Final Paper
Modeling the Evolution of Trade in Small-Scale Societies Jennifer Shi Santa Fe Institute Research Experiences for Undergraduates
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I.
Introduction In several archaeology and anthropology papers, such as those by Cobb (1993, 2001), trade is discussed as an important mechanism for the development of complex relationships between ancient small-scale societies. Trade networks gave these societies access to desired natural resources to produce goods for their members, as well as the ability to specialize production through differentiation of labor. However, exchange between nearby societies also facilitated the emergence of inequality both within and between these societies (Earle 1987; Cobb 1993; Cobb 2001). One aspect of exchange that desires further examination is the variation in economic outcomes of trade, which results from differences between societies and their relations with one another. In this paper, I develop and explore an agent-based model of the exchange of goods in small-scale societies. This model examines the production and trade of two goods between two societies. I apply economic models of production and utility to determine trade outcomes, looking at how exchange varies with different characteristics for each society. First, I examine a broad range of microeconomic literature to find applicable approaches to modeling each society’s production, utility, and trade for two goods. Next, I incorporate these models into simulations written in MATLAB, to examine the effects of varying production functions in an agent-based model of trade for two goods between two societies. Then, I apply this model for multiple time steps to show the development of specialization in two societies as a result of trade.
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II.
Economic Principles and Models of Trade Modeling Production of Goods In order to determine how much of two goods a society can produce, economists use a model of production possibilities to graphically display each society’s production possibilities curve (Rittenburg 2009). A production possibilities curve is a graphical representation of the alternative combinations of two goods an economy can produce (Rittenburg 2009), with its limited resources. For my model, I assume that each society can produce only two goods, and that quantities for factors of production and technology available are fixed (Rittenburg 2009). The absolute value of the slope of each production function is known as the marginal rate of transformation (Pindyck 2005), abbreviated as MRT. For simplicity, each society in my model has a linear production function, which has constant returns to scale and a constant MRT. The MRT is the rate at which the production of one good can be redirected into the production of the other good (Pindyck 2005). If A and B are variables for units of Good A and Good B, respectively, produced by a society; b is the y-intercept of a linear production function;and a is the x-intercept of a linear production function, then:
In Figure 1 below, the marginal rate of transformation for Society 1 is 3/4, meaning it must give up producing 3 units of Good B in order to produce 4 more units of Good A. The MRT for Society 2 is 4/3, meaning it must give up producing 4 units of Good B in order to produce 3 more units of Good A.
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Figure 1: Society 1’s production possibilities curve is in blue, with marginal rate of transformation = 3/4. Society 2’s production possibilities curve is in magenta, with marginal rate of transformation = 4/3.
From Figure 1, one can determine both absolute and comparative advantage. When looking at the production of Good A, Society 1 has absolute advantage over Society 2, because it can produce a higher output of Good A than Society 2. Likewise, Society 2 has absolute advantage in producing Good B, because it can produce a higher output of Good B than Society 2. One can determine each society’s comparative advantage in production by comparing both societies’ marginal rates of transformation. In Figure 1, Society 1 has an MRT of 3/4, while Society 2 has and MRT of 4/3. Society 1 has comparative advantage in producing Good A, since it can produce Good A more efficiently than Society 2 can, and Society 2 has comparative advantage in producing Good B. Even if one society has absolute advantage in 4
producing both goods, comparative advantage can still be determined by comparing the MRT of each society’s production function, which represents the efficiency of production for each good (Blaumol 2009). As long as the two societies have different MRTs, each society has comparative advantage in producing one of the two goods. For determining whether trade occurs, I focus on comparative advantage, not absolute advantage. Modeling Utility of Goods When determining the optimum production amount along the production function, each society should consider its individual utility function. The utility function I use to model each society’s utility is the Cobb-Douglas utility function, which depends on A (the quantity of Good A), and B (the quantity of Good B): , Marginal utility, the extra utility that a consumer gets from consuming the last unit of a good, is the slope of the utility function as we hold the quantity of the other good constant (Gibson 2009; Perloff 2012):
The marginal rate of substitution (MRS) is the rate at which a consumer is willing to give up one good in exchange for another good while maintaining the same level of utility (Pindyck 2005). The MRS corresponds to the slope of the indifference curve (Gibson 2009), which can be found as the marginal utility of Good A divided by the marginal utility of Good B
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:
In order to find the production point that maximizes each society’s individual utility function, one must find the point at which the utility curve is tangent to the production function (Ahern 2004; Perloff 2012). To find the point of tangency, set the marginal rate of transformation equal to the marginal rate of substitution for each society:
Find the marginal utilities of
Set
, to solve for
, and solve for B:
Set B equal to production function:
Solve for optimum A and B: Society 1: Society 2:
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:
Figure 2: Based on each society’s production and utility functions, the individual optimum production points are S1(A,B) = (5, 11.25), and S2(A,B) = (11.25, 5).
