Neoclassical Trade Theories PDF

Neoclassical trade theories Lesson #5 Economics of International Trade [EM1057] Prof. A. Paradiso Copyrighted work. Do not copy, cite, or distribute without permission of the author. Content...

Neoclassical trade theories Lesson #5 Economics of International Trade [EM1057] Prof. A. Paradiso Copyrighted work. Do not copy, cite, or distribute without permission of the author. Contents 1 Introduction 2 2 Characteristics of neoclassical economics 3 3 The neoclassical interpretation of Ricardo's theory of comparative advantage 5 3.1 Theoretical formulation of Ricardo's theory: Assumptions............. 5 3.2 Ricardo's model in details.............................. 6 3.2.1 Supply side.................................. 6 3.2.2 Demand side................................. 12 3.2.3 Equilibrium.................................. 14 3.2.3.1 Autarky equilibrium........................ 14 3.2.3.2 Free trade equilibrium....................... 17 3.3 Ricardian theory of international trade: Neoclassical version vs. original version 20 3.4 Extensions of the neoclassical Ricardian model and empirical evidence...... 20 3.4.1 The generic case of N goods and M countries............... 20 3.4.2 The empirical evidence............................ 22 3.4.2.1 Test on Ricardian's hypothesis................... 22 3.4.2.2 Revealed comparative advantage................. 23 4 The neoclassical model of international trade 24 4.1 The neoclassical model: Assumptions........................ 25 4.2 Neoclassical's model in details............................ 26 4.2.1 Supply side.................................. 27 4.2.2 Demand side................................. 29 4.2.3 Equilibrium.................................. 30 4.2.3.1 Autarky equilibrium........................ 30 4.2.3.2 Free trade equilibrium....................... 33 1  4.3 Neoclassical theorems................................. 35 4.3.1 FPE theorem................................. 36 4.3.2 SS theorem.................................. 37 4.3.3 RY theorem.................................. 38 4.3.4 HO theorem.................................. 38 4.4 Extensions of the Neoclassical Model and Empirical Evidence.......... 39 4.4.1 The General Case of N Goods, M Countries, and H Factors: The HOV Model..................................... 39 4.5 Empirical Tests of the HOV Model......................... 41 4.5.1 Leontief (1953)................................ 42 4.5.2 Baldwin (1971)................................ 42 4.5.3 Leamer (1980, 1984)............................. 42 4.5.4 Bowen, Leamer, and Sveikauskas (1987).................. 42 4.5.5 Treer (1993, 1995).............................. 43 4.5.6 Davis and Weinstein (2001)......................... 43 4.5.7 Summary of Empirical Findings....................... 43 5 Conclusions 43 A The Distribution of Capital and Labor Between the Two Sectors 47 1 Introduction The classical economists, exemplied by Smith and Ricardo, asserted that free international trade plays a pivotal role in fostering economic growth and development. According to Smith, international trade serves as a catalyst for market expansion, facilitating the division of labor and engendering a self-sustaining process of growth over time. The signicance of international trade lies in its role as the primary driver of growth in Western economies, representing the `rst mover' in their developmental process, as envisioned by Smith. In his perspective, the trajectories of economic development and international trade theories are seamlessly integrated. Similarly, for Ricardo, international trade assumes a paramount role in driving economic growth, particularly in economies characterized by diminishing returns in the agricultural sec- tor, such as England. In these contexts, foreign trade becomes the crucial element preventing the decline of the prot rate. As in Smith's framework, development and international trade theories are interconnected in Ricardo's analysis. However, in contrast to the dynamic nature of the development process, the theory of international trade in Ricardo is static. This introduces coherence challenges in Ricardo's theoretical framework, given the intrinsic dierences in the dynamic development process and the static nature of international trade theory. Towards the end of the 19th century, the neoclassical school replaced the classical school as the dominant economic thought. Their economic perspective was radically dierent from 2 the classical school. Changes in the methodological approach, the subject, and the tools of economic analysis also had a signicant impact on the study of international economics. While the classical school aimed to study the forces that allowed the development and growth of a country, the neoclassical school focused on studying the optimal allocation of scarce resources applicable to alternative uses. The shift in the goal (or the subject of study) had important consequences in the way economic phenomena were studied. No longer focused on economic aggregates (social classes), but rather on individuals behaving rationally by maximizing a certain mathematical function subject to a constraint. This `simple' way of explaining agents' actions allows for mathematical modeling of these actions. The economic system becomes the sum of individual behaviors, and the equilibrium of the system is calculated by balancing demand with supply. The equilibrium of this system is dened precisely as `general' because it represents an overall balance where the quantity demanded equals the quantity supplied across all markets in the economy. One characteristic of traditional general equilibrium economic models is their lack of dy- 1 namics. These models are fundamentally static as they focus on a specic point in time, representing a simultaneous equilibrium between supply and demand for all goods and factors of production within an economy. The chapter is organized as follows. Section 2 describes the characteristics of the neo- classical school, highlighting the dierences with the classical school. Section 3 reinterprets Ricardo's theory of comparative advantage from a neoclassical perspective. Section 4 presents the neoclassical model of international trade. Section 5 concludes. 2 Characteristics of neoclassical economics The neoclassical school diers from the classical school in various aspects: 1. Object of study: The classical economists are concerned with studying the conditions for the reproduction of an economic system, while neoclassical economists focus on how the economic system optimally utilizes scarce resources. 2. Methodological approach: The classical school analyzes economic phenomena that, by their nature, evolve over time, such as growth. In contrast, the neoclassical school focuses on static equilibria, where the temporal dimension is not relevant. 3. Subjects of the economy: The classical school studies the behavior of aggregates or groups of subjects, such as social classes. Neoclassical economists, on the other hand, analyze the behavior of individual actorsconsumers and rms. Aggregates are considered as the sum of individual behaviors. 1 The term `traditional' is used to emphasize that, in the last 30 years, dynamic neoclassical models known as DSGE (Dynamic Stochastic General Equilibrium) have been developed, which consider changes over time. This implies that prices, quantities produced and consumed, as well as choices made by consumers and rms, can vary over time. 3  4. Relationship between subjects: Classical economists see social classes in a conictual relationship, from which capital shares to invest are derived. Neoclassical individuals behave collaboratively, determining system equilibrium through the interplay of supply and demand. 5. Behavior of subjects: Social classes follow social conventions or historical traditions. Neo- classical individuals act rationally, seeking to maximize their interests subject to avail- ability constraints. From the discussion, it is evident that the classical school regards economics as a social and historical science, requiring multidisciplinary approaches. In contrast, neoclassical economists view economics as a quantitative science, formalizing individual behaviors mathematically. Moreover, let us consider the concept of equilibrium according to the two schools. While 2 classical economists focus on the distributive equilibrium of pre-determined output, neoclas- sical economists achieve equilibrium by equalizing production with the overall demand of the economy. This equilibrium is termed `general' because it implies that the equality between output and demand holds across all markets in the economy. Neoclassical equilibrium is static and enduring over time, indicating that investments do not uctuate over time as the capital remains constant. However, as the name `neoclassical school' suggests, they must have inherited something from the previous school. This inheritance is particularly represented by: 1. Say's Law. Both schools adhere to the idea that supply generates its own demand, establishing a causality from supply to demand. 