International Journal of Development and Conflit 7(2017) 1 11 DECOMPOSING HE GAINS FROM RADE IN HE PRESENCE OF IME-CONSUMING CONSUMPION BINH RAN-NAM Shool of axation and Business Law, UNSW Sydney, NSW 2052, Australia Shool of Business and Management, RMI University Vietnam, Ho Chi Minh City, Vietnam We examine the deomposition of the gains from trade when onsumption is time onsuming in a simple open eonomy setting. While trade remains welfare improving, the soures of trade gainfulness differ from those in onventional trade models. In partiular, the onventionally defined exhange (onsumption) and speialisation (prodution) gains vanish. here are, however, positive gains from time realloation (away from prodution toward onsumption) and speialisation assoiated with this time realloation. Keywords: Gossen, Beker, Consumption time onstraint, Gains from trade, ime realloation JEL Classifiation: F10, F11, D11 1. Introdution and Context In a previously published artile in this journal (ran-nam, 2011) it has been argued that the theory of time alloation dates bak at least to Gossen, a Prussian ivil servant, whose lifetime work was ontained in a single book published in 1854. his book was a treatise on eonomi laws (i.e., laws ditated by nature) and moral laws (i.e., rules of human ondut). In his work, Gossen (1854; 1983) saw that what is ultimately sare is time alone. In his view, even in the land of Cokaigne where ommodities are freely available in unlimited quantities, there will still be an eonomising problem. Gossen s book wasrather hastily ompleted, very diffiult to read and also a ommerial failure (Georgesu-Roegen, 1983). As a result, his ideas were little known both within and outside Germany. When his work was brought to the attention of fathers of the Marginal Revolution, Jevons (1879: pp. xxxv xlvi) and Walras (1885) praised Gossen s ontribution to the theory of marginal utility but they did not pay attention to his primary emphasis on the onstraint of time. For a long time, Gossen s ontributions to eonomi theory were largely ignored by mainstream eonomists. he author is grateful to Professors Ngo Van Long and Murray C. Kemp for their invaluable omments. he usual aveat applies. Email: b.tran-nam@unsw.edu.au Copyright Binh ran-nam, Liensed under the Creative Commons Attribution-NonCommerial Liense 3.0
2 Binh ran-nam Muh later, Beker (1965) proposed a general theory of time alloation in whih households at as produtive agents who ombine time and market goods to generate vetors of basi ommodities. In introduing the onept of a household prodution funtion, Beker generalised the Gossenian onsumption time onstraint. Sine Gossen s book was obsure to English-speaking eonomists until the early 1980s, it is not surprising that Beker did not refer to the work of Gossen.Sadly, this neglet has surprisingly persisted until today, despite the availability of Blitz English translation of Gossen s book in 1983 with a very omprehensive introdution by Georgesu-Roegen (1983). Gossen s onsumption time onstraint has been mainly disussed in the ontext of losed eonomies; see, for example, Georgesu Roegen (1983), Niehans (1990) and Steedman (2002). In the ase of open eonomies, Kemp (2009) demonstrated that the normative theory of trade, inluding the well-known Grandmont MFadden (1972) and Kemp Wan (1972) propositions, survive the inorporation of a Gossenian time onstraint. More reently, ran-nam (2012) investigated the pattern of and gains from trade in a Riardian model inorporating a onsumption time onstraint. While trade remains potentially gainful, little is known about the soure of gains from trade in the presene of time-onsuming onsumption. In partiular, do the exhange (onsumption) and speialisation (prodution) gains remain valid when onsumption is itself time-onsuming? While this question has been partly onsidered in the small-ountry ase of a Riardian world (ran-nam,2012), it is unlear whether the findings an be extended beyond the one-fator, linear prodution tehnologies ase. he prinipal aim of this paper is to deompose the gains from trade in a Heksher Ohlin Samuelson (HOS) model under the assumption that the at of onsumption itself takes time. o fous on the deomposition of trade gains and for ease of graphial illustration, the analysis is based on a simple model of two ountries, two goods and two fators with representative agents. However, it will soon be apparent the basi reasoning of the analysis in this simple world would ontinue to hold in the more omplex ases with higher dimensions or ertain types of heterogenous agents. he problem under study is not only theoretially interesting but also empirially relevant. Speifially, there has been growing anedotal evidene that workers, espeially those in developed nations, feel inreasingly stressed beause they are time poor (see, for example, Shulte, 2014). Workers have been spending more time at work so that they do not have suffiient time for leisure and onsumption. his suggests that it is no longer sensible to dismiss the importane of balaning between working and onsumption. he remainder of this paper is organised as follows. In Setion I we onsider how the onsumption time onstraint an be interpreted and formulated mathematially. Setion 2 examines a simple losed eonomy with two goods, two fators, representative agents and time-onsuming onsumption. A simple HOS model is then analysed in Setion 3. It is shown that the exhange (onsumption) and speialisation (prodution) gains, as onventionally defined in the literature, both vanish. here are, however, a time alloation gain arising from realloating more labour time to onsumption, and a speialisation gain assoiated with this realloation. Setion 4 onludes.