In Figure 2, the intercepts of the production functions are: . Society 1 has a utility function, and Society 2 has a utility function,
, where , where
, so optimum production points are:
Modeling the Exchange of Goods Without trade, each society produces at its individual optimum production point. However, trade allows each society to reach points beyond its individual production possibilities curve, attaining a higher utility than without trade.
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In order to exchange two goods, both societies must agree on a common price ratio
for exchange, where
is the price of Good A, and
is the price of
Good B. If the two societies agree to trade, the price ratio must be in between the two societies’ marginal rates of transformation in order for both societies to benefit, meaning that:
. Trade Outcomes
trade curve trade curve
Figure 3: Each society’s trade curve, at price ratio
= 1.1, is shown by the dashed line that intersects the individual optimum production point of that society.
Figure 3 shows dashed lines for possible trade outcomes in each society, at an arbitrarily determined price ratio,
= 1.1. For Society 1, trading Good A for
Good B can increase its utility, since there exists a point on its trade curve (the blue dashed line) with higher utility than that of the initial optimum production point.
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Society 2, in turn, benefits from trading Good B for Good A. Each society’s optimum point after trade is shown in Figure 4.
trade curve trade curve
Figure 4: With trade, each society can increase its utility by moving along its trade curve toward its optimum outcome with trade. Society 1 must trade A=1.19 for B=1.31, to reach its optimum; Society 2 must trade B=0.60 for A=0.66, to reach its optimum.
Determining Amounts for Trade In many cases, the two societies engaged in trade have different optimum trade amounts. For example, in Figure 4, Society 1 must trade 1.19 units of Good A for 1.31 units of Good B in order to reach its optimum point with trade, while Society 2 only needs to trade 0.66 units of Good B for 0.60 units of Good A in order to reach its trade optimum. This conflict of interest would likely result in a bargaining situation to
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determine the quantity of Good A (from 0.60 and 1.19) to trade for a quantity of Good B (from 0.66 and 1.31), at the price ratio
= 1.1.
In my model, I use the lower boundary values for the trade vector, so Society 1 exports 0.60 units of Good A to Society 2, and Society 2 exports 0.66 units of Good B to Society 1, shown in Figure 5.
trade curve trade curve
Figure 5: The societies must agree on the same trade values at the price ratio
= 1.1. In my model, I use the lower boundary values, A=0.60 and B=0.66, as the agreed-upon values for trade. S1 exports A, and S2 exports B. After trade, the each society reaches the point on its trade curve denoted by the asterisk.
III.
Methods Agent-based Simulation of Trade for Two Goods in Two Societies In my initial MATLAB model, I simulate trade between two societies (1 and 2) producing the same two goods (A and B). For each simulation run, I randomly assign each society a linear production function and a Cobb-Douglas utility function, in 10
which the exponent (
) is randomly assigned. The price ratio is also
determined randomly within the range: Varying Production Functions In my first analysis, I look at the effect on trade outcomes of varying one society’s production function, keeping all else constant. The simulations show that, for a fixed production function for one society, fixed utility functions for each society, and a fixed price ratio for trade, there are various production functions for the second society that will result in trade, and some production functions that will not result in trade. Production functions that result in trade are those in which both societies can increase their utilities through a common trade amount. If two societies, as shown in Figure 5, agree to trade the amount that Society 2 needs to trade in order to reach its trade optimum, Society 1 also increases its utility through trade, although it does not reach its trade optimum. Production functions that do not result in trade are cases in which both societies have the same MRTs. If both production functions have the same slope, then their price ratio for trade will equal their shared MRT, meaning that their optimum point with trade would be the same point as their optimum without trade. As a result, the two societies have no benefit from trade, and therefore remain self-sufficient, assuming that their indifference to trade results in no trade. Multiple Time Steps When expanding the model to multiple time steps, each society must consider its production for the next time period, assuming that trade will occur again. The
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direction for the change in production for the next time step depends on which good each society is better off exporting. In Figure 6, Society 1 can export Good A in order to receive a greater amount of Good B than it would be able to produce on its own when producing the after-trade quantity of Good A. If the societies behave to maximize their utilities under the assumption that trade will occur, then for the next time step, Society 1 should produce more of Good A in order to receive an even greater amount of Good B, since Society 1 has comparative advantage in producing Good A. Likewise, Society 2 should produce more of Good B to export in exchange for Good A from Society 1.
trade curve trade curve
Figure 6: For the next time step, each society will shift production (indicated by the green arrows) toward producing the good in which it has comparative advantage. S1 produces more of Good A, and S2 produces more of Good B.