2. Laissez-faire. Both schools advocate for non-intervention in the economy. They believe that state intervention does not improve or may even worsen economic conditions. 3. Principle of diminishing returns. The neoclassical school adopts Ricardo's idea of dimin- ishing returns in agriculture and applies it to all sectors. With the increasing use of a factor, holding others constant, the additional quantity produced of a good increases but at a diminishing rate. In the context of international trade, the main changes will involve: 1. The formalization in mathematical terms of supply and demand functions in dierent markets. 2. A static analysis of equilibrium in which investments play no role in increasing the capital endowment in the system. 3. The use of the comparative statics tool to assess equilibriums both in autarky and in an open economy. 2 Itis worth recalling, in light of section 1 of chapter 4, that output is pre-determined based on production conditions expressed by equation (2). 4  In this process of `transformation' in approach, methodology, and tools used to study trade aspects, it is pertinent to begin by examining how Ricardian theory can be translated into neoclassical terms. This translation has the advantage of highlighting conceptual and method- ological dierences between the two schools of thought, allowing for a better understanding of the theoretical foundations of international trade in the neoclassical framework. 3 The neoclassical interpretation of Ricardo's theory of comparative advantage The neoclassical reformulation of Ricardo's theory of comparative advantage necessitates the modeling of supply and demand. It is important to emphasize that demand was not part of Ricardo's original theoretical framework (this is why Ricardo does not provide a determination of the terms of trade, as discussed in Section 3.2.1 of chapter 4). This reformulation has also led to other theoretical implications. While for Ricardo, the source of comparative advantage was exogenously determined by natural factors, the neoclassical reformulation attributes this source to dierent technologies among countries. 3.1 Theoretical formulation of Ricardo's theory: Assumptions Before delving into the formalization, let us highlight the assumptions of the neoclassical Ri- cardian model: 1. There are two countries (A and B ), each producing two nal goods (computers C and wheat G). 2. There are no barriers to trade between the two countries. This implies that the two goods are `tradable'. 3. In the production of goods, only labor is employed. Capital does not exist. Therefore, the model under consideration is a single-factor model: labor. 4. The production functions of the two goods have xed coecients, so the production per worker (i.e., productivity) is constant and does not change with the scale of production. 5. Technology diers across countries and goods. However, the goods produced between the two countries are homogeneous. 6. Labor is perfectly mobile within the country but completely immobile (i.e., non-tradable) across borders. 7. Perfect competition characterizes both industries in both countries and there are no trans- portation and transaction costs. 8. Each nation employs all available labor. 5  9. The individual worker is compensated based on their cost of utilization in the production process, which corresponds to the wage rate w. 10. Residents in all nations have dierent tastes and preferences, implying that community indierence curves for A and B are distinct. 11. Trade between nations is perfectly balanced: imports are equal to exports in value. Based on these assumptions, we can make the following observations: 1. Assumption #6, #7 and #9 implies that wage rates (w ) equalize only among sectors within the domestic economy (i.e., wCA = wG A = wA ̸= wCB = wG B = wB ). 2. Assumption #8 implies that all resources are utilized. This result is a consequence of the acceptance of the so-called Say's Law by Ricardo (and most authors of the classical school). A consequence of this law is that entrepreneurs always employ all available resources because they know there will always be a demand capable of `balancing' the generated supply. 3. A xed-coecient production function (assumption #4) is a type of production function that satises the condition of constant unit production cost with respect to the volume of production. This characteristic, as we will see in detail shortly when we formally analyze this function, has important implications for the model's outcomes. 3.2 Ricardo's model in details Building upon the assumptions outlined in section 3.1, it is possible to construct a hypothetical model of a global economy with two countries, in which only dierences in labor productivity and labor costs between countries inuence trade. In this model, the crucial factor that creates conditions for international trade lies precisely in the observed dierence in labor productivity between the two countries. According to the neoclassical view of the Ricardo model, this dierence is attributed to distinct technologies between countries. Hence, the source of the dierence in production costs between countries, which Ricardo attributed to natural factors, is now determined by technological factors. Both sources of dierence, however, remain exogenous. 3.2.1 Supply side We have already mentioned in section 3.1 that the production functions in the Ricardian model are linear (with xed coecients). Formally, we can write: X i = di · Li , (1) 6 where Xi is the quantity produced by a generic country i (for i = A, B ), di is its production i 3 coecient, and L is the labor employed. The linearity of technologies has some important implications. The productivities, both average and marginal, coincide and are equal to the production coecients: Xi ∆X i di = =. (2) Li ∆Li Here, di represents the quantity of Xi produced on average by each worker, and it is also the i additional quantity of X produced by the last worker. As you can see, productivities are constant and do not depend on the level of production. The reciprocal of the production coecient also has economic signicance. The ratio 1/di represents the amount of labor required to produce one unit of good X i. In other words, 1 = di · (Li /X i ) ⇒ 1/di = Li /X i. The assumption of perfect competition in the labor market (as well as in the goods market) i implies that the marginal cost (denoted as M CX ) must equal the marginal revenue represented by the unit price of the good PXi : PXi = M CX i = ai · w i , (3) where ai ≡ 1/di , and wi indicates the unit labor cost. What we have derived applies to the production of both goods, computers and wheat, by countries A and B, with the caveat that production techniques (and thus production coecients) vary from sector to sector and from country to country. For this reason, to avoid confusion from now on, we will denote the quantity of computers produced by country i (for i = A, B ) i i as C and the quantity of wheat produced by country i (for i = A, B ) as G. The respective production functions can thus be written as follows: C i = diC · LiC (4) Gi = diG · LiG , where LiC and LiG indicate the quantities of labor employed in the production of C and G in country i. At this point, we need to introduce the concept of the Production Possibility Frontier (PPF), which represents all the combinations of goods that can be eciently produced by a country using all available resources. The PPF takes into account the country's resource constraints and production technology, emphasizing that all resources are fully utilized within the considered economic system, held constant throughout the analysis. Based on equation (4), we know that the quantity of labor used in the production of Ci is i i equal to LC. So, if the total available labor quantity is L̄ , then the amount of labor that will i i be used in the production of G must be equal to L̄ − LiC. The quantity produced of Gi will 3 Equation (1) represents the `single-factor' version of the `multi-factor' xed-coecient production function, akin to the one presented in chapter 