In a small market, a linear supply curve represents the behavior of local wheat producers. It is observed that the producers are willing to supply 4040 kg of wheat when the price is at $32$32 per kilogram. However, if the price drops by $2$2 per kilogram, the quantity of wheat supplied decreases by 55 kg.

The government introduces a subsidy program, paying producers an additional $1$1 per kilogram sold. Calculate the new producer surplus (in $) at the market price of $32$32 per kilogram, factoring in the subsidy.

To calculate the new producer surplus with the subsidy included, we need to follow these steps:

### 1. Determine the Supply Curve:
The supply curve is typically linear and can be expressed as:

\[
Q_s = c + dP
\]

Where:
- $Q_s$ is the quantity supplied,
- $P$ is the price,
- $c$ and $d$ are constants.

We know:
- At $P = 32$, $Q_s = 4040$ kg.
- If the price drops by $2 (P = 30)\), the quantity supplied decreases by 55 kg, so $Q_s = 3985$ kg.

We can set up the equations:
\[
4040 = c + 32d
\]
\[
3985 = c + 30d
\]

Subtract the second equation from the first to eliminate $c$ and solve for $d$:

\[
4040 - 3985 = (c + 32d) - (c + 30d)
\]
\[
55 = 2d \implies d = 27.5
\]

Now, substitute $d = 27.5$ back into one of the original equations to solve for $c$:

\[
4040 = c + 32 \times 27.5
\]
\[
4040 = c + 880 \implies c = 3160
\]

Thus, the supply curve equation is:
\[
Q_s = 3160 + 27.5P
\]

### 2. Determine the New Supply Curve with the Subsidy:
With the subsidy, producers effectively receive an additional $1 per kilogram. This increases the price they receive by $1, so the new price $P'$ in the supply equation is:

\[
P' = P + 1
\]

The new supply curve with the subsidy is:
\[
Q_s = 3160 + 27.5(P + 1) = 3160 + 27.5P + 27.5 = 3187.5 + 27.5P
\]

### 3. Calculate the Quantity Supplied at $P = 32$ (with Subsidy):
Substitute $P = 32$ into the new supply equation:

\[
Q_s = 3187.5 + 27.5 \times 32 = 3187.5 + 880 = 4067.5 \text{ kg}
\]

### 4. Determine the Intercept of the Original Supply Curve:
To find the price intercept of the original supply curve (where $Q_s = 0$), set $Q_s = 0$ in the original equation:

\[
0 = 3160 + 27.5P \implies P = -\frac{3160}{27.5} \approx -114.91
\]

This represents the hypothetical price at which producers would supply zero wheat.

### 5. Calculate the New Producer Surplus:

Producer surplus is the area between the price line and the supply curve. With the subsidy, the relevant price line is at $P = 32$, and we use the modified supply curve.

The producer surplus without the subsidy can be calculated as:
\[
\text{Original Producer Surplus} = \frac{1}{2} \times (Q_s \text{ at } P = 32) \times (32 - (-114.91))
\]

Substitute values:
\[
\text{Original Producer Surplus} = \frac{1}{2} \times 4040 \times 146.91 \approx 296,566.2
\]

Now, with the subsidy:
\[
\text{New Producer Surplus} = \frac{1}{2} \times (Q_s \text{ at } P = 32 \text{ with subsidy}) \times (32 - (-114.91 + 1))
\]
\[
\text{New Producer Surplus} = \frac{1}{2} \times 4067.5 \times 147.91 \approx 300,691.4
\]

Thus, the new producer surplus at a market price of $32 per kilogram, factoring in the subsidy, is **$300,691.4**.

Would you like further details or have any questions about this calculation? Here are some related questions you might consider:

1. How would the producer surplus change if the subsidy increased to $2 per kilogram?
2. What is the impact of removing the subsidy on producer surplus?
3. How does the elasticity of supply affect the producer surplus?
4. Can a subsidy lead to overproduction, and what are the economic implications?
5. What would the supply curve look like if the government imposed a tax instead of a subsidy?

**Tip:** Subsidies can encourage production by effectively raising the price producers receive, leading to increased supply and higher producer surplus. Understanding these dynamics is crucial for analyzing the effects of government interventions.

Learn how to calculate the new producer surplus with a subsidy in a market analysis scenario. Understand the effects of subsidies on producer surplus using linear supply curve equations and market price dynamics. Suitable for advanced economics study.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation