Question
Suppose the market for demand for apple can be described by the following equation:
Demand: P = 10 - Q
Supply: P = Q - 4
a. What is the equilibrium price and quantity
b. What is the consumer surplus and producer surplus at the equilibrium price
c. Suppose government imposes a tax of K1 per unit to reduce apple consumptions die to health concerns and raise government revenues:
i. What will be the new equilibrium quantity be?
ii. What price will the buyer pay? What amount per unit will the seller receive
iii. What is the new producer and consumer surplus?
iv. What is the government revenue ?
What is the dead weight loss?
Demand: P = 10 - Q
Supply: P = Q - 4
a. What is the equilibrium price and quantity
b. What is the consumer surplus and producer surplus at the equilibrium price
c. Suppose government imposes a tax of K1 per unit to reduce apple consumptions die to health concerns and raise government revenues:
i. What will be the new equilibrium quantity be?
ii. What price will the buyer pay? What amount per unit will the seller receive
iii. What is the new producer and consumer surplus?
iv. What is the government revenue ?
What is the dead weight loss?
Answers
GPT-4o mini
To solve the problem, we will follow each part step by step.
### a. Equilibrium Price and Quantity
To find the equilibrium price and quantity, we set the demand and supply equations equal to each other.
1. Demand: \( P = 10 - Q \)
2. Supply: \( P = Q - 4 \)
Setting them equal:
\[
10 - Q = Q - 4
\]
Rearranging gives:
\[
10 + 4 = Q + Q
\]
\[
14 = 2Q \quad \Rightarrow \quad Q = 7
\]
Now substituting \( Q = 7 \) back into either the demand or supply equation to find \( P \):
Using the demand equation:
\[
P = 10 - 7 = 3
\]
Thus, the equilibrium price is **P = 3** and the equilibrium quantity is **Q = 7**.
### b. Consumer Surplus and Producer Surplus
1. **Consumer Surplus (CS)**: This is the area above the price and below the demand curve up to the equilibrium quantity.
- The maximum price consumers are willing to pay when \( Q = 0 \) is \( P = 10 \).
- The equilibrium price is \( P = 3 \).
- CS = (1/2) * base * height = (1/2) * Q * (Max Price - Equilibrium Price)
- \( CS = \frac{1}{2} \times 7 \times (10 - 3) = \frac{1}{2} \times 7 \times 7 = 24.5 \)
2. **Producer Surplus (PS)**: This is the area below the price and above the supply curve up to the equilibrium quantity.
- The minimum price producers are willing to accept when \( Q = 0 \) is \( P = -4 \).
- The equilibrium price is \( P = 3 \).
- PS = (1/2) * Q * (Equilibrium Price - Min Price)
- \( PS = \frac{1}{2} \times 7 \times (3 - (-4)) = \frac{1}{2} \times 7 \times 7 = 24.5 \)
### c. Impact of a Government Tax of K1
The tax will affect the supply curve, effectively shifting it up by K1. The new supply equation after tax will be:
\[
P = Q - 4 + 1 \quad \Rightarrow \quad P = Q - 3
\]
#### i. New Equilibrium Quantity
Setting the new supply equation equal to the demand equation:
\[
10 - Q = Q - 3
\]
Rearranging gives:
\[
10 + 3 = 2Q \quad \Rightarrow \quad 13 = 2Q \quad \Rightarrow \quad Q = 6.5
\]
#### ii. Price Paid by Buyer and Received by Seller
**Price Paid by Buyer (Pb)**: Substitute the new equilibrium quantity into the demand equation:
\[
P_b = 10 - 6.5 = 3.5
\]
**Price Received by Seller (Ps)**:
\[
P_s = P_b - K1 = 3.5 - 1 = 2.5
\]
#### iii. New Producer and Consumer Surplus
1. **New Consumer Surplus (CS)**:
\( CS = \frac{1}{2} \times 6.5 \times (10 - 3.5) = \frac{1}{2} \times 6.5 \times 6.5 = 21.125 \)
2. **New Producer Surplus (PS)**:
\( PS = \frac{1}{2} \times 6.5 \times (2.5 - (-4)) = \frac{1}{2} \times 6.5 \times 6.5 = 21.125 \)
#### iv. Government Revenue
Government revenue from the tax can be calculated as:
\[
\text{Revenue} = \text{Tax} \times \text{Quantity Sold} = 1 \times 6.5 = K6.5
\]
### Deadweight Loss
The deadweight loss (DWL) is the loss of economic efficiency that occurs when the equilibrium quantity is not achieved. It can be calculated as the area of the triangles formed by the decrease in quantity due to the tax.
DWL = \( \frac{1}{2} \times \text{Tax} \times \text{Change in Quantity} \)
The change in quantity is \( 7 - 6.5 = 0.5 \).
