Economics 105                Berger     

                      Homework #3-Answers

 

                            Table 1

 

                                       TAX = $3    SUB = $3

   P     Qd(SR)      Qd(LR)      Qs(0)   Qs(TAX)     Qs(SUB)     

  $14      20           *         200     140        260   

   13      30           *         180     120        240   

   12      40           0         160     100        220   

   11      50          20         140      80        200   

   10      60          40         120      60        180       

    9      70          60         100      40        160        

    8      80          80          80      20        140        

    7      90         100          60       0        120        

    6     100         120          40       *        100

    5     110         140          20       *         80        

    4     120         160           0       *         60        

    3     130         180           *       *         40         

 

 1. Use the numbers in Table 1 to calculate the following:

 

a. Compute the Qs(TAX) when the initial tax burden of $3 is assigned to the producer.

 

The easiest method is to choose any Qs such as 100 which has a supply price of $9. Add $3 tax to that supply price for a total price of $12. Then the new supply curve Qs(TAX) has 100 units at the supply price of $12.

 

 

 

 

 

 

 

 

 

 

 

b. Compute the following:

 

(1) ED(SR) and ES(TAX) between $11 and $9.

 

              (70-50)/(70+50)      20/120

ED(SR) =  -  ─────────────────  =  ───────

                            ($9-$11)/($9+$11)     $2/$20

 

  = 5/3 = 1.67

 

                            (80-40)/(80+40)       40/120

ES(TAX) =  ───────────────────  =  ───────

                           ($11-$9)/($11+$9)      $2/$20

 

   = 10/3 = 3.33

 

     (2) ED(LR) and ES(TAX) between $10 and $9.

 

              (60-40)/(60+40)      20/100

 ED(LR) =  -  ─────────────────  =  ───────

                            ($9-$10)/($9+$10)     $1/$19

 

  = 19/5 = 3.8

 

 

                            (60-40)/(60+40)       20/100

ES(TAX) =  ───────────────────  =  ───────

                           ($10-$9)/($10+$9)      $1/$19

 

   = 19/5 = 3.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 2. Draw a graph of the SR demand curve and the two supply curves, being sure to label all relevant points with numbers (Note: You should not use every number in the table; instead you should label the numbers where the supply and demand curves touch the relevant axes, the equilibrium points, etc.).

                

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.   Draw a graph of the LR demand curve and the two supply curves, being sure to label all relevant points with numbers.

 

 

 4. Where is the unit elastic point on the SR and LR demand curves? Explain.

 

 

The unit elastic point can be found in either of two

ways: (1) find the maximum total revenue or (2) draw the demand curve touching both axes and divide these end points by two.          

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 5. Use the numbers in Table 1 to do the following problems concerning the final burden of a tax.

 

a. Compute the final burden of the tax for producers (PB) and consumers (CB) for the SR demand curve and indicate these areas on a graph. Explain what you find and compare your results to the elasticity computations in 1.b.(1).

CB(SR):  ($10-$8)60 = $120

 

PB(SR):  ($8-$7)60 = $60

 

PB(SR) + CB(SR) = $180 = Tax Revenue

 

A larger share of the burden of the tax (2/3) is borne by consumers while a smaller share (1/3) is borne by producers. That is, producers have shifted 2/3 of the burden of the tax to consumers. This is consistent with the elasticity computations which showed the ED = 1.67 and ES = 3.33; when ES > ED, consumers will bear a larger share of the burden of the tax (they have the more inelastic curve). 

 

 

 

 

 

 

 

b. Compute the final burden of the tax for producers and consumers for the LR demand curve and indicate these areas on a graph. Explain what you find and compare your results to the ones you calculated for the SR demand curve and to the elasticity computations in 1.b.ii.

 

CB(LR): ($9.50-$8)50 = $75

 

PB(LR): ($8-$6.50)50 = $75

 

PB(LR) + CB(LR) = $150

 

The final tax burden is shared equally in the long-run. This is consistent with the elasticity computations: ED(LR) = 3.4 = ES(TAX). The total tax revenue collected in the LR is smaller which implies that some consumers were able to substitute away from the taxed good in the LR. That also accounts for the more elastic demand curve. Substitution has allowed consumers to lower their tax burden by $45 ($120 - $75) and shift $15 ($75 - $60) of the tax burden back to producers.

 

 

 

 

 

 

 

 

 6. Using the numbers in Table 1, compute the Welfare Losses (WLs) for the SR demand curve and the LR demand curve and indicate these areas on the two graphs. Explain what you find.

 

WL(SR) = ½($10-$7)(80-60) = $30

WL(LR) = ½($9.50-$6.50)(80-50) = $45

 

This implies that the misallocation of resources (WL) that occurs when a tax is applied (driving resources into producing goods of a lower value in other parts of the economy) is worse in the LR than in the SR. The economic explanation for this arises from the possibility for substitution by consumers: in the LR, they can substitute more effectively for the taxed good than they can in the SR. This also implies that any tax will become less effective at raising revenue over the LR because of these improved substitution possibilities. Check: TaxRev(SR) > TaxRev(LR).

