Basic Analysis of Autarky and Free Trade Models


 Cecil Jackson
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1 Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently ll the trnsctions tke plce within the domestic mrket under closed economy. The utrky equilibrium price is determined when the totl domestic supply of the commodity is equl to the totl domestic demnd. Autrky model for prticulr commodity cn be represented by the following equtions. (1) Q P α X, 0 s (2) Q P β Z, 0 (3) Qs Qd d The eqution (1) denotes tht quntity supplied (Q s ) is function of its own price (P) nd vector of supply shifters (X) such s input prices nd policy vrible. These vribles re clled supply shifters becuse chnges in ny of these vribles shift the supply curve up or down depending on the sign of the corresponding coefficient in the prmeter vectorα 2. The elements in the prmeter α 2 cn be positive or negtive depending on the vrible in the supply shifter vector X. The eqution (2) expresses the behviorl reltionship tht quntity demnded (Q d ) is function of its own price nd vector of demnd shifters (Z) such s income nd prices of complements nd substitutes. Chnges in demnd shifter will shift the demnd curve up or down depending on the sign of the corresponding coefficients in vectorβ 2. The elements in the prmeter vector β 2 cn be
2 2 positive or negtive depending on the vribles in the demnd shifter Z. The eqution (3) entils tht for this commodity mrket to ttin equilibrium, quntity supplied must be equl to quntity demnded. The endogenous vribles in this model re quntity supplied, quntity demnded, nd price. The exogenous vribles re supply shifters nd demnd shifters. A grphicl representtion of the utrky model in QP spce is illustrted in Figure 2.1. Note tht the supply nd demnd curves re drwn by ssuming X nd Z re zero. The supply curve in QP spce hs the required positive slope ( 1). The verticl intercept is negtive ( 0 ) to meet the provision tht quntity will be supplied only fter the price reches certin minimum positive level. Chnges in ny of the supply shifters will shift the supply curve up or down depending on the sign of the corresponding coefficients in α 2. The demnd curve in QP spce hs positive verticl intercept ( 0 ) nd negtive slope ( 1) s one would expect. Chnges in ny of the demnd shifters will shift the demnd curve up or down depending on the sign of the corresponding coefficients in β 2. By equting the quntity supplied nd quntity demnded s in (3) we cn solve for equilibrium prices ( P) (4) 0 0 β2z α2x P 1 1
3 3 Note tht the equilibrium price is expressed in terms of prmeters nd supply nd demnd shifters. For P to be positive, the numertor in equtions (4) hs to be positive, since the denomintor is positive s specified in the model. The equilibrium quntity Q is obtined by substituting the eqution for P either in (1) or (2) nd solving the resulting expression (5) Q αβ αβ αβ Z βα X β α 1 1 Note tht the equilibrium quntity is lso expressed in terms of prmeters nd supply nd demnd shifters. Since the denomintor is positive, for Q to be economiclly meningful, the numertor lso hs to be positive. Refer to Figure 2.1 to understnd how the determintion of P nd Q is illustrted grphiclly. A grphicl representtion of the utrky model in PQ spce, in ccordnce with the convention, is illustrted in Figure 2.2. It will be useful for lter nlysis to express the supply nd demnd equtions in price dependent forms corresponding to the curves in the PQ spce s in Figure 2.2. Thus, the price dependent supply nd demnd equtions re respectively, (6) α0 Qs 2 P s α X. α α α (7) P d 0 Qd β2 Z 1 1 1
4 4 where P s nd P d re supply nd demnd prices, respectively, nd ll other vribles re s defined before. The equilibrium condition is stted s (8) P s = P d. 0 / 1 The supply curve hs the positive slope 1/ 0 1, nd the verticl intercept is positive stisfying the provision tht the price hs to be sufficiently positive for ny quntity supply to be forthcoming, ssuming X is zero. Chnges in the supply shifters (X) will shift the supply curve up or down depending on the signs of the corresponding coefficients in α 2 / 1. The demnd curve hs the required negtive slope 1 1/ 0, nd the verticl intercept is positive t 0 / 1. Chnges in the demnd shifters will shift the demnd curve up or down depending on the sign of the corresponding coefficients in 2 / 1. The intersection of supply nd demnd curves, which is equivlent to setting the Q s = Q d s in (3) or P s = P d s in (8), determines the equilibrium price ( P ) nd quntity ( Q ). The significnce of the restrictions tht P nd Q re positive cn be demonstrted using either Figure 2.1 or 2.2. For P nd Q to be positive, the equilibrium point should be locted bove the horizontl xis, which requires tht the numertors in (4) nd (5) hve to be positive, given tht the denomintors re positive. Though the mterils presented in this section re elementry, they will be useful in nlyzing the comprtive sttics of policy impcts in the lter chpters.
