Question

TARIFFS PARTIAL EQUILIBRIUM DUE: Suppose we have the following demand and supply functions (taken from Assignment #2). HOME FOREIGN Demand P100-2Q SupplyPQ DemandP- 200 2Q SupplyP-4Q 1:Two-country model with IMPORT TARIFFS: use the functions above (1 point) a) Calculate the autarchy prices in each country. Who imports? Exports? (1 point) b) (2 point) c) (1 points)d) Suppose the importer imposes an import tax of $2 per unit. Calculate the new equilibrium world price. What is Home's and Foreign's domestic prices? Find the change in Consumer and Producer Surplus in each country. Recall that import tariffs need to be included in social surplus (EG: SS-CS + PS revenue) Does the importer gain from the tariff. Does the exporter? Show and explain fully Give the intuitiorn (5 points)e) Find the importer's optimal tariff. Show your work. 2:Two-country model with EXPORT TARIFFS: use the functions above (2 points)a) (2 points)b) (1 points)c) (5 points)d) Suppose the exporter imposes an export tax of $2 per unit. Calculate the new equilibrium world price. What are the tariff-ridden domestic prices? Find the change in Consumer and Producer Surplus in each country. Recall that export tariffs need to be included in social surplus (EG: SS- CS+PS +revenue) Does the importer gain from the tariff. Does the exporter? Show and explain fully Contrast with the importer's tariff. Find the exporter's optimal tariff. Show your work. 3:Two-country model with import and export tariffs: use the functions above. Suppose the exporter imposes an export tax of $2 per unit and the importer imposes an import tax of $2 per unit. b) d) (3 points) Calculate the new equilibrium world price and domestic prices Does the importer gain from the tariff war? Does the exporter? Show and explain full (7 points)

Answer 1

ANSWER 1
Given :

HOME –
Demand : P = 100 – ½ Q               Supply : P = ½ Q
or              Q = 200 – 2 P                               Q = 2 P

FOREIGN –
Demand : P = 200 – 2 Q                Supply : P = ¼ Q
or              Q = 100 – ½ P                              Q = 4 P

1. a. Autarchy price refers to the market clearing price in a closed economy, that is, the price at which domestic demand equals domestic supply, for each country.
HOME –

     Demand = Supply

⇒ 200 – 2 P = 2 P

⇒ 200 = 4 P

⇒ $50 = P

FOREIGN –

     Demand = Supply

⇒ 100 – ½ P = 4 P

⇒ 100 = 4 P + ½ P

⇒ 100 = 9/2 P

⇒ 200/9 = P
⇒ $22.22 = P

Given that there is no transportation cost and no trade restriction,
if P_H = Prices in Home Country & P_F = Prices in Foreign Country

Then, when P_H > P_F ⇒ Home country imports until P_H = P_F.
           when P_H < P_F ⇒ Home country exports until P_H = P_F.

Here, P_H = $50 & P_F = $22.22,

⇒ P_H > P_F ⇒ Home country imports and Foreign country exports until P_H = P_F.

1. b. If the importer (Home) imposes a tax of t = $2/unit, the cost for the exporter (Foreign) increases. Now, exporter will not be willing to sell overseas unless the price difference between the two markets is at least $2/unit.

HOME - Demand: P = 100 - 1/2 Q or Q = 200 - 2 P Supply: P = 1/2 Q or Q = 200 - 2 P FOREIGN - Demand: P = 200 - 2 Q or Q = 100 - 1/2 P Supply: P = 1/4 Q Q = 4 P Autarchy price refers to the market clearing price in a closed economy, that is, the price at which domestic demand equals domestic supply, for each country. HOME - Demand = Supply 200 - 2 P = 2 P 200 = 4 P $50 = P FOREIGN - Demand = Supply 100 - 1/2 P = 4 P 100 = 4 P + 1/2 P 100 = 9/2 P 200/9 = P $22.22 = P Given that there is no transportation cost and no trade restriction, if P_H = Prices in Home Country & P_F = Prices in Foreign Country Then, when P_H > P_F Home country imports until P_H = P_F. when P_H < P_F

The above figure depicts this effect. In the figure, MD denotes the Home Demand Import Curve, which is the excess of how much Home consumers demand over what Home producers supply, that is, Home Demand - Home Supply. On the other hand, XS denotes the Foreign Export Supply Curve, which is the excess of how much Foreign producers supply over what Foreign consumers demand, that is, Foreign Supply - Foreign Demand.