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I assume that each society determines its production amount independently, although it may be influenced by what the society learns about its trading partner after engaging in trade. The decision of how much to change production also depends on each society’s attitude toward trade. A society that is more willing to engage in trade (or risk-loving) would make greater changes toward specialization, while a more cautious (risk-averse) society would take smaller steps toward specialization, if specializing at all. The risk of shifting production toward specialization is that if no trade occurs, the society is worse off than it was before specializing. There are many ways to determine new production quantities for the next time step. In my model, I use the largest amount that a society is willing to trade in the previous time step to determine new production quantities. In each of my simulation runs, both societies use this method for determining step size, although the model could be modified to incorporate different methods for each society to determine step size, depending on the characteristics of each society and the relationship between the societies.
IV.
Results Development of Specialization When incorporating multiple time steps, shown in Figure 7, each society moves its production toward greater specialization in the good for which they have comparative advantage. Society 1 changes production to produce more of Good A, while Society 2 changes production to produce more of Good B. In the last panel of Figure 7, both societies are completely specializing in the good for which they have comparative advantage.
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trade curve trade curve
trade curve trade curve
trade curve trade curve
Figure 7: The panel on the top left shows the shift in production after one time step; on the top right, production continues to shift toward specialization after several more time steps; in the bottom panel, S1 completely specializes in A, and S2 completely specializes in B. At complete specialization, both societies reach their maximum utilities after trade.
As each society moves toward complete specialization, its utility without trade decreases, while its utility after trade increases. In the last time step, each society has an individual utility of zero without trade, but with trade, both of their utilities are maximized. This result demonstrates a trade-off between producing for selfsufficiency versus producing with the intention of trade, since a society must move away from its individual optimum production point in order to specialize for trade.
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Role of Utility Functions Even though societies with comparative advantage in a good will shift production toward producing more of that good, complete specialization is not optimum for all situations. For example, if both societies value the same good at a much higher value than the other good, which is reflected in their utility functions, the society with comparative advantage in producing the less-valued good may not reach complete specialization because when it produces too much of the less-valued good, its utility after trade decreases with greater specialization. In the case where: a1 = 20; b1 = 20; a2 = 15; b2 = 25; U1 (A, B) = U2 (A, B) = A0.9 B 0.1 both societies value Good A more than Good B; looking at the results in Table 1 of the multiple-time-step simulation for this initial condition, Society 2, which has comparative advantage in producing Good B, does not reach complete specialization in producing Good B, because its utility after trade reaches its maximum at time t = 2, where it still produces some of Good A. Table 1: Selected results of the multiple-time-step simulation with the above-stated conditions. A1 is the amount of Good A produced by Society 1; U1 is Society 1’s utility after trade; A2 is the amount of Good A produced by Society 2; U2 is Society 2’s utility after trade. Society 1 specializes completely in the production of A, while Society 2 does not completely specialize in producing B, since its utility is higher when producing both A and B.
Time (t)
Units of A1
U1 after trade
Units of A2
U2 after trade
0
18.00
14.50
13.50
11.42
1
18.51
14.62
13.07
11.48
2
19.41
14.82
12.32
11.59
3
20.00
14.94
10.93
11.49
4
20.00
14.91
12.32
11.59
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These results show that the utility of Society 2 (U2 after trade) first increases, then decreases with time, as production moves toward specializing in Good B. As a result of this decrease in utility in the third time step, Society 2 moves production back to its point in the previous time step, since the utility was higher at that point. One can also see that Society 1 reaches complete specialization in Good A fairly quickly, after the third time step, but as it remains at complete specialization, its utility decreases in the next time step due to the change in Society 2’s production. Knowing one’s utility function is useful for determining a trade partner, because a society that prefers Good A over Good B is likely to benefit more from developing a trade relationship with a society with the opposite preferences (preferring B over A). Additionally, when a society searches for its optimum production point with trade, its trade outcomes are influenced by the other society’s production decisions. Since both societies are shifting their production to maximize trade utility at the same time, each society’s maximum after trade is influenced by what the other society produces and wants to trade in the corresponding time step. Role of Production Functions Differences between two societies’ production functions also affect whether or not complete specialization occurs. If Society 1 produces both goods at a much larger scale than Society 2 (meaning Society 1 has a much larger area under its production possibilities curve than Society 2), then Society 2 may not be capable of producing the amount of Good B that Society 1 desires to exchange when it completely specializes in Good A. As a result, while Society 2 benefits from completely specializing in Good B, Society 1 is better off producing both A and B.
