Spring 004 Eon 455 Harvey Lapan Eon 455 Answers - Problem Set 4 1. Consier a small ountry (Belgium with the ollowing eman an supply urves or orn: Supply = 4P s ; Deman = 1000 Assume Belgium an import steel at a given worl prie o: P s = 15. Further, assume that Belgium imposes a tari o t per unit o import. Ps a Show how: omesti prie, onsumption an proution hange as t inreases. Also, alulate how onsumer surplus, prouer surplus, an government tari revenue hange as t inreases. Given the worl prie, the tari inlusive prie o imports in Belgian is given by: P = 15 + t ; this will be the omesti prie i it is less than the autarky prie (i.e., i t 75. Note that or t > 75 the tari is prohibitive, there are no imports, an the omesti prie equals the autarky prie o 00. Thus, assuming t < 75, we have: P = 15 + t ; D= 1000 P = 875 t; S = 4P = 500 + 4 t; M = D S = 375 5t Consumption an imports all, an proution rises, as t inreases. The hanges in onsumer an prouer surplus are seen on the iagram on the next page. The inrease in prouer surplus is given by area {15,A,A,(15+t}, whereas the erease in onsumer surplus is given by: {15,B,B,(15+t}. Thus: 1 1 TR = tm = t 375 5t = 375t 5t is tari revenue. PS = t ( 500 + 500 + 4t = 500t + t ; CS = t ( 875 + 875 t = 875t + ( t Thus, it is easily seen that prouer surplus inreases with the tari, an onsumer surplus ereases (or t < 75, whereas tari revenue inreases with the tari or t < 37.5, an then ereases thereater. Overall: Welare = TR + PS + CS = ( 5 t so that the tari lowers overall welare. (ii t>75, the tari is prohibitive, no trae ours an omesti prie is 00. b Compare the omesti equilibrium when t=5 to the ase where there is no tari, but there is an import quota o 50 units. From part (a, with t=5, imports M = 375 5t = 50. Thus, a quota o 50 an a tari o 5 have iential eets on omesti prie, onsumption, proution an imports. The only possible ierene is the tari revenue (whih is 6,50 uner the tari. Uner the quota, importers make 5 on eah unit importe an hene will earn exess proits o 6,50, unless the quota lienses are autione o, in whih ase the two poliies are iential.
P S 00 15+t A B 15 A E F B D 500 500+4t 875-t 875 Q Figure 1 There is one omesti prouer o steel in Belgium. In orer to answer this part (an espeially, we nee to know the irm s ost urve. Sine the supply urve, uner ompetition is the (inustry marginal ost urve, it makes sense to use the supply urve (really, the inverse supply urve as the marginal ost. Hene: = 4 = ( 4 or: MC ( Q ( Q 4 s Q Ps Ps Qs s =. s While we on t nee it, you an get the ost urve by integrating the marginal ost. Hene: = ( 8 + = ( 4 TC Q Q MC Q Q s s s s where is a onstant o integration. As a reerene point, we alulate the monopoly solution assuming no imports are allowe (the autarky monopoly solution: ( 1000 ( 8 π = PQ TC Q = Q Q Q sine eman is Q = ( 1000 P P = ( 1000 Q π == MR MC = ( 1000 Q ( Q 4 = 0 Q = M ( 4000 9 ; P = M ( 1000 QM = ( 5000 9 > 00 Q So, note that the ompetitive equilibrium prie woul be 00 (i the inustry were ompetitive, while the monopoly equilibrium results in higher prie an lower output. Now we onsier how the opportunity to trae hanges this. Uner a tari, beause o the threat o imports, the monopolist an never harge a prie above ( 15 t whih is the prie o importe steel. Hene, the threat o imports ats as a prie eiling or the +,