4, under the assumption that A is set to one. 7 therefore be equal to: Gi = diG · (L̄i − LiC ) = diG · L̄i − diG · LiC. (5) From the production of C i, we know that LiC = aiC · C i , so by substituting, we have: L̄i aiC Gi = − ·C i. (6) aiG aiG |{z} |{z} vertical intercept slope The equation (6) represents the PPF for the economy under consideration (country A or B) as it represents the potentially producible quantities of both G and C while utilizing all available resources (labor). Graphically, equation (6) can be represented as depicted in Figure 1. The vertical intercept i (L̄ /aiG ) represents the maximum value of grain produced in country i when country i does i not produce computers. Similarly, the horizontal intercept (L̄ /aiC ) represents the maximum value of computers produced in country i when country i does not produce grain. The PPF is a straight line with a negative slope because the slope is negative. The absolute value of the slope of the PPF provides us with a measure of the opportunity cost of producing computers instead of wheat: |aiC /aiG |. i i Naturally, the reciprocal of this ratio (|aG /aC |) provides us with the opportunity cost of producing wheat instead of computers. Figure 1: Production possibilities frontier Specifying the countries, three potential cases that may arise are represented in Figure 2: Case #1: A has an absolute advantage in producing computers (LA /aA B B G > L /aG ), whereas B has an absolute advantage in producing wheat (LA /aA B B C < L /aC ). We can also see that A has a comparative advantage in producing computers because the absolute value of the A A B B slope of its PPF is lower than that of B's PPF (|aC /aG | < |aC /aG |). Obviously, the inverse A A B B is also true: B exhibits a comparative advantage in producing wheat (|aG /aC | > |aG /aC |). A A Case #2: A has an absolute advantage in producing computers (L /aG > LB /aB G ), while 8  A A there is no advantage for either of the two countries in producing wheat (L /aC = LB /aB C ). A A A exhibits a relative advantage in computer production (|aC /aG | < |aB B C /aG |). Case #3: A has an absolute advantage in the production of both goods (LA /aA B B G > L /aG A A B B and L /aC > L /aC ). A still exhibits a comparative advantage in computer production A A B B (|aC /aG | < |aC /aG |). Figure 2: Three cases At this point, it is appropriate to introduce some dening terms. The opportunity cost of producing one good rather than another is also referred to as the Marginal Rate of Transfor- mation (MRT). So, in our case, we have: aiC M RTCi = OCCi = = abs. value of the slope of PPF of country i aiG (7) ai M RTGi = OCGi = iG = reciprocal of the abs. value of the PPF slope of country i. aC Here OCCi (OCGi ) represents the opportunity cost of producing good C (G) instead of good G (C ), and M RTCi (M RTGi ) refers to the marginal rate of transformation when considering 4 producing C (G) rather than G (C ). Since the unit prices of goods are equal to the marginal costs (see (3)), condition (7) can be rewritten as: wi · aiC M CCi PCi M RTCi = = = wi · aiG M CGi PGi (8) w i · ai M CGi PGi M RTGi = i G = =. w · aiC M CCi PCi 4 Incomparative advantage models, opportunity costs and marginal rates of transformation between two goods are equivalent concepts and refer to the same idea. Both express how much of one good must be sacriced to obtain an additional unit of the other good. Therefore, in practical terms, they represent the same trade-o relationship between the two goods in production. 9 The MRT equals: (1) the ratio of unit labor requirements; (2) the ratio of marginal costs; (3) the relative price of one good respect to the other. Based on these denitions, we are now able to establish the conditions that must be met for it to be advantageous for both countries to engage in trade with each other. The condition that must hold is that in autarky (i.e., in the absence of international trade), the following equation is valid: PCA PCB M RTCA B = A ̸= M RTC = B. (9) PG PG Of course, if (9) holds, then it also holds that M RTGA ̸= M RTGB. Returning to the three cases outlined above, it is clear that in all of them we have: PCA PCB M RTCA = < M RT B C =. (10) PGA PGB Therefore, A has a relative advantage in computer production, while B has a relative advantage in wheat production. In all three cases, the conditions for international trade are met. However, the most interesting case is case #3, in which even though A has an absolute advantage in the production of both goods, there are conditions for international trade because B has a relative advantage in the production of the good (G) for which A has a relative disadvantage. In the end, trade will benet both countries. This is the case considered by 5 Ricardo. From autarky to free trade So, if condition (10) holds, then it is advantageous for both countries to engage in a trade relationship. We already know from this condition that country A, having a comparative advantage in the production of good C, nds it benecial to trade C for G, and vice versa for country B. Two questions arise naturally: At what exchange rate do the countries trade goods? What quantity do they actually trade? On both points, we know that Ricardo did not provide an answer. He simply asserted that if the international exchange rates, dened as [PC /PG ]t , fell within the range of values:  t PCA PC PCB < < , (11) PGA PG PGB then both countries would benet from international trade. In practice, Ricardo reasoned as follows. He assumed that the international exchange rate (terms of trade) was exogenously xed at a value satisfying (11). In this neoclassical reinter- pretation of Ricardo's thought, this implies the following. Since technologies are linear (i.e., production is characterized by xed coecients), what holds true for the rst computer and the rst unit of wheat also holds true for all subsequent units. Therefore, if the terms of trade between the two goods in the international market are more favorable than the internal terms 5 In fact, he envisioned these two countries as England and Portugal, with the latter having an absolute advantage in the production of both goods. His question was: even in this unfavorable scenario, under what conditions can England nd it advantageous to trade with Portugal in order to improve its economic situation and increase its well-being? For more details on this point see section 3 in chapter 4. 10 of trade (i.e., if the exchange ratio between the goods in the international market is better than the ratio determined internally by the available technology), then it is advantageous for a country to specialize completely in the production of the good in which it has a competitive 6 advantage. Figure 3 helps us illustrate this. The production point for each country corresponds to the point where the use of production factors is total. So, country A's production is located at A A B B L̄ /a C , and country B's production is at L̄ /a G. International trade takes place along a line with an intermediate slope (gray) between the production frontiers (PPF) of the two countries. Figure 3: Terms of trade These new lines represent the national income line, which in autarky coincides with the PPF (because in closed economy equilibrium, the quantities produced internally are sold at autarkic prices), while in an open economy, they do not coincide because the quantity produced of the good is based on technical coecients, but then the exchange (sales) occurs based on interna- 7 tional exchange ratios. The slope of the income line, after international trade, corresponds t to −[PC /PG ]. So, the income actually obtained does not coincide with the output produced internally. From the graph, it can be seen that for any terms of trade that satisfy (11), the income lines are more favorable to both countries because they allow them to obtain more of both goods compared to the case of autarky. The contribution of John Stuart Mill was crucial in describing the mechanism for forming the terms of trade (see section 3.2.1 of chapter 4). To determine the quantities actually exchanged, we need to explicitly specify the preferences of consumers in the two countries. 6 However, for this situation to occur, the condition explained in footnote 9 must be satised. 7 See section 3.2.3.2 for more details. 11 3.2.2 Demand side We consider the residents of country A and Country B as two distinct yet unique `communities.' Each community comprises residents with identical preferences and tastes, which we will depict using 'community indierence curves.' The community indierence curves exhibit the following properties: 1. Each point along an indierence curve represents a specic combination of computers and wheat consumed by the community. 