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Impact of Price Ratio on Utility When determining a price ratio for trade, each society desires to give up as little as it can for as much as it can get in exchange. This means that each society prefers to exchange goods at a price ratio that is as different from its own MRT as possible, which allows for a greater increase in that society’s utility. Figure 8 shows that when the production function has MRT = 1 and completely specializes in producing Good B, the price ratio that extends further out on the x-axis (
= 0.5), shown by the
magenta dotted line, increases the society’s utility more than a price ratio ( 0.75) that extends less on the x-axis, shown by the magenta dashed line.
Figure 8: The magenta dashed line shows price ratio = 0.75, with a maximum utility shown by the black dashed line. The magenta dotted line shows price ratio = 0.5, with a maximum utility shown by the black dotted line. The maximum utility is higher for the price ratio that extends further out from the production possibilities curve, which in this case is when = 0.5.
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This result shows that finding a trading partner with a production function with an MRT as different as possible from its own, which allows for a larger range of potential price ratios, is an important factor for trade outcomes. Since the two societies must agree on the same price ratio, it is also important to consider what kind of bargaining or decision-making process the two societies engage in to reach their agreement, an area in which I hope to explore in future research for this model.
V.
Conclusions Under ideal conditions in which production and utility functions remain constant over time and the price ratio remains the same, societies that benefit from trade will develop some degree of specialization based on comparative advantage. However, when applying these models to small-scale societies, one must consider the uncertainties and risks they face in their future conditions for production and trade. These non-optimal circumstances influence how societies determine their next production point, based on expectations of what will be most beneficial to them for the next time step. Since these societies may have concerns about immediate risk and uncertainty that outweigh the desire to optimize for the future, they may not take the simple, direct path towards specialization, or may not even specialize at all. If a society faces low likelihood of trading with another partner, then it is better off being more self-sufficient and producing at or near its optimum production point without trade. However, if a society has formed a strong trade network with another society that also greatly benefits from their exchange of goods, they together may go to greater lengths to ensure that trade occurs, once deciding to specialize. In addition,
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societies that rely strongly on trade may anticipate that their trade partners can assist them when faced with adverse conditions for production or trade. In my future research, I plan to examine the factors in my current model—price ratio and amount of each good traded—that can be determined through bargaining; to incorporate the probability of trade at each time step; and to vary production functions at each time step, based on changes in production conditions. After incorporating these pieces to my model, I hope to apply this model to archaeological data to explain different economic outcomes in various small-scale societies.
Acknowledgments I would like to thank my mentors, Anne Kandler and Laura Fortunato, for all of their guidance in this research project. I would also like to thank John Miller, Sam Bowles, and Eric Rupley, for their input toward my model; the National Science Foundation for funding; and the Santa Fe Institute for hosting me as an undergraduate researcher for the summer.
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References Ahern, Kenneth R. “Economics 101, Winter 2004, TA Lecture 5: General Equilibrium.” Baumol, William J., and Alan S. Blinder. Economics: Principle and Policy, 11th ed. Mason, OH, USA: South-Western Cengage Learning, 2009. Cobb, Charles R. “Archaeological Approaches to the Political Economy of Nonstratified Societies.” Archaeological Method and Theory, Vol. 5 (1993), pp. 43-100: Springer. Cobb, Charles R. “Specialization, Exchange, and Power in Small-Scale Societies and Chiefdoms.” From Quarry to Cornfield: The Political Economy of Mississippian Hoe Production, Ch. 2. Tuscaloosa, AL, USA: University of Alabama Press, 2001. pp 19-46. Earle, Timothy K. “Chiefdoms in Archaeological and Ethnohistorical Perspective.” Annual Review of Anthropology, Vol. 16 (1987), pp. 279-308: Annual Reviews. Gibson, Bill. “Cobb-Douglas Functions.” 14 Aug 2009. Perloff, Jeffrey M. Microeconomics, 6th ed. Boston, MA, USA: Addison-Wesley, 2012. Pindyck, Robert S., and Daniel L. Rubinfield. Microeconomics, 5th ed. Pearson Education, 2005. Rittenburg, Libby, and Timothy Tregarthen. Principles of Microeconomics, v. 1.0. Flat World Knowledge, Inc., 2009.