In the absence of tax, price would be equalised to P_W (henceforth referred to as Price in the world market) in both Home and Foreign, as shown by point 1 in the middle panel.

With the import tax being imposed in, the exporters will not trade unless P_H exceeds P_F by at least t. If no exports are coming in, an excess demand will be created in Home and an excess supply will be created in Foreign, which leads to a price wedge between Home and Foreign prices. The excess demand pressure increases the prices in Home country from P_H to P_T and the excess supply decreases the prices in Foreign country from P_F to P_T*. The volume of trade declines from Q_W ( free trade level of trade ) to Q_T ( volume of trade with tax ).

In the figure, D₂ – S₂ = Q_T.
                      D₁ – S₁ = Q_W.


The intersection of the world demand and world supply gives the equilibrium world price, P_W. At equilibrium level,
                                  Imports by Home = Exports by Foreign
⇒      Home Demand – Home Supply = Foreign Supply – Foreign Demand
⇒ Home Demand + Foreign Demand = Foreign Supply + Home Supply
⇒                                 World Demand = World Supply

Hence, we get,
(200 – 2 P_W) + (100 – ½ P_W) = 2P_W + 4P_W
                        300 – 5/2 P_W = 6P_W
                              $600/17 = P_W
                            or $35.29 = P_W

Now, let Q_W = Equilibrium level of Quantity traded in World Market,
Therefore, in terms of quantity,
(100 – ½ Q_W) + (200 – 2Q_W) = ¼ Q_W + ½ Q_W
                         300 – 5/2 Q_W = ¾ Q_W
                                          300 = 13/4Q_W
                                  1200/13 = Q_W
                                   or 92.31 = Q_W

At Q_W, Home Import Demand = Foreign Export Supply, that is, MD = XS
when P_T – P_T* = T, that is, P_T – P_T* = $2.

Home Import Demand = Home Demand – Home Supply
                                  MD = 200 – 2 P_T – 2 P_T
                                          = 200 – 4 P_T
                             or, MD = 100 – ½ Q_T – ½ Q_T
                                          = 100 – Q_T

Foreign Export Supply = Foreign Supply – Foreign Demand
                                   XS = 4P – 100 + ½ P_T*
                                        = 9/2 P_T* – 100

At Q_W, Home Import Demand = Foreign Export Supply, that is, MD = XS
when P_T – P_T* = T, that is, P_T – P_T* = $2.
⇒ P_T = $2 + P_T*

⇒              200 – 4 P_T = 9/2 P_T* – 100
⇒ 200 – 4 (2 + P_T*) = 9/2 P_T* – 100
⇒     200 – 8 – 4 P_T* = 9/2 P_T* – 100
⇒               192 +100 = 9/2 P_T* + 4 P_T*
⇒                         292 = 17/2 P_T*
⇒                 $584/17 = P_T*
                   or $34.35 = P_T*

As we know, P_T = $2 + P_T*
⇒ P_T = $2 + $584/17
         = $618/17 or $36.35

1. c. As seen in part (b), import tax raises the prices in Home from P_H to P_T and lowers the prices in Foreign from P_F to P_T*.

HOME
As shown in the diagram below, the demand in Home decreases from D₁ to D₂, and the supply increases from S₁ to S₂.




D₁ = 200 – 2P_W = 200 – 2(600/17) = 2200/17
      = 129.41.
D₂ = 200 – 2 P_T = 200 – (618/17) = 2164/17
      = 127.29.
S₁ = 2 P_W = 2 (600/17) = 1200/17
     = 70.58.
S₂ = 2 P_T = 2 (618/17) = 1236/17
     = 76.98.