2. Each point along an indierence curve indicates a level of utility derived from consump- tion. 3. Points on the same indierence curve provide the same level of utility, while points above (or below) yield a higher (or lower) level of utility. 4. There exist an innite number of indierence curves in each graphical space. 5. Indierence curves are smooth and convex toward the origin, implying that each commu- nity faces diminishing marginal utility from consuming computers and wheat. 6. The shape and position of indierence curves dier between countries because tastes and preferences vary among countries. A utility function that satises the properties described above is the Cobb-Douglas type function, expressed in Equation (12), where the parameter i δCi (δG ) represents the income share spent on computers (wheat) in country i: i i U i = [CCi ]δC · [CGi ]δG with i δG = 1 − δCi < 1. (12) Figure 4: Indierence curves for A and B Figure 4 displays a possible graphical representation of utility functions of the type (12) for countries A and B. Each utility curve represents various combinations of goods C and G that provide the same level of satisfaction. To achieve a higher level of satisfaction, positioned higher on the graph, one needs a greater quantity of goods to consume. The more goods one can consume, the higher the achievable level of satisfaction. Countries A and B have dierent 12 preferences due to dierent values of the parameter δC. For example, in Figure 4 we assumed A B that country A has δC = 0.6, while country B has δG = 0.4. Looking at points A and B on nation A's indierence curve in Figure 4, it can be observed that they represent dierent combinations of computers and wheat but still provide the same level of satisfaction because they are on the same indierence curve. Based on what has been mentioned, residents of country A, to move, for example, from point A to point B, are willing to accept ∆G as compensation for the loss of ∆C. The rate at which they are willing to substitute A for B is equal to ∆C/∆G, which represents the slope of the line connecting the two points. As the segment narrows and the two points approach the limit, they converge at point C as shown in Figure 5. The slope of the line passing through point C on the indierence curve measures the `instantaneous' rate of substitution known as the Marginal Rate of Substitution (MRS). This term is dened as `instantaneous' because it reects people's willingness to ex- change one good for another at the `margin,' which means in small increments. In other words, MRS is the rate at which people are willing to trade a small (innitesimal) amount of one good for another without altering their overall level of satisfaction, keeping the margin of utility constant. The absolute value of the slope of the indierence curve represents the marginal rate of substitution (M RSC , where the subscript C indicates that we are calculating the slope of the indierence curve with good C on the horizontal axis of the graph, i.e., we are calculating how much G must change to compensate for a loss of utility due to a reduction in good C ): M UC δC CC M RSC = |slope ind. curve| = = · , (13) M UG 1 − δC CG where M UC and M UG are the marginal utility of C and G, respectively. Figure 5: Marginal rate of substitution for Nation A Analytically, the consumer's objective is to maximize (12), taking into account the income level I and the prices of the nal goods PC and PC. This is achieved by strategically choosing the consumption levels of the two goods that optimize this outcome. In formula: max U = [CC ]δC · [CG ]1−δC s.t. I = w · L = PC · C C + PG · C G , (14) CC ,CG where, for simplicity, we have omitted the country i index. 13  To solve this problem we dene the Lagrangean L, by using the Lagrange multiplier λ: L = [CC ]δC · [CG ]1−δC + λ · [I − PC · CC + PG · CG ]. (15) Deriving the two rst order conditions ∂L/∂CC = ∂L/∂CG = 0, after some manipulations, we get: CG∗ 1 − δC P C ∗ = ·. (16) CC δC PG where CG∗ and CC∗ are the optimal consumption values of G and C (i.e., the quantities consumed that maximize utility), respectively. Since PG · CG∗ = [(1 − δC )/δC ] · PC · CC∗ , substituting this result in the income constraint, we get: 1 − δC I = PC · CC∗ + · PC · CC∗. (17) δC After some manipulations, it is easy to nd that: PC · CC∗ = δC · I = δC · w · L, (18) from which: 1 δC · L CC∗ = δC · w · L · = , (19) PC aC where we used the (3). In a completely analogous manner, it must hold that: (1 − δC ) · L CG∗ =. (20) aG 3.2.3 Equilibrium In this section, we calculate the equilibrium of the neoclassical version of the Ricardian model in two scenarios: autarky equilibrium and open economy equilibrium. The equilibrium is discussed both in mathematical terms and through simulation (numerically simulating the analytically obtained model). Additionally, the simulation results are used to generate graphs, enhancing the intuitive presentation of the ndings. 3.2.3.1 Autarky equilibrium In autarky, the quantity produced coincides with the quantity consumed, as there is no pos- sibility to buy and/or sell goods with foreign countries. For this reason, the PPF coincides with the income line in a closed economy, and the terms of trade are equal to the ratio of the 14 production technical coecients. This implies that: δC · L C∗ = aC (21) (1 − δC ) · L G∗ = , aG ∗ where the symbol   denotes optimal production values that satisfy consumers' optimal pref- erences. Thus, they represent equilibrium values. It is also interesting to note that by dening ϕC ≡ LC /L (i.e., the share of workers employed in sector C for country i), it is possible to rewrite (1) (referring to the production of good C ) as follows: 1 C= · ϕC · L. (22) aC By equating this equation with (19), it is immediate to nd that: ϕC = δC , (23) which means that, in equilibrium, the share of income consumed on good C corresponds to the share of workers employed in sector C. The equilibrium in the neoclassical version of the Ricardian theory of comparative advan- tages can be numerically represented as shown in Table 1. In the table, we have provided the values of exogenous variables and the values of endogenous variables. Graphically, the equilibrium can be represented as illustrated in Figure 6. It corresponds to the point where the following condition occurs in each country: aC M RTC = |slope PPF| = = |slope inc. line| = M RSC aG (24) M UC δC C∗ PC = |slope ind. curve| = = · C∗ =. M UG 1 − δC CG PG This condition occurs at point A for country A and at point B for country B. The line that starts from the origin and passes through the equilibrium point represents the optimal consumption path. It represents the optimal combination of consumption of the two goods as economic 8 conditions change, especially in the relative prices of domestic and imported goods. As evident from the values in Table 1, country A has a comparative advantage over country B in producing good C (aA A B B C /aG = 0.73 < aC /aG = 2.67), and vice versa for good G (aA A G /aC = 1.36 < aB B G /aC = 0.37). Therefore, in the absence of barriers, the conditions for international trade exist. 8 In other words, the optimal consumption path corresponds to the following equation: CC = (aC /aG ) · [δC /(1 − δC )] · CG. 15 Table 1: Neoclassical version of Ricardo's theory of comparative advantage: Autarky equilibrium for country A and country B Countries Exogenous variables A B L 7.00 3.00 Share of income spent on C (δC ) 0.60 0.40 Production parameter of C (dC ) 1.50 1.20 Production parameter of G (dG ) 1.10 3.20 Unit labor cost of C (aC = 1/dC ) 0.67 0.83 Unit labor cost of G (aG = 1/dG ) 0.91 0.31 Endogenous variables Price C/ G (PC /PG = aC /aG ) Price 0.73 2.67 Labor input for C (LC = ϕC · L) 4.20 1.20 Labor input for G (LG = (1 − ϕC ) · L) 2.80 1.80 Production of C (C = dC · LC ) 6.30 1.44 Production of G (G = dG · LG ) 3.08 5.76 Consumption of C (CC = (δC · L)/aC ) 6.30 1.44 Consumption of G (CG = [(1 − δC ) · L]/aG ) 3.08 5.76 δ 1−δC Utility (welfare) level ([CC ] C · [CG ] ) 4.73 3.31 Income in PC (I/PC = L/aC ) 10.50 3.60 Income in PG (I/PG = L/aG ) 7.70 9.60 Auxiliary variables ϕC (= δC ) 0.60 0.40 Figure 6: Graphical representation of autarky equilibrium in neoclassical version of Ricardo's theory of comparative advantage 16 3.2.3.2 Free trade equilibrium The international trade equilibrium is also characterized by the maximization of utility and prot. However, the main dierence compared