monopolist. I 15 + t 00, the monopolist s output i hosen where prie equals marginal ost - will not aet prie, an hene he will proue where marginal ost equals prie: the same outome as or a ompetitive inustry. Hene: s t 75: P = 15 +t; Q = 500 + 4 t; D= 875 t; M = 375 5t 0 I 15 + t > 00, then imports are riven to zero an the irm gets to exerise some monopoly power. However, the threat o imports prevents the irm rom exerising its ull monopoly power until the prie o imports rises to the autarky monopoly prie. Note that even when imports are zero the threat o imports still limits the monopolist s priing strategy an hene aets its output: 3875 5000 t > 75 P > 00; M = 0, Q = D = 1000 P = 875 t 9 9 M t ( 3875 9 P= P = ( 5000 9 ; Q= D= ( 4000 9 ( As seen in lass, a quota has a ierent eet than a tari. For example, i the quota is set at zero (imports are banne, then the monopoly prie ours whereas uner a tari there are a range o taris that result in no imports but lower pries than the autarky monopoly prie. In general, i we let L represent the quota limit, omesti prie is: P ( 1000 L ( 8 = ( 1000 L ( π = Q where Q is the omesti irm s output. Sine imports are ixe, the irm an rive up prie by lowering proution. Its proits are given by: π = PQ Q Q Q Q 8 Q = 1000 L Q Q 4 = 0 4000 4L 4000 + 5L 5 1000 L Q ( L = ; D( L = ( Q+ L = ; P( L = ( 1000 L Q = = 555.55.55L 9 9 9 Note that when L = 0 the monopoly solution emerges. It is interesting to note that when the import quota is set at the ree trae level (375, the quota leas the monopolist to reue output an raise prie (to 347. whereas uner peret ompetition a quota at the ree trae level woul have no eet. t eto ompare the quota equilibrium ( to the tari ( when t 75, we have uner a tari: 75 M( t = ( 375 5t or t = ( 75 ( M 5 whih implies: Pt ( = ( 15 + t = 00 ( M( t 5 Comparing prie, as a untion o imports, uner the quota to that uner the tari, it is apparent that i the two poliies result in the same level o imports, the quota will result in higher omesti prie an lower omesti output (uner monopoly. For example, at t=5, imports are 50 an prie is 150; at L=50, prie is 416.66.., well above the prie with a tari. Thus, the quota leas to higher proits or the irm an larger losses or onsumers (it also leas to greater government revenue, i the quotas are autione o, sine the gap between worl an omesti prie is larger. As to whih poliy is better, it epens on your objetives; i your goal is to give the irm high proits, the quota oes that better! However, i the goal is minimizing the eaweight loss rom the poliy, the tari is learly better sine the aitional reution o omesti output an onsumption uner the quota reates urther ineiieny.. Next, onsier the ase o two large ountries: 3
US: Deman = 360 4 us ; Supply us us = where is the prie o omputers in the US; P P China: Deman = 180 ; Supply 4 = where is the prie o omputers in China; P P P P a Assuming ree trae (no taris, in the equilibrium prie an quantities trae. The US import eman (M is given by: M = D us S us = 360 6P us The Chinese export supply (X is given by: X = S D = 6P 180 us where P is the prie in China an P is the prie in the US. Note that uner autarky the US prie is 60, an the Chinese prie is 30 (see igure, sine exports (an imports in eah ountry will be zero us uner these pries. Uner ree trae we have: P = P = P ( stans or ree trae an thus: M = X 360 6P = 6P 180 P = 45 Thus, exports (imports equal 90 uner ree trae. Domesti onsumption an supply is oun by substituting bak into the eman an supply urves. For the US an China, uner ree trae: US: us us D = 360 4P = 180; S = P = 90 ; China: D = 180 P = 90; S = 4P = 180 b Show how a US import tari o 1 aets the volume o trae, pries in China an the US, an welare in eah ountry. Who pays or the US tax? Explain. Sine part ( asks you to answer