Math Problem Statement

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.

Solution

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:

Qs=c+dPQ_s = c + dP

Where:

  • QsQ_s is the quantity supplied,
  • PP is the price,
  • cc and dd are constants.

We know:

  • At P=32P = 32, Qs=4040Q_s = 4040 kg.
  • If the price drops by $2 (P = 30)), the quantity supplied decreases by 55 kg, so Qs=3985Q_s = 3985 kg.

We can set up the equations: 4040=c+32d4040 = c + 32d 3985=c+30d3985 = c + 30d

Subtract the second equation from the first to eliminate cc and solve for dd:

40403985=(c+32d)(c+30d)4040 - 3985 = (c + 32d) - (c + 30d) 55=2d    d=27.555 = 2d \implies d = 27.5

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

4040=c+32×27.54040 = c + 32 \times 27.5 4040=c+880    c=31604040 = c + 880 \implies c = 3160

Thus, the supply curve equation is: Qs=3160+27.5PQ_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 PP' in the supply equation is:

P=P+1P' = P + 1

The new supply curve with the subsidy is: Qs=3160+27.5(P+1)=3160+27.5P+27.5=3187.5+27.5PQ_s = 3160 + 27.5(P + 1) = 3160 + 27.5P + 27.5 = 3187.5 + 27.5P

3. Calculate the Quantity Supplied at P=32P = 32 (with Subsidy):

Substitute P=32P = 32 into the new supply equation:

Qs=3187.5+27.5×32=3187.5+880=4067.5 kgQ_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 Qs=0Q_s = 0), set Qs=0Q_s = 0 in the original equation:

0=3160+27.5P    P=316027.5114.910 = 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=32P = 32, and we use the modified supply curve.

The producer surplus without the subsidy can be calculated as: Original Producer Surplus=12×(Qs at P=32)×(32(114.91))\text{Original Producer Surplus} = \frac{1}{2} \times (Q_s \text{ at } P = 32) \times (32 - (-114.91))

Substitute values: Original Producer Surplus=12×4040×146.91296,566.2\text{Original Producer Surplus} = \frac{1}{2} \times 4040 \times 146.91 \approx 296,566.2

Now, with the subsidy: New Producer Surplus=12×(Qs at P=32 with subsidy)×(32(114.91+1))\text{New Producer Surplus} = \frac{1}{2} \times (Q_s \text{ at } P = 32 \text{ with subsidy}) \times (32 - (-114.91 + 1)) New Producer Surplus=12×4067.5×147.91300,691.4\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.

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Math Problem Analysis

Mathematical Concepts

Supply and Demand
Linear Functions
Producer Surplus

Formulas

Linear supply curve equation Q_s = c + dP

Theorems

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Suitable Grade Level

Advanced Economics