5 5 IMPORT DEMAND AND EXPORT SUPPLY SCHEDULES Consider country with supply nd demnd curves for prticulr commodity, s depicted in PQ spce in Figure 2.3. The utrky equilibrium price of the commodity is t P. For prices below P domestic demnd will exceed domestic supply, nd this country would like to import to meet ll of its domestic demnd. The excess or import demnd (ED) of this commodity, depicted in Figure 2.3b, is the horizontl difference between domestic demnd nd supply curves for ech price below P. The curve in Figure 2.3b mesures import demnd t vrious prices below P. The excess demnd curve is negtively sloped becuse s the price flls frther below the P, import demnd increses. At the utrky equilibrium price the import demnd is zero, since the domestic supply stisfies the domestic demnd exctly. To trnslte the excess demnd into mthemticl sttements, consider the equtions in (1) nd (2) s the domestic supply nd demnd function of this importing country, which re rewritten here: (1) Q P α X, 0 s (2) β Z Q P, 0. d The excess demnd function is derived s (9) Q Q Q P β Z α X. ed d s The price dependent form of the excess demnd function, corresponding to the curve in Figure 2.3b, is (10) P ed 0 0 Qed β2z α2x where P ed is the excess demnd price. Thus, the intercept in Figure 2.3b is /, which is positive; the slope in 1/ , which is negtive. 1 1
6 6 Consider the exporting country with supply nd demnd curves for prticulr commodity, s depicted in Figure 2.4. The utrky equilibrium price of the commodity is t P. At the utrky equilibrium price, the export supply is zero since the domestic demnd bsorbs ll the domestic supply. For prices bove P the domestic supply will exceed the domestic demnd. The excess or export supply (ES) for this commodity, depicted in Figure 2.4b, is the horizontl difference between the domestic supply nd demnd curves t ech price bove P. This curve cptures the quntity of exports t vrious prices bove P. The excess supply curve is positively sloped becuse s the price rises frther bove P, this country will be ble to supply more to the foreign mrkets. As in excess demnd function, the excess supply cn lso be put in mthemticl terms. Consider the following domestic supply (eqn. 1) nd demnd (eqn. 2) functions of n exporting country. (1) Q P α X, 0 s (2) β Z Q P, 0. d The excess supply function is derived s
7 Q Q Q P α X β Z. (11) es s d The price dependnt form of the excess supply function, corresponding to Figure 2.4b, is (12) P es 0 0 Qes α2x β2z where P es is the excess supply price. Thus, the intercept in Figure 2.4b is /, which is positive implying tht exports will not occur, ssuming X nd Z re 0, unless the price exceeds utrky equilibrium price. The slope of the excess supply cure is positive t1/ 1 1. It is importnt to note tht country cn switch from n importer when world prices re below the utrky equilibrium price t some period to n exporter when world prices re bove the utrky equilibrium price t other periods. This possibility is illustrted in Figure 2.5. Domestic demnd nd supply re plotted in Figure 2.5; the utrky equilibrium price is t P. For prices below P, this country imports, nd the corresponding import demnd schedule is the AB portion of the curve shown in the
8 8 positive qudrnt of Figure 2.5b. For prices bove P, this country exports, nd the corresponding export supply schedule is the CA portion of the curve shown in the negtive qudrnt of Figure 2.5b. Thus, in Figure 2.5b, imports re mesured long the positive prt of the xxis nd the exports re mesured long the negtive prt of the x xis. In mthemticl terms, the import demnd eqution (9) cn lso be considered s the export supply function when Q ed is negtive, i.e., Q d < Q s. The cse where country switched from n exporter to n importer is illustrted in Figure 2.6. For prices bove P, this country exports, nd the corresponding excess supply schedule is the AB portion of the curve shown in the positive qudrnt of Figure 2.6b. For prices below P, this country imports, nd the corresponding import demnd schedule is the CA portion of the curve shown in the negtive qudrnt of Figure 2.6b. Thus, in Figure 2.6b, exports re mesured long the positive prts of the xxis nd imports re mesured long the negtive prt of the xxis. In mthemticl terms, the export supply eqution (11) cn lso be considered s import demnd function if Q es is negtive, i.e., Q s < Q d. A shortfll in domestic production cn mke n exporting country
