \][/tex]
Given [tex]\( Q_d = 12 - P \)[/tex]:
[tex]\[ \frac{dQ_d}{dP} = -1 \][/tex]
Therefore,
[tex]\[ \text{PED} = -1 \times \left( \frac{2.4}{9.6} \right) = -0.25 \][/tex]
For supply elasticity:
[tex]\[ \text{PES} = \left( \frac{dQ_s}{dP} \right) \left( \frac{P}{Q_s} \right) \][/tex]
Given [tex]\( Q_s = 8.4 + 0.5P \)[/tex]:
[tex]\[ \frac{dQ_s}{dP} = 0.5 \][/tex]
Therefore,
[tex]\[ \text{PES} = 0.5 \times \left( \frac{2.4}{9.6} \right) = 0.125 \][/tex]
### iv. Determine the Nature and Interpret the Elasticity of Demand and Supply
The nature of the price elasticity of demand and supply can be deduced based on the computed elasticities:
- Elasticity of Demand (PED): [tex]\( -0.25 \)[/tex]
Since [tex]\( | -0.25 | < 1 \)[/tex], the demand is inelastic. This means that the quantity demanded is relatively unresponsive to price changes around the equilibrium.
- Elasticity of Supply (PES): [tex]\( 0.125 \)[/tex]
Since [tex]\( 0.125 < 1 \)[/tex], the supply is inelastic. This indicates that the quantity supplied is relatively unresponsive to price changes around the equilibrium.
#### Interpretation:
- Inelastic Demand: Consumers are not very sensitive to price changes. For example, a price increase will result in a proportionally smaller decrease in the quantity demanded.
- Inelastic Supply: Producers are not very sensitive to price changes. For instance, a price increase will result in a proportionally smaller increase in the quantity supplied.
Thus, both the demand and supply for this market are inelastic at the equilibrium price and quantity.
### i. Calculate the Market Clearing Price and Quantity
The market clearing price (equilibrium price) is determined where the quantity demanded equals the quantity supplied ([tex]\( Q_d = Q_s \)[/tex]).
Set the demand function equal to the supply function:
[tex]\[ 12 - P = 8.4 + 0.5P \][/tex]
Solving for [tex]\( P \)[/tex]:
[tex]\[ 12 - 8.4 = 0.5P + P \][/tex]
[tex]\[ 3.6 = 1.5P \][/tex]
[tex]\[ P = \frac{3.6}{1.5} \][/tex]
[tex]\[ P = 2.4 \][/tex]
This is the equilibrium price. Plug it back into either the demand or supply function to find the equilibrium quantity.
Using the demand function [tex]\( Q_d = 12 - P \)[/tex]:
[tex]\[ Q_d = 12 - 2.4 \][/tex]
[tex]\[ Q_d = 9.6 \][/tex]
Hence, the market clearing price is [tex]\( P = 2.4 \)[/tex] and the market clearing quantity is [tex]\( Q = 9.6 \)[/tex].
### ii. Determine Surplus or Shortage at Price Levels of 2 Birr and 3 Birr
At [tex]\( P = 2 \)[/tex]:
- Quantity demanded: [tex]\( Q_d = 12 - 2 = 10 \)[/tex]
- Quantity supplied: [tex]\( Q_s = 8.4 + 0.5(2) = 9.4 \)[/tex]
Surplus/Shortage:
[tex]\[ Q_s - Q_d = 9.4 - 10 = -0.6 \][/tex]
Since [tex]\( Q_d > Q_s \)[/tex], there is a shortage of 0.6 units at the price of 2 birr.
At [tex]\( P = 3 \)[/tex]:
- Quantity demanded: [tex]\( Q_d = 12 - 3 = 9 \)[/tex]
- Quantity supplied: [tex]\( Q_s = 8.4 + 0.5(3) = 9.9 \)[/tex]
Surplus/Shortage:
[tex]\[ Q_s - Q_d = 9.9 - 9 = 0.9 \][/tex]
Since [tex]\( Q_s > Q_d \)[/tex], there is a surplus of 0.9 units at the price of 3 birr.
### iii. Compute Price Elasticity of Demand and Supply at the Equilibrium Level
The price elasticity of demand (PED) and price elasticity of supply (PES) at the equilibrium level can be calculated using the equilibrium price ([tex]\( P = 2.4 \)[/tex]) and quantity ([tex]\( Q = 9.6 \)[/tex]).
For demand elasticity:
[tex]\[ \text{PED} = \left( \frac{dQ_d}{dP} \right) \left( \frac{P}{Q_d} \right) \][/tex]
Given [tex]\( Q_d = 12 - P \)[/tex]:
[tex]\[ \frac{dQ_d}{dP} = -1 \][/tex]
Therefore,
[tex]\[ \text{PED} = -1 \times \left( \frac{2.4}{9.6} \right) = -0.25 \][/tex]
For supply elasticity:
[tex]\[ \text{PES} = \left( \frac{dQ_s}{dP} \right) \left( \frac{P}{Q_s} \right) \][/tex]
Given [tex]\( Q_s = 8.4 + 0.5P \)[/tex]:
[tex]\[ \frac{dQ_s}{dP} = 0.5 \][/tex]
Therefore,
[tex]\[ \text{PES} = 0.5 \times \left( \frac{2.4}{9.6} \right) = 0.125 \][/tex]
### iv. Determine the Nature and Interpret the Elasticity of Demand and Supply
The nature of the price elasticity of demand and supply can be deduced based on the computed elasticities:
- Elasticity of Demand (PED): [tex]\( -0.25 \)[/tex]
Since [tex]\( | -0.25 | < 1 \)[/tex], the demand is inelastic. This means that the quantity demanded is relatively unresponsive to price changes around the equilibrium.
- Elasticity of Supply (PES): [tex]\( 0.125 \)[/tex]
Since [tex]\( 0.125 < 1 \)[/tex], the supply is inelastic. This indicates that the quantity supplied is relatively unresponsive to price changes around the equilibrium.
#### Interpretation:
- Inelastic Demand: Consumers are not very sensitive to price changes. For example, a price increase will result in a proportionally smaller decrease in the quantity demanded.
- Inelastic Supply: Producers are not very sensitive to price changes. For instance, a price increase will result in a proportionally smaller increase in the quantity supplied.
Thus, both the demand and supply for this market are inelastic at the equilibrium price and quantity.
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