to the autarky case is that now the income available to consumers no longer coincides with the PPF. The reason is simple: the PPF is constructed based on the technical production coecients, which in autarky also determine relative prices. It is worth noting that in autarky, the ratio of product prices is independent of nominal wages, as follows: PCi /PGi = (wi · aiC )/(wi · aiG ). In the presence of international trade, relative prices are inuenced by the exchange activity between the two countries and are no longer xed by the technical production coecients. In the example discussed so far, following international trade, each country specializes in the 9 production of the good for which it has a comparative advantage over the other country. Thus, A specializes in the production of C, and B in the production of G. Therefore, the following relationships must hold: PC∗ = wA · aC (25) and PG∗ = wB · aG , (26) ∗ where now the symbol   indicates that the price values are those determined following inter- national trade. Taking the ratio, it is evident that the terms of trade depend on the ratio of wages in the two countries: PC∗ w A aC = ·. (27) PG∗ w B aG Analytically, the ratio of nominal wages that ensures equilibrium in the international mar- kets of the two goods is obtained by solving this system: CCA − C A = C B − CCB | {z } | {z } Excess demand for C in A Excess supply of C in B (28) CCG − C G = CG − CG. | {z } | {z C} Excess demand for G in A Excess supply of G in B In practice, it is required that in each market, the excess demand is oset by an excess supply. Given that there are only two markets and that if one good is not produced, all resources are employed in the production of the other good, then if one market for one good is in equilibrium, the other must be as well. Therefore, the condition to determine the ratio of nominal wages in the two countries that guarantees equilibrium can be derived by studying only one of the two markets. This condition, known as Walras's law, states that in the presence of n markets, if n − 1 are in equilibrium, the n-th one is as well. It is named after the economist who formulated 9 For full specialization to occur, the quantity produced by each country of the good for which it has a comparative advantage must cover the demand of both countries. Since, for any generic good X , the maximum quantity produced by a country corresponds to LX /aX , this condition depends on both the size of the country and production techniques. In our example, this condition is satised. 17 this theorem. Let us focus on the market for good C. Since each country specializes in the production of the good in which it has a comparative advantage, we know that good C is entirely produced by country A. Therefore, the equilibrium condition for this market can be written as follows: CCA − C A = −CCB. (29) Substituting the appropriate values, we immediately nd that: A 1 B 1 L · · (1 − δCA ) = δCB · wB · L · A , (30) aC w · aC from which: A wA L · (1 − δCA ) = B. (31) wB L · δCB According to (31), the wage ratio depends on the ratio of available labor forces and a combi- nation of consumer preferences in the two countries. Based on this result, it is possible to determine the values of all other variables in the case of free trade, as represented in Table 2. As observed, country A specializes in the production of good C, and country B in the production of good G. In the end, the welfare of both countries has increased compared to the autarky case, indicating that international trade has brought benets to all. In an open economy, given the changed terms of trade in international prices, the income line is no longer aligned with the PPF. Specically, the income line in a closed economy for country A corresponds to: A IA PA L aA A CGA = A − CA · CCA = A − C · CC , (32) PG PG aG aA G while in an open economy, it corresponds to: A ! wA wA      A L aC CGA = · − · · CCA. (33) wB aB G wB aB G From the comparison, it is immediately noticeable that both the vertical intercept and the slope have changed. However, it is interesting to note that despite this, setting the consumption of good Gto zero, the horizontal intercept (i.e., the maximum level of computers consumable) A A remains unchanged and determined by L /aC. A similar discussion applies to country B, with the caveat that now the vertical intercept (i.e., the maximum level of consumable wheat) remains unchanged. In the gure 7, we represent the free trade equilibrium from a graphical perspective. As seen, the income line now diverges from the PPF, and the equilibrium consumption combination 18 corresponds to the point where the indierence curve is tangent to the income line, i.e. where: PC∗ |slope inc. line| = = M RSC = |slope ind. curve|. (34) PG∗ Graphically, this point corresponds to point A(C) for country A and point B(C) for country B. The equilibrium production, with complete specialization, corresponds to the point where all resources (labor) are invested in the production of the good for which there is a comparative advantage. This point corresponds to point A(P ) for country A and point B(P ) for country B. Table 2: Neoclassical version of Ricardo's theory of comparative advantage: Free trade equilibrium for country A and country B Countries Exogenous variables A B L 7.00 3.00 Share of income spent on C (δC ) 0.60 0.40 Production parameter of C (dC ) 1.50 1.20 Production parameter of G (dG ) 1.10 3.20 Unit labor cost of C (aC = 1/dC ) 0.67 0.83 Unit labor cost of G (aG = 1/dG ) 0.91 0.31 Endogenous variables A A B Wage in A/ Wage in B (w /wB = L · (1 − δCA )/(δCB · L )) 0.43 0.43 A B A B Price C / Price G (PC /PG = (w /w ) · (aC /aG )) 0.91 0.91 Labor input for C (LC = ϕC · L) 7.00 0.00 Labor input for G (LG = (1 − ϕC ) · L) 0.00 3.00 Production of C (C = dC · LC ) 10.50 0.00 Production of G (G = dG · LG ) 0.00 9.60 Consumption of C (CC = (δC · w · L)/PC ) 6.30 4.20 Consumption of G (CG = [(1 − δC ) · w · L]/PG ) 3.84 5.76 δ 1−δC Utility (welfare) level ([CC ] C · [CG ] ) 5.17 5.08 Income in PC (I/PC = w · L/PC ) 10.50 10.50 Income in PG (I/PG = w · L/PG ) 9.60 9.60 Auxiliary variables ϕC (= δC ) 1.00 0.00 Figure 7: Graphical representation of free trade equilibrium in neoclassical version of Ricardo's theory of comparative advantage 19 3.3 Ricardian theory of international trade: Neoclassical version vs. original version In analyzing the principle of comparative advantage according to Ricardo, we have adopted the neoclassical framework. We provided both an algebraic and a numerical representation (based on the latter), which form the basis for a theory of international trade. This theory, which we termed neoclassical Ricardian, identies the determinants of compar- ative advantage and, consequently, international trade, in the technological dierences among countries; dierences that result in varying labor productivities. While this theory is formally impeccable, it does not fully reect Ricardo's original thinking, both in terms of the determi- nants of international trade and the purpose of international trade. We recall that he saw the `natural conditions' as the source of international trade, and 'natural conditions' do not neces- sarily imply dierent technology. Furthermore, he conceived international trade as a `tool' to prevent the fall of the prot rate and, consequently, to promote the growth process of a country. In contrast, in the neoclassical version, international trade primarily functions to enhance the well-being (utility) of consumers and the availability of goods in an economy. 3.4 Extensions of the neoclassical Ricardian model and empirical ev- idence In this section, we will explore extensions of the neoclassical Ricardian model and the associated empirical evidence. We will begin by demonstrating how to generalize the model to handle N goods and M countries. Subsequently, we will discuss key empirical ndings and delve further into the topic by introducing an extension based on the so-called Balassa Index, which highlights the conditions of revealed comparative advantage. 