Now,
Area (a + b) = (S₂ – 0) (P_T – P_W) = 1236/17 (618/17 – 600/17) = 22248/289
                      = 76.98.
        Area b = ½ ( S₂ – S₁) (P_T – P_W) = ½ (1236/17 – 1200/17) (618/17 – 600/17) = 324/298
                     = 1.12.
        Area c = (D₂ - S₂) (P_T – P_W) = (2164/17 – 1236/17) (618/17 – 600/17) = 16704/289
                     = 57.79.
        Area d = ½ (D₁ – D₂) (P_T – P_W) = ½ (2200/17 – 2164/17) (618/17 – 600/17) = 324/289
                      = 1.12
Area (c + e) = tax rate x (D₂ - S₂) = 2 x (2164/17 – 1236/17) = 1856/17
                       = 109.18

1. Consumer Surplus : Consumer surplus is given by the area below demand curve and above price line. Hence, increase in price lowers the consumer surplus by an area of a+b+c+d, as shown in the figure.

Change in Consumer Surplus = Area (a + b + c + d)
                                                    = 22248/289 + 16704/289 + 324/289
                                                    = 39276/289
                                                    = 135.9

2. Producer Surplus : Producer Surplus is the area below price line and above the supply curve. Hence, increase in prices leads to an increase in producer surplus by an amount of area a, as shown in the figure.


Change in Producer Surplus = Area (a + b) – Area b
                                                  = 22248/289 – 324/289
                                                  = 21924/289
                                                  = 75.86

3. Revenue : Home Government receives revenue when an import tax is imposed, the value of which is given by tax rate times the import volume.

Here, Import Volume = D₂ – S₂
Revenue = tax rate x Import Volume = Area (c + e) = 1856/17 = 109.18.

Net Cost of Import Tax = Consumer Loss – Producer Gain – Government Revenue
                                             = (a + b +c + d) – a – (c + e)
                                             = b + d – e  

Thus,
Total Change in Social Surplus = e – (b + d)
                                                      = 1856/17 – 324/289 – 324/289
                                                      = 30904/289
                                                      = 106.93
FOREIGN
As shown in the diagram below, the demand in Foreign increases from D₁* to D₂*, and the supply decreases from S₁* to S₂*.

D₁* = 100 – ½ P_W = 100 – ½ (600/17) = 1400/17
       = 82.35
D₂* = 100 – ½ P_T* = 100 – ½ (584/17) = 1408/17
       = 82.82.
S₁* = 4 P_W = 4 (600/17) = 2400/17
      = 141.18.
S₂* = 4 P_T* = 4 (584/17) = 2336/17
      = 137.41.

Now,
      Area c = ½ ( D₂* - D₁*) (P_W – P_T*) = ½ ( 1408/17 – 1400/17) (600/17 – 584/17) = 64/289
                  = 0.22.
     Area d = (S₂* - D₂*) (P_W – P_T*) = (2336/17 – 1408/17) (600/17 – 584/17) = 14848/17
                  = 51.38.
     Area e = ½ ( S₁* - S₂*) (P_W – P_T*) = ½ (2400/17 – 2336/17) (600/17 – 584/17)
                   = 1.77.
Area (a + b) = Area ( a + b + c ) – Area c
                    = (D₂* – 0) (P_W – P_T*) – Area c = 1408/17 (600/17 – 584/17) – 64/289 = 22464/289
                    = 77.73.

1. Consumer Surplus : Consumer surplus is given by the area below demand curve and above price line. Hence, decrease in price increases the consumer surplus by an area of a+b, as shown in the figure.

Change in Consumer Surplus = Area (a + b)
                                                    = 22464/289
                                                    = 77.73.