this or the general ase o any tari, it is easier to start by assuming the tari is t, rather than 1, an then substitute in or t=1. Thus, with any tari, we have: P tax is on US imports. Equilibrium requires: us us M = 360 6 P + t = X = 6P 180 540 6t = 1P P = 45.5 t, P = 45 +.5t us = P +t (whih means when t=1, the US prie is 1 units higher than the Chinese prie sine the So, when t=1, the Chinese prie alls by 6 an the US prie rises by 6. Note that, even though the US imposes the tax, China pays some o the tax (beause the prie it reeives or its exports alls. As a result o the tax, the volume o trae alls. Substituting the equilibrium prie bak yiels exports (imports o: ( = ( = 6 ( 180 = 6( 45.5 180 = 90 3t; or t=1, us M t X t P t t X t = 1 = 54 Clearly, Chinese welare alls (they reeive lower pries or exports an reeive no tari revenue, whereas the impat on the US is ambiguous. In terms o Figure, the loss to China is the area o the trapezoi {45,E,B,[45-.5t]} with t=1. At t=1, this trapezoi has height 6, one base is (90, the other is 54, so the area (or loss to China is 43. For the US, onsumers lose (higher pries, prouers gain (higher pries, an the government gains tari revenue. Overall, the US private setor loses area {45,E,A,[45+.5t]}=43. However, the US government gains the tari revenue, measure by area: {[45-.5t], B,A,[45+.5t]=154=648. So, or t=1, the US gains 16, China loses 43, an the overall loss is area AEB = 16. 4
(ii the tari were remove, but an import quota o 54 replae it, then the volume o trae woul be the same uner the two poliies. Hene, pries in eah ountry woul be the same uner the quota as they ha been uner the tari. The only possible ierene is that the tari revenue now beomes proits or US importers, who buy at the Chinese prie (39 an sell at the U.S. prie (51. I the US autions o the quota lienses, the two poliies are iential P 60 45+(.5t A China X 45 E F 45-(.5t B 30 US M 90-3t 90 Figure Q (How oes a US import tari o t aet pries, trae volumes an welare? (i How oes the tari aet China s welare (sum o onsumer an prouer surplus? The analysis was isusse in part (b, an the igure above illustrates these hanges. China, ue to lower export pries (an no tari revenue is hurt. The total loss in China is the area {45,E,B,[45-.5t]}; the US private setor loses area {45,E,A,[45+.5t]}, while the US government gains the tari revenue, measure by area: {[45-.5t], B,A,[45+.5t]}. Thus: China loses: (.5t( 90 1.5t US private setor loses: ( t (.5 90 1.5 US government tari revenue = ( t t ( 90 3t t 90 3 t.5t 90 1.5t = 45t 9 4 t > 0 i t < 0 Net US gain = ( Overall welare hange = Net US gain plus Chinese loss = t t ( t( t 45.5.5 90 1.5 = 1.5t 5
So, even i the US gains rom the tari, it gains less than China loses, so there is an overall ineiieny ue to the tari. In terms o the igure, the ineiieny (eaweight loss is the triangle: {A,B,E}. Finally, to alulate the hanges in onsumer an prouer surplus or China (an the US you must go bak to the eman an supply urves (not just the export supply or import eman urves. While the igures are not rawn, the metho o ining the surplus is the same as above. Hene: For China: 1 1 CS = ( P( D0 + D1 = (.5t 90 + [ 90 + t] =.5t 90 +.5t 1 1 PS = ( P( S0 + S1 = (.5t 180 + [ 180 t] =.5t 180 t PS + CS =.5t 90 1.5t Net Change in Chinese welare= (iithe impat on US welare, summarize above is given by: 1 1 CS = P D + D =.5 t 180 + 180 t =.5 t 180 t ( 0 1 ( [ ] 1 1 PS = P S + S =.5 t 90 + 90 + t =.5 t 90 +.5 t ( 0 1 ( [ ] Change Private Setor Welare (US = CS PS =.5t ( 90 1.5t Tari Revenue = ( t iexports = t ( 90 3t +, as above t 90 3 t.5t 90 1.5t = t 45.5t =.5t 0 t > 0 i t < 0 Net US gain = The US an gain rom this tari beause it aets worl prie; speiially, by restriting imports the tari lowers the worl prie o US imports (even though omesti prie rises. This hange in worl prie has a positive eet on the US eonomy (an a negative eet on the Chinese eonomy, an thus rom the US perspetive the improve terms o trae may oset the ineiieny the tari auses. However, overall worl welare (sum o US an Chinese surplus must all. Thus:.5t 0 t.5t 90 1.5t = 1.5t < 0, as explaine earlier. Net US gain + Net Chinese gain = (iiifin the tari that maximizes US welare. From above, the gain in US welare is:.5t( 0 t G t =. Taking the erivative: G =.5( 0 t. For t small, G (US welare inreases with the tari; when t is large enough G t G ereases as t inreases urther. The value o t that maximizes G is oun by setting = 0, as the t seon orer onition is easily seen to hol. This implies: t = 10 is the optimal tari or the US. 6
(ivi the U.S. eliminates its import tari, but China imposes an export tari o the same magnitue, the private setor losses are as above, but the tari revenue is transerre to China. Thus, the US woul lose area {45,E,A,[45+.5t]}, while China woul gain {45,F,A,[45+.5t]}, while losing {B,F,E} or the same worl loss o {A,B,E}. Numerially: Net Gain to China = ( t( t + t( t = t( t Net Gain to US = (.5t( 90 1.5t < 0.5 90 1.5 90 3.5 90 4.5 > 0 or 0< t < 0 3. The purpose o the question is to show that, i the goal is to inrease omesti proution, proution subsiies are more eetive (less ostly than taris. To illustrate this, use the moel o problem 1, an assume the government s goal is to inrease proution to 600 units. a To inrease omesti proution to 600 with a tari requires a omesti prie o 150, hene a tari o 5. Using the results rom Problem 1, omesti onsumption alls to 850. The welare impat (t=5 are: PS = 500t + t = 13, 750 CS t ( t TR t t = 375 5 = 6, 50 is tari revenue Net Welare Change = 1,56.5 = 875 + = 1,56.5 b With a proution subsiy o 15, the onsumer prie remains at 15 (the worl prie an onsumers are unaete. The prie reeive by prouers inreases to 150, so the hange in prouer surplus is the same as (a. However, instea o tari revenue there is the ost to taxpayers o the subsiy. Hene, with a proution subsiy: PS = 500t + t = 13, 750 Tax Revenue = Net Welare Change = 1,50 5 600 = 15,000 ; The proution subsiy has no impat on onsumers, whereas the tari oes. Clearly, the proution subsiy is more eiient sine it oes not involve a loss to onsumers (in terms o Figure 1, the inrease in prouer surplus is area {15,A,A,[15+t]}, an the ost o the subsiy is {15,E,A,[15+t]}, so the eaweight loss is area {A,E,A}. With the tari there is the aitional eaweight loss o {B,B,F}, ue to the hange in onsumption. An import tari raises the prie to both onsumers an prouers; hene, it ats like a tax on onsumers (they pay more an as a subsiy to prouers (they reeive more or their output. A iret proution subsiy has the same impat on prouers, but oes not aet onsumers; a tax on onsumption (regarless o where the goo is proue woul not aet the prie prouers reeive (given the worl prie, but woul raise onsumer pries. Hene, the tari has the same impat as both subsiizing omesti proution an taxing omesti onsumption. 4. Given the proution untions: 7