9 9 become n importing country. Similrly, bumper crop cn switch country from importer to exporter sttus. A clssic exmple is Indin whet trde. Between 1970 nd 1991, Indi imported whet in 12 yers nd exported in 10 yers. LARGE AND SMALL COUNTRY ASSUMPTIONS In the literture, discussions bout lrge nd smll countries re encountered frequently. The impcts of trde policies on lrge nd smll trding countries differ considerbly. Since we nlyze the lrge nd smll country cses throughout the text, it is importnt t this juncture to understnd the differences between the lrge nd smll trding ntions. First, let us exmine the importing countries. LARGE AND SMALL IMPORTING COUNTRIES A lrge importing country imports lrge quntities of prticulr commodity nd plys mjor role in the world mrket. Consequently, lrge importing country cn behve s monopsony nd lter the import price by chnging its import demnd. This implies tht the excess supply, which origintes from the exporting country, fced by this importing country is positively sloped (see Figure 2.7). Thus, the lrge importing country
10 10 cn exercise its monopsony power to lower (rise) the import price by decresing (incresing) the import demnd s shown in Figure 2.7. The eqution (11) cn be used to represent the excess supply fced by this country. A smll importing country imports only smll quntity of prticulr commodity nd plys n insignificnt role in the world mrket. Consequently it cn not lter the import price nd behves s price tker. This implies tht smll importing country fces perfectly elstic excess supply curve, just like competitive firm being one of numerous smll firms behves s price tker in the input mrket by fcing perfectly elstic input supply. Thus, chnges in the import demnd by the smll importing country do not lter the import price s shown in Figure 2.8. Since the excess supply is perfectly elstic nd import prices re constnt t, sy P, the excess supply cn be represented s P = C, where C is constnt. LARGE AND SMALL EXPORTING COUNTRIES A lrge exporting country exports lrge quntities of prticulr commodity nd it plys mjor role in the world mrket. Consequently, lrge exporting country cn
11 11 behve s monopoly nd lter the import price by chnging its export supply. This implies tht the import demnd, which origintes from the importing country, fced by this exporting country is negtively sloped (Figure 2.9). Thus, the lrge exporting country cn exercise its monopoly power to lower (rise) the export price by incresing (decresing) the export supply s shown in Figure 2.9. The eqution (9) cn be used to represent the import demnd fced by this country. A smll exporting country mens it exports only smll quntity of prticulr commodity nd plys n insignificnt role in the world mrket. Consequently it is ineffective in ltering the export price nd behves s price tker. This implies tht this country fces perfectly elstic excess demnd curve, just like competitive firm being one of numerous smll firms behves s price tker in the product mrket by fcing perfectly elstic demnd. Thus, chnges in the export supply by the smll exporting country do not lter the export price shown in Figure Since the excess demnd is perfectly elstic nd export prices re constnt t, sy P, the excess demnd cn be represented s P = D, where D is constnt.
12 12 TRADE AND WORLD PRICE DETERMINATION Trde in prticulr commodity tkes plce between two countries becuse of price differences. The price difference is cused by the differences in the demnd nd supply conditions in ech country. Thus, prices will be lower in country where there is n excess supply reltive to prices in country where there is n excess demnd. Under these conditions, the commodity will be exported by the country where the price is lower nd imported by the country where price is higher. To formlly exmine trde between two countries, consider countries A nd B, whose supply nd demnd functions re drwn in Figure The utrky equilibrium price in the country A is P ; t ny price bove P, there will be surplus or excess supply of tht commodity. Consequently, country A would like to export this commodity; the export supply (ES ) originting from this country is drwn in the middle digrm. The utrky equilibrium price in country B is P b ; t ny price below P b, there will be shortge or excess demnd for tht commodity. Consequently, country B would like to import this commodity; the import demnd (ED b ) rising from this country is drwn in the middle digrm. It should be noted tht Figure 2.11 is drwn ssuming tht currencies in the two countries re the sme, nd thus, the prices plotted long the verticl xes re in the sme units. This ssumption of single currency will be mintined until chpter 15 in Houck, where we discuss the effects of exchnge rte chnges on trde.