Deomposing the Gains from radein the Presene of ime-consuming Consumption 3 2. he Consumption ime Constraint Apart from the literature assoiated with the work of Gossen and Beker (1960), remaining eonomi models, inluding the labour leisure hoie model or theory of trade with variable labour supply (see, for example, Kemp and Jones, 1962; Martin and Neary, 1980, Woodland, 1982; Mayer, 1991), all fail to reognise that onsumption takes time. Even when the importane of a onsumption time onstraint is reognised, there is still a debate about how time should be treated. It is an input or merely a ontext? Several authors (see, for example, Winston, 1982: 164; Steedman, 2001: 5) have argued that time is a ontext in Gossen ase while it is an input in Beker ase. In this respet, the approahes of Gossen and Beker are said to be fundamentally different. But regardless of how one may interpret the role of time in onsumption, it an be shown below that both approahes are similar, at least from a mathematial formulation point of view. he fat that onsumption takes time adds another onstraint to the onsumer hoie problem. Sine Gossen did not express his idea in mathematial form, it is somewhat unlear how his onsumption time onstraint should be formulated. A seemingly plausible n way is to start with L i = g i (X i ) and L L where X i refers to the amount of the i-th good i 1 i i= purhased, L to the number of time units required to onsume X i, L is the total number of time units available for onsumption, g i (0) = 0 and g i (0) is stritly inreasing (i = 1, 2,, n). hus, the simplest formulation of the Gossen onsumption onstraint, as adopted by n Niehans (1995) and Steedman (2002), is that ax L where a i stands for the number of i i= 1 time units required to onsume one unit of the i-th good (i = 1, 2,, n). Expressed in this way, it is not diffiult to see that the Gossenian approah an be viewed as a speial ase of Beker s prodution funtion approah, C i = h i (X i, L i ) and n Li L, where C i refers to the amount of the i-th basi ommodity (i = 1, 2,, n). i= 1 More speifially, if all h i take the simple Leontief form, i.e., C i = min{x i, L i /a i } (i = 1, 2,, n), then the Bekerian model simplifies into the Gossenian model. he approah adopted in this paper is both Gossenian and Bekerian. It is Gossenian in the sense that all onsumption tehnologies take the simple Leontief form. But it is also Bekerian in the sense that time is formally treated as a labour input to be expended in the onsumption proess. 3. he Closed Eonomy We begin by desribing the losed eonomy and deriving its autarki equilibrium. he eonomy is populated by idential agents who are endowed with labour and apital where labour (measured in time units) an be alternatively alloated between prodution and onsumption 1 while apital is employed in prodution only. For eah agent, onsuming and working annot be undertaken simultaneously. Aggregate endowments of labour and apital are denoted by L and K respetively. i 1 he at of onsumption an be broadly interpreted to inlude searh, purhase, preparation and onsumption.
4 Binh ran-nam In the output market, prie-taking, ompetitive firms produe two private goods with the aid of two essential inputs, labour and apital. he aggregate prodution funtions are written as Q i = F i (L i, K i ) where Q i is the output of the i-th setor, and L i and K i are the amounts of labour and apital employed in setor i respetively (i = 1, 2). It is assumed that F i (0, 0) = 0 and F i is a twie differentiable funtion that exhibits onstant returns to sale, positive, diminishing marginal produts, and stritly dereasing marginal rate of tehnial substitution along any isoquant. hus, F i is onave. he problem faing the produer of good i is to maximise P i Q i WL i RK i by the hoie of (L i, K i ) subjet to the prodution onstraint and the non-negativity of inputs where P i is the prie of good i (i = 1, 2), and Wand R are the wage and rental rates respetively. Let X i denote the quantity of the onsumption good i that the onsumer purhases, and let C i denote the quantity of the final onsumption good i that the onsumer wishes to enjoy. o transform X i into C i, the onsumer needs to use another input, alled onsumption time. he Leontief onsumption tehnology is assumed, i.e., C i = min{x i, L i / a i } where L i refers to the number of time units required to onsume X i,a i (> 0) is the tehnologial oeffiient assoiated with good i(i = 1, 2). For example, suppose a 1 = 3, then if the onsumer wants C 1 = 1, he/she needs to hoose X 1 = 1 and L 1 = 3. Note that information searh, time spent on purhasing, et, an be inorporated into the tehnologial oeffiients of this Leontief onversion tehnology. Sine agents are idential it is possible to speak of a soial utility funtion. he utility funtion is summarised as U(C 1, C 2 ) where U is supposed to be homotheti and twie differentiable with positive marginal utility and stritly diminishing marginal rate of substitution along any indifferene urve. hus, U is stritly quasi-onave. It is also assumed that MRS 21 dc 2 /dc 1 (0) as C 1 /C 2 0 ( ). he onsumer s problem is to maximise U(C 1, C 2 ) by the hoie of (C 1, C 2 ) subjet n to the time onstraint Li L and finanial onstraint where the finanial onstraint i= 1 varies depending on whether the eonomy is losed or open. wo remarks deserve mention. First, while pooled onsumption (eating or wathing V together) is allowed for, onsuming jointly does not give an agent more satisfation than onsuming alone. Seondly, a vast majority of leisure-related ativities, suh as reading a book, or skiing, involves ombining intermediate goods with sare onsumption time. Suh ative leisure as well as passive leisure (where neither apital nor a ommodity is required) an be straightforwardly aommodated in the present model. In this simple eonomy, the ompetitive equilibrium an be obtained as the solution to the entral planner s problem. In the absene of international trade, fousing on the real side of the eonomy, the model onstraints an be expressed in the form of inequalities as follows: F i (L i, K i ) Q i 0, i = 1, 2 (1) Q i X i 0, i = 1, 2 (2) min{x i, L i / a i } C i 0, i = 1, 2 (3) L L 1 L2 L1 L 2 0 (4)
Deomposing the Gains from radein the Presene of ime-consuming Consumption 5 K K 1 K 2 0 (5) i i i i i i K 0, L 0, L 0, Q 0, X 0, C 0, i= 1, 2. (6) where (2) and (3) an be thought of as the finanial a nd onsumption t ime onstraints respetively. Any vetor y {C 1 1, C 2, X 1, X 2, Q 1, Q 2,K 1, K 2, L 1, L 2, L, L 2 } that satisfies (1) (6) is said to be feasible. We define the set S of feasible alloations as S {y 12 + suh that (1) to (6) hold} (7) By a theorem in akayama (1974) on onave programming, if all inequality onstraints are expressed in the form z j (.) 0, j = 1, 2,, m, and if eah of these z j is a onave funtion (possibly linear), 2 then the feasible set is a onvex and non-empty set. Next, we define the projetion of the feasible set S into the feasible final-goods spae (C 1, C 2 ). Call this set S C. S C {(C 1, C 2 ) 2 + suh that y S} (8) It is well-known that the projetion of a onvex set in + m 2 into + is itself a onvex set. S C is therefore also a non-empty and onvex set. he maximisation of a stritly quasi-onave funtion U(C 1, C 2 ) over a onvex set S C yields a unique solution ( C1, C 2 ) (akayama, 1974). his result an be summarised as Proposition 1. here exists one and only one autarki equilibrium in this eonomy. he upper boundary of S C an be expressed as C 2 = f(c 1 ) where f is onave and stritly dereasing. he funtion f aptures information about resoure endowments, and prodution and onsumption tehnologies. he graph of f is the lous of all maximal onsumption points of the eonomy under autarky and an thus be thought of as the autarki onsumption possibility frontier (CPF). C Let LM L L be the amount of labour time devoted to manufaturing. We define the prodution possibility set S Q {(Q 1, Q 2 ) 2 + :Q i F i (L i, K i ), L 1 +L 2 L M and K 1 +K 2 K }. (9) he upper boundary of the set is the prodution possibility frontier, and an be represented by the stritly dereasing and onave funtion Q2 = ψ ( Q1; LM, K). We are partiularly interested in the speifi urve Q2 = ψ ( Q1 ; LM, K), where 2 2 C C C C LM L L and L L1 + L2 = ax i i = ac i i refers to the equilibrium time i= 1 i= 1 devoted to onsumption. his urve generates the relevant relative supply urve Q 2 /Q 1, as an inreasing funtion of their relative prie p P 2 /P 1. Sine we know X2/ X1 = C2/ C1, we an pin down the equilibrium relative supply, Q / Q = X / X, under autarky. his point 2 1 2 1 2. It is well-known that the funtion min{.,.} is a onave funtion.
6 Binh ran-nam on the relative supply urve determines the equilibrium relative prie p P2/ P1. Without loss of generality, let P 1 = 1 so that p = P2. Given the relative prie p, we an now work bakward to find the equilibrium fator pries in terms of good one, W and R, by using the onditions that the prie of eah good is equal to its unit ost 1 (W, R) = 1 and 2 (W, R) = p. he autarki equilibrium an be illustrated graphially by noting that, for given any value LM (0, L), the PPF, Q2 = ψ ( Q1; LM, K), and the onsumption-time budget line, Q 2 = [L L M aq 1 1 ]/a 2, an be plotted. hese two urves may not interset at all or interset(at one or two points)or tangential in the positive quadrant. he intersetion and tangential ases are illustrated in Figures 1 and 2, respetively, where the urve P P stands for the PPF and the line for the onsumption-time budget. When P P lies everywhere below (above), agents devote too little (muh) time to working so that after onsuming all inome they still have some surplus time (or they do not have suffiient time to onsume all inome). his an be referred to as an inome-poor (time-poor) situation. Neither an inome-poor nor a time-poor alloation an be optimal beause eonomi agents an always onsume more of both goods by devoting more (less) time to prodution. hus, only points of intersetion (K in Figure 1-a or K and K in Figure 1-b) or tangeny (K in Figure 2) an be maximal onsumption points under autarky. As L M varies from 0 to L, the lous of suh points traes out the CPF mentioned previously. Good 2 P K O Figure 1-a: One-intersetion ase P Good 1 Good 2 P K K O Figure 1-b: wo-intersetion ase P Good 1
Deomposing the Gains from radein the Presene of ime-consuming Consumption 7 Good 2 P K O Figure 2: angential ase P Good 1 4. he rading World Now we turn our attention to international trade. For simpliity, the world is supposed to onsist of two ountries, alled home (H) and foreign (F). Eah ountry produes two idential private goods with the aid of two essential inputs, labour and apital, as desribed in Setion 2. It is assumed that labour, apital and final onsumption goods annot be traded, but trade in produed goods is free, ostless and balaned. Suppose that the autarki equilibrium relative prie of produed goods in H, p differs from that in F, p F. hen there is an inentive for the two ountries to trade. Let the world terms of trade (relative prie of intermediate two in terms of intermediate good one) be denoted by p w. he finanial onstraint (2) in the losed eonomy ase of H now beomes the balane of trade onstraint: (Q 1H X 1H ) + p w (Q 2H X 2H ) 0 (2) Given p w, it is possible to determine