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1
I.
Introduction In several archaeology and anthropology papers, such as those by Cobb (1993, 2001), trade is discussed as an important mechanism for the development of complex relationships between ancient small-scale societies. Trade networks gave these societies access to desired natural resources to produce goods for their members, as well as the ability to specialize production through differentiation of labor. However, exchange between nearby societies also facilitated the emergence of inequality both within and between these societies (Earle 1987; Cobb 1993; Cobb 2001). One aspect of exchange that desires further examination is the variation in economic outcomes of trade, which results from differences between societies and their relations with one another. In this paper, I develop and explore an agent-based model of the exchange of goods in small-scale societies. This model examines the production and trade of two goods between two societies. I apply economic models of production and utility to determine trade outcomes, looking at how exchange varies with different characteristics for each society. First, I examine a broad range of microeconomic literature to find applicable approaches to modeling each society’s production, utility, and trade for two goods. Next, I incorporate these models into simulations written in MATLAB, to examine the effects of varying production functions in an agent-based model of trade for two goods between two societies. Then, I apply this model for multiple time steps to show the development of specialization in two societies as a result of trade.
2
II.
Economic Principles and Models of Trade Modeling Production of Goods In order to determine how much of two goods a society can produce, economists use a model of production possibilities to graphically display each society’s production possibilities curve (Rittenburg 2009). A production possibilities curve is a graphical representation of the alternative combinations of two goods an economy can produce (Rittenburg 2009), with its limited resources. For my model, I assume that each society can produce only two goods, and that quantities for factors of production and technology available are fixed (Rittenburg 2009). The absolute value of the slope of each production function is known as the marginal rate of transformation (Pindyck 2005), abbreviated as MRT. For simplicity, each society in my model has a linear production function, which has constant returns to scale and a constant MRT. The MRT is the rate at which the production of one good can be redirected into the production of the other good (Pindyck 2005). If A and B are variables for units of Good A and Good B, respectively, produced by a society; b is the y-intercept of a linear production function;and a is the x-intercept of a linear production function, then:
In Figure 1 below, the marginal rate of transformation for Society 1 is 3/4, meaning it must give up producing 3 units of Good B in order to produce 4 more units of Good A. The MRT for Society 2 is 4/3, meaning it must give up producing 4 units of Good B in order to produce 3 more units of Good A.
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Figure 1: Society 1’s production possibilities curve is in blue, with marginal rate of transformation = 3/4. Society 2’s production possibilities curve is in magenta, with marginal rate of transformation = 4/3.
From Figure 1, one can determine both absolute and comparative advantage. When looking at the production of Good A, Society 1 has absolute advantage over Society 2, because it can produce a higher output of Good A than Society 2. Likewise, Society 2 has absolute advantage in producing Good B, because it can produce a higher output of Good B than Society 2. One can determine each society’s comparative advantage in production by comparing both societies’ marginal rates of transformation. In Figure 1, Society 1 has an MRT of 3/4, while Society 2 has and MRT of 4/3. Society 1 has comparative advantage in producing Good A, since it can produce Good A more efficiently than Society 2 can, and Society 2 has comparative advantage in producing Good B. Even if one society has absolute advantage in 4
producing both goods, comparative advantage can still be determined by comparing the MRT of each society’s production function, which represents the efficiency of production for each good (Blaumol 2009). As long as the two societies have different MRTs, each society has comparative advantage in producing one of the two goods. For determining whether trade occurs, I focus on comparative advantage, not absolute advantage. Modeling Utility of Goods When determining the optimum production amount along the production function, each society should consider its individual utility function. The utility function I use to model each society’s utility is the Cobb-Douglas utility function, which depends on A (the quantity of Good A), and B (the quantity of Good B): , Marginal utility, the extra utility that a consumer gets from consuming the last unit of a good, is the slope of the utility function as we hold the quantity of the other good constant (Gibson 2009; Perloff 2012):
The marginal rate of substitution (MRS) is the rate at which a consumer is willing to give up one good in exchange for another good while maintaining the same level of utility (Pindyck 2005). The MRS corresponds to the slope of the indifference curve (Gibson 2009), which can be found as the marginal utility of Good A divided by the marginal utility of Good B
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:
In order to find the production point that maximizes each society’s individual utility function, one must find the point at which the utility curve is tangent to the production function (Ahern 2004; Perloff 2012). To find the point of tangency, set the marginal rate of transformation equal to the marginal rate of substitution for each society:
Find the marginal utilities of
Set
, to solve for
, and solve for B:
Set B equal to production function:
Solve for optimum A and B: Society 1: Society 2:
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:
Figure 2: Based on each society’s production and utility functions, the individual optimum production points are S1(A,B) = (5, 11.25), and S2(A,B) = (11.25, 5).