Q = L K Q = L implies: TC WL WQ a 1 1 ; Sine proit maximization implies prie equals marginal ost: P = MC = W an P MC W ( Q K I goo F is to be proue, we must have it is not possible to have = =, TC ( = WL = W Q K = = or: Q = ( P P > W P = W (i P K W will want to hire more labor. But this will bi up the wage until < W, then goo F will not be proue; sine there will be exess proits in prouing F, an F prouers P = W. Conentrating on an interior solution (where both goos are proue we have: Supply: Q = ( PK W = ( PK P = ( ρ K = ρ where: ρ ( P P Q ( L L ( Q L L K K = = = ρ 4K = 500 ρ, sine L = 500, K = 4 ( This solution requires ρ 10 5 ; i ρ > 10 5 only goo C is proue. For preenes: U = D 45 ( D + D ( U D ( U D Deman: ( P P = 45 D = ρ. Hene: D = 45 ρ; D = ( I P ( 45 ρ P ; emans are oun by setting MRS= prie ratio: ; ρ 45 (Note that i the relative prie o C is too high, eman or C will be zero; an i inome is too low, eman or F will be there. For simpliity, we ous on ases where both goos are onsume. The autarky equilibrium thus satisies: A Q = D ρ = 45 ρ, an hene the autarky prie: ρ = 15. W P ( W P = ( W P ρ = ( 115 = or ( W P = 1, Only relative prie matters beause oubling output pries will ouble the wage rate. Hene, marginal ost an prie both ouble, so supply is unhange. On the eman sie, inome oubles, but pries also ouble, so eman is unhange. The real question is: what oes a prie o 15 mean. There is no absolute stanar or pries: rather it is 15 units o goo F per unit o goo C. w w W b As given in the question, the worl pries are suh that: P = 1, P = 8, an hene ρ = 8. Thus, uner ree trae the ountry woul import goo C. Uner ree trae, with omesti pries equal worl pries, we have: 8
Q = ρ = 16, Q = ( 500 ρ = 436, D = 45 ρ = 37 Imports o goo C: M = D Q = 1. Balane o trae equilibrium implies: PX = PM X = ρm = 8 1 = 168, Hene, omesti onsumption o F (proution less exports is: 436 168 = 68 A 5% import tari results in the ollowing omesti pries: w P = P 1+ t = 10; P = P = 1 hene: ρ = 10 w As long as goo F is still proue the omesti nominal wage is unhange (W =, so the real wage in terms o goo F is unhange. Sine the omesti prie o C rises, the real wage in terms o C alls an hene labor loses rom this tari (slightly ierent rom the speii ator moel beause there is no speii ator in setor F. The real return to apital will rise, o ourse (as preite by the speii ator moel: ( K ( P ( L K 1 = L = ( Q K R= P Q. From earlier: an supply o C is given by: = ( ; hene: = an: ( 4 implying: ( Q ρ K L ( ρ K 4K L K = ( ρ = ( ( an: ( RP = ( ρ 4, ( RP ( ρ = 4 R P ρ so: Thus, inreasing the omesti prie o C (an its relative prie raises the real return to apital, whether measure in terms o the import or export goo. Finally, omesti proution an onsumption are oun by substitution with ρ =10 : Q = ρ = 0;, Q = 500 ρ = 400 D = 45 ρ = 35 an thus: M = 35 0 = 15. From the balane o trae, sine worl prie is unhange: w X = ρ M = 8 15 10; omesti onsumtion o F: D = Q X = 80 = I the import tari is remove, but an export tari o 5%, base on the omesti prie, is impose we have: w w P P t 1.5P 1.5P 1 or: P = 45 =.8; P = P = 8 = = = ( an ρ = P P = 8.8 = 10 So, the relative prie is the same with the two plans. Naturally, the nominal wage alls to.8 but the real wage (in terms o its purhasing power o either C or F is the same as in part (, with the import tari. Similarly, or apital, the nominal return R inreases sine ρ inreases, as oes the real return in terms o either goo. Finally, sine proution an onsumption epen only on relative pries, all outputs, emans, an relative pries are the same as in. There is no ierene, in terms o the real eonomi impat, between an import an export tari o omparable magnitues. e With an import tari o 5% an an export subsiy o 5%, both base on the worl prie, we have: s w w P = 8i 1+ t = 10; P = P + subsiy = P i (1 +.5 = 1.5 an: ρ = ( P P = 8. P 9
