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taxes and supply and demand

The supply curve for a certain industry is given by S(p) = pāˆ’2 and demand by D(p) = 30 āˆ’ 3p.
 
NEW PART TO QUESTION: What is the total tax revenue and the producer and consumer surplus under a $2 tax? What is the deadweight loss (DWL)?  
 
1. If the tax doubles from $2 to $4 per unit, by what factor does the DWL change? (It is possible to answer this question without recalculating the DWL, by geometric intuition regarding the DWL triangle.)
 
2. Suppose, before any tax was passed, the supplier believed the market would not change any time soon, and locked in labor contracts which commit irrevocably to the current (no-tax) quantity of supply. What would the new supply curve be? Draw it and write its equation.
 
3. With the new supply curve as in (8), who would bear a $2 per unit tax? What would the deadweight loss be?

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Andy C. | Math/Physics TutorMath/Physics Tutor
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1)
 
Formula for the deadweight loss is DWL = 0.5 * (P2 - P1) * (Q2-Q1) where
  P1 is the original price, P2 is the price after the tax is imposed, 
  Q1 is the original demand at the original price, Q2 is the new demand quantity after the tax.
 
 
The supply is zero when the price is $2. So no vendors are willing to sell the product for $2.
Likewise, the demand is zero when p is $10. So no customers are willing to pay $10 for the product.
 
Suppose the price of the product paid by the vendor is $N which includes the $2 tax. Without the tax, the
price of the product paid by the vendor is $N - 2. When the tax is doubled, the price of the product
paid by the vendor is $n-2 + 4 = $n + 2. So the price increases by $2.
 
The vendors are happy to raise the price by $2, so S(p+2) = (p+2)-2 = p. The supply will increase by 2 units.
The problem is that $2 could have been used for other things, perhaps more worthwhile. And there's more bad news!
The demand will decrease D(p+2) = 30 - 3(p +2) = 30 - 3p - 6 = 24 - 3p. The demand decreases by 6.
 
So in terms of price p, the DWL = 0.5 * (p+2 - p) * ( (24 - 3p) - (30 - 3p)) 
                                               = 0.5 * 2 * ( 24 - 3p - 30 + 3p)
 
                                              = 0.5 * 2 * (-6) = -6
 
2) DOUBLE WHAMMY! Now the poor supplier has to pay ALL the labor contracts AND twice the taxes.
    That will raise the price another $2.
    Replacing p with p+2 --> S(p+2) = p. But the taxes raise it another $2. So S(p) = p+2.
 
3) Here we go again.    Guess who eat the taxes?????  You and me, and the rest of the consumers. Anyways....
         The supply this time is P+2 - (p-2) = p +2 - p + 2 = 4 <--- the original price was p-2
         THe demand is going to get even worse because of the price increase: D(p+4) = 30 - 3(p+4) =
                                                                                                                              30 - 3p - 12
                                                                                                                              18 - 3p
 
         The new demand - the old demand this time is 18 - 3p - (30 - 3p) = 18- 3p - 30 + 3p = -12
 
         DWL = 0.5 * 4 * -12 = -24
 
 
 
SMH

Comments

I have found the supply and demand surplus in your other post.