In Figure 2, the intercepts of the production functions are: . Society 1 has a utility function, and Society 2 has a utility function,
, where , where
, so optimum production points are:
Modeling the Exchange of Goods Without trade, each society produces at its individual optimum production point. However, trade allows each society to reach points beyond its individual production possibilities curve, attaining a higher utility than without trade.
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In order to exchange two goods, both societies must agree on a common price ratio
for exchange, where
is the price of Good A, and
is the price of
Good B. If the two societies agree to trade, the price ratio must be in between the two societies’ marginal rates of transformation in order for both societies to benefit, meaning that:
. Trade Outcomes
trade curve trade curve
Figure 3: Each society’s trade curve, at price ratio
= 1.1, is shown by the dashed line that intersects the individual optimum production point of that society.
Figure 3 shows dashed lines for possible trade outcomes in each society, at an arbitrarily determined price ratio,
= 1.1. For Society 1, trading Good A for
Good B can increase its utility, since there exists a point on its trade curve (the blue dashed line) with higher utility than that of the initial optimum production point.
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Society 2, in turn, benefits from trading Good B for Good A. Each society’s optimum point after trade is shown in Figure 4.
trade curve trade curve
Figure 4: With trade, each society can increase its utility by moving along its trade curve toward its optimum outcome with trade. Society 1 must trade A=1.19 for B=1.31, to reach its optimum; Society 2 must trade B=0.60 for A=0.66, to reach its optimum.
Determining Amounts for Trade In many cases, the two societies engaged in trade have different optimum trade amounts. For example, in Figure 4, Society 1 must trade 1.19 units of Good A for 1.31 units of Good B in order to reach its optimum point with trade, while Society 2 only needs to trade 0.66 units of Good B for 0.60 units of Good A in order to reach its trade optimum. This conflict of interest would likely result in a bargaining situation to
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determine the quantity of Good A (from 0.60 and 1.19) to trade for a quantity of Good B (from 0.66 and 1.31), at the price ratio
= 1.1.
In my model, I use the lower boundary values for the trade vector, so Society 1 exports 0.60 units of Good A to Society 2, and Society 2 exports 0.66 units of Good B to Society 1, shown in Figure 5.
trade curve trade curve
Figure 5: The societies must agree on the same trade values at the price ratio
= 1.1. In my model, I use the lower boundary values, A=0.60 and B=0.66, as the agreed-upon values for trade. S1 exports A, and S2 exports B. After trade, the each society reaches the point on its trade curve denoted by the asterisk.
III.
Methods Agent-based Simulation of Trade for Two Goods in Two Societies In my initial MATLAB model, I simulate trade between two societies (1 and 2) producing the same two goods (A and B). For each simulation run, I randomly assign each society a linear production function and a Cobb-Douglas utility function, in 10
which the exponent (
) is randomly assigned. The price ratio is also
determined randomly within the range: Varying Production Functions In my first analysis, I look at the effect on trade outcomes of varying one society’s production function, keeping all else constant. The simulations show that, for a fixed production function for one society, fixed utility functions for each society, and a fixed price ratio for trade, there are various production functions for the second society that will result in trade, and some production functions that will not result in trade. Production functions that result in trade are those in which both societies can increase their utilities through a common trade amount. If two societies, as shown in Figure 5, agree to trade the amount that Society 2 needs to trade in order to reach its trade optimum, Society 1 also increases its utility through trade, although it does not reach its trade optimum. Production functions that do not result in trade are cases in which both societies have the same MRTs. If both production functions have the same slope, then their price ratio for trade will equal their shared MRT, meaning that their optimum point with trade would be the same point as their optimum without trade. As a result, the two societies have no benefit from trade, and therefore remain self-sufficient, assuming that their indifference to trade results in no trade. Multiple Time Steps When expanding the model to multiple time steps, each society must consider its production for the next time period, assuming that trade will occur again. The
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direction for the change in production for the next time step depends on which good each society is better off exporting. In Figure 6, Society 1 can export Good A in order to receive a greater amount of Good B than it would be able to produce on its own when producing the after-trade quantity of Good A. If the societies behave to maximize their utilities under the assumption that trade will occur, then for the next time step, Society 1 should produce more of Good A in order to receive an even greater amount of Good B, since Society 1 has comparative advantage in producing Good A. Likewise, Society 2 should produce more of Good B to export in exchange for Good A from Society 1.
trade curve trade curve
Figure 6: For the next time step, each society will shift production (indicated by the green arrows) toward producing the good in which it has comparative advantage. S1 produces more of Good A, and S2 produces more of Good B.