2. Producer Surplus : Producer Surplus is the area below price line and above the supply curve. Hence, decrease in prices leads to decrease in producer surplus by an amount of area (a + b + c + d + e), as shown in the figure.


Change in Producer Surplus = Area (a +b) + Area c + Area d + Area e
                                                  = 22464/289 + 64/289 + 14848/289 + 512/289
                                                  = 37888/289
                                                  = 131.1.

3. Revenue : Foreign Government receives no revenue when an import tax is imposed by Home country, hence Foreign Government revenue is ZERO.

Thus,
Change in Total Social Surplus = Gain from Consumer Surplus – Loss in Production Surplus
                                                      = (a + b) – (a + b + c + d + e)
                                                      = - (c + d + e)
                                                      = - (64/589 + 14848/589 + 572/589)
                                                      = - 15424/289
                                                      = - 53.37.

1. d. HOME
From part (c), we know that,
Net Cost of Import Tax = Consumer Loss – Producer Gain – Government Revenue
                                             = (a + b +c + d) – a – (c + e)
                                             = b + d – e
or, Total Change in Social Surplus = e – (b + d)


Areas b and d are 2 triangles that measure the efficiency loss resulting from -
1. Production distortion loss due to excess production.
2. Consumption distortion loss due to too little consumption.

Rectangle e represents the gains from terms of trade, which result due to decline in export prices from P_W to P_T*.
Terms of trade in the ratio of a country’s export prices to its import prices. Reduction in foreign export prices implies that domestic import prices fall, and therefore terms of trade for Home country improve.

Here, loss from b & d = 324/289 + 324/289 = 648/289

Gain from e = 1856/17.

Total Change in Social Surplus = e – (b + d)
                                                      = 1856/17 – 324/289 – 324/289
                                                      = 30904/289
                                                      = 106.93
Hence, the terms of trade gain to Home is higher than the efficiency loss due to production and consumption distortions. Thus, the importer gains from import tax.


FOREIGN
From part (c), we know that,
Change is Total Social Surplus = Gain from Consumer Surplus – Loss in Production Surplus
                                                      = (a + – (a + b + c + d + e)
                                                      = - (c + d + e)

Areas e and c are 2 triangles that measure the efficiency loss resulting from -
1. Production distortion loss due to too little production.
2. Consumption distortion loss due to excess consumption.

Rectangle d represents the negative terms of trade effect, which results due to decline in export prices from P_W to P_T*.

Change in Total Social Surplus = Gain from Consumer Surplus – Loss in Production Surplus
                                                      = (a + – (a + b + c + d + e)
                                                      = - (c + d + e)
                                                      = - (64/589 + 14848/589 + 572/589)
                                                      = - 15424/289
                                                      = - 53.37.

The loss from producer loss outweighs gain from consumer surplus, hence, the exporter does not gain from the imposition of import tax.

1. d. For optimal tax rate,
                  Marginal gain from terms of trade = Efficiency loss from Consumption &
                                                                                    Production distortions
                                                                     Area e = Area ( b + d )
                                           Area ( c + e ) – Area c = Area b + Area d
optimal tax rate x (D₂ - S₂) – (D₂ - S₂) (P_T – P_W) = ½ ( S₂ – S₁) (P_T – P_W) + ½ (D₁ – D₂) (P_T – P_W)
                           t* x (D₂ - S₂) – (D₂ - S₂) (P_T – P_W) = ½ (P_T – P_W) [( S₂ – S₁) + (D₁ – D₂)]
                                        (D₂ - S₂) [ t* – (P_T – P_W) ] = ½ (P_T – P_W) [( S₂ – D₂) + (D₁ – S₁)]

                                    In part (b), we saw that, D₂ – S₂ = Q_T.
                                                                               D₁ – S₁ = Q_W.
Using above mentioned expressions,
                                                  ⇒   (Q_T) [ t* – (P_T – P_W) ] = ½ (P_T – P_W) [(– Q_T) + Q_W]
                                                 
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The above expression gives the optimal tax rate for the importer.