Thus, DWL = \( \frac{1}{2} \times 1 \times 0.5 = 0.25 \).
### Summary of Results
- **Equilibrium Price**: 3
- **Equilibrium Quantity**: 7
- **Consumer Surplus**: 24.5
- **Producer Surplus**: 24.5
- **New Equilibrium Quantity after Tax**: 6.5
- **Price Paid by Buyer**: 3.5
- **Price Received by Seller**: 2.5
- **New Consumer Surplus**: 21.125
- **New Producer Surplus**: 21.125
- **Government Revenue**: K6.5
- **Deadweight Loss**: 0.25
### a. Equilibrium Price and Quantity
To find the equilibrium price and quantity, we set the demand and supply equations equal to each other.
1. Demand: \( P = 10 - Q \)
2. Supply: \( P = Q - 4 \)
Setting them equal:
\[
10 - Q = Q - 4
\]
Rearranging gives:
\[
10 + 4 = Q + Q
\]
\[
14 = 2Q \quad \Rightarrow \quad Q = 7
\]
Now substituting \( Q = 7 \) back into either the demand or supply equation to find \( P \):
Using the demand equation:
\[
P = 10 - 7 = 3
\]
Thus, the equilibrium price is **P = 3** and the equilibrium quantity is **Q = 7**.
### b. Consumer Surplus and Producer Surplus
1. **Consumer Surplus (CS)**: This is the area above the price and below the demand curve up to the equilibrium quantity.
- The maximum price consumers are willing to pay when \( Q = 0 \) is \( P = 10 \).
- The equilibrium price is \( P = 3 \).
- CS = (1/2) * base * height = (1/2) * Q * (Max Price - Equilibrium Price)
- \( CS = \frac{1}{2} \times 7 \times (10 - 3) = \frac{1}{2} \times 7 \times 7 = 24.5 \)
2. **Producer Surplus (PS)**: This is the area below the price and above the supply curve up to the equilibrium quantity.
- The minimum price producers are willing to accept when \( Q = 0 \) is \( P = -4 \).
- The equilibrium price is \( P = 3 \).
- PS = (1/2) * Q * (Equilibrium Price - Min Price)
- \( PS = \frac{1}{2} \times 7 \times (3 - (-4)) = \frac{1}{2} \times 7 \times 7 = 24.5 \)
### c. Impact of a Government Tax of K1
The tax will affect the supply curve, effectively shifting it up by K1. The new supply equation after tax will be:
\[
P = Q - 4 + 1 \quad \Rightarrow \quad P = Q - 3
\]
#### i. New Equilibrium Quantity
Setting the new supply equation equal to the demand equation:
\[
10 - Q = Q - 3
\]
Rearranging gives:
\[
10 + 3 = 2Q \quad \Rightarrow \quad 13 = 2Q \quad \Rightarrow \quad Q = 6.5
\]
#### ii. Price Paid by Buyer and Received by Seller
**Price Paid by Buyer (Pb)**: Substitute the new equilibrium quantity into the demand equation:
\[
P_b = 10 - 6.5 = 3.5
\]
**Price Received by Seller (Ps)**:
\[
P_s = P_b - K1 = 3.5 - 1 = 2.5
\]
#### iii. New Producer and Consumer Surplus
1. **New Consumer Surplus (CS)**:
\( CS = \frac{1}{2} \times 6.5 \times (10 - 3.5) = \frac{1}{2} \times 6.5 \times 6.5 = 21.125 \)
2. **New Producer Surplus (PS)**:
\( PS = \frac{1}{2} \times 6.5 \times (2.5 - (-4)) = \frac{1}{2} \times 6.5 \times 6.5 = 21.125 \)
#### iv. Government Revenue
Government revenue from the tax can be calculated as:
\[
\text{Revenue} = \text{Tax} \times \text{Quantity Sold} = 1 \times 6.5 = K6.5
\]
### Deadweight Loss
The deadweight loss (DWL) is the loss of economic efficiency that occurs when the equilibrium quantity is not achieved. It can be calculated as the area of the triangles formed by the decrease in quantity due to the tax.
DWL = \( \frac{1}{2} \times \text{Tax} \times \text{Change in Quantity} \)
The change in quantity is \( 7 - 6.5 = 0.5 \).
Thus, DWL = \( \frac{1}{2} \times 1 \times 0.5 = 0.25 \).
### Summary of Results
- **Equilibrium Price**: 3
- **Equilibrium Quantity**: 7
- **Consumer Surplus**: 24.5
- **Producer Surplus**: 24.5
- **New Equilibrium Quantity after Tax**: 6.5
- **Price Paid by Buyer**: 3.5
- **Price Received by Seller**: 2.5
- **New Consumer Surplus**: 21.125
- **New Producer Surplus**: 21.125
- **Government Revenue**: K6.5
- **Deadweight Loss**: 0.25