3.4.1 The generic case of N goods and M countries One of the main limitations of the theoretical Ricardian model lies in the fact that it is designed to study interactions between only two countries. Obviously, reality is more complex, as each country produces more than two goods and has relations with many other countries. Therefore, it becomes necessary to extend the model to consider these two aspects. To do this, we need to reason in terms of monetary wages. From section 3.2.3.2, we know that in the presence of free trade, it is necessary that: aA C w A · aA C aB C A < B < , (35) aG B w · aG aB G where wA e wB are the monetary wages in A and B, respectively. It is easy to show that, by multiplying everything by (aB A G /aC ), the relationship can be rewritten as follows: aB G wA aB C < <. (36) aA G w B a A C 20 It is clear that wages in the two countries are expressed in their respective currencies, so for a proper comparison, both should be expressed in the same currency. Therefore, the above relationship can be re-written as follows: aB G wA aB C A < < , (37) aG B w ·E aA C where E is the exchange rate between the two countries. This way, the wage in country B is expressed in the currency of country A. Based on (37), a reduction in the terms of trade wA /(wB ·E) implies greater competitiveness for country A in the production of good C and lower competitiveness in the production of good G compared to country B. This means that A specializes more in the production and export of C and tends to import more of good G. This reasoning can be easily extended to the case of N goods. Let us arrange the ratios between the technical coecients of various sectors in ascending order: aB 1 aB 2 aBN < < · · · <. (38) aA 1 a A 2 a A N For all sectors where the ratio of technical coecients is less than the ratio of wages in the two countries, country A (B) is an importer (exporter) of those goods. For all the remaining sectors where the ratio of technical coecients is greater than the ratio of wages, country A (B) is an exporter (importer) of those goods. The case with M countries can be readily developed by assuming that one of the two countries is an `aggregate' of several countries. For example, country B can be considered as the group of G7 countries. In all these cases, it is necessary to construct the technical coecients and wages appropriately. A simple approach is to take the weighted average of coecients and wages (the latter after converting them into a common currency). Weights can be determined, for example, by the size of the country. By doing so, we can state that for any generic country K belonging to the G7 group and for any generic sector J for which the following condition holds: aG7−K J wK < , (39) aK 1 wG7−K aG7−K J and wG7−K represent the weighted average of the technical coecient of sector J and the wage at the G7 group level, excluding country K, country K has a competitive disadvantage compared to the remaining G7 group regarding the production of good J. Therefore, it is expected that K is a net importer of good J compared to the rest of the G7 group. An increase in the wage ratio only worsens the competitiveness of country K (and thus increases the imports of J). 21 3.4.2 The empirical evidence In this section, we will rst examine empirical studies directly aimed at testing the Ricardian hypothesis of the model, and subsequently, we will analyze an alternative empirical approach in which comparative advantage is revealed by examining export ow data between countries. 3.4.2.1 Test on Ricardian's hypothesis In the literature, various studies have empirically tested the hypothesis of the Ricardian model, which posits that countries tend to produce and export goods for which relative labor produc- tivity is higher than relative wages. MacDougall (1951) The author focused his analysis on 1937 data concerning the United States and the United Kingdom across 25 sectors, employing the method known as the `third- countr' method. Instead of examining direct bilateral trade between the two selected countries, MacDougall investigated the relationship between the ratio of sectoral labor productivities in the two countries and the ratio of quantities exported by these sectors, expressed as a percentage of the rest of the world (the so-called third country). MacDougall chose to express the exports of the United States and the United Kingdom in relation to the rest of the world rather than studying direct bilateral trade between them for two fundamental reasons. Firstly, in the 1937 sample, the majority of sectoral exports, both British and American, were directed towards the third market, thus limiting direct trade between the two countries. Secondly, the presence of trade taris, with relatively higher taris in the United States, led to asymmetric bilateral trade, with high US taris completely osetting the British comparative advantage in many sectors. Therefore, considering the ratio of exports to the rest of the world helped avoid distortions caused by these specic conditions and allowed for a more accurate analysis of the relationship between labor productivity and exports. The author found that, on average, the ratio of American wages to British wages was two to one. Consistent with expectations, he demonstrated that...where American output per worker was more than twice the British, the United States had, in general, the bulk of the export market, while for products in which it was less than twice as high, the bulk of the market was held by Britain (MacDougall, 1951). Stern (1962) and Balassa (1963) These studies, utilizing data from the 1950s, conrmed the ndings of MacDougall (1951). Stern (1962) revealed that average American wages were approximately 3.4 times higher than British average wages. Consequently, it is anticipated that the relative exports (American/British) would surpass unity whenever American productivity exceeds 3.4 times that of the UK. In twenty out of twenty-four industries, the observed results align with the expected outcome, indicating that whenever U.S. sectoral productivity exceeds 3.4 times that of the UK, US exports are relatively higher than those of the UK. Similar results are found by Balassa (1963). 22 Golub (1994) and Golub and Hsieh (2000) Golub (1994) conducted a statistical analysis on the impact of relative unit labor costs on sectoral trade balances. The ndings revealed that changes in sectoral trade balances over time can be eectively explained by the evolution of unit labor costs. This observation suggests a conformity of trade patterns to comparative advantage. The countries involved in this analysis are part of the G7 group during the period 19701989. In a more recent study, Golub and Hsieh (2000) investigated the same eect across a broader spectrum of countries using a cross-sectional methodology. The results consistently oer robust support for the Ricardian model. 3.4.2.2 Revealed comparative advantage Given that much information about countries necessary for empirically measuring compara- tive advantages is lacking (such as wages or labor productivity, especially over extended time horizons), some authors have opted to directly examine the international statistics between countries and, based on these results, deduce a theory of comparative advantages. The following schema can help understand the relationship between the theoretical notion of comparative advantage and the measures of comparative advantage that one aims to calculate 10 using this approach: EC → CA → T P C → RCA. Economic conditions (EC ) in various countries determine the pattern of comparative advan- tages (CA). This pattern, in turn, shapes the pattern of international trade, production, and consumption (T P C ) among the dierent countries. Indices constructed based on TPC can be used to indicate comparative advantage and are referred to as `revealed comparative advantage' (RCA). The index most frequently used for this scope was introduced by Balassa (1965) and is therefore known as the Balassa index. If we denote a generic country with the symbol K, with K representing the world economy or a group of countries, S denoting a specic sector (out of a total of S), and t representing the time period, the Balassa index is expressed as follows: ExKSt ExKSt ExKt ExKSt ExKt ExKSt ExKt ExKSt BIKSt|K = ExKSt = · = · = ExKt , (40) ExKt ExKt ExKSt ExKSt ExKt ExKt where ExKSt denotes the exports (Ex) of a specic country (K ) in a particular sector (S ) during a given period (t). This index serves as a sectoral relative export measure in terms of the share of world exports. It can also be represented as the country's share of sectoral exports (ExKSt /ExKSt ) normalized by a weighted sum of export shares in all sectors, which is equivalent to ExKt /ExKt. Before calculating the Balassa index, we need to decide which country's exports to include and which countries to use as reference countries. Once these elements are identied, an index greater than one implies that the chosen country (K ) demonstrates a revealed comparative 10 This diagram is taken from Ballance et al. (1987). 23 advantage in industry S. This indicates that the industry holds more signicance for the exports of country K than for those of the reference countries. In other words, country K exhibits a sectoral share exceeding that of the overall countries K. Essentially, a country exporting substantial quantities of a specic good compared to others is presumed to possess a comparative advantage in that sector. There are several freely available online databases that calculate revealed comparative ad- vantage indexes for dierent countries at the sectoral level and for various time periods. We highlight two of them. The rst is produced by the French institute CEPII (http://www. cepii.fr/CEPII/en/bdd_modele/bdd_modele_item.asp?id=26), and the second is produced by Prof. Charles van Marrewijk (http://www2.econ.uu.nl/users/marrewijk/research/balassa. htm). 4 The neoclassical model of international trade As discussed in the previous section, the reinterpretation of Ricardo's theory in a neoclassical framework, using the tools of the marginalist school, led to the identication of comparative advantage in the dierence in technology between countries, overturning Ricardo's original idea that pointed to diverse natural conditions (such as climate) as the origin of comparative advantage. While this result is correct within the context of a logical-mathematical approach, it has limitations in explaining trade between countries with similar levels of technological develop- ment. For instance, when examining the trade ow between the United States and the United Kingdom, as analyzed by various scholars as described in section 3.4.2, it is evident that in almost all commodity sectors, the technological level between the United States and the United Kingdom is practically identical. It is indeed challenging to imagine, for example, that pro- ductivity in the production of milk or automobiles in the US is dierent from that in the UK. According to the theory, under these circumstances, signicant trade between these countries should not occur. Consequently, many scholars have sought to extend the Ricardian model from a single factor to a multi-factor model, shifting the focus from international diversity in `production functions' to diversity in factor endowments. In this kind of `extension', the considered production factors include both capital and labor. The resulting model is known as the neoclassical model of international trade (or simply the neoclassical model for brevity). In the neoclassical model, industries can have dierent technologies, which means that some industries will be capital-intensive (using a higher capital-to-labor ratio), while others will be more labor-intensive (using a lower capital-to-labor ratio). It is important to note that an industry is not inherently capital or labor-intensive; it mainly depends on the adopted technology and the initial endowment of capital and labor resources. The chosen capital-to- labor ratio reects the choices made by rms, which, in turn, depend on the prices of labor and capital. 24  There are two fundamental consequences when dealing with capital and labor in this manner. Firstly, resource endowments play a crucial role in determining trade patterns, as does labor and capital productivity. Secondly, transitioning from an autarky trade regime to free trade involves reallocating resources among industries. The composition of a nation's production will change during this shift from autarky to free trade. This means that some individuals will benet from it while others will face negative consequences. It cannot be claimed that a regime change from autarky to free trade is always `welfare-enhancing' unless certain highly simplied assumptions are made, which we will discuss later. In this chapter, we will continue to use the computer and wheat sectors, as well as nations A and B, to illustrate the neoclassical trade theory. Since we now have two factors of production (capital and labor) instead of just one, we can refer to this illustration as a `2 × 2 × 2' model: two goods, two nations, and two factors of production. 4.1 The neoclassical model: Assumptions The neoclassical model is based on these assumptions, some of which are borrowed from the classical model: 1. There are two countries (A and B), each producing two nal goods (computers C and wheat G). 2. There are no barriers to trade between the two countries. 3. Two factors are employed in the production of both goods: capital (K ) and labor (L). The productivity of both factors (and therefore their costs) inuences the nal price of the goods. This happens because now capital is a physical asset and no longer just a wage advance. 4. The technology diers between sectors. The technological dierence can be expressed in terms of the dierent proportion in which rms in the two sectors employ capital and labor. The computer sector uses a higher ratio of capital to labor (capital-intensive), while the wheat sector uses a lower ratio of capital to labor (labor-intensive). Formally: KC /LC > KG /LG. 5. The technology is such that there are no economies of scale, and there are diminishing marginal returns. This means that now the marginal cost is no longer xed and varies with the quantity produced of each good. 6. The factor endowments in the two countries are xed and dierent. In other words, country A has a dierent combination of labor and capital compared to country B. 7. The rms in each country have access to the same technology, so KC /LC and KG /LG is identical in both sectors of both countries. Of course, this also implies that KC /LC > KG /LG holds in both countries. 25  8. Labor and capital are perfectly mobile within the country but perfectly immobile (i.e., non-tradable) between nations. 9. Perfect competition characterizes both industries in both countries and there are no trans- portation and transaction costs. 10. All factors are fully utilized in both countries. 11. The individual worker and the individual unit of capital employed in the production process are paid at the unit wage rate w and the interest rate r. 12. The wage rate (w ) and the interest rate (r ) are given. In absence of international trade, unit factor costs dier between countries and depend on factor endowments. A country with a higher K/L will have a lower r/w ratio and vice versa. 13. Residents in all nations have identical tastes and preferences. This implies that community indierence curves for A and B are identical. 14. Trade between nations is perfectly balanced: imports are equal to exports in value. Based on these assumptions, we can make some observations: 1. The cost of capital usage (equal to the interest rate r) in the neoclassical model is a concept distinct from the prot rate. The cost of usage is a kind of `rent' that the entrepreneur must pay to use the capital stock in the production process. In the classical model, the capitalist is also the entrepreneur who owns the capital, and everything obtained from the sale of the product (prot) is reinvested entirely in new capital. If the prots earned exceed the costs of the consumed capital in the production process, the economic system can grow. In the neoclassical context, capital is not owned by the entrepreneur but belongs to the capitalist who receives precisely r for each unit of capital lent. Prots, in the `classical' sense, are implicitly equal to zero. Capital is exogenously available and remains intact without being consumed. Therefore, there is no need for any prot to replenish it. 2. The w/r ratio inuences capitalists' choices on where to invest. If, indeed, w/r is very high, many capitalists will prefer to invest in more capital-intensive production techniques. 3. In each country, aggregate income corresponds to the sum of workers' and capitalists' incomes: L · w + K · r. 4.2 Neoclassical's model in details Let us try to put together all the components of this model using the assumptions described in section 4.1. We start from the supply side, move on to the demand side, and nally, determine the equilibrium both in autarky and under free trade conditions. One consideration: to simplify the notation as much as possible, we omit, when appropriate and without generating ambiguity, 26 the superscript i to indicate the reference country, knowing well that the results we present apply to both countries (given the dierences in parameters and/or other variables). 