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I assume that each society determines its production amount independently, although it may be influenced by what the society learns about its trading partner after engaging in trade. The decision of how much to change production also depends on each society’s attitude toward trade. A society that is more willing to engage in trade (or risk-loving) would make greater changes toward specialization, while a more cautious (risk-averse) society would take smaller steps toward specialization, if specializing at all. The risk of shifting production toward specialization is that if no trade occurs, the society is worse off than it was before specializing. There are many ways to determine new production quantities for the next time step. In my model, I use the largest amount that a society is willing to trade in the previous time step to determine new production quantities. In each of my simulation runs, both societies use this method for determining step size, although the model could be modified to incorporate different methods for each society to determine step size, depending on the characteristics of each society and the relationship between the societies.
IV.
Results Development of Specialization When incorporating multiple time steps, shown in Figure 7, each society moves its production toward greater specialization in the good for which they have comparative advantage. Society 1 changes production to produce more of Good A, while Society 2 changes production to produce more of Good B. In the last panel of Figure 7, both societies are completely specializing in the good for which they have comparative advantage.
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trade curve trade curve
trade curve trade curve
trade curve trade curve
Figure 7: The panel on the top left shows the shift in production after one time step; on the top right, production continues to shift toward specialization after several more time steps; in the bottom panel, S1 completely specializes in A, and S2 completely specializes in B. At complete specialization, both societies reach their maximum utilities after trade.
As each society moves toward complete specialization, its utility without trade decreases, while its utility after trade increases. In the last time step, each society has an individual utility of zero without trade, but with trade, both of their utilities are maximized. This result demonstrates a trade-off between producing for selfsufficiency versus producing with the intention of trade, since a society must move away from its individual optimum production point in order to specialize for trade.
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Role of Utility Functions Even though societies with comparative advantage in a good will shift production toward producing more of that good, complete specialization is not optimum for all situations. For example, if both societies value the same good at a much higher value than the other good, which is reflected in their utility functions, the society with comparative advantage in producing the less-valued good may not reach complete specialization because when it produces too much of the less-valued good, its utility after trade decreases with greater specialization. In the case where: a1 = 20; b1 = 20; a2 = 15; b2 = 25; U1 (A, B) = U2 (A, B) = A0.9 B 0.1 both societies value Good A more than Good B; looking at the results in Table 1 of the multiple-time-step simulation for this initial condition, Society 2, which has comparative advantage in producing Good B, does not reach complete specialization in producing Good B, because its utility after trade reaches its maximum at time t = 2, where it still produces some of Good A. Table 1: Selected results of the multiple-time-step simulation with the above-stated conditions. A1 is the amount of Good A produced by Society 1; U1 is Society 1’s utility after trade; A2 is the amount of Good A produced by Society 2; U2 is Society 2’s utility after trade. Society 1 specializes completely in the production of A, while Society 2 does not completely specialize in producing B, since its utility is higher when producing both A and B.
Time (t)
Units of A1
U1 after trade
Units of A2
U2 after trade
0
18.00
14.50
13.50
11.42
1
18.51
14.62
13.07
11.48
2
19.41
14.82
12.32
11.59
3
20.00
14.94
10.93
11.49
4
20.00
14.91
12.32
11.59
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These results show that the utility of Society 2 (U2 after trade) first increases, then decreases with time, as production moves toward specializing in Good B. As a result of this decrease in utility in the third time step, Society 2 moves production back to its point in the previous time step, since the utility was higher at that point. One can also see that Society 1 reaches complete specialization in Good A fairly quickly, after the third time step, but as it remains at complete specialization, its utility decreases in the next time step due to the change in Society 2’s production. Knowing one’s utility function is useful for determining a trade partner, because a society that prefers Good A over Good B is likely to benefit more from developing a trade relationship with a society with the opposite preferences (preferring B over A). Additionally, when a society searches for its optimum production point with trade, its trade outcomes are influenced by the other society’s production decisions. Since both societies are shifting their production to maximize trade utility at the same time, each society’s maximum after trade is influenced by what the other society produces and wants to trade in the corresponding time step. Role of Production Functions Differences between two societies’ production functions also affect whether or not complete specialization occurs. If Society 1 produces both goods at a much larger scale than Society 2 (meaning Society 1 has a much larger area under its production possibilities curve than Society 2), then Society 2 may not be capable of producing the amount of Good B that Society 1 desires to exchange when it completely specializes in Good A. As a result, while Society 2 benefits from completely specializing in Good B, Society 1 is better off producing both A and B.