         

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 7. Repeat the above exercises for the case of a specific subsidy to producers (Note: This applies to 1.a., 2, 3, 5-omit the comparison to the elasticity calculation-and 6). Instead of the final burden of the tax for consumers and producers substitute the final gains from the subsidy for consumers (CG) and producers (PG).

 

1.a. See calculations above in Table 1.

 

2. Graph SR demand and supply curves with subsidy.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. Graph LR demand and two supply curves with subsidy.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.a. CG(SR) = Consumer Gains Short-Run = ΔCS(SR)

 

Two Methods

 

(1) Old CS(SR) = $320

                   New CS(SR) = ½($16-$6)100 = $500

                   ΔCS = $500-$320 = $180 = CG

 

     (2) CG(SR) = ($8-$6)80 + ½($8-$6)(100-80)

                          = $160 + $20 = $180

 

    PG(SR) = Producer Gains Short-Run = ΔPS(SR)

 

Two Methods

 

(1) Old PS(SR) = $160

    New PS(SR) = ½($9-$4)100 = $250

    ΔPS(SR) = $250-$160 = $90 = PG(SR)

 

     (2) PG(SR) = ($9-$8)80 + ½($9-$8)(100-80)

                          = $80 + $10 = $90

 

 

 

 

 

 

 

 

5.b. CG(LR) = Consumer Gain Long-Run

 

(1) Old CS(LR) = $160

                   New CS(LR) = ½($12-$6.50)110 = $302.50

                   ΔCS(LR) = $302.50-$160 = $142.50 = CG(LR)

 

     (2) CG(LR) = ($8-$6.50)80 + ½($8-$6.50)(110-80)

                          = $120 + $22.50 = $142.50

 

PG(LR) = Producer Gain Long-Run

 

(1) Old PS(LR) = $160

         New PS(LR) = ½($9.50-$4)110 = $302.50

                   ΔPS(LR) = $302.50 - $160 = $142.50 = PG

 

     (2) PG(LR) = ($9.50-$8)80 + ½($9.50-$8)(110-80)

                          = $120 + $22.50 = $142.50

 

 

 

 

 

 

 

 

 

 

 

 

6.  SUB(SR) = Subsidy Cost to Government in Short-Run

 

              (1) SUB(SR) = ($9-$6)100 = $300

 

                   WL(SR) = ½($9-$6)(100-80) = $30

 

                   Alternative Calculation of WL

 

                   WL(SR) = SUB(SR) - CG(SR) - PG(SR)

                          = $300 - $180 - $90 = $30

 

              (For CG(SR) and PG(SR) see 5.a.)

 

(2) SUB(LR) = ($9.50 - $6.50)110 = $330

 

                   WL(LR) = ½($9.50-$6.50)(110-80) = $45

 

                   Alternative Calculation of WL

 

                   WL(LR) = SUB(LR) - CG(LR) - PG(LR)

                          = $330 - $142.50 - $142.50 = $45

 

                   (For CG(LR) and PG(LR) see 5.b.)

 

 

 

 

 

 

Briefly, economic analysis reveals a welfare loss from subsidies. Such a loss occurs because resources are diverted from a higher valued use in other parts of the economy to this market where they have a lower valued use. This welfare loss and the accompanying subsidy are both larger in the long-run because consumers have an incentive to substitute the subsidized good for other non-subsidized goods.

        

Note also that the size of the subsidy increases in the Long-Run.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 8. Do the following problems using the concept of cross elasticity.

 

a. Graph and explain what happens in the legal and illegal markets for alcohol when the alcohol tax in the legal market is increased (assume the cross elasticity of demand is +5).

 

Since legal and illegal alcohol are substitutes, the increase in price of legal alcohol (because of the tax increase) causes a decrease in QD of legal alcohol. This, in turn, causes an increase in demand for illegal alcohol as those on the margin of switching move into the illegal market. With an increase in demand for illegal alcohol, there is an increase in the quantity demanded at each price. Hence an increase in the price of legal alcohol leads to an increase in the QD of illegal alcohol at each price (a shift).

 

 

 

 

 

 

 

 

 

 

 

 

                     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b. Graph and explain what happens in the automobile and gasoline markets when new regulations require the installation of air bags in all new cars (an increase in the cost of a new car). Explain what the cross elasticity of demand should be between these two markets.

 

ED(ab) should be negative since gasoline and automobiles are complements. That is, an increase in the cost of producing autos shifts the supply curve to the left, causing the price of autos to rise. As the Qd of autos decreases, the demand for gasoline will decrease (shift to the left). This means that the Qd of gasoline decreases at each price of gasoline. An increase in the price of autos therefore causes a decrease in the Qd of gasoline at each price (a shift).