13 13 If the trde is llowed between these two countries, country A will export nd country B will import until excess supply of A, ES, equls the excess demnd of B, ED b, Thus, the free trde equilibrium or world price, denoted P, is determined when excess supply nd excess demnd re equl. This condition lso implies tht t P, totl world (A+B) supply equls totl world (A + B) demnd. This condition is shown below mthemticlly. Free trde converts the two seprte domestic mrkets in A nd B into single globl mrket. Also, free trde will equlize, ssuming zero trnsport cost, the prices in both countries. Thus, with free trde the previling price in A is P, the quntity supplied nd demnded t P re Q s nd Qd nd the quntity of excess supply is The price in B is lso P, the quntity supplied nd demnded t P re b Qs nd Q es. b Q d, nd the quntity of excess demnd is Q b ed. From the middle digrm, it should be cler tht Q es is equl to Q ed b. The mthemticl nlysis of free trde equilibrium is crried out next. As we proceed with mthemticl nlysis, reders cn improve their understnding by relting the equtions with the grphs in Figure Let country A s supply nd demnd be (13) QS 0 1P α2x 0, 1 0 (14) Qd 0 1P β2z 0, 1 0 nd the excess supply of country A is derived s (15) es S d Q Q Q P α X β Z The vrible definitions re s defined before, except now the superscript refers to country A.
14 14 Similrly, country B s supply nd demnd re (16) b b b b b b QS 0 1P α2 X 0 b, 1 b 0 (17) b b b b b b Qd 0 1P β2z 0 b, 1 b 0 nd the excess demnd of country B is derived s b b b b b b b b (18) ed d S b 2 b b 2 b Q Q Q P α X β Z. The vrible definitions re s defined before, except now the superscript b refers to country B. The free trde equilibrium is obtined by equting excess supply nd excess demnd, (19) Q es Q b ed nd setting P = P b = P. In figure 2.11, this free trde equilibrium is t point f. Note tht the equlity of excess supply nd demnd implies tht (20) s d d s' Q Q Q Q or (20 ) b b s s d d Q Q Q Q. Thus, free trde equilibrium is obtined by equting totl world supply nd totl world demnd. In this system of twocountry free trde models, there re seven equtions  from eqn 13, to eqn19, nd seven endogenous vribles  b Q s, Q d, Q es, Q s, Q d, Q ed, nd P. First we cn solve for free trde equilibrium price P, by setting P = P b = P from eqn 19.
15 15 (21) P b b α X β Z α X β Z As one would expect, P is determined by the prmeters nd the supply nd demnd shifters in both the countries. More specificlly, the fctors tht increse the supply (demnd) in both countries will depress (boost) the world price. On the other hnd, the fctors tht decrese the supply (demnd) in both countries will boost (depress) the world price. Since the denomintor in eqn. (21) is positive, the numertor hs to be positive for P in order to be economiclly meningful. It is worth referring to figure 2.11 to know how the price determintion in eqn (21) is illustrted grphiclly. P cn be substituted for P in the supply nd demnd equtions (13) nd (14) to obtin the quntity supplied nd demnded in country A under free trde: (22) Q s nd b b b b b b α2X α X 1β2Z 1β2Z b b β2Z β Z 1α2X 1α2X (23) Qd The quntities supplied nd demnded in country A under free trde re lso functions of prmeters nd supply nd demnd shifters. More specificlly, the fctors tht increse the domestic demnd in both countries nd domestic supply in country A will ugment the quntity supplied in A, becuse the demnd fctors boost the world price nd domestic supply shifters shift the supply to the right, which cuses n increse in quntity supplied in country A. On the other hnd, fctors tht increse the supply in country B
16 16 depress the world price which reduces the quntity supplied in A. Fctors tht increse the domestic supply in both countries nd domestic demnd country A will expnd the quntity demnded in country A, becuse the supply fctors lower the world price nd domestic demnd shifters shift the demnd to the right, which cuses n increse in quntity demnded in country A. On the other hnd, fctors tht increse the demnd in country B boost the world price which reduces the quntity demnded in A. Similr discussions for the fctors tht hve negtive effects on demnd nd supply cn be esily extended. For quntity supplied nd demnded to be positive, the numertors in eqns. (22) nd (23) should be positive. The equilibrium quntity of