Suppose the market for donuts is perfectly competitive and defined by the following:

Demand: P = 18-3Q

Supply: P = 2 + Q

At the competitive market equilibrium, price elasticity of supply is

which is

Suppose the government decides to discourage the consumption of donuts by imposing a tax of $4/donut.

The tax will generate $

in tax revenue and will decrease the price received by donut producers by $

Let's work through the problem step by step.

### 1. **Finding the Competitive Market Equilibrium**

The equilibrium in a perfectly competitive market occurs where the quantity demanded equals the quantity supplied. This means setting the demand equation equal to the supply equation:

\[ P_{\text{demand}} = P_{\text{supply}} \]

Given the equations:

- **Demand**: $ P = 18 - 3Q $
- **Supply**: $ P = 2 + Q $

Setting them equal:

\[ 18 - 3Q = 2 + Q \]

Solve for $ Q $:

\[ 18 - 2 = Q + 3Q \]

\[ 16 = 4Q \]

\[ Q_{\text{equilibrium}} = 4 \]

Now, plug $ Q = 4 $ back into either the demand or supply equation to find the equilibrium price $ P $:

\[ P = 18 - 3(4) = 18 - 12 = 6 \]

So, the **equilibrium price** is $ P_{\text{equilibrium}} = 6 $.

### 2. **Price Elasticity of Supply at Equilibrium**

Price elasticity of supply ($ E_s $) is calculated using the formula:

\[ E_s = \frac{P}{Q} \times \frac{dQ}{dP} \]

From the supply equation $ P = 2 + Q $, we can find the derivative $ \frac{dQ}{dP} $:

\[ \frac{dQ}{dP} = 1 \]

Now, substituting the equilibrium values $ P = 6 $ and $ Q = 4 $:

\[ E_s = \frac{6}{4} \times 1 = \frac{3}{2} = 1.5 \]

So, the **price elasticity of supply** at the equilibrium is **1.5**.

### 3. **Tax Impact on the Market**

The government imposes a tax of $4 per donut. This tax will shift the supply curve upward by $4. The new supply equation becomes:

\[ P_{\text{new supply}} = (2 + Q) + 4 = 6 + Q \]

To find the new equilibrium, set the new supply equation equal to the demand equation:

\[ 18 - 3Q = 6 + Q \]

Solve for $ Q $:

\[ 18 - 6 = Q + 3Q \]

\[ 12 = 4Q \]

\[ Q_{\text{new equilibrium}} = 3 \]

Now, substitute $ Q = 3 $ back into the new supply equation to find the price paid by consumers:

\[ P_{\text{new supply}} = 6 + 3 = 9 \]

The price producers receive after the tax is $ P_{\text{producers}} = 9 - 4 = 5 $.

### 4. **Tax Revenue**

Tax revenue is calculated as:

\[ \text{Tax Revenue} = \text{Tax per unit} \times \text{Quantity sold} \]

\[ \text{Tax Revenue} = 4 \times 3 = 12 \]

### 5. **Decrease in Price Received by Producers**

The initial price producers received was $6, and after the tax, they receive $5. So, the decrease in price received by producers is:

\[ \text{Decrease in Price} = 6 - 5 = 1 \]

### Summary:
- **Price Elasticity of Supply**: 1.5
- **Tax Revenue**: $12
- **Decrease in Price Received by Producers**: $1

Do you want details or have any questions? 

Here are 8 related questions:

1. How does the tax impact the overall market efficiency in a perfectly competitive market?
2. What would happen if the tax was imposed on consumers instead of producers?
3. Can you explain the concept of tax incidence and how it relates to this problem?
4. How would a subsidy instead of a tax affect the supply curve?
5. What if the demand was perfectly inelastic? How would that change the outcome?
6. How would you calculate the deadweight loss from the tax?
7. What is the significance of price elasticity of supply in determining the burden of the tax?
8. How would a larger or smaller tax affect the quantity of donuts sold?

**Tip**: In a perfectly competitive market, the burden of a tax is shared between consumers and producers depending on the relative elasticities of supply and demand.

Explore how price elasticity of supply impacts competitive markets with a detailed example of a tax on donuts. Learn about equilibrium, tax revenue calculation, and the impact on producers. Suitable for advanced college-level economics students.

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