4.2.1 Supply side Let us now try to understand what type of production function satises assumption #5 in section 4.1. One type of production function capable of meeting this assumption is the Cobb- Douglas production function. For a generic good X, production is given by: αX X = A · KX · L1−α X X , 0 < αX < 1, (41) where the parameter αX is a measure of the capital intensity of the production process for X. From this production function, it is easy to show that the marginal productivity of labor (MPL) and capital (MPK) is decreasing by calculating the second derivatives with respect to 11 each input (labor and capital) of the production function: ∂X αX = M P L = [A · KX · L−α X X ] · (1 − αX ) ∂LX ∂X αX −1 = M P K = [A · KX · L1−α X X ] · αX ∂KX (42) ∂ 2X αX −(αX +1) = [A · KX · LX ] · (1 − αX ) · (−αX ) < 0 ∂L2X ∂ 2X αX −2 1−αX 2 = [A · KX · LX ] · αX · (αX − 1) < 0. ∂KX At this point, we are ready to address the cost minimization problem, which can be formu- lated as follows: αX min w · LX + r · KX s.t. X = KX · L1−α X X = 1, (43) LX ,KX where the problem is represented in terms of a generic good X and the minimization of one unit of good X. This latter aspect is related to the property of constant returns to scale. In fact, if (Γ · KX )αX · (Γ · LX )1−αX = Γ · X , then the minimization of one unit of good X is exactly the same as the minimization of Γ units of good X. It is just a matter of scale. For our purposes, it is convenient to set up the problem relative to the production of just one unit of good X. We dene the Lagrangean L as: αm L = w · LX + r · KX + λ · (1 − Km · L1−α m m ), (44) where λ is the Lagrange multiplier. Imposing the conditions ∂(L)/∂KX = ∂(L)/∂LX = 0, after some manipulations we arrive 11 `Diminishingmarginal returns' and `decreasing marginal productivity' are two dierent ways of expressing the same concept in economics. Both refer to the situation where adding an additional unit of one input (while holding the other input constant) in a production process results in a progressively smaller increase in output. 27 to: ∗ KX αX w ∗ = · , (45) LX 1 − αX r ∗ where KX and L∗X are the optimal values of capital and labor employed to produced one unit of output. Substituting into the production function needed to produce one unit of X (i.e., αX 1−αX X= KX · LX = 1), we get:  1−αX ∗ αX w KX = · 1 − αX r  −αX (46) ∗ αX w LX = ·. 1 − αX r Substituting this value into the cost equation of the rm (costX = w · L + r · K ), we arrive at: ∗ cost = γX · w1−αX · rαX , (47) ∗ where costX is the optimal cost sustained to produce one unit of output X and −αX γX ≡ αX · (1 − αX )−(1−αX ). (48) Under perfect competition we have that marginal costs equal marginal revenue, so that we have: PX = cost∗X = γX · w1−αX · rαX. (49) Now we have all the information needed to build the PPF, which, as a reminder, represents the combination of all potentially producible levels of the two goods by a nation, using its available resources fully and eciently. The construction follows these steps: 1. Construction of the isoquant for the production of goods C and G. The isoquant repre- sents all possible combinations of factors K and L to produce a certain level of output. Fixing the output levels of C and G at C and G, the isoquant equations for C and G are:  1−αC αC w C= · 1 − αC r  −αG (50) αG w G= ·. 1 − αG r 2. Determining the absolute value of the slope of the isoquant, which is dened as dKX /dLX = |M P L/M P K|, for each good X = C, G. In formula: dKC 1 − αC KC = · dLC αC LC (51) dKG 1 − αG KG = · dLG αG LG 3. Ensuring that the absolute values of the slopes of the isoquants are equal: |(1 − αC )/αC · 28  (KC /LC )| = |(1 − αG )/αG · (KG /LG )| and substituting the values for KG and LG with those derived from the factor constraint, namely K − KC and L − LC respectively. After some manipulations, we get: αC · (1 − αC ) · K̄ · LC KC =. (52) αG · (1 − αC ) · L̄ + (αC − αG ) · LC 4. Fix the values of K , L, αC , and αG. 5. Start from an arbitrary value of LC and, using (52), determine the capital value in the production of C that takes into account the factor constraints between sectors. 6. Insert the found values of KC and LC into the production function for C to determine the produced quantity of C. 7. Based on the factor constraint, once KC and LC are determined, retrieve the values of capital and labor used in the production of G. 8. Substitute the values of capital and labor factors used in C into the production function for C to determine the actual quantity of computers produced. 9. Repeat steps 58 for a large range of values of LC. The PPF is represented in gure 8. In contrast to the Ricardian PPF, which is a straight line, the neoclassical PPF is curved (concave to the origin). Nevertheless, similar to the Ricardian case, the slope of the PPF still reects the opportunity cost of substituting computers with wheat. The Marginal Rate of Transformation (M RTC ) quanties the rate at which computers need to be sacriced to produce an additional unit of wheat while maintaining the overall production level constant. It represents the absolute value of the slope of the PPF at a particular combination of goods being produced. Since factor markets are perfectly competitive, M RTC is calculated as M CC /M CG , where M CC represents the marginal cost of computer production, and M CG represents the marginal cost of wheat production. Only in the limiting case where αC = αG will the PPF appear as a straight line, similar to the Ricardian case. However, even in this case, the slope of the PPF will be constant, but this is because the relative marginal costs of producing the two goods are always equal to each other, and not because they are constant, as in the Ricardian case. In other words, the PPF appears as a straight line only when the relative marginal costs between the goods are constant. 4.2.2 Demand side The problem for a consumer aiming to maximize utility (as described by equation (12)), given the income level I and the prices of nal goods PC and PC , by selecting the consumption levels CC and CG is precisely specied as in (14). 29  Figure 8: Productivity possibility frontiers, impact of changing αC for αG = 0.1 The solution to this maximization problem yields the result in (16). It is clear that there is no dierence on the demand side compared to what has already been discussed in the Ricardian model in the neoclassical version (see section 3.2.2). The only aspect that needs to be emphasized is that now, at least in a closed economy, there is no longer any equality between the PPF and the income line. This is because the PPF, due to the characteristics of the production function as analyzed in section 4.2.1, appears as a concave curve toward the origin. 4.2.3 Equilibrium Just as done in section 3.2.3, in this section we derive the equilibrium in a closed economy and an open economy. The equilibrium is mathematically derived and discussed using simulations. These simulations will then be used to generate graphs that will help the reader interpret the results even more intuitively. 4.2.3.1 Autarky equilibrium In a closed economy, we know that the condition that must hold is that the quantity produced is equal to the quantity consumed. However, in autarky, both goods are produced. Therefore, the problem consists of optimally allocating the available quantities of the two inputs across the two sectors. The rst step is to keep in mind that, in every equilibrium, all available factors must be 30 used by both sectors so that the following condition holds: LC + LG = L; KC + KG = K. (53) Once we know the total amount of labor (L) and capital (K ), it is enough to know the amount of labor and capital employed in sector C to know the remaining amount of both factors employed in sector G. Thus, in equilibrium, the following must hold: αC KC∗  ∗ I C = · ϕC · L = CC∗ = δC · , (54) L∗C PC ∗ where the symbol   indicates the equilibrium level, and ϕC , the share of total available labor employed in sector C, is dened in Appendix A. Substituting equation (45) and equation (49) into equation (54), we have:   αC αC w 1  w  αC 1 · · ϕC · L = δC · I · · ·. (55) 1 − αC r γC r w Since I = r · K + w · L = r · (K + (w/r) · L), we can write:  αC αC w  w  1  w  αC r · · ϕC · L = δC · K · · L · · ·. (56) 1 − αC r r γC r w Substituting into equation (56) the corresponding values of ϕC and γC (corresponding to (89) and (48), respectively), after some manipulations, we can write: 1 − αG K αG w  w  · ·L− · · L = δC · K · · L. (57) αC − αG L αC − αG r r At this point, solving for w/r, we arrive at the following formulation: w K = γau · , (58) r L dove   (1 − αG ) − δC · (αC − αG ) γau ≡. (59) αG + δC · (αC − αG ) According to equation (58), the ratio w/r is a function of factor endowments and parameters. Most importantly, γau is equal between the two countries ba