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Impact of Price Ratio on Utility When determining a price ratio for trade, each society desires to give up as little as it can for as much as it can get in exchange. This means that each society prefers to exchange goods at a price ratio that is as different from its own MRT as possible, which allows for a greater increase in that society’s utility. Figure 8 shows that when the production function has MRT = 1 and completely specializes in producing Good B, the price ratio that extends further out on the x-axis (
= 0.5), shown by the
magenta dotted line, increases the society’s utility more than a price ratio ( 0.75) that extends less on the x-axis, shown by the magenta dashed line.
Figure 8: The magenta dashed line shows price ratio = 0.75, with a maximum utility shown by the black dashed line. The magenta dotted line shows price ratio = 0.5, with a maximum utility shown by the black dotted line. The maximum utility is higher for the price ratio that extends further out from the production possibilities curve, which in this case is when = 0.5.
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This result shows that finding a trading partner with a production function with an MRT as different as possible from its own, which allows for a larger range of potential price ratios, is an important factor for trade outcomes. Since the two societies must agree on the same price ratio, it is also important to consider what kind of bargaining or decision-making process the two societies engage in to reach their agreement, an area in which I hope to explore in future research for this model.
V.
Conclusions Under ideal conditions in which production and utility functions remain constant over time and the price ratio remains the same, societies that benefit from trade will develop some degree of specialization based on comparative advantage. However, when applying these models to small-scale societies, one must consider the uncertainties and risks they face in their future conditions for production and trade. These non-optimal circumstances influence how societies determine their next production point, based on expectations of what will be most beneficial to them for the next time step. Since these societies may have concerns about immediate risk and uncertainty that outweigh the desire to optimize for the future, they may not take the simple, direct path towards specialization, or may not even specialize at all. If a society faces low likelihood of trading with another partner, then it is better off being more self-sufficient and producing at or near its optimum production point without trade. However, if a society has formed a strong trade network with another society that also greatly benefits from their exchange of goods, they together may go to greater lengths to ensure that trade occurs, once deciding to specialize. In addition,
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societies that rely strongly on trade may anticipate that their trade partners can assist them when faced with adverse conditions for production or trade. In my future research, I plan to examine the factors in my current model—price ratio and amount of each good traded—that can be determined through bargaining; to incorporate the probability of trade at each time step; and to vary production functions at each time step, based on changes in production conditions. After incorporating these pieces to my model, I hope to apply this model to archaeological data to explain different economic outcomes in various small-scale societies.
Acknowledgments I would like to thank my mentors, Anne Kandler and Laura Fortunato, for all of their guidance in this research project. I would also like to thank John Miller, Sam Bowles, and Eric Rupley, for their input toward my model; the National Science Foundation for funding; and the Santa Fe Institute for hosting me as an undergraduate researcher for the summer.
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References Ahern, Kenneth R. “Economics 101, Winter 2004, TA Lecture 5: General Equilibrium.” Baumol, William J., and Alan S. Blinder. Economics: Principle and Policy, 11th ed. Mason, OH, USA: South-Western Cengage Learning, 2009. Cobb, Charles R. “Archaeological Approaches to the Political Economy of Nonstratified Societies.” Archaeological Method and Theory, Vol. 5 (1993), pp. 43-100: Springer. Cobb, Charles R. “Specialization, Exchange, and Power in Small-Scale Societies and Chiefdoms.” From Quarry to Cornfield: The Political Economy of Mississippian Hoe Production, Ch. 2. Tuscaloosa, AL, USA: University of Alabama Press, 2001. pp 19-46. Earle, Timothy K. “Chiefdoms in Archaeological and Ethnohistorical Perspective.” Annual Review of Anthropology, Vol. 16 (1987), pp. 279-308: Annual Reviews. Gibson, Bill. “Cobb-Douglas Functions.” 14 Aug 2009. Perloff, Jeffrey M. Microeconomics, 6th ed. Boston, MA, USA: Addison-Wesley, 2012. Pindyck, Robert S., and Daniel L. Rubinfield. Microeconomics, 5th ed. Pearson Education, 2005. Rittenburg, Libby, and Timothy Tregarthen. Principles of Microeconomics, v. 1.0. Flat World Knowledge, Inc., 2009.
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