So, these poliies raise the nominal pries o all goos by 5%, an hene will inrease the nominal wage by 5%, but relative pries will be unaete an there will be no real eet (on onsumption, proution, trae o these poliies. In essene, they are like a evaluation o the ountry s urreny. EXTRA CREDIT PROBLEM 5. Same set-up as problem #: US: Deman = 360 4 us ; Supply = us China: Deman P P = 180 ; Supply = 4 P The US import eman (M is given by: M = D S = 360 6P The Chinese export supply (X is given by: X = S D = 6P 180 China has an export tari o, P us us us θ US has an import tari o t. Hene: us P = P + ( θ + t a Using earlier results (sine only sum o two taris matters you get equilibrium prie an quantity trae as a untion o the ombine tari: ( θ ( θ us M = 360 6 P + t+ = X = 6P 180 540 6 t+ = 1P us ( θ ( θ ( θ ( θ ( θ ( θ ( θ ( θ P = 45.5 t+, P = 45 +.5 t+ us us us us us D = 360 4P = 180 t+ ; S = P = 90 + t+ ; M = 90 3 t+ D = 180 P = 90 + t+ ; S = 4P = 180 t+ ; X = 90 3 t+ w us The worl prie is the prie outsie both the US an China i.e., P = P t = P + θ = 45 +.5( θ t The US import tari rives own the worl prie (helping the importing ountries an the Chinese export tari rives up the worl prie (helping the exporting ountries. b As earlier, US private setor loses ue to higher omesti prie; alulation is exatly same as earlier, exept it epens on the sum o the two taris. Similarly, Chinese private setor loses ue to erease in omesti prie; again, it is the same as earlier, exept it epens on the same o the two taris. 1. US private setor loses: (.5[ t+ θ ]( 90 1.5[ t+ θ ]. US Tari Revenue: t 90 3( t+ θ 3. Net Change in US welare = ( θ [ θ] { 45( t θ ( 3 4( t+ θ( 3t θ } ( [ θ ] t 90 3 t+.5 t+ 90 1.5 t+ = 4. China private setor loses: (.5[ t+ θ ]( 90 1.5[ t+ θ ] 5. China Tari Revenue: θ 90 3( t + θ 6. Net Change in China s welare = θ ( θ [ θ] { 45( θ t ( 3 4( t+ θ( 3θ t } ( [ θ ] 90 3 t+.5 t+ 90 1.5 t+ = 10
7. Net Change in Worl Welare (#3 + #6 = ( 3( t + θ The US hooses t to maximize its welare (gain, given θ. From (#3: { 45( θ ( 3 4( + θ( 3 θ } t t t t t = 10 ( θ 3 Note when θ = 0 this gives you the optimal US tari rom Q#. ( t θ ( t θ ( t θ = 45 3 4 3 3 4 3 + = 45 3 3 + = 0 Next, hoose the Chinese optimal export tari, θ, given the US tari, t. From (#6: { 45( θ ( 3 4( + θ( 3θ } t t t θ θ = 10 ( t 3 ( θ t ( t θ ( θ t = 45 3 4 3 3 4 3 + = 45 3 3 + = 0 e Using the results rom ( an ( an solving simultaneously: ( t θ 3 + = 30 3θ + = 30 tˆ = ˆ θ = 7.5, ( t This is the Nash equilibrium or this tari game. It is possible or both ountries to be worse o; we know one must be worse o an, i the untions are airly similar (as they are here then it is likely that both ountries are worse o. China s tari beneits it, but inlits more amage on the US; the US tari beneits the US but inlits more amage on China. Together, the two taris may (in equilibrium hurt eah ountry yet neither ountry has the inentive to unilaterally remove its tari. We an use equations #3 an #6 above to see the net impat on eah ountry: Net Change in US welare = { ( t θ ( t+ θ( t θ } = = Net Change in China s welare = { ( θ t ( t θ( θ t } sine t = θ = 7.5. 45 3 4 3 3 4 15 15 168.75 45 3 4 + 3 = 3 4 15 15 = 168.75 Both ountries lose rom the use o their optimal tari, given that the other ountry also uses its optimal tari. g The prisoner s ilemma is that both woul gain by ooperating, but neither has the unilateral inentive to hange poliy. Only oorinate ations through a treaty, or example, an solve the problem an lea to ree trae, where both parties gain ompare to this equilibrium. 11