exports of country A under free trde cn be obtined by either substituting P in the excess supply eqn. (15) or by subtrcting quntity demnded (eqn. 23) from quntity supplied eqn. (22) (24) Q es b b b b b b b b α X β Z α X β Z The exports of country A depend on the prmeters nd supply nd demnd shifters in both the countries. More specificlly, the fctors tht increse the domestic supply in country A will increse the quntity of exports becuse these fctors expnd the supply by shifting the domestic supply curve to the right. The positive domestic demnd shifters in country A will reduce the quntity of exports becuse these fctors increse the domestic demnd, by shifting the demnd curve to the right, thereby crowding out the vilbility of the commodity for export. The positive supply shifters in country B will
17 17 reduce the exports of country A becuse more of this commodity is now vilble in country B s the supply shifts to the right nd the excess demnd shifts to the left. The fctors tht increse the domestic demnd in country B will expnd the quntity of exports of country A becuse more of this commodity is demnded in country B s the demnd curve shifts to the right nd the excess demnd shifts to the right. Similr discussions for the fctors tht hve negtive effects on demnd nd supply cn be esily extended. Refer to figure 2.11 to understnd how determintion of grphiclly. Q es is illustrted P cn be substituted for P b in the supply nd demnd equtions (16) nd (17) to obtin the quntity supplied nd demnded in country B under free trde: (25) Q b s b b b b α2X α X 1β2Z 1β2Z nd b b b b β2Z β Z 1α2X 1α2X (26) b Qd The quntity supplied nd demnded in country B under free trde re functions of prmeters nd supply nd demnd shifters. More specificlly, the fctors tht increse the domestic demnd in both countries nd domestic supply in country B will ugment the quntity supplied in country B, becuse the demnd fctors boost the world price nd domestic supply shifters shift the supply to the right, which helps to increse the quntity supplied in country B. On the other hnd, fctors tht increse the supply in country A depress the world price which reduces the quntity supplied in B. The fctors tht
18 18 increse domestic supply in both countries nd the domestic demnd in country B will expnd the quntity demnded in country B, becuse the supply fctors lower the world price nd domestic demnd shifters shift the demnd to the right, which helps to increse the quntity demnded in country B. On the other hnd, fctors tht increse the demnd in country A boost the world price which reduces the quntity demnded in B. Similr discussions for the fctors tht hve negtive effects on demnd nd supply cn be esily extended. For quntity supplied nd demnded to be positive, the numertors in ( eqns. 25 nd 26) should be positive. The equilibrium quntity of imports of country B under free trde cn be obtined by either substituting P in the excess demnd eqn. (18) or by subtrcting quntity supple (eqn. 25) from quntity demnded (eqn. 26) (27) Q b ed b b b b b b b b α X β Z α X β Z The imports of country B depend on the prmeters nd supply nd demnd shifters in both the countries. More specificlly, the fctors tht increse the domestic demnd in country B will increse the quntity of imports becuse these fctors expnd the demnd by shifting the domestic demnd curve nd lso the excess demnd curve to the right. The positive domestic supply shifter in country B will reduce the quntity of imports becuse these fctors increse the domestic supply, by shifting the supply curve to the right, thereby reducing the need for imports, i.e., the excess demnd shifts to the left. The positive supply shifters in country A will increse the imports by country B becuse more
19 19 of this commodity is now vilble in the world mrket s country A s supply shifts to the right. The fctors tht increse the domestic demnd in country A will reduce the quntity of imports by country B becuse more of this commodity is demnded in country A s the demnd curve shifts to the right nd less is vilble in the world mrket, i.e., the excess supply shifts to the left. Similr discussions for the fctors tht hve negtive effects on demnd nd supply cn be esily extended. Refer to figure 2.11 to understnd how determintion of Q es is illustrted grphiclly.
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