the exess demand of eah output. his an be easily done if H s post-trade prodution is not ompletely speialised. When both goods are produed, the prie-equals-ost onditions uniquely determine the fator pries W H and R H : W H = W H (p w ) and R H = R H (p w ).hen the virtual national inome of H, whih inludes the imputed value w w w of labour used in onsumption ativities, is YH ( p, LH, KH) = WH( p ) LH + RH( p ) KH. he pries of the two final onsumption goods are the world pries plus opportunity osts of onsumption, i.e., π 1H = 1 + a 1H W H (p w ) = p 1H (p w ) and p 2H = p w + a 2H W H (p w ) = p 2H (p w ). It is well known in the trade literature that the ompensating-variational measure of gains from trade an be deomposed into onsumption (exhange) gain and prodution (speialisation) gain(see, for example, Bhagwati and Srinivasan, 1983: 167 168). Exhange gain refers to the gain that a trading ountry an enjoy when, in free trade, it was onstrained to produe at the autarki prodution bundle whereas onsumption was allowed to be at international pries. Speialisation gain then refers to the additional gain a trading ountry an enjoy from being allowed to shift prodution under free trade from the autarki equilibrium, H
8 Binh ran-nam to the post-trade equilibrium aording to the priniple of omparative advantage. In the present model, exhange gain an be interpreted as the gain that would arue if the home ountry ontinued to alloate L MH to prodution and produe the autarki equilibrium bundle ( X1H, X 2H). Under free trade, the home ountry an now afford a bundle whih lies beyond its autarki equilibrium PPF and belongs to an indifferene urve that is higher than the autarki equilibrium one. However, holding the amount of time alloated to onsumption onstant at L CH, agents in the home ountry are unable to have suffiient time to fully onsume that new bundle. his is depited in Figure 3 where any bundle along the dotted ray A h B or A h C (representing world terms of trade) is finanially affordable but not time feasible to the home ountry. hus, there is no exhange gain in the present model. In fat, more generally, there is no exhange gain for any given alloation of time to onsumption. Good 2 B P A h C P Good 1 Figure 3: Neither exhange gain nor (onventional) speialisation gain from trade. In general, the home ountry would beome time poor under trade if the autarki level of time alloation between onsumption and prodution was maintained so that neither exhange gain nor speialisation gain would be possible. hus, for trade to be gainful, there must be first a realloation of labour time away from prodution toward onsumption. his would eliminate, or at least lessen, time poverty and then allow a realloation of inputs between the two produtive setors aording to the priniple of omparative advantage. In this sense, there is a speialisation gain assoiated with time realloation between onsumption and prodution. If a trading ountry devotes less time to prodution, its PPF shifts downward relative to the autarki PPF. By produing at a point where the marginal rate of transformation is equal to the world terms of trade and trading at international pries, the home ountry an finanially afford a bundle that lies beyond its autarki equilibrium PPF. At the same time, having more time available for onsumption, eonomi agents in the home ountry now also have suffiient time to fully onsume this bundle. Note that the post-trade prodution point
Deomposing the Gains from radein the Presene of ime-consuming Consumption 9 is onsistent with the theory of omparative advantage in the sense that if ph > ( < ) p, the home ountry will export good two (one). he above argument is illustrated in Figure 4. In this Figure, H is assumed to be relatively more labour-abundant than F. H s endowments of apital and labour time are KH = 200 and LH = 400 respetively whereas for F, K F = 320 and L H = 300. For simpliity, we also assume that a 1 = a 2 = 1 for both H and F. Further, good one is assumed to be always more labour intensive than good two. Good 2 w 200 180 170 Y 100 Af B Ah 30 X 30 100 170 180 200 Good 1 Figure 4: ime alloation and speialisation gains from trade (a 1 = a 2 = 1) L = 400; K = 200; L = 300; K = 320 H H F F A A A A MH CH MF CF L MH L CH L MF L CF Autarky: L = 220; L = 180; L = 120; L = 180 Post-trade: = 200; = 200; = 100; = 200 H s autarki equilibrium is at point A H where 120 units of good one and 60 units of good two are produed. hus 180 units of labour will be needed in onsuming these outputs. H s A labour employment in manufaturing under autarky is L MH = 400 180 = 220. Its PPF is A Q2H ψ ( Q1 H, LMH, KH ) = ψ(q 1H ; 220, 200). F s autarki equilibrium is at point A F where 60 units of good one and 120 units of good two are produed. hus 180 units of labour will also be needed in onsuming these outputs. F s labour employment in manufaturing under A A autarky is L MF = 300 180 = 120. Its PPF is Q2F ψ ( Q1 F, LMF, KF ) = ψ(q 1F ; 120, 320). As drawn, H s autarki welfare is the same as that of F (note that both ountries have the same autarki onsumption-time budget line).
10 Binh ran-nam Now allow both ountries to trade. H will export good one (whih it has a omparative advantage) and import good two, and F will export good two and import good one. Both ountries beome wealthier and an attain the higher level of welfare as depited by point B in Figure 4. At B eah ountry onsumes 100 units of good one and 100 units of good two. his means eah of them needs 200 units of labour time for post-trade onsumption. his also implies that, as a result of trade, less labour is alloated to manufaturing in both ountries: L MH = 400 200 = 200 and L MF = 300 200 = 100. heir PPFs are shifted downward (not drawn) to Q2H = ψ ( Q1 H ; LMH, KH ) = ψ(q 1H ; 200, 200) and Q2F ψ ( Q1 F, L M F, K F ) = ψ(q 1F ; 100, 320). H and F post-trade prodution points are X(170, 30) and Y(30, 170) respetively. In summary, we may now state Proposition 2. here is neither exhange gain nor (onventional) speialisation gain in this simple trading world in whih onsumption takes time. However, there is a time alloation gain (under whih a trading nation will devote more time to onsumption relative to autarky) and a speialisation gain assoiated with this time realloation. 4. Conluding Remarks he present paper has examined the gains from trade in a simple HOS world when onsumption takes time. It began by onsidering the equilibrium of a losed eonomy with representative agents, two goods, two fators and Leontief onsumption tehnologies. It was shown that key results of the onventional model would arry over to the present model with the traditional PPF being replaed by the CPF whih an be defined as the lous of all maximal onsumption points under autarky. urning to international trade, the introdution of time-taking onsumption implies substantial modifiation to the onventional deomposition of the gains from trade. his is beause trade in produed goods an take plae but not trade in time. While trade in goods expands agents finanial affordability, their onsumption time onstraints remain. As a result, both the exhange and speialisation gains, as onventionally defined in the literature, vanish in the presene of a time-taking onsumption. here are, however, positive gains from time alloation (shifting labour away from prodution toward onsumption) and speialisation assoiated with that time realloation. For simpliity of analysis and ease of graphial illustration, many simplifiation assumptions have been made inluding the number of goods and ountries, onstant time rate of onsumption time and representative agents. It should be apparent that the assumptions onerning the dimensionality of the model are not essential beause the reasoning of the model remains valid when there are more than two goods or two ountries. Similarly, assuming variable rates of onsumption, the results of the model also ontinue to hold if the aggregate onsumption-time budget urve is onave. Heterogeneity of eonomi agents an be aommodated in a limited way as follows. If all agents have idential homotheti preferenes and